Mathmatics ofSound Analysis PDF generated using the open source mwlib toolkit. See http://code.pediapress.com/ for more information. PDF generated at: Sun, 28 Aug 2011 21:24:54 UTC Contents Articles Fast Fourier transform Discrete Hartley transform Discrete Fourier transform Fourier analysis Sine Trigonometric functions Complex number Microphone practice Wave 1 8 11 25 32 46 66 83 86 References Article Sources and Contributors Image Sources, Licenses and Contributors 100 102 Article Licenses License 104 Fast Fourier transform 1 Fast Fourier transform A fast Fourier transform (FFT) is an efficient algorithm to compute the discrete Fourier transform (DFT) and its inverse. "The FFT has been called the most important numerical algorithm of our lifetime (Strang, 1994)." (Kent & Read 2002, 61) There are many distinct FFT algorithms involving a wide range of mathematics, from simple complex-number arithmetic to group theory and number theory; this article gives an overview of the available techniques and some of their general properties, while the specific algorithms are described in subsidiary articles linked below. A DFT decomposes a sequence of values into components of different frequencies. This operation is useful in many fields (see discrete Fourier transform for properties and applications of the transform) but computing it directly from the definition is often too slow to be practical. An FFT is a way to compute the same result more quickly: computing a DFT of N points in the naive way, using the definition, takes O(N2) arithmetical operations, while an FFT can compute the same result in only O(N log N) operations. The difference in speed can be substantial, especially for long data sets where N may be in the thousands or millions—in practice, the computation time can be reduced by several orders of magnitude in such cases, and the improvement is roughly proportional to N / log(N). This huge improvement made many DFT-based algorithms practical; FFTs are of great importance to a wide variety of applications, from digital signal processing and solving partial differential equations to algorithms for quick multiplication of large integers. The most well known FFT algorithms depend upon the factorization of N, but (contrary to popular misconception) there are FFTs with O(N log N) complexity for all N, even for prime N. Many FFT algorithms only depend on the fact that is an th primitive root of unity, and thus can be applied to analogous transforms over any finite field, such as number-theoretic transforms. Since the inverse DFT is the same as the DFT, but with the opposite sign in the exponent and a 1/N factor, any FFT algorithm can easily be adapted for it. Definition and speed An FFT computes the DFT and produces exactly the same result as evaluating the DFT definition directly; the only difference is that an FFT is much faster. (In the presence of round-off error, many FFT algorithms are also much more accurate than evaluating the DFT definition directly, as discussed below.) Let x0, ...., xN-1 be complex numbers. The DFT is defined by the formula Evaluating this definition directly requires O(N2) operations: there are N outputs Xk, and each output requires a sum of N terms. An FFT is any method to compute the same results in O(N log N) operations. More precisely, all known FFT algorithms require Θ(N log N) operations (technically, O only denotes an upper bound), although there is no known proof that better complexity is impossible. To illustrate the savings of an FFT, consider the count of complex multiplications and additions. Evaluating the DFT's sums directly involves N2 complex multiplications and N(N − 1) complex additions [of which O(N) operations can be saved by eliminating trivial operations such as multiplications by 1]. The well-known radix-2 Cooley–Tukey algorithm, for N a power of 2, can compute the same result with only (N/2) log2 N complex multiplies (again, ignoring simplifications of multiplications by 1 and similar) and N log2N complex additions. In practice, actual performance on modern computers is usually dominated by factors other than arithmetic and is a complicated subject (see, e.g., Frigo & Johnson, 2005), but the overall improvement from O(N2) to O(N log N) remains. so Winograd can be used to obtain minimal-multiplication FFTs and is often used to find efficient algorithms for small factors. a tradeoff no longer favorable on modern processors with hardware multipliers. into cyclotomic Another polynomial viewpoint is exploited by the Winograd algorithm. (The Rader-Brenner and QFT algorithms were proposed for power-of-two sizes. this comes at the cost of many more additions. unfortunately. is based on interpreting the FFT as a recursive factorization of the polynomial . 0. The most well-known use of the Cooley–Tukey algorithm is to divide the transform into two pieces of size at each step. expresses a DFT of prime size as a cyclic convolution of (composite) size . but this time of the same size (which can be zero-padded to a power of two and evaluated by radix-2 Cooley–Tukey FFTs. exploiting the existence of a generator for the multiplicative group modulo prime . For with coprime and . Bruun's algorithm applies to arbitrary even composite sizes. and therefore require few (if any) multiplications. 1984) that those two authors had independently re-invented an algorithm known to Carl Friedrich Gauss around 1805 (and subsequently rediscovered several times in limited forms). because the Cooley–Tukey algorithm breaks the DFT into smaller DFTs. Another prime-size FFT is due to L. but it was later discovered (Heideman & Burrus. Cooley and J. in particular.Fast Fourier transform 2 Algorithms Cooley–Tukey algorithm By far the most common FFT is the Cooley–Tukey algorithm. one can use the Prime-Factor (Good-Thomas) algorithm (PFA). it also re-expresses a DFT as a convolution. Other FFT algorithms There are other FFT algorithms distinct from Cooley–Tukey. respectively (and other variants such as the split-radix FFT have their own names as well). such as those described below. Winograd showed that the DFT can be computed with only irrational multiplications. via the identity . leading to a proven achievable lower bound on the number of multiplications for power-of-two sizes. In particular. for example). Also. it was later superseded by the split-radix variant of Cooley–Tukey (which achieves the same multiplication count but with fewer additions and without sacrificing accuracy).) Bruun's algorithm. along with O(N) multiplications by complex roots of unity traditionally called twiddle factors (after Gentleman and Sande. reducing multiplications at the cost of increased additions and reduced numerical stability. to factorize the DFT similarly to Cooley–Tukey but without the twiddle factors. W. This is a divide and conquer algorithm that recursively breaks down a DFT of any composite size N = N1N2 into many smaller DFTs of sizes N1 and N2. or −1. most traditional implementations rearrange the algorithm to avoid explicit recursion. Bluestein. Rader's algorithm. The Rader-Brenner algorithm (1976) is a Cooley–Tukey-like factorization but with purely imaginary twiddle factors. it can be combined arbitrarily with any other algorithm for the DFT. I. 1966). . here into real-coefficient polynomials of the form and . Although the basic idea is recursive. which factorizes polynomials—these often have coefficients of 1. These are called the radix-2 and mixed-radix cases. W. Indeed. Winograd also makes use of the PFA as well as an algorithm by Rader for FFTs of prime sizes. based on the Chinese Remainder Theorem. This method (and the general idea of an FFT) was popularized by a publication of J. and is sometimes called the chirp-z algorithm. but any factorization can be used in general (as was known to both Gauss and Cooley/Tukey). Tukey in 1965. and is therefore limited to power-of-two sizes. Algorithms that recursively factorize the DFT into smaller operations other than DFTs include the Bruun and QFT algorithms. but it is possible that they could be adapted to general composite . which can then be computed by a pair of ordinary FFTs via the convolution theorem (although Winograd uses other convolution methods). a tight multiplications required by an FFT. sometimes called the "arithmetic complexity" (although in this context it is the exact count and not the asymptotic complexity that is being . the count of arithmetic operations is usually the focus of such questions. no published FFT algorithm has achieved fewer than complex-number additions (or their equivalent) for power-of-two . 1990). it is possible to express an even-length real-input DFT as a complex DFT of half the length (whose real and imaginary parts are the even/odd elements of the original real data). although this is not a tight bound because extra additions are required as part of complex-number multiplications.g. (This argument would imply that at least real additions are required. but it has not proved popular. Pan (1986) proved an lower bound assuming a bound on a measure of the FFT algorithm's "asynchronicity". For the case of power-of-two . saving roughly a factor of two in time and memory. A tight lower bound is not known on the number of required additions. in which case the outputs satisfy the symmetry and efficient FFT algorithms have been designed for this situation (see e. although actual performance on modern-day computers is determined by many other factors such as cache or CPU pipeline optimization. these algorithms require too many additions to be practical. Sorensen. Moreover. that no additive identities in the roots of unity are exploited). Following pioneering work by Winograd (1978). the input data for the DFT are purely real. Papadimitriou (1979) argued that the number of complex-number additions achieved by Cooley–Tukey algorithms is optimal under certain assumptions on the graph of the algorithm (his assumptions imply. In particular. 1987).Fast Fourier transform 3 FFT algorithms specialized for real and/or symmetric data In many applications. and many open problems remain. but the generality of this assumption is unclear. Morgenstern proved an lower bound on the addition count for algorithms where the multiplicative constants have bounded magnitudes (which is true for most but not all FFT algorithms). It can be shown that only multiplications are required to compute a DFT of power-of-two length lower bound is known for the number of real irrational real . Unfortunately. There are further FFT specializations for the cases of real data that have even/odd symmetry. although no algorithms with lower complexity are known.) Thus far. It is not even rigorously proved whether DFTs truly require (i. at least on modern computers with hardware multipliers. It was once believed that real-input DFTs could be more efficiently computed by means of the discrete Hartley transform (DHT). but it was subsequently argued that a specialized real-input DFT algorithm (FFT) can typically be found that requires fewer operations than the corresponding DHT algorithm (FHT) for the same number of inputs. although lower bounds have been proved under some restrictive assumptions on the algorithms. explicit algorithms that achieve this count are known (Heideman & Burrus. 1986. among other things. Alternatively. order or greater) operations.g.. In 1973. One approach consists of taking an ordinary algorithm (e. DCTs/DSTs can also be computed via FFTs of real data combined with O(N) pre/post processing. Duhamel. Bruun's algorithm (above) is another method that was initially proposed to take advantage of real inputs. Cooley–Tukey) and removing the redundant parts of the computation. A third problem is to minimize the total number of real multiplications and additions. even for the simple case of power of two sizes. Instead of directly modifying an FFT algorithm for these cases.e. followed by O(N) post-processing operations. in which case one can gain another factor of (roughly) two in time and memory and the DFT becomes the discrete cosine/sine transform(s) (DCT/DST). Computational issues Bounds on complexity and operation counts A fundamental question of longstanding theoretical interest is to prove lower bounds on the complexity and exact operation counts of fast Fourier transforms. Moreover. no tight lower bound has been proven. Since 1968. complex-data FFTs are so closely related to algorithms for related problems such as real-data FFTs. e. A wavelet-based approximate FFT by Guo and Burrus (1996) takes sparse inputs/outputs (time/frequency localization) into account more efficiently than is possible with an exact FFT. discrete cosine transforms. since it is based on the compressibility (rank deficiency) of the Fourier matrix itself rather than the compressibility (sparsity) of the data. The upper bound on the relative error for the Cooley–Tukey algorithm is O(ε log N). Lundy and Van Buskirk. Cooley–Tukey. the lowest published count for power-of-two was long achieved by the split-radix FFT algorithm. Even the "exact" FFT algorithms have errors when finite-precision floating-point arithmetic is used. that any improvement in one of these would immediately lead to improvements in the others (Duhamel & Vetterli. discrete Hartley transforms. and it is not unusual for incautious FFT implementations to have much worse accuracy.e. Again. however. Such algorithms trade the approximation error for increased speed or other properties. (1999) achieves lower communication requirements for parallel computing with the help of a fast multipole method. To verify the correctness of an FFT implementation. however. (1995). . if they use inaccurate trigonometric recurrence formulas. rigorous guarantees can be obtained in O(N log N) time by a simple procedure checking the linearity. are very sensitive to the accuracy of the twiddle factors used in the FFT (i. with an error that can be made arbitrarily small at the expense of increased computations. are intrinsically less stable. most FFT algorithms. because it is the simplest.g. e. which requires real multiplications for . however. This was recently reduced to (Johnson and Frigo. i. compared to O(εN3/2) for the naïve DFT formula (Gentleman and Sande. have excellent numerical properties as a consequence of the pairwise summation structure of the algorithms. with rms errors growing as O(√N) for the Cooley–Tukey algorithm (Welch. 1969). but these errors are typically quite small. Some FFTs other than Cooley–Tukey. In fixed-point arithmetic. A few "FFT" algorithms have been proposed. These results. and fixed-point FFT algorithms involve rescaling at each intermediate stage of decompositions like Cooley–Tukey. Only the Edelman algorithm works equally well for sparse and non-sparse data. impulse-response. In fact.Fast Fourier transform 4 considered). neglecting floating-point errors). Most of the attempts to lower or prove the complexity of FFT algorithms have focused on the ordinary complex-data case. 1995). 2007). being only O(ε √log N) for Cooley–Tukey and O(ε √N) for the naïve DFT (Schatzman. and time-shift properties of the transform on random inputs (Ergün. 2007. 1996). where ε is the machine floating-point relative precision. Accuracy and approximations All of the FFT algorithms discussed below compute the DFT exactly (in exact arithmetic. For example. 1966). the root mean square (rms) errors are much better than these upper bounds. the trigonometric function values). 1990). However. an approximate FFT algorithm by Edelman et al. and so on. however.e. that compute the DFT approximately. such as the Rader-Brenner algorithm. the finite-precision errors accumulated by FFT algorithms are worse. Another algorithm for approximate computation of a subset of the DFT outputs is due to Shentov et al.g. even achieving this accuracy requires careful attention to scaling in order to minimize the loss of precision. is performed element-wise. Other. (This may also have cache benefits. hence the name. vector-radix-2 divides all of the dimensions by two). and then perform the one-dimensional FFTs along the direction. Still. In particular.) The simplest case of vector-radix is where all of the radices are equal (e. known as the row-column algorithm (after the two-dimensional case. the multidimensional DFT transforms an array summations (over with a -dimensional vector of indices for each ). Yet another variation is to perform matrix transpositions in between transforming subsequent dimensions. In more than two dimensions.e.g. and still has is not even) (see Frigo and Johnson. Equivalently. More generally. and this algorithm corresponds to first performing the FFT of all the rows and then of all the columns (or vice versa).Fast Fourier transform 5 Multidimensional FFTs As defined in the multidimensional DFT article. which view the transform in terms of convolutions and polynomial products. more complicated. methods include polynomial transform algorithms due to Nussbaumer (1977). . i. is essentially a row-column algorithm. so that the transforms operate on contiguous data. where is the total number of data points transforms of size . where the division by a set of . and so on (or actually. This method is easily complexity. although all of them have complexity. but this is not necessary. There are other multidimensional FFT algorithms that are distinct from the row-column algorithm. any ordering will work). below). 2005). For example. there are In two dimensions. . then along the shown to have the usual FFTs is: dimension. which is a generalization of the ordinary Cooley–Tukey algorithm where one divides the transform dimensions by a vector of radices at each step. nested as defined . Perhaps the simplest non-row-column FFT is the vector-radix FFT algorithm. it is often advantageous for cache locality to group the dimensions recursively. this remains a straightforward variation of the row-column algorithm that ultimately requires only a one-dimensional complexity. the can be viewed as an matrix. so the complexity of the sequence of transformed. it is simply the composition of a sequence of sets of one-dimensional DFTs. an asymptotically optimal cache-oblivious algorithm consists of recursively dividing the dimensions into two groups and that are transformed recursively (rounding if FFT algorithm as the base case. a three-dimensional FFT might first perform two-dimensional FFTs of each planar "slice" for each fixed . This compositional viewpoint immediately provides the simplest and most common multidimensional DFT algorithm. this is especially important for out-of-core and distributed memory situations where accessing non-contiguous data is extremely time-consuming. etcetera. See Duhamel and Vetterli (1990) for more information and references. one simply performs a sequence of one-dimensional FFTs (by any of the above algorithms): first you transform along the dimension. That is. performed along one dimension at a time (in any order). Vector radix with only a single non-unit radix at a time. "Gauss and the history of the fast Fourier transform". W. Introduction to Algorithms. and S. doi:10. "Algorithms meeting the lower bounds on the multiplicative complexity of lengthDFTs and their connection with practical algorithms".1145/225058. Various groups have also published "FFT" algorithms for non-equispaced data. Toledo. html • Brenner. 2005.. Opt. E. The Future Fast Fourier Transform? (doi:10. 19 (90): 297–301.1994. C.1984. Michael T. Leiserson. Computing 20: 1094–1114. N. along with an algorithm conjectured (but not proven) to have O(N2 log2 N) complexity.1117/12. Burrus. Burrus. Vetterli. IEEE ASSP Magazine 1 (4): 14–21. C. 265–327.389994).1109/TASSP. Sig. and Clifford Stein. ohiou. The Fast Fourier Transform.255236). P. 34 (1): 91–95. 27th ACM Symposium on the Theory of Computing: 407–416. M. C. Acoust. P. M. References [1] http:/ / www.1109/29. or NDFT.1162805. 1866. Rivest. 2nd. Gentleman and G.1986. Fast transforms: Algorithms.1464352 H. (1984). T. James W. C. Cormen.Fast Fourier transform 6 Other generalizations An O(N5/2 log N) generalization to spherical harmonics on the sphere S2 with N2 nodes was described by Mohlenkamp (1999). D. which itself is often computed only approximately). Göttingen: Königliche Gesellschaft der Wissenschaften. New York: Prentice-Hall • Cooley. Tukey. "An algorithm for the machine calculation of complex Fourier series". A.. Especially chapter 30. 2825: 250–259. MIT Press and McGraw-Hill.1164785. S.org/ fftw-paper-ieee. New York: Academic Press. "Fast Fourier transforms—for fun and profit. IEEE Acoustics. O. doi:10. Burrus. (2002).1109/MASSP.1137/S1064827597316266). Signal Processing 19: 259–299. S. Proc. " The Design and Implementation of FFTW3 (http://fftw." Proc. Guo and C. Johnson. Guo. Speech & Signal Processing 24 (3): 264–266. IEEE Trans. (1965). Charles E. AFIPS 29: 563–578. The Quick Discrete Fourier Transform (doi:10. • Brigham. G. H. Soc. A. Acoust. Acoust. (2001). Comput. math. D. "On the number of multiplications necessary to compute a lengthDFT". S." • Duhamel. Duhamel and M.225167). Edelman. Frigo and S. (1976). Proc. & K." Proceedings of the IEEE 93: 216–231. McCorquodale. ISBN 0-262-03293-7. but rather some approximation thereof (a non-uniform discrete Fourier transform. Rao. Eng. • Thomas H. doi:10.1162257. Carl Friedrich Gauss. Burrus. • • • • • • • • • • • Heideman.pdf). Rader. 1994. Sci. 1996. John W. Sig. M. R.. 1966.1976. Sande. 38 (9): 1504–151. Funda Ergün. Math. Sidney (1986). Speech and Sig. Sitton.1145/1464291. Fast Fourier transforms: a tutorial review and a state of the art (doi:10. Speech. 2001. SIAM J. Johnson. edu/ ~mjm/ research/ libftsh.. "A New Principle for Fast Fourier Transformation". F. doi:10. A spherical-harmonic algorithm with O(N2 log N) complexity is described by Rokhlin and Tygert (2006). Pierre (1990). ed.. G. Such algorithms do not strictly compute the DFT (which is only defined for equispaced data). applications. Fast approximate Fourier transform via wavelets transform (doi:10.1109/TASSP. 1982. Elliott. Proc. Speech.1109/ICASSP.1090/S0025-5718-1965-0178586-1. IEEE Conf. SPIE Intl. Mohlenkamp also provides an implementation in the libftsh library [1].60070. "Nachlass: Theoria interpolationis methodo nova tractata. 1999.1016/0165-1684(90)90158-U). Ronald L. . Proc. H. 1990. doi:10. doi:10. "Polynomials and the FFT." 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Tasche. doi:10. G. D. ISBN 978-0-521-88068-8 • Rokhlin. Accuracy of the discrete Fourier transform and the fast Fourier transform (http:// portal.1137/050623073.1162035.1987.acm. Information Proc. Wavelets. Mark (2006). SIAM J.321761. Flannery. M.1969. Heute. IEEE Trans. 2007. Hossen.org/citation. doi:10. Burrus.html#pg=600). Cites Strang. • Shentov. Speech Sig. BP (2007). interpretations and extensions". Steidl. JSTOR 2006266 7 . • Nussbaumer.. The trade-off between the additive complexity and the asyncronicity of linear and bilinear algorithms (doi:10. ISBN 0-7693-0112-6.pdf)".1049/el:19770280. Comput..1090/S0025-5718-1978-0468306-4. Sci.). T. New York: Cambridge University Press. (1999). • Morgenstern. Modern Sampling Theory: Mathematics and Applications (Birkhauser). Processing 35 (35): 849–863. ACM 20 (2): 305–306. J. See also Sorensen.1145/321752. Vetterling. Heideman. "On computing the discrete Fourier transform". Computation 32 (141): 175–199. S. (1977). 82. V. and Signal Processing 35 (9): 1353–1353.. V. (1969). Heideman. H.. • Christos H. Computing 27 (6): 1903–1928.1109/TASSP. Fourier Anal.edu/~mjm/ research/MOHLEN1999P. Acoustic Analysis of Speech.J.pdf). L. "Chapter 12. Charles (2002). S. M.1016/0165-1684(94)00103-7.1016/0020-0190(86)90035-9). doi:10. Numerical Recipes: The Art of Scientific Computing (3rd ed. "Real-valued fast Fourier transform algorithms".. C. 250-255.math. Discrete Hartley transform A discrete Hartley transform (DHT) is a Fourier-related transform of discrete. xN-1 are transformed into the N real numbers H0.com/support/library/downloads/articles/ DP-30 Years of FFT. It was subsequently argued. A historical review of hardware FFT devices. Pascal. and code.html) • Fast Fourier Transforms (http://cnx. .. C++. Although these conventions occasionally vary between authors. invertible function H : Rn -> Rn (where R denotes the set of real numbers). Markus Pueschel.. Definition Formally.usf.net/fasttransforms/fft. the DHT is the discrete analogue of the continuous Hartley transform. the discrete Hartley transform is a linear. .fftw. Johnson (2008). L. V. HN-1 according to the formula .ntu.pdf)".com/ModelRiskHelp/ index.pdf) • FFT Textbook notes.htm#Aggregate_distributions/Aggregate_modeling_-_Fast_Fourier_Transform_FFT_method. etc. and Steven G.. The combination is sometimes denoted . links.html) • FFT programming in C++ — Cooley–Tukey algorithm. with no intrinsic involvement of complex numbers.pitt.. Sidney Burrus. Ivan Selesnick.php) GPL Licensed multilanguage (VBA. Videos (http://numericalmethods. " 30 years of FFT Analyzers (http://www. periodic data similar to the discrete Fourier transform (DFT). The N real numbers x0.librow. Sidney Burrus. PPTs. the DHT was originally proposed by R. • Links to FFT code and information online.com/articles/article-10) • Online documentation.edu. Hartley in 1942.multi-instrument.eng. Bracewell in 1983 as a more efficient computational tool in the common case where the data are purely real.org/links. 30th anniversary issue).org/content/col10550/). • FFT Basics and Case Study Using Multi-Instrument (http://www. Connexions online book edited by C.edu/~kirk/cs1501/animations/FFT. • ALGLIB FFT Code (http://www.html) at Holistic Numerical Methods Institute. Matteo Frigo.. N. Its main distinction from the DFT is that it transforms real inputs to real outputs.html) • National Taiwan University – FFT (http://www.de/fxt/) • Using FFT to construct aggregate probability distributions (http://www. Just as the DFT is the discrete analogue of the continuous Fourier transform. (http://www.cs. book. that specialized FFT algorithms for real inputs or outputs can ordinarily be found with slightly fewer operations than any corresponding algorithm for the DHT (see below)...dataphysics.com/doc/D1002/ FFT_Basics_and_Case_Study_using_Multi-Instrument_D1002.htm) • Sri Welaratna.tw/cml/dsp/training/coding/transform/ fft.alglib.edu/topics/fft. Because there are fast algorithms for the DHT analogous to the fast Fourier transform (FFT).Fast Fourier transform 8 External links • Fast Fourier Algorithm (http://www. Sound and Vibration (January 1997. with analogous applications in signal processing and related fields. however.) numerical analysis and data processing library. and should be contrasted with the that appears in the DFT definition (where i is the imaginary unit). with chapters by C. As with the DFT.jjj. (http://www. they do not affect the essential .vosesoftware. the overall scale factor in front of the transform and the sign of the sine term are a matter of convention. introduced by R. (http://www.cmlab.csie. As with the DFT. is simply the DHT of Hk multiplied by 1/N. On present-day computers.256. therefore.) In particular. the discrete Hartley transform is a linear operator. This count doesn't include the division by 2. For real inputs xn. suppose that the vectors X. and z respectively. all of length N. up to an overall scale factor. and Z denote the DHT of x. 9 Properties The transform can be interpreted as the multiplication of the vector (x0. performance is determined more by cache and CPU pipeline considerations than by strict operation counts. a cyclic convolution z = x*y of two vectors x = (xn) and y = (yn) to produce a vector z = (zn). (1987) and Duhamel & Vetterli (1987). Stanford placed this patent in the public domain in 1994 (Bracewell.646. issued in 1987 to Stanford University. from Cooley-Tukey to Prime-Factor to Winograd (Sorensen et al. which can be absorbed e. Nearly every FFT algorithm. (However. not including the costs of the transforms below. the inverse transformation. because the pairwise operation above requires 8 real-arithmetic operations compared to the 6 of a complex multiplication. at least when applied to power-of-two sizes N. into the 1/N normalization of the inverse DHT. highly optimized real-input FFT libraries are available from many . and vice versa. neither appears to have a substantial a priori speed advantage (Popovic and Sevic. the DHT is equivalent to computing the DFT of xn multiplied by 1+i. etcetera). have not yet been investigated in the context of the DHT. the DHT transforms a convolution into a simple combination of pairs of real frequency components...g. which allows one to recover the xn from the Hk. This FHT algorithm. the DHT analogue of the Cooley-Tukey algorithm is commonly known as the Fast Hartley Transform (FHT) algorithm. xN-1) by an N-by-N matrix. that compute the same result in only O(N log N) operations. however. 1985) to Bruun's (Bini & Bozzo. y.. then taking the real part of the result. just as the DFT transforms a convolution into a pointwise multiplication of complex numbers (pairs of real and imaginary parts). 1995). . evaluating the DHT definition directly would require O(N2) arithmetical operations (see Big O notation). Thus. a fast algorithm for the DHT (see below) yields a fast algorithm for convolution. they argued that a DHT of power-of-two length can be computed with.) Fast algorithms Just as for the DFT. The latter authors obtained what appears to be the lowest published operation count for the DHT of power-of-two sizes. a few of the more exotic FFT algorithms. As mentioned above. has a direct analogue for the discrete Hartley transform. Since FHT and real-input FFT algorithms have similar computational structures. is the subject of the United States patent number 4. (Note that this is slightly more expensive than the corresponding procedure for the DFT. DHT algorithms are typically slightly less efficient (in terms of the number of floating-point operations) than the corresponding DFT algorithm (FFT) specialized for real inputs (or outputs). at best. and a slight difference in arithmetic cost is unlikely to be significant.. This was first argued by Sorensen et al. the DHT is its own inverse (involutary). Y. the DFT output Xk has a real part (Hk + HN-k)/2 and an imaginary part (HN-k . 2 more additions than the corresponding number of arithmetic operations for the real-input DFT. The inverse DHT then yields the desired vector z. As a practical matter. In this way. 1994). such as the QFT. and was first described by Bracewell in 1984. In particular. Then the elements of Z are given by: where we take all of the vectors to be periodic in N (XN = X0. becomes a simple operation after the DHT. employing a split-radix algorithm (similar to the split-radix FFT) that breaks a DHT of length N into a DHT of length N/2 and two real-input DFTs (not DHTs) of length N/4. Conversely. 1993).. There are fast algorithms similar to the FFT. The DHT can be used to compute the DFT. That is. In this way. The matrix is invertible.Hk)/2.Discrete Hartley transform properties of the transform. " J." Computers in Physics 9 (4). • H. 10 References • • • • • R. Hong and M. N. New York. because the redundancies are hidden behind intricate permutations and/or phase rotations in those algorithms. 849–863 (1987).g. In contrast. 818–824 (1987). Jones. 217–227 (1982). Speech Sig. • S. Hartley. • H. 1010–1018 (1984). Bracewell. R. Soc. IEEE 72 (8). a non-DHT-based adaptation of Rader's algorithm for real-input DFTs is also possible (Chu & Burrus. • Matteo Frigo and Steven G. Acoust." IEEE Transactions on Acoustics. 2005). A. T. Opt. "Improved Fourier and Hartley transform algorithms: application to cyclic convolution of real data." Computers & Mathematics (with Applications) 26 (9). 35–52 (1993). V." Proc. Speech Sig. Sorensen. 2178-2182 (1994). Heideman. Speech. 1231–1238 (1985). Processing ASSP-35 (6). V." IEEE Trans." Proc. D. S. D. Processing ASSP-35. 373–379 (1995). T. • D." Proc. Burrus. • Mark A." IEEE Trans. Acoust. a standard prime-size FFT algorithm. "A prime factor FTT [sic] algorithm using distributed arithmetic. despite the existence of O(N log N) complex-data algorithms for such cases.Discrete Hartley transform sources (e. Bini and E. (1988). 144–150 (1942). N. Speech Sig. R. and Signal Processing 30 (2). Am. "Discrete Hartley transform. "On computing the discrete Hartley transform. 400-412 (1994). S. Bracewell. Duhamel. "A new look at the comparison of the fast Hartley and Fourier transforms. Jones. • J. Sorensen. On the other hand. Johnson. N. pdf . O'Neill. "The fast Hartley transform. Vetterli and P. 1986). whereas highly optimized DHT libraries are less common. • Miodrag Popović and Dragutin Šević. References [1] http:/ / fftw. Rader's algorithm. "A more symmetrical Fourier analysis applied to transmission problems. "Computing with the Hartley Transform. Signal Processing 42 (8). Bracewell. L." IEEE Trans." IEEE Trans. Press. can be directly applied to the DHT of real data for roughly a factor of two less computation than that of the equivalent complex FFT (Frigo and Johnson. L. Burrus." Proc. Byte 13(4):293-300. M. IEEE 93 (2). L. Acoust. 216–231 (2005). R. "Basefield transforms with the convolution property. Bozzo. the redundant computations in FFTs due to real inputs are more difficult to eliminate for large prime N. IEEE 82 (3). • Pierre Duhamel and Martin Vetterli. 1982). IRE 30. "Fast discrete transform by means of eigenpolynomials. "Faster than Fast Fourier". org/ fftw-paper-ieee. 1832–1835 (1983). and C. and M. The Hartley Transform (Oxford Univ. 73 (12). N. from CPU vendors such as Intel). On the other hand. R. Chu and C. "Real-valued fast Fourier transform algorithms. Burrus. C. "The Design and Implementation of FFTW3 [1]. Processing ASSP-33 (4). V. Heideman. Bracewell. . or simply the DFT.) is a primitive N'th root of unity. For the same reason. of the original function (which is often a function in the time domain). we are making extensive use of Euler's formula to express sinusoids in terms of complex exponentials.. while the with frequency . Such inputs are often created by sampling a continuous function. making the DFT ideal for processing information stored in computers. The terminology is further blurred by the (now rare) synonym finite Fourier transform for the DFT. Formally. It transforms one function into another. the inverse DFT cannot reproduce the entire time domain. the DFT is widely employed in signal processing and related fields to analyze the frequencies contained in a sampled signal. whereas "FFT" refers to a specific family of algorithms for computing DFTs. . In particular. it only evaluates enough frequency components to reconstruct the finite segment that was analyzed. unless the input happens to be periodic (forever). the discrete Fourier transform (DFT) is a specific kind of discrete transform. 1969) but has the same initialism.Discrete Fourier transform 11 Discrete Fourier transform In mathematics. which are much easier to manipulate.. By writing the equations in this form. In the same way. like a person's voice. The DFT computes the cycles per sample. But the DFT requires an input function that is discrete and whose non-zero values have a limited (finite) duration. there is a clear distinction: "DFT" refers to a mathematical transformation or function.. Therefore it is often said that the DFT is a transform for Fourier analysis of finite-domain discrete-time functions. (This expression can also be written in terms of a DFT matrix. The inverse discrete Fourier transform (IDFT) is given by A simple description of these equations is that the complex numbers different sinusoidal components of the input "signal" IDFT shows how to compute the as a sum of sinusoidal components represent the amplitude and phase of the from the . and to perform other operations such as convolutions or multiplying large integers. The input to the DFT is a finite sequence of real or complex numbers (with more abstract generalizations discussed below). FFT algorithms are so commonly employed to compute DFTs that the term "FFT" is often used to mean "DFT" in colloquial settings. which is called the frequency domain representation. respectively: and phase from the complex modulus and argument of . . if this is not actually true. used in Fourier analysis. regardless of how it is computed. to solve partial differential equations. when scaled appropriately it becomes a unitary matrix and the Xk can thus be viewed as The transform is sometimes denoted by the symbol . XN−1 by the DFT according to the formula: where i is the imaginary unit and coefficients of x in an orthonormal basis. we obtain the sinusoid amplitude . Unlike the discrete-time Fourier transform (DTFT). as in or or . The sinusoidal basis functions of the decomposition have the same properties.... Definition The sequence of N complex numbers x0. xN−1 is transformed into the sequence of N complex numbers X0. Using the DFT implies that the finite segment that is analyzed is one period of an infinitely extended periodic signal. a window function has to be used to reduce the artifacts in the spectrum. which apparently predates the term "fast Fourier transform" (Cooley et al. A key enabling factor for these applications is the fact that the DFT can be computed efficiently in practice using a fast Fourier transform (FFT) algorithm. by writing in polar form. The only requirements of these conventions are that the DFT and IDFT have opposite-sign exponents and that the product of their normalization factors be 1/N. The Plancherel theorem and Parseval's theorem If Xk and Yk are the DFTs of xn and yn respectively then the Plancherel theorem states: where the star denotes complex conjugation. linear transformation with C denoting the set of complex numbers.Discrete Fourier transform 12 where atan2 is the two-argument form of the arctan function. In other words. . Note that the normalization factor multiplying the DFT and IDFT (here 1 and 1/N) and the signs of the exponents are merely conventions. A normalization of for both the DFT and IDFT makes the transforms unitary. the DFT is often thought of as a matched filter: when looking for a Properties Completeness The discrete Fourier transform is an invertible. an N-dimensional complex vector has a DFT and an IDFT which are in turn N-dimensional complex vectors. Orthogonality The vectors form an orthogonal basis over the set of N-dimensional complex vectors: where is the Kronecker delta. Equivalently. (The convention of a negative sign in the exponent is often convenient because it means that a "positive frequency" frequency of +1. This orthogonality condition can be used to derive the formula for the IDFT from the definition of the DFT. is the amplitude of . one correlates the incoming signal with a frequency of −1. for any N > 0. Parseval's theorem is a special case of the Plancherel theorem and states: These theorems are also equivalent to the unitary condition below. which has some theoretical advantages.) In the following discussion the terms "sequence" and "vector" will be considered interchangeable. and differ in some treatments. but it is often more practical in numerical computation to perform the scaling all at once as above (and a unit scaling can be convenient in other ways). and is equivalent to the unitarity property below. Discrete Fourier transform 13 Periodicity If the expression that defines the DFT is evaluated for all integers k instead of just for the resulting infinite sequence is a periodic extension of the DFT. then Similarly. periodically). At those values.N-1]: which is the convolution of the sequence with a periodically extended sequence defined by: . where the subscript is interpreted modulo N (i. The periodicity can be shown directly from the definition: . Note that we can also extend the limits of m to infinity. where p is any integer. Mathematically. it can be shown that the IDFT formula leads to a periodic extension. if : is replaced by shift of the input . it is 1. a circular corresponds to multiplying the output represents the vector x then if then and Circular convolution theorem and cross-correlation theorem The convolution theorem for the continuous and discrete time Fourier transforms indicates that a convolution of two infinite sequences can be obtained as the inverse transform of the product of the individual transforms. It can therefore be replaced by an infinite sum of Kronecker delta functions. With sequences and transforms of length N. with the understanding that the x and y sequences are defined as 0 outside [0.e. and we continue accordingly. The shift theorem Multiplying by a linear phase for some integer m corresponds to a circular shift of the output by a linear phase.. periodic with period N. Similarly. a circularity arises: The quantity in parentheses is 0 for all values of m except those of the form . [1] Convolution theorem duality It can also be shown that: which is the circular convolution of and . then is real as well. if the are real numbers. changing to ) without changing the interpolation property. notice that the Nyquist component This interpolation is not unique: aliasing implies that one could add N to any of the complex-sinusoid frequencies (e. For even N. however. convolutions can be used to efficiently compute DFTs via Rader's FFT algorithm and Bluestein's FFT algorithm. it consists of sinusoids whose frequencies have the smallest possible magnitudes: the interpolation is bandlimited. similar to the inverse DFT formula. Furthermore. In contrast. . is handled specially. it can be shown that: which is the cross-correlation of and operations for an output efficiency of A direct evaluation of the convolution or correlation summation (above) requires sequence of length N. is typical because it has two useful properties. for N odd.Discrete Fourier transform 14 Similarly. The choice above. Two such methods are called overlap-save and overlap-add. Second. its use is a common mistake. but giving different values in between the points. and is not generally real-valued for real . An indirect method. Methods have also been developed to use circular convolution as part of an efficient process that achieves normal (non-circular) convolution with an or sequence potentially much longer than the practical transform size (N). Trigonometric interpolation polynomial The trigonometric interpolation polynomial for N even . the most obvious trigonometric interpolation polynomial is the one in which the frequencies range from 0 to (instead of roughly to as above). satisfies the interpolation property for .g. First. can take advantage of the the fast Fourier transform (FFT) to achieve much better performance. This interpolation does not minimize the slope. using transforms. where the coefficients Xk are given by the DFT of xn above. and all of the properties of a rigid rotation can be found in the unitary DFT. For the special case . which are always of the coordinate system. the DFT becomes a unitary transformation. The inverse transform is then given by the inverse of the above matrix: With unitary normalization constants matrix: . this implies that the length of a vector is preserved as well—this is just Parseval's theorem: . or as described below. the DFT can be expressed as a Vandermonde matrix: where is a primitive Nth root of unity. defined by a unitary where det() is the determinant function. a unitary transformation can be thought of as simply a rigid rotation The orthogonality of the DFT is now expressed as an orthonormality condition (which arises in many areas of mathematics as described in root of unity): If is defined as the unitary DFT of the vector then and the Plancherel theorem is expressed as: If we view the DFT as just a coordinate transformation which simply specifies the components of a vector in a new coordinate system. In a real vector space. then the above is just the statement that the dot product of two vectors is preserved under a unitary DFT transformation. The determinant is the product of the eigenvalues.Discrete Fourier transform 15 The unitary DFT Another way of looking at the DFT is to note that in the above discussion. Consider the unitary form defined above for the DFT of length N. although it was later shown to have been equivalent to a problem solved by Gauss (Dickinson and Steiglitz. Define swap( ) as with its real and imaginary parts swapped—that is. The multiplicity depends on the value of N modulo 4. that is involutary—that is. where This matrix satisfies the matrix polynomial equation: This can be seen from the inverse properties above: operating operating satisfy the equation: twice gives the original data in reverse order. +i.) First. and are the subject of ongoing research. In particular.Discrete Fourier transform 16 Expressing the inverse DFT in terms of the DFT A useful property of the DFT is that the inverse DFT can be easily expressed in terms of the (forward) DFT. whereas the eigenvectors are complicated.) Third.A closely related involutary transformation (by a factor of (1+i) /√2) is factors in cancel the 2. (For example. the inverse transform is the same as the forward transform with the real and imaginary parts swapped for both input and output. for Second. the subscripts are interpreted modulo N. For real inputs . closely related to the DFT. a unitary matrix is never defective. in computations. This means that the eigenvalues Therefore. (Note that there are N independent eigenvectors. is clearly its own inverse: .) The problem of their multiplicity was solved by McClellan and Parks (1972). the real part of . which is also involutary. if then swap( ) is . they have some multiplicity. Then That is. since the is none other than the discrete Hartley transform. thus. which is sometimes preferable because it requires no modification of the data values. or −i. and is given by the following table: . not unique. which is its own inverse. so four times gives back the original data and is thus the identity matrix. 1988).. via several well-known "tricks". a variant of this conjugation trick. Equivalently. swap( ) equals . it is often convenient to only implement a fast Fourier transform corresponding to one transform direction and then to get the other transform direction from the first. we can compute the inverse DFT by reversing the inputs: (As usual. −1. Eigenvalues and eigenvectors The eigenvalues of the DFT matrix are simple and well-known. the eigenvalues of are the fourth roots of unity: is +1. 1982). matrix. we have . The multiplicity Since there are only four distinct eigenvalues for this gives the number of linearly independent eigenvectors corresponding to each eigenvalue. involves swapping real and imaginary parts (which can be done on a computer simply by modifying pointers). up to a normalization (Duhamel et al. The conjugation trick can also be used to define a new transform. one can also conjugate the inputs and outputs: . size N 4m λ = +1 m+1 λ = −1 λ = -i λ = +i m m m+1 m+1 m m m m−1 m m 4m + 1 m + 1 4m + 2 m + 1 4m + 3 m + 1 m+1 m Otherwise stated. 1972. the natural orthogonal eigenfunctions are the Hermite functions. Hanna et al. but it converges rapidly. Moreover. 1997). 2008). Since periodic summation of the function means discretizing its frequency spectrum and discretization means periodic summation of the spectrum. the following is an eigenvector of DFT: • For DFT period N = 2L = 4K. Various researchers have proposed different choices of eigenvectors. A straightforward approach is to discretize the eigenfunction of the continuous Fourier transform. 1982. selected to satisfy useful properties like orthogonality and to have "simple" forms (e. 2005).. Dickinson and Steiglitz. however. McClellan and Parks.g. Grünbaum. the following is an eigenvector of DFT: • The choice of eigenvectors of the DFT matrix has become important in recent years in order to define a discrete analogue of the fractional Fourier transform—the DFT matrix can be taken to fractional powers by exponentiating the eigenvalues (e. the discretized and periodically summed Gaussian function yields an eigenvector of the discrete transform: • . The "best" choice of eigenvectors to define a fractional discrete Fourier transform remains an open question. where K is an integer. 1982. Rubio and Santhanam.g. Candan et al. Gurevich and Hadani. where K is an integer. the characteristic polynomial of is: No simple analytical formula for general eigenvectors is known. 2000.Discrete Fourier transform 17 Multiplicities of the eigenvalues λ of the unitary DFT matrix U as a function of the transform size N (in terms of an integer m). Two other simple closed-form analytical eigenvectors for special DFT period N were found (Kong. namely the Gaussian function. A closed form expression for the series is not known... Atakishiyev and Wolf. 2004. .. the eigenvectors are not unique because any linear combination of eigenvectors for the same eigenvalue is also an eigenvector for that eigenvalue. 2008): For DFT period N = 2L + 1 = 4K +1. so various discrete analogues of these have been employed as the eigenvectors of the DFT. For the continuous Fourier transform. such as the Kravchuk polynomials (Atakishiyev and Wolf. 1997. the Shannon entropies are defined as: and and the Hirschman uncertainty principle becomes[2] : The equality is obtained for equal to translations and modulations of a suitably normalized Kronecker comb of will then be proportional to a [2] period A where A is any exact integer divisor of N. Using Euler's formula. the "DC" element is purely real. However. Therefore. with an associated probability mass function constructed from the transformed variable: For the case of continuous functions P(x) and Q(k). the Heisenberg uncertainty principle states that: where and are the variances of and respectively. as they often are in practical applications. because the uncertainty will not be shift-invariant. The real-input DFT If are real numbers. In this case. The probability mass function suitably translated Kronecker comb of period B=N/A. . and one obtains the complete information by only looking at roughly half of the outputs . In the discrete case.[2] . and for even N the "Nyquist" element is also real. the interpolating trigonometric polynomial can then be interpreted as a sum of sine and cosine functions.Discrete Fourier transform 18 Uncertainty principle If the random variable is constrained by: then may be considered to represent a discrete probability mass function of n. with the equality attained in the case of a suitably normalized Gaussian distribution. The subscripts are interpreted modulo N. The Hirschman uncertainty principle is expressed in terms of the Shannon entropy of the two probability functions. an analogous uncertainty principle is not useful. the Hirschman uncertainty will have a useful analog for the case of the DFT. then the DFT obeys the symmetry: The star denotes complex conjugation. so there are exactly N non-redundant real numbers in the first half + Nyquist element of the complex output X. the DFT output for real inputs is half redundant. Although the variances may be analogously defined for the DFT. This is and . Such shifted transforms are most often used for symmetric data. ) and vice-versa for is known as an odd-time odd-frequency discrete Fourier transform (or O DFT). Another interesting choice is . The solution is broken up into plane waves. While the ordinary DFT corresponds to a periodic signal in both produces a signal that is anti-periodic in frequency domain ( . the specific case of 2 time and frequency domains. 2005)[3] The discrete Fourier transform can be viewed as a special case of the z-transform. the multidimensional DFT expresses the input as a superposition of plane waves. also called the shifted DFT or offset DFT. or multidimensional sinusoids. evaluated on the unit circle in the complex plane. Multidimensional DFT The ordinary DFT transforms a one-dimensional sequence or array variable n. where we define as d-dimensional vectors of indices from 0 to .Discrete Fourier transform 19 Generalized/shifted DFT It is possible to shift the transform sampling in time and/or frequency domain by some real shifts a and b. all four of its eigenvalues (see above) have equal multiplicities (Rubio and Santhanam. and has analogous properties to the ordinary DFT: Most often. The multidimensional DFT of a multidimensional array variables for in is defined by: that is a function of exactly one discrete that is a function of d discrete where more compactly expressed as above and the d output indices run from in vector : notation. This is sometimes known as a generalized DFT (or GDFT). This decomposition is of great importance for everything from digital image processing (two-dimensional) to solving partial differential equations. which is called the centered DFT (or CDFT). when N is a multiple of four. The direction of oscillation in space is . The amplitudes are . which we define as where the division is defined as to be performed element-wise. The inverse of the multi-dimensional DFT is. respectively. and the sum denotes the set of nested summations above. The multidimensional DFT can be computed by the composition of a sequence of one-dimensional DFTs along each dimension. to represent different boundary symmetries. In the two-dimensional case the independent DFTs of the rows (i. given by: As the one-dimensional DFT expresses the input as a superposition of sinusoids. shifts of (half a sample) are used.e.. analogous to the one-dimensional case. and for real-symmetric data they correspond to different forms of the discrete cosine and sine transforms. The centered DFT has the useful property that. more general z-transforms correspond to complex shifts a and b above. Thus. along ) are . • As already noted. the DFT outputs have a conjugate symmetry similar to the one-dimensional case above: where the star again denotes complex conjugation and the ). Spectral analysis When the DFT is used for spectral analysis. which is manifested as a loss of detail (aka resolution) in the DTFT. A final source of distortion (or perhaps illusion) is the DFT itself. When the available data (and time to process it) is more than the amount needed to attain the desired frequency resolution. which generally entails a type of distortion called aliasing. the time-samples of some signal sequence usually represents a finite set of uniformly-spaced . which is a particular implementation used in conjunction with the fast Fourier transform (FFT) algorithm. . So there is a practical limit to the benefit that can be obtained from a fine-grained DFT. The real-input multidimensional DFT For input data consisting of real numbers. • The procedure is sometimes referred to as zero-padding. The inefficiency of performing multiplications and additions with zero-valued "samples" is more than offset by the inherent efficiency of the FFT. The conversion from continuous time to samples (discrete-time) changes the underlying Fourier transform of x(t) into a discrete-time Fourier transform (DTFT). Alternatively the columns can be computed first and then the rows. -th subscript is again interpreted modulo (for Applications The DFT has seen wide usage across a large number of fields. where t represents time. That can be mitigated by increasing the resolution of the DFT. This approach is known as the row-column algorithm. Choice of an appropriate sub-sequence length is the primary key to minimizing that effect. There are also intrinsically multidimensional FFT algorithms. a standard technique is to perform multiple DFTs. Similarly. The order is immaterial because the nested summations above commute. averaging the magnitude components of the multiple DFTs is a useful procedure to reduce the variance of the spectrum (also called a periodogram in this context). the conversion from a very long (or infinite) sequence to a manageable size entails a type of distortion called leakage. two examples of such techniques are the Welch method and the Bartlett method. the general subject of estimating the power spectrum of a noisy signal is called spectral estimation. we only sketch a few examples below (see also the references at the end). Choice of an appropriate sample-rate (see Nyquist frequency) is the key to minimizing that distortion. That procedure is illustrated in the discrete-time Fourier transform article. which is a function of a continuous frequency domain. leakage imposes a limit on the inherent resolution of the DTFT. An algorithm to compute a one-dimensional DFT is thus sufficient to efficiently compute a multidimensional DFT. because it is just a discrete sampling of the DTFT.Discrete Fourier transform computed first to form a new array form the final result . Then the independent DFTs of y along the columns (along 20 ) are computed to . If the desired result is a power spectrum and noise or randomness is present in the data. for example to create a spectrogram. a fast Fourier transform. All applications of the DFT depend crucially on the availability of a fast algorithm to compute discrete Fourier transforms and their inverses. ." This can be rewritten as a cyclic convolution by taking the coefficient vectors for a(x) and b(x) with constant term first. The ordinary product expression for the coefficients of c involves a linear (acyclic) convolution. then appending zeros so that the resultant coefficient vectors a and b have dimension d > deg(a(x)) + deg(b(x)). in the Fourier representation. One then uses the inverse DFT to transform the result back into the ordinary spatial representation.g. differentiation is simple—we just multiply by i n. each is transformed. is often chosen for the transform operation. d should be chosen as the smallest integer greater than the sum of the input polynomial degrees that is factorizable into small prime factors (e. which is limited to composite sizes. The decompressor computes the inverse transform based on this reduced number of Fourier coefficients. Where c is the vector of coefficients for c(x). 3.. Then. deg(a(x)) + deg(b(x)) of the coefficient vector With a fast Fourier transform. The advantage of this approach is that it expands the signal in complex exponentials einx. Thus. and then the Fourier coefficients of high frequencies. which are eigenfunctions of differentiation: d/dx einx = in einx. depending upon the FFT implementation)..) A linear differential equation with constant coefficients is transformed into an easily solvable algebraic equation. where indices do not "wrap around. and the convolution operator is defined so But convolution becomes multiplication under the DFT: Here the vector product is taken elementwise. Due to its simplicity and speed.Discrete Fourier transform 21 Data compression The field of digital signal processing relies heavily on operations in the frequency domain (i. the discrete cosine transform or sometimes the modified discrete cosine transform. Such an approach is called a spectral method. a choice similar to that in the trigonometric interpolation section above should be used. In this case. that the choice of n is not unique due to aliasing. for the method to be convergent. and 5. which are assumed to be unnoticeable. For example. Thus the coefficients of the product polynomial c(x) are just the terms 0. (Note. Polynomial multiplication Suppose we wish to compute the polynomial product c(x) = a(x) · b(x).) Partial differential equations Discrete Fourier transforms are often used to solve partial differential equations. . 2. where again the DFT is used as an approximation for the Fourier series (which is recovered in the limit of infinite N). are discarded.e. several lossy image and sound compression methods employ the discrete Fourier transform: the signal is cut into short segments. the resulting algorithm takes O (N log N) arithmetic operations. on the Fourier transform). however. (Compression applications often use a specialized form of the DFT. the Cooley–Tukey FFT algorithm.. In other words. as detailed in the sequel. Since either or is is a Dirichlet larger than one and thus warrants fast convergence of one of the two series. where W is an odd integer. . kernel) Discretization and periodic summation of the scaled Gaussian functions for . one may generalize the DFT by either changing the target (taking values in a field other than the complex numbers). After polynomial multiplication. or the domain (a group other than a finite cyclic group). for large you may choose to compute the frequency spectrum and convert to the time domain using the discrete Fourier transform. and is a sinc-like function (specifically. be suggestively written This latter may to emphasize that this is a complex vector space whose coordinates are indexed by the n-element set From this point of view. a sequence of n complex numbers can be thought of as an element of n-dimensional complex space or equivalently a function from the finite cyclic group of order n to the complex numbers. 22 Some discrete Fourier transform pairs Some DFT pairs Note Shift theorem Real DFT from the geometric progression formula from the binomial theorem is a rectangular window function of W points centered on n=0. Generalizations Representation theory The DFT can be interpreted as the complex-valued representation theory of the finite cyclic group. Integers can be treated as the value of a polynomial evaluated specifically at the number base. Derivation as Fourier series The DFT can be derived as a truncation of the Fourier series of a periodic sequence of impulses.Discrete Fourier transform Multiplication of large integers The fastest known algorithms for the multiplication of very large integers use the polynomial multiplication method outlined above. More narrowly still. a relatively low-complexity carry-propagation step completes the multiplication. one may generalize the DFT to representation theory generally. or more narrowly to the representation theory of finite groups. with the coefficients of the polynomial corresponding to the digits in that base. N − 1} → C. L. For this reason. pp. AFIPS Spring Joint Computing Conf. shift. ….2: The DFT and FFT.). Such properties include the completeness. J. . Rabiner and C. convolution. Notes [1] T. R. Özaydin. • Cormen. Plancherel/Parseval. Introduction to Algorithms (Second ed. xN−1 of complex numbers. Jr. Alternatives As with other Fourier transforms. "Entropy-Based Uncertainty Measures for . 1972. • Oppenheim. see comparison of the discrete wavelet transform with the discrete Fourier transform. Stockham. edu/ w/ images/ 9/ 95/ 2002-26. References • Brigham. Victor. "Chapter 8: The Discrete Fourier Transform" (http://www. "Chapter 30: Polynomials and the FFT". the standard DFT is seen as the Fourier transform on a cyclic group. Thomas H.. which can be viewed as a function {0. MIT Press and McGraw-Hill. Alan V. (1999). see number-theoretic transform and discrete Fourier transform (general). Oran (1988).: Prentice Hall. For details. For more information. at the expense of greater difficulty representing frequency.dspguide.. Upper Saddle River.com/ch8/1. Other finite groups The standard DFT acts on a sequence x0. From the point of view of time–frequency analysis. vol. IEEE Transactions on Signal Processing 53 (8): 2690. E. Charles E. Reprinted in Digital Signal Processing. orthogonality. As wavelets have location as well as frequency. SPTM-P12. In this framework. and Buck. htm). . usc. Tomasz. pdf). 1.J. Schafer. • Smith. Discrete-time signal processing. [3] Santhanam. III. Calif.: California Technical Publishing. there are various alternatives to the DFT for various applications. The Scientist and Engineer's Guide to Digital Signal Processing (Second ed. 1385-1388. Murad (2005). and Clifford Stein (2001). ….. and such generalizations are commonly called number-theoretic transforms (NTTs) in the case of finite fields. and With a Hirschman Optimal Transform for " (http:/ / redwood.4). 822–848. (1999). periodicity. N. The fast Fourier transform and its applications. editors. [2] DeBrunner. and unitarity properties above.Discrete Fourier transform 23 Other fields Many of the properties of the DFT only depend on the fact that denoted or (so that is a primitive root of unity. which act on functions G → C where G is a finite group. Rivest. pp.). Englewood Cliffs. N. pp.. Leiserson. Ronald L. Balu. esp. while the multidimensional DFT is a Fourier transform on a direct sum of cyclic groups. Proceedings of the 32nd IEEE International Conference on Acoustics. The analog of the DFT is the discrete wavelet transform (DWT). Retrieved 2011-06-23. as well as many FFT algorithms. Rader. ISBN 0-262-03293-7. G. ISBN 0-13-307505-2. "Discrete Gauss-Hermite functions and eigenvectors of the centered discrete Fourier transform" (http:/ / thamakau. R. Santhanam. Thalanayar S. a key limitation of the Fourier transform is that it does not include location information. Steven W.J." in 1966 Proc. San Diego. and Signal Processing (ICASSP 2007. edu/ Proceedings/ ICASSP 2007/ pdfs/ 0301385. W. berkeley. ISBN 0-9660176-3-3.. and thus has difficulty in representing transients. M. "High-speed convolution and correlation. Joseph P. Speech. section 30. R. only frequency information. 830–838.: Prentice Hall. prominent among which are wavelets. New York: IEEE Press. Havlicek. the discrete Fourier transform can be defined by using roots of unity in fields other than the complex numbers. they are better able to represent location. sometimes ). pdf). x1. Przebinda. The multidimensional DFT acts on multidimensional sequences. ISBN 0-13-754920-2. which can be viewed as functions This suggests the generalization to Fourier transforms on arbitrary finite groups. coded in C and under General Public License (GPL) (http://www. IEEE Trans. "The finite Fourier transform".1495. "The finite harmonic oscillator and its applications to sequences. and Waleed Abd El Maguid Ahmed (2004). A. 1972. Welch (1969).1109/29. Audio Electroacoustics 17 (2): 77–85.1972. doi:10.1016/j. "Eigenvalues and eigenvectors of the discrete Fourier transformation".14.1162036. • Shamgar Gurevich and Ronny Hadani (2009). Audio Electroacoust. • Magdy Tawfik Hanna. Lewis. W. • C. M. and J. Cooley. B. IEEE Sig. doi:10. Circ. "The discrete fractional Fourier transform". Parks (1972). Lett. "Eigenvectors and functions of the discrete Fourier transform". Applied and Computational Harmonic Analysis 27 (1): 87–99. Processing 36 (2): 285–286. Math.Ozaktas (2000). • Shamgar Gurevich.843762. Anal.. Opt.1162342. Dickinson and Kenneth Steiglitz (1982).com/fourierseries2/DFT_tutorial. "On the diagonalization of the discrete Fourier transform". doi:10.1364/JOSAA. "Analytic Expressions of Two Discrete Hermite-Gaussian Signals".org) • The DFT “à Pied”: Mastering The Fourier Transform in One Day (http://www. 24 External links • Interactive flash tutorial on the DFT (http://www. (Note that this paper has an apparent typo in its table of the eigenvalue multiplicities: the +i/−i columns are interchanged. Vargas-Rubio and Balu Santhanam (2005). Ronny Hadani. and is easily confirmed numerically.html) • Fast implementation of the DFT . "On the multiangle centered discrete fractional Fourier transform". arXiv:0808.909865. I 51 (11): 2245–2254.edu/~jos/mdft/ mdft. IEEE Trans. IEEE Trans.2008. Acoust. arXiv:0808. McClellan and T. "Fractional Fourier-Kravchuk transform". IEEE Transactions on Information Theory 54 (9): 4239–4253. Speech and Sig. Soc. Smith III (http://ccrma.1163843.N.926440. and P. Etcheto (1988). Syst.com/admin/ dft-a-pied/) . doi:10. J. 88 (2): 355–363.1109/TASSP.836850. J.2008. Circuits and Systems –II: Express Briefs.3281. Appl.fourier-series. preprint at.11. "Hermite-Gaussian-like eigenvectors of the discrete Fourier transform matrix based on the singular-value decomposition of its orthogonal projection matrices". doi:10. Kong (2008). • Natig M.stanford. Piron. IEEE Trans.Discrete Fourier transform • P.003.dspdimension.2005. H. doi:10..1982.1519. 12 (4): 273–276. • Bradley W. The correct table can be found in McClellan and Parks. doi:10. M. • Juan G.2007. Processing 30 (1): 25–31. doi:10.1016/0022-247X(82)90199-8. • J.) • F. IEEE Trans. doi:10. M.1109/78.1969. doi:10. A. 55 (1): 56–60. Grünbaum (1982).1109/TAU.839980. • F. on Signal Processing 48 (5): 1329–1337. Acoust. doi:10.1109/TIT. "The eigenvectors of the discrete Fourier transform". Duhamel. Candan. doi:10.fftw. communication and radar". P.001467. A 14 (7): 1467–1477.1109/LSP. 20 (1): 66–74. preprint at.html) • Mathematics of the Discrete Fourier Transform by Julius O.1109/TCSI. Atakishiyev and Kurt Bernardo Wolf (1997). Am.1109/TCSII. "On computing the inverse DFT". Speech and Sig. Nabila Philip Attalla Seif. Kutay and H.2004. Proc. and Nir Sochen (2008). • J. IEEE Trans.acha. IEEE Trans.1109/TAU. which means that this representation transforms linear differential equations with constant coefficients into ordinary algebraic ones (Evans 1998). Each transform used for analysis (see list of Fourier-related transforms) has a corresponding inverse transform that can be used for synthesis. the subject of Fourier analysis encompasses a vast spectrum of mathematics. cryptography. combinatorics. (Conte & de Boor 1980) Fourier transformation is also useful as a compact representation of a signal. and most generally via Pontryagin duality)(Rudin 1990). The Fourier components of each square are rounded to lower arithmetic precision. • The transforms are usually invertible. The decomposition process itself is called a Fourier transform. while the operation of rebuilding the function from these pieces is known as Fourier synthesis. Today. and weak components are eliminated entirely. probability theory. oceanography. • By the convolution theorem. number theory. and the general field is often known as harmonic analysis. numerical analysis. so that the remaining components can be stored very compactly. partial differential equations. option pricing. who showed that representing a function by a trigonometric series greatly simplifies the study of heat propagation. In mathematics. Moreover. diffraction. signal processing. are unitary as well (a property known as Parseval's theorem or. . Fourier analysis is named after Joseph Fourier. the behavior of a linear time-invariant system can be analyzed at each frequency independently. and other areas. the original concept of Fourier analysis has been extended over time to apply to more and more abstract and general situations. In image reconstruction. and then inverse-transformed to produce an approximation of the original image. with proper normalization. JPEG compression uses a variant of the Fourier transformation (discrete cosine transform) of small square pieces of a digital image. acoustics. as the Plancherel theorem. imaging. Fourier transforms turn the complicated convolution operation into simple multiplication.Fourier analysis 25 Fourier analysis In mathematics. more generally. Applications Fourier analysis has many scientific applications — in physics. The transform is often given a more specific name which depends upon the domain and other properties of the function being transformed. For example. • The exponential functions are eigenfunctions of differentiation. geometry. Fourier analysis is a subject area which grew from the study of Fourier series. the process of decomposing a function into simpler pieces is often called Fourier analysis. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions. each Fourier-transformed image square is reassembled from the preserved approximate components. Therefore. In the sciences and engineering. the term Fourier analysis often refers to the study of both operations. statistics. optics. This wide applicability stems from many useful properties of the transforms: • The transforms are linear operators and. • The discrete version of the Fourier transform (see below) can be evaluated quickly on computers using fast Fourier transform (FFT) algorithms. which means that they provide an efficient way to compute convolution-based operations such as polynomial multiplication and multiplying large numbers (Knuth 1997). stripe artifacts from strip aerial photography. and reversing the transformation. • Image processing to remove periodic or anisotropic artifacts such as jaggies from interlaced video. The resultant transforms can be seen as special cases or generalizations of each other. • X-ray crystallography to reconstruct a crystal structure from its diffraction pattern. • Cross correlation of similar images for co-alignment. (Continuous) Fourier transform Most often. the unqualified term Fourier transform refers to the transform of functions of a continuous real argument. The different names usually reflect different properties of the function or data being analyzed. See Fourier transform for even more information. F(ν) → ƒ(t) conventions for amplitude normalization and frequency scaling/units transform properties tabulated transforms of specific functions • an extension/generalization for functions of multiple dimensions. or wave patterns from radio frequency interference in a digital camera. • Digital radio reception with no superheterodyne circuit. concentrating them for easier detection and/or removal. • Removal of unwanted frequencies from an audio recording (used to eliminate hum from leakage of AC power into the signal. A large family of signal processing techniques consist of Fourier-transforming a signal.Fourier analysis 26 Applications in signal processing When processing signals. seismic waves. such as time (t). to eliminate the stereo subcarrier from FM radio recordings). • Many other forms of spectroscopy also rely upon Fourier Transforms to determine the three-dimensional structure and/or identity of the sample being analyzed. when pressed. The more common variants are shown below. • Noise gating of audio recordings to remove quiet background noise by eliminating Fourier components that do not exceed a preset amplitude. to reveal the two component tones and therefore which button was pressed. such as images . Some examples include: • Telephone dialing. including Infrared and Nuclear Magnetic Resonance spectroscopies. such as audio. some of which have different names. radio waves. as in a modern cell phone or radio scanner. and even images. light waves. Variants of Fourier analysis Fourier analysis has different forms. In this case the Fourier transform describes a function ƒ(t) in terms of basic complex exponentials of various frequencies. • Fourier transform ion cyclotron resonance mass spectrometry to determine the mass of ions from the frequency of cyclotron motion in a magnetic field. • Generation of sound spectrograms used to analyze sounds. the Fourier transform is given by the complex number: Evaluating this quantity for all values of ν produces the frequency-domain function. are each a sum of two separate tones (frequencies). ν. including: • • • • the inverse transform. In terms of ordinary frequency. • Equalization of audio recordings with a series of bandpass filters. manipulating the Fourier-transformed data in a simple way. Fourier analysis can isolate individual components of a compound waveform. Fourier analysis can be used to separate (or analyze) the telephone signal. the touch-tone signals for each telephone key. If information is lost. and the S[n] sequence are the Fourier coefficients. including the historical development. aside from possible convergence issues. in that it is the reconstruction of the original function that was transformed into the Fourier series coefficients. It can represent either: (a) a periodic function. The Fourier series is analogous to the inverse Fourier transform. When the coefficients are derived from a function. It follows that if ƒ(t) is τ-periodic (case a). Applications of the DTFT are not limited to sampled functions. including: • • • • • the inverse transform normalized frequency units windowing (finite-length sequences) transform properties tabulated transforms of specific functions . So it cannot represent any arbitrary function. See Fourier series for more information. or (b) a function that is defined only over a finite-length interval (or compact support). It can be applied to any discrete sequence. The Fourier series is a periodic function. A useful "discrete-time" function can be obtained by sampling a "continuous-time" function.u+τ ]. The amplitude and phase of a sinusoid can be combined into a single complex number. then only an approximation to the original transform. s(t). called a Fourier coefficient. The identity above is a result of the Poisson summation formula. ƒ(t) as follows: then.Fourier analysis 27 Fourier series A Fourier series is a representation of a function in terms of a summation of a potentially infinite number of harmonically-related sinusoids or complex exponential functions with different amplitudes and phases. s(t) will equal ƒ(t) in the interval [u. the discrete-time Fourier transform (DTFT) provides a useful frequency-domain transform. s(nT). can be obtained by looking at one period of the periodic function: which is the DTFT. See Discrete-time Fourier transform for more information on this and other topics. The general form of a Fourier series is: where τ is the period (case a) or the interval length (case b). Then the values produced by the Fourier series outside the finite interval are irrelevant. Discrete-time Fourier transform (DTFT) For functions of an integer index. The DTFT is also equivalent to the Fourier transform of a "continuous" function that is constructed by using the s[n] sequence to modulate a Dirac comb. S(f). s(t) and ƒ(t) are equal everywhere. which produces a sequence. for integer values of n and some time-interval T. Fourier analysis 28 Discrete Fourier transform (DFT) When s[n] is periodic. The term DFS. the DFT computes more closely-spaced samples of one period of S1/T(ƒ). • When N is larger than the non-zero portion of s[n]. and so the entire sequence can be described by just N coefficients. That is frequently done to provide an interpolated view of the DTFT. This sequence is N-periodic. See Discrete Fourier transform for much more information. analogous to case b (above) of the Fourier series. The . But a discrete subset of its values is sufficient to reconstruct/represent the (finite) portion of s[n] that was analyzed. When s[n] is not periodic. but that is not a requirement for the inverse transforms to work as shown. S1/T(ƒ) is another Dirac comb function. which makes it a practical and important transformation on computers. is sometimes used instead of DFT to convey that s[n] is periodic. known most often as the DFT and sometimes as the discrete Fourier series (DFS). • The term DFT is ambiguous in the sense that it does not tell us whether the inverse transform is valid for all n or just for a sequence of length N. i. and the is the summation of sequence In the columns labeled Finite duration or periodic the transforms are most useful when sequence are restricted to duration . over any interval of length N. but its non-zero portion has finite duration (N). S1/T(ƒ) is continuous and finite-valued. including: • • • • the inverse transform transform properties applications tabulated transforms of specific functions Summary This section consolidates the information above. using a consistent notation to emphasize the relationships between the four transforms. Notation: is a non-periodic function with Fourier transform is a time-interval of N samples taken at sample-intervals of time T. mentioned above. And that subset is again the DFT. And the integral formula for the coefficients simplifies to: for all integer values of k. The DFT can be computed using a fast Fourier transform (FFT) algorithm. whether the original s[n] sequence is periodic or finite. modulated by the coefficients of a Fourier series.e.[1] The DFT also has an inverse transform that reproduces the periodic s[n] sequence. known as zero-padding. is a periodic summation of is a periodic summation of is a periodic summation of discrete sequence is the integral of over any interval of length P. with period N. a fact that is often misinterpreted. which relates Fourier transforms and convolutions. As alternatives to the Fourier transform. there. so there is no loss of generality. the ) is always the inverse of these discrete-frequency transforms. See also the Pontryagin duality for the generalized underpinnings of the Fourier transform. while the Fourier transform has perfect frequency resolution. a function (of time) is a representation of a signal with perfect time resolution. with the wavelet analog of the (continuous) Fourier transform being the continuous wavelet transform. Fourier series) is seen by considering periodic function ( Continuous-time transforms Any duration (continuous frequency) Transform Finite duration or periodic (discrete frequencies) 29 to be the function being transformed. but no time information. the Gabor transform or fractional Fourier transform. in time–frequency analysis. or can use different functions to represent signals. Time–frequency transforms In signal processing terms.Fourier analysis periodic case (e. Inverse e Any duration (continuous frequency) Finite duration or periodic (discrete frequencies) Fourier transforms on arbitrary locally compact abelian topological groups The Fourier variants can also be generalized to Fourier transforms on arbitrary locally compact abelian topological groups. the Fourier transform takes functions on a group to functions on the dual group. . one uses time–frequency transforms to represent signals in a form that has some time information and some frequency information – by the uncertainty principle. there is a trade-off between these.g. These can be generalizations of the Fourier transform. but no frequency information. This treatment also allows a general formulation of the convolution theorem. such as the short-time Fourier transform. as in wavelet transforms and chirplet transforms. In either case. which are studied in harmonic analysis. Any periodic function can be represented as a periodic summation of another function. it takes a function from the time domain into the frequency domain. introducing the Fourier series. the transform has a standard interpretation as the frequency spectrum of the signal. in the case of a Fourier series or discrete Fourier transform. while Lagrange's work was a sine-only series (a form of discrete sine transform). They can equally be applied to analyze spatial frequencies. the sinusoids are harmonics of the fundamental frequency of the function being analyzed.Fourier analysis 30 History A primitive form of harmonic series dates back to ancient Babylonian mathematics. That is.[5] Technically. it is a decomposition of a function into sinusoids of different frequencies. variants of the discrete Fourier transform were used by Alexis Clairaut in 1754 to compute an orbit. which is the DFT of order 3.[4] The subsequent development of the field is known as harmonic analysis.[3] which has been described as the first formula for the DFT. and temporal frequencies. . the Fourier transform often takes a time series or a function of continuous time. Historians are divided as to how much to credit Lagrange and others for the development of Fourier theory: Daniel Bernoulli and Leonhard Euler had introduced trigonometric representations of functions. The first fast Fourier transform (FFT) algorithm for the DFT was discovered around 1805 by Carl Friedrich Gauss when interpolating measurements of the orbit of the asteroids Juno and Pallas.[2] In modern times. Clairaut's work was a cosine-only series (a form of discrete cosine transform). using what would today be called samples. This justifies their use in branches such diverse as image processing.[5] An early modern development toward Fourier analysis was the 1770 paper Réflexions sur la résolution algébrique des équations by Lagrange. and indeed for nearly any function domain. When the function ƒ is a function of time and represents a physical signal. although that particular FFT algorithm is more often attributed to its modern rediscoverers Cooley and Tukey. and maps it into a frequency spectrum..[6] Euler and Lagrange both discretized the vibrating string problem. and Carl Friedrich Gauss used trigonometric series to study the heat equation. which in the method of Lagrange resolvents used a complex Fourier decomposition to study the solution of a cubic:[7] Lagrange transformed the roots into the resolvents: where ζ is a cubic root of unity. heat conduction and automatic control. in computing the coefficients of a trigonometric series for a vibrating string.[4] so Fourier's contribution was mainly the bold claim that an arbitrary function could be represented by a Fourier series. notably Jean le Rond d'Alembert. whose crucial insight was to model all functions by trigonometric series.[6] [8] Interpretation in terms of time and frequency In signal processing.[4] and Lagrange had given the Fourier series solution to the wave equation. The magnitude of the resulting complex-valued function F at frequency ω represents the amplitude of a frequency component whose initial phase is given by the phase of F. but the breakthrough development was the 1807 paper Mémoire sur la propagation de la chaleur dans les corps solides by Joseph Fourier. where they were used to compute ephemerides (tables of astronomical positions). Fourier transforms are not limited to functions of time. and is also an early instance of representation theory. a true cosine+sine DFT was used by Gauss in 1805 for trigonometric interpolation of asteroid orbits.[4] and in 1759 by Joseph Louis Lagrange. A number of authors. S. H. Birkhäuser. Lis. and C. ISBN 978 0 52145718 7. Brack. S. and B. Cambridge University Press.M1) [4] Briggs. com/ books?id=KVeXG163BggC& pg=PA501) [8] Terras.W. google. T. Boca Raton. com/ books?id=-B2TA669dJMC& pg=PA30#PPA31. Elementary Numerical Analysis (Third ed. CRC Press.. Fourier analysis on finite groups and applications (http:/ / books.3. com/ books?id=H5smrEExNFUC& pg=PA11) Neugebauer. ISBN 978 0 89871342 8. Henson.. google. New York: McGraw Hill. Introduction to Fourier Analysis on Euclidean Spaces. since a Fourier series is a sum of sinusoids. Fourier Analysis on Groups. ISBN 978-048622332-2. com/ ?id=fye--TBu4T0C).). (4). p. and G. Palombi. William L. com/ ?id=KVeXG163BggC). A. . ISBN 0-8493-2876-4 • Rudin. arXiv:physics/0310126 [3] Terras. E. SIAM. Partial Differential Equations. The evolution of applied harmonic analysis: models of the real world (http:/ / books. google. Weiss (1971). org/ xpls/ abs_all. Heck. p. 11 (http:/ / books.M1) [6] Heideman. dspguide. Wiley-Interscience. com/ books?id=coq49_LRURUC& pg=PA2#PPA2. . p. The Scientist and Engineer's Guide to Digital Signal Processing (http://www. Basic algebra (http:/ / books. ieee. ISBN 0-691-08078-X . com/ ?id=-B2TA669dJMC). S. Inc.Fourier analysis 31 Notes [1] We note that DFS is actually a misnomer.). com/ books?id=fye--TBu4T0C& pg=PA62) Rota.." IEEE ASSP Magazine. ISBN 978 0 52145718 7. Gian-Carlo. The DFT: an owner's manual for the discrete Fourier transform (http:/ / books. Handbook of Integral Equations.M. ISBN 978 0 81764125 2.. Section 4. Johnson.V. Birkhäuser. American Mathematical Society.). Dover Publications. google. (1998). [2] Prestini. ISBN 978 0 81763866 5.305: Addison-Wesley Professional. " Gauss and the history of the fast Fourier transform (http:/ / ieeexplore. com/ books?id=coq49_LRURUC& pg=PA2#PPA4. (1999). Van Emden (1995).M1) References • Conte.. ISBN 9780849382758 • Kamen. (1997). The Art of Computer Programming Volume 2: Seminumerical Algorithms (3rd ed. com/ ?id=H5smrEExNFUC). ISBN 978 0 89871342 8. San Diego. Cambridge University Press. William L. Springer. com/ ?id=JVhTtVA2zr8C) (2 ed. Manzhirov (1998). google. jsp?arnumber=1162257). not the sequence of coefficients. p. Henson. 30 (http:/ / books. 501 (http:/ / books. Princeton University Press. google. (2001). ISBN 047152364X • Smith. . D. google. google. google. google. google. Anthony W. Calif. D. ISBN 0070662282 • Evans. 4 (http:/ / books. .com/pdfbook. com/ ?id=coq49_LRURUC). google. 14–21 (1984) [7] Knapp. Fabrizio (1997). 2 (http:/ / books. . ISBN 978 0 81763248 9. .M1) [5] Briggs. google. google. ISBN 0201896842 • Polyanin. Donald E. Steven W. E. Otto (1969) [1957].htm) (Second ed.3. L. Carl (1980). com/ books?id=-B2TA669dJMC& pg=PA30#PPA30. p. p. google. 62 (http:/ / books.. Fourier analysis on finite groups and applications (http:/ / books. (2006).: California Technical Publishing. M. ISBN 0-13-017293-6 • Knuth.C: Discrete Fourier transforms.. Matthias. Audrey (1999). The DFT : an owner's manual for the discrete Fourier transform (http:/ / books. SIAM.). Burrus. Elena (2004). and A. Van Emden (1995). com/ ?id=-B2TA669dJMC). ISBN 3540761241 • Howell. Analyzing shell structure from Babylonian and modern times. 1. com/ ?id=coq49_LRURUC). 31 (http:/ / books. pg. Walter (1990). Principles of Fourier Analysis.D. The Exact Sciences in Antiquity (http:/ / books. Audrey (1999). Indiscrete thoughts (http:/ / books. Brack-Bernsen. Kenneth B.. CRC Press. de Boor. . ISBN 0-9660176-3-3 • Stein. p. "Fundamentals of Signals and Systems Using the Web and Matlab". archive.ipmnet.html) by Steven Lehar. Lectures 7-15 make use of it.edu/~slehar/fourier/fourier. by Alan Peters Sine Sine Basic features Parity Domain Codomain Period Specific values At zero Maxima Minima Specific features Root Critical point Inflection point Fixed point kπ kπ-π/2 kπ 0 0 ((2k+1/2)π.Fourier analysis 32 External links • Tables of Integral Transforms (http://eqworld.-1) odd (-∞.1) ((2k-1/2)π. • An Intuitive Explanation of Fourier Theory (http://cns-alumni.∞) [-1. .1] 2π Variable k is an integer. • Lectures on Image Processing: A collection of 18 lectures in pdf format from Vanderbilt University. (http://www.ru/en/auxiliary/aux-inttrans. Lecture 6 is on the 1.bu.org/details/ Lectures_on_Image_Processing).and 2-D Fourier Transform.htm) at EqWorld: The World of Mathematical Equations. jya-ardha. Trigonometric functions are commonly defined as ratios of two sides of a right triangle containing the angle.[1] The word "sine" comes from a Latin mistranslation of the Arabic jiba. and average temperature variations throughout the year. the sine function gives the ratio of the length of the opposite side to the length of the hypotenuse. The sine function is commonly used to model periodic phenomena such as sound and light waves.Sine 33 In mathematics. Surya Siddhanta). if the hypotenuse is twice as long. To define the trigonometric functions for an acute angle A. the angle x is given in radians (π = 180°). according to the triangle postulate the inside angles of every triangle total 180° (π radians). The hypotenuse is always the longest side of a right-angled triangle. etc. sine gives the ratio of the length of the side opposite to an angle to the length of the hypotenuse. Also they are out of phase by 90°: = . The derivative of is . the two non-right angles total 90° (π/2 radians). allowing their extension to arbitrary positive and negative values and even to complex numbers. cos and sin give the respective x. Therefore respective trigonometric functions. y coordinates on a unit circle. in this case side a. The angle A (having measure α) is the angle between the hypotenuse and the adjacent line. in a right-angled triangle. the position and velocity of harmonic oscillators. so are the other sides. The three sides of the triangle are named as follows: The sine function graphed on the Cartesian plane. • The opposite side is the side opposite to the angle we are interested in (angle A). In this graph. or between the hypotenuse and the "adjacent" side (cosine) or between the "opposite" and the "adjacent" side (tangent). For example. sunlight intensity and day length. And for a given angle. The function sine can be traced to the jyā and koṭi-jyā functions used in Gupta period Indian astronomy (Aryabhatiya. via translation from Sanskrit to Arabic and then from Arabic to Latin. The sine and cosine functions are related in multiple ways. In our case . Therefore. depending only on the size of the angle. The sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse. in this case side h. the sine function is a function of an angle. In ordinary Euclidean geometry. In a right triangle. Right-angled triangle definition For any similar triangle the ratio of the length of the sides remains the same. Sine is usually listed first amongst the trigonometric functions. which is a transliteration of the Sanskrit word for half the chord. The following definition applies to such angles. More modern definitions express them as infinite series or as solutions of certain differential equations.[2] For the angle α. in this case side b. so each of these angles must be greater than 0° and less than 90°. start with any right triangle that contains the angle A. • The adjacent side is the side that is in contact with (adjacent to) both the angle we are interested in (angle A) and the right angle. express those ratios: between the hypotenuse and the "opposite" side to an angle A in question (see illustration) in the case of sine function. and can equivalently be defined as the lengths of various line segments from a unit circle. • The hypotenuse is the side opposite the right angle. This can also be achieved by requiring certain symmetries and that sine be a periodic function. Unlike the definitions with the right triangle or slope. respectively. the angle can be extended to the full set of real arguments by using the unit circle. making an angle of θ with the positive half of the x-axis. The point's distance from the origin is always 1. 0) in the Cartesian coordinate system. • Rise is equal to sin θ multiplied by the length of the line segment In contrast. intersect the unit circle. Let a line through the origin. cosine is used for the telling the run from the angle.y) on the circle of unit radius at an obtuse angle θ > π/2 . Relation to the unit circle In trigonometry. Relation to slope The trigonometric functions can be defined in terms of the rise. and when it has a length of 1 it is also equivalent to the radius of the unit circle. Point P(x. and slope of a line segment relative to some horizontal line. run.Sine 34 Note that this ratio does not depend on the size of the particular right triangle chosen. The xand y-coordinates of this point of intersection are equal to cos θ and sin θ. The radius has a length of 1. The unit circle. as long as it contains the angle A. The line segment is the equivalent of the hypotenuse in the right-triangle. The variable t is an angle measure. • When the length of the line segment is 1. since all such triangles are similar. Arctan is used for telling the angle from the slope. a unit circle is the circle of radius one centered at the origin (0. and tangent is used for telling the slope from the angle. Illustration of a unit circle. sine takes an angle and tells the rise • Sine takes an angle and tells the rise per unit length of the line segment. but also sin(π) = 0. Cosecant gives the ratio of the length of the hypotenuse to the length of the opposite side: Inverse The inverse function of sine is arcsine (arcsin or asin) or inverse sine (sin−1). k is some integer: . the reciprocal of sin(A) is csc(A). It follows that the arcsine function is also multivalued: arcsin(0) = 0. sin(0) = 0. sin(2π) = 0 etc. the function may be restricted to its principal branch. for each x in the domain the expression arcsin(x) will evaluate only to a single value.Sine 35 Identities Exact identities (using radians): These apply for all values of . Arcsin is the inverse of sin. or cosec(A). i. called its principal value. When only one value is desired. but also arcsin(0) = π. it is not an exact inverse function but a partial inverse function. etc. Animation showing the graphing process of y = sin x (where x is the angle in radians) using a unit circle Reciprocal The reciprocal of sine is cosecant. As sine is multivalued.e. With this restriction. The usual principal values of the arcsin(x) function graphed on the cartesian plane. For example. arcsin(0) = 2π. Sine in terms of the other common trigonometric functions: The four quadrants of a Cartesian coordinate system. Other trigonomic functions It is possible to express any trigonometric function in terms of any other (up to a plus or minus sign. or using the sign function).Sine Arcsin satisfies: 36 and Calculus For the sine function: The derivative is: The antiderivative is: C denotes the constant of integration. fθ Using plus/minus (±) fθ= ± per Quadrant I II + + III IV + - Using sign function (sgn) fθ= cos + + cot + + - - + tan + - - + + - - + . using the Cartesian coordinate system. Inflection Maxima Root. it is positive in the first quadrant. or use . Quadrant 1st Quadrant 2nd Quadrant 3rd Quadrant 4th Quadrant Degrees Radians Value Sign Monotony Convexity increasing concave decreasing concave decreasing convex increasing convex Points between the quadrants. . The basic relationship between the sine and the cosine can also be expressed as the Pythagorean trigonometric identity: where sin2x means (sin(x))2. get the value using the fact the sine function has a period of 360° (or 2π rad): . k is an integer. Properties relating to the quadrants Over the four quadrants of the sine function is as follows.Sine 37 sec + + - + - - + Note that for all equations which use plus/minus (±). Inflection Minima For arguments outside of those in the table. The quadrants of the unit circle and of sin x. Degrees Radians 0 ≤ x < 2π 0 Radians sin x 0 1 0 -1 Point type Root. for example. it can be shown that the derivative of sine is cosine and the derivative of cosine is the negative of sine. . Euler's formula) are. Using the reflection from the calculated geometric derivation of the sine is with the 4n + k-th derivative at the point 0: The sine function (blue) is closely approximated by its Taylor polynomial of degree 7 (pink) for a full cycle centered on the origin. In most branches of mathematics beyond practical geometry. again. so Mathematically important relationships between the sine and cosine functions and the exponential function (see. the number of radians is y = πx /180. This gives the following Taylor series expansion at x = 0. angles are generally measured in radians. elegant when the functions' arguments are in radians and messy otherwise. which is obtained by omitting the factorials in the denominator. One can then use the theory of Taylor series to show that the following identities hold for all real numbers x (where x is the angle in radians) :[3] If x were expressed in degrees then the series would contain messy factors involving powers of π/180: if x is the number of degrees.Sine 38 Series definition Using only geometry and properties of limits. A similar series is Gregory's series for arctan. Sine 39 Continued fraction The sine function can also be represented as a generalized continued fraction: The fixed point iteration xn+1 = sin xn with initial value x0 = 2 converges to 0. B and C: This is equivalent to the equality of the first three expressions below: where R is the triangle's circumradius. The continued fraction representation expresses the real number values. b. This is a common situation occurring in triangulation. a technique to determine unknown distances by measuring two angles and an accessible enclosed distance. It can be proven by dividing the triangle into two right ones and using the above definition of sine. The law of sines is useful for computing the lengths of the unknown sides in a triangle if two angles and one side are known. of the sine function. both rational and irrational. . and c and angles opposite those sides A. Law of sines The law of sines states that for an arbitrary triangle with sides a. 707106781186548 0. sin θ). The angles are given in degrees and radians. together with the corresponding intersection point on the unit circle.258819045102521 0. (cos θ.866025403784439 0.5 0.965925826289068 1 A memory aid (note it does not include 15° and 75°): . x (angle) Degrees Radians Grads 0° 180° 15° 165° 30° 150° 45° 135° 60° 120° 75° 105° 90° 0 0g 200g 162⁄3g 1831⁄3g 331⁄3g 1662⁄3g 50g 150g 662⁄3g 1331⁄3g 831⁄3g 1162⁄3g 100g 1 Exact 0 sin x Decimal 0 0.Sine 40 Values sin(x) Some common angles (θ) shown on the unit circle. Sine 41 x in degrees x in radians 0° 0 30° π/6 45° π/4 60° π/3 90° π/2 90 degree increments: x in degrees 0° 90° 180° 270° 360° x in radians 0 π/2 0 1 π 0 3π/2 -1 2π 0 Other values not listed above: A019812 [4] A019815 [5] A019818 [6] A019821 [7] A019827 [8] A019830 [9] A019833 [10] A019836 [11] A019842 [12] A019845 [13] A019848 [14] A019851 [15] For angles greater than 2π or less than −2π, simply continue to rotate around the circle; sine periodic function with period 2π: for any angle θ and any integer k. The primitive period (the smallest positive period) of sine is a full circle, i.e. 2π radians or 360 degrees. Sine 42 Relationship to complex numbers Sine is used to determine the imaginary part of a complex number given in polar coordinates (r,φ): An illustration of the complex plane. The imaginary numbers are on the vertical coordinate axis. the imaginary part is: r and φ represent the magnitude and angle of the complex number respectively. i is the imaginary unit. z is a complex number. Although dealing with complex numbers, sine's parameter in this usage is still a real number. Sine can also take a complex number as an argument. Sine with a complex argument The definition of the sine function for complex arguments z: Domain coloring of sin(z) over (-π,π) on x and y axes. Brightness indicates absolute magnitude, saturation represents imaginary and real magnitude. Sine 43 sin(z) as a vector field where i 2 = −1. This is an entire function. Also, for purely real x, For purely imaginary numbers: It is also sometimes useful to express the complex sine function in terms of the real and imaginary parts of its argument: Usage of complex sine sin z is found in the functional equation for the Gamma function, . Which in turn is found in the functional equation for the Riemann zeta-function, As a holomorphic function, sin z is a 2D solution of Laplace's equation: the trigonometric functions as they are in use today were developed in the medieval period.. tang..Sine 44 Complex graphs Sine function in the complex plane real component imaginary component magnitude Sin-1 in the complex plane real component imaginary component magnitude History While the early study of trigonometry can be traced to antiquity.. In a paper published in 1682. also defining them as infinite series and presenting "Euler's formula".. as well as the near-modern abbreviations sin. Surya Siddhanta). this work was finished by Rheticus' student Valentin Otho in 1596. 'cos'. The chord function was discovered by Hipparchus of Nicaea (180–125 BC) and Ptolemy of Roman Egypt (90–165 AD). The Opus palatinum de triangulis of Georg Joachim Rheticus. The function sine (and cosine) can be traced to the jyā and koṭi-jyā functions used in Gupta period Indian astronomy (Aryabhatiya. and 'tan' is by the 16th century French mathematician Albert Girard.[17] Leonhard Euler's Introductio in analysin infinitorum (1748) was mostly responsible for establishing the analytic treatment of trigonometric functions in Europe. with tables for all six trigonometric functions. and cosec.. Leibniz proved that sin x is not an algebraic function of x.[16] Roger Cotes computed the derivative of sine in his Harmonia Mensurarum (1722). sec.[1] The first published use of the abbreviations 'sin'. these were further promulgated by Euler (see below). cot.[18] . was probably the first in Europe to define trigonometric functions directly in terms of right triangles instead of circles. via translation from Sanskrit to Arabic and then from Arabic to Latin. a student of Copernicus. cos. Springer. This can lead to very different results for different algorithms in special circumstances. e. wikipedia. or range of input values accepted. accuracy. Some CPU architectures have a built-in instruction for sin. org/ wiki/ Oeis%3Aa019836 [12] http:/ / en. edu/ people/ rickey/ hm/ CalcNotes/ default. org/ wiki/ Oeis%3Aa019848 [15] http:/ / en. including the Intel x86 FPU. org/ wiki/ Oeis%3Aa019818 [7] http:/ / en. John Wiley & Sons. org/ wiki/ Oeis%3Aa019827 [9] http:/ / en. 14/31 (http:/ / www. along with other trigonometric functions. Notes [1] Boyer. In programming languages. wikipedia. A History of Mathematics (Second ed. was to pre-calculate a table of sin values. chapter 3. org/ wiki/ Oeis%3Aa019821 [8] http:/ / en. the word sine derives from the Sanskrit word for chord. the most widely-used standard for floating-point computation.[21] Algorithms for calculating sin may be balanced for such constraints as speed. [17] " Why the sine has a simple derivative (http:/ / www.Sine 45 Etymology Etymologically. math. htm) by V. sinus (which means "bosom" or "bay" or "fold") [19] [20] The English form sine was introduced in the 1590s. math. Inc. wikipedia.1. ac. This allowed results to be looked up from a table rather than being calculated in real time. jp/ ~bjorner/ ae-is-budapest/ talks/ Sept20pm2_Zimmermann. [21] Grand Challenges of Informatics. There is no standard algorithm for calculating sin.1. A history of mathematics. math. ﺟــﻴــﺐabbreviated jb .). org/ wiki/ Oeis%3Aa019842 [13] http:/ / en. A once common programming optimization. usma. sidebar 6. p210. is widely available across programming languages and platforms. in Historical Notes for Calculus Teachers (http:/ / www. usma. (1991). regarding the etymology. "jb" was interpreted as the word jaib . 210. p210. [3] See Ahlfors. Paul Zimmermann. says nothing on the topic of calculating trigonometric functions such as sin. org/ wiki/ Oeis%3Aa019812 [5] http:/ / en. Carl B. wikipedia. edu/ people/ rickey/ hm/ CalcNotes/ Sine-Deriv. Frederick Rickey (http:/ / www. September 20. jiva*(jya being its more popular synonym). usma. ﺟــﻴــﺐwhich means "bosom". With modern CPU architectures this method typically offers no advantage. sin is usually either a built in function or found within the language's standard math library. for example a value per degree. wikipedia.g. IEEE 754-2008. Software implementations The sin function. pages 43–44. 2006 – p. pdf) . wikipedia. edu/ people/ rickey/ ) [18] See Boyer (1991). A history of mathematics. wikipedia. pdf)". org/ wiki/ Oeis%3Aa019845 [14] http:/ / en. [20] Victor J Katx. wikipedia. when the Arabic text was translated in the 12th century into Latin by Gerard of Cremona. jaist. sin(1022). org/ wiki/ Oeis%3Aa019830 [10] http:/ / en. The translator used the Latin equivalent for "bosom". wikipedia. such as for very large inputs. This was transliterated in Arabic as jiba . sidebar 6. wikipedia. wikipedia. [2] Victor J Katx. org/ wiki/ Oeis%3Aa019833 [11] http:/ / en. Elements of the History of Mathematics. org/ wiki/ Oeis%3Aa019851 [16] Nicolás Bourbaki (1994). p. [19] See Maor (1998). [4] http:/ / en. ﺟــــﺐSince Arabic is written without short vowels. ISBN 0-471-54397-7. portability. org/ wiki/ Oeis%3Aa019815 [6] http:/ / en. used especially in 3D graphics.. wikipedia. The hypotenuse is always the longest side of a right-angled triangle. for instance. The most familiar trigonometric functions are the sine. and then derive these relations. and average temperature variations through the year. In this use. and physics. in navigation. A common use in elementary physics is resolving a vector into Cartesian coordinates. engineering. • The adjacent side is the side having both the angles of interest (angle A and right-angle C). Especially with the last four. in this case side h. start with any right triangle that contains the angle A. where a triangle is formed by a ray originating at the origin and making some angle with the x-axis. cosine. That is. and can equivalently be defined as the lengths of various line segments from a unit circle. among many other applications. Trigonometric functions are commonly defined as ratios of two sides of a right triangle containing the angle. More modern definitions express them as infinite series or as solutions of certain differential equations. and tangent. Trigonometric functions have a wide range of uses including computing unknown lengths and angles in triangles (often right triangles). The three sides of the triangle are named as follows: • The hypotenuse is the side opposite the right angle. If the hypotenuse is twice as long. It is these ratios that the trigonometric functions express. Trigonometric functions are important in the study of triangles and modeling periodic phenomena. The sine and cosine functions are also commonly used to model periodic function phenomena such as sound and light waves. there are six basic trigonometric functions. sunlight intensity and day length. allowing their extension to arbitrary positive and negative values and even to complex numbers.Trigonometric functions 46 Trigonometric functions In mathematics. and the tangent function gives the slope (y-component divided by the x-component). so are the sides. the sine of the angle gives the length of the y-component (rise) of the triangle. these relations are often taken as the definitions of those functions. or by other means. In the context of the standard unit circle with radius 1. tabulated here with equations that relate them to one another. They are used to relate the angles of a triangle to the lengths of the sides of a triangle. More precise definitions are detailed below. the trigonometric functions (also called circular functions) are functions of an angle. in this case side a. trigonometric functions are used. . In modern usage. the cosine gives the length of the x-component (run). for any similar triangle the ratio of the hypotenuse (for example) and another of the sides remains the same. Right-angled triangle definitions The notion that there should be some standard correspondence between the lengths of the sides of a triangle and the angles of the triangle comes as soon as one recognizes that similar triangles maintain the same ratios between their sides. the position and velocity of harmonic oscillators. but one can define them equally well geometrically. in this case side b. To define the trigonometric functions for the angle A. • The opposite side is the side opposite to the angle we are interested in (angle A). The trigonometric functions are summarized in the following table and described in more detail below. π − θ. π − θ. For example. and if the range of θ is extended to additional rotations. this behavior repeats periodically with a period 2π.90°) as expressed in interval notation. Therefore. π + θ. the figure shows sin θ for angles θ. The value of the sine repeats itself apart from sign in all four quadrants. The angle θ is the angle between the hypotenuse and the adjacent line – the angle at A in the accompanying diagram. (Top): Trigonometric function sinθ for selected angles θ. The following definitions apply to angles in this 0° – 90° range. so each of these angles must be in the range of (0°. (Bottom) Graph of sine function versus angle. in a right-angled triangle. and 2π − θ depicted on the unit circle (top) and as a graph (bottom).Trigonometric functions 47 In ordinary Euclidean geometry. according to the triangle postulate the inside angles of every triangle total 180° (π radians). or by requiring certain symmetries and that they be periodic functions. They can be extended to the full set of real arguments by using the unit circle. Function Sine Cosine Tangent Abbreviation sin cos tan (or tg) Description opposite / hypotenuse adjacent / hypotenuse opposite / adjacent adjacent / opposite hypotenuse / adjacent hypotenuse / opposite Identities (using radians) Cotangent cot (or ctg or ctn) Secant Cosecant sec csc (or cosec) . and 2π − θ in the four quadrants. Angles from the top panel are identified. π + θ. the two non-right angles total 90° (π/2 radians). thus angles are being measured in radians. and tangent The sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse. "moving" means depending on θ. ("Fixed" in this context means not moving as θ changes. In our case The sine. cosine. and secant functions of an angle constructed geometrically in terms of a unit circle. tan θ goes from 0 to ∞.Trigonometric functions 48 Sine.) Thus. tangent. The secant and tangent functions rely on a fixed vertical line and the sine function on a moving vertical line. sin θ goes from 0 to 1. as θ goes from 0 up to a right angle. . The number θ is the length of the curve. and sec θ goes from 1 to ∞. or cosec(A). is the reciprocal of sin(A).e.use horizontal lines where the others use vertical lines. cotangent. In our case The tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side (called so because it can be represented as a line segment tangent to the circle). i. Note that this ratio does not depend on size of the particular right triangle chosen.e. since all such triangles are similar. and cosecant functions of an angle θ constructed geometrically in terms of a unit circle. as long as it contains the angle A. the ratio of the length of the adjacent side to the length of the opposite side: . i. The functions whose names have the prefix co.Trigonometric functions 49 The cosine.e. i. the ratio of the length of the hypotenuse to the length of the adjacent side: The cotangent cot(A) is the reciprocal of tan(A). The cosecant csc(A). the ratio of the length of the hypotenuse to the length of the opposite side: The secant sec(A) is the reciprocal of cos(A).[1] In our case The acronyms "SOHCAHTOA" and "OHSAHCOAT" are commonly used mnemonics for these ratios. Reciprocal functions The remaining three functions are best defined using the above three functions. The cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse. e. if the line segment has length 5. Cosine takes an angle and tells the run when the length of the line is 1. The unit circle definition does. run is second. With a segment length of 1 (as in a unit circle) the following correspondence of definitions exists: 1. Sine takes an angle and tells the rise when the length of the line is 1. indeed it relies on right triangles for most angles.Trigonometric functions 50 Slope definitions Equivalent to the right-triangle definitions the trigonometric functions can be defined in terms of the rise. This shows the main use of tangent and arctangent: converting between the two ways of telling the slant of a line. (Note that the arctangent or "inverse tangent" is not to be confused with the cotangent. and tells the rise when the run is 1. angles and slopes. The three main trigonometric functions are commonly taught in the order sine. permit the definition of the trigonometric functions for all positive and negative arguments. some common angles. From the Pythagorean theorem the equation for the unit circle is: The unit circle In the picture. measured in radians. Measurements in the counterclockwise direction are positive angles and measurements in the clockwise direction are negative angles. The slope is commonly taught as "rise over run" or rise/run. . not just for angles between 0 and π/2 radians. the circle of radius one centered at the origin. For instance. and slope of a line segment relative to some horizontal line. i. The unit circle definition provides little in the way of practical calculation. it does affect rise and run. 3. tangent. rise is first. which is cosine divided by sine. 2. Sine is first. run. however. Tangent takes an angle and tells the slope. Cosine is second. Tangent is the slope formula that combines the rise and run. just multiply the sine and cosine by the line length. To adjust and find the actual rise and run when the line does not have a length of 1. cosine. It also provides a single visual picture that encapsulates at once all the important triangles. are given.) While the length of the line segment makes no difference for the slope (the slope does not depend on the length of the slanted line).. the run at an angle of 7° is 5 cos(7°) Unit-circle definitions The six trigonometric functions can also be defined in terms of the unit circle. e. the radius is equal to the hypotenuse and has length 1. The smallest positive period of a periodic function is called the primitive period of the function. 2π radians or 360 degrees. π radians or 180 degrees. making an angle of θ with the positive half of the x-axis. . for any angle θ and any integer k. i. Above. • The primitive period of the tangent or cotangent is only a half-circle. For angles greater than 2π or less than −2π. so we have sin θ = y/1 and cos θ = x/1. only sine and cosine were defined directly by the unit circle. The triangle in the graphic enforces the formula.e. or cosecant is also a full circle.Trigonometric functions Let a line through the origin. The xand y-coordinates of this point of intersection are equal to cos θ and sin θ. The primitive period of the sine or cosine is a full circle. but other trigonometric functions can be defined by: So : • The primitive period of the secant. The unit circle can be thought of as a way of looking at an infinite number of triangles by varying the lengths of their legs but keeping the lengths of their hypotenuses equal to 1. Note that these values can easily be memorized in the form 51 but the angles are not equally spaced. i. The values for 15°. 2π radians or 360 degrees. respectively. intersect the unit circle. 54° and 75° are slightly more complicated. i. simply continue to rotate around the circle.e. sine and cosine are periodic functions with period 2π: The sine and cosine functions graphed on the Cartesian plane. • DE is exsec(θ) = sec(θ) − 1 (the portion of the secant outside. • The graph of the tangent function also has a vertical asymptote at θ = (k + 1/2)π. Cotangent (dotted) • sec(θ) = OE and csc(θ) = OF are segments of secant lines (intersecting the circle at two points). or ex. respectively. • cos(θ) is the horizontal distance OC. • The function changes slowly around angles of kπ. because the function approaches infinity as θ approaches (k + 1/2)π from the left and minus infinity as it approaches (k + 1/2)π from the right. Trigonometric functions: Sine. • Its θ-intercepts correspond to those of sin(θ) while its undefined values correspond to the θ-intercepts of cos(θ). all of the basic trigonometric functions can be defined in terms of a unit circle centered at O (as shown in the picture to the right). a definition introduced in India[2] (see history).[3] ) . and the basic trigonometric identities can also be proven graphically. cot(θ) is another tangent segment. sin(θ) is AC (half of the chord). Cosecant (dotted). hence the word tangent for this function. and can also be viewed as projections of OA along the tangent at A to the horizontal and vertical axes. and versin(θ) = 1 − cos(θ) is CD. • tan(θ) is the length of the segment AE of the tangent line through A. the θ-intercepts of the cosine function. • In particular. Tangent. AF. but changes rapidly at angles close to (k + 1/2)π. (Many similar constructions are possible. Cosine. and similar such geometric definitions were used historically. for a chord AB of the circle. where θ is half of the subtended angle. All of the trigonometric functions of the angle θ can be constructed geometrically in terms of a unit circle centered at O. Alternatively.Trigonometric functions 52 The image at right includes a graph of the tangent function. • From these constructions. the circle). Secant (dotted). it is easy to see that the secant and tangent functions diverge as θ approaches π/2 (90 degrees) and that the cosecant and cotangent diverge as θ approaches zero. from the foundations of the real number system. These identities are sometimes taken as the definitions of the sine and cosine function. Other series can be found. Bn is the nth Bernoulli number. Combining these two series gives Euler's formula: cos x + i sin x = eix.) One can then use the theory of Taylor series to show that the following identities hold for all real numbers x:[4] The sine function (blue) is closely approximated by its Taylor polynomial of degree 7 (pink) for a full cycle centered on the origin. all angles are measured in radians. in Fourier series). and generally in calculus. since the theory of infinite series can be developed. see also the significance of radians below.Trigonometric functions 53 Series definitions Using only geometry and properties of limits. The differentiability and continuity of these functions are then established from the series definitions alone. it can be shown that the derivative of sine is cosine and the derivative of cosine is the negative of sine. They are often used as the starting point in a rigorous treatment of trigonometric functions and their applications (e.. and En (below) is the nth Euler number. Tangent . independent of any geometric considerations.g. (Here.[5] For the following trigonometric functions: Un is the nth up/down number. it can be . They have the same Taylor series. have a combinatorial interpretation: they enumerate alternating permutations of finite sets of odd cardinality. called the "secant numbers". the numerators.[9] By combining the expressed as an absolutely convergent series: -th with the -th term. and so the trigonometric functions are defined on the complex numbers using the Taylor series above.[7] Cotangent From a theorem in complex analysis. there is a unique analytic continuation of this real function to the domain of complex numbers.Trigonometric functions When this series for the tangent function is expressed in a form in which the denominators are the corresponding factorials. There is a series representation as partial fraction expansion where just translated reciprocal functions are summed up. have a combinatorial interpretation: they enumerate alternating permutations of finite sets of even cardinality. such that the poles of the cotangent function and the reciprocal functions match:[8] This identity can be proven with the Herglotz trick. called the "tangent numbers". the numerators.[6] Cosecant 54 Secant When this series for the secant function is expressed in a form in which the denominators are the corresponding factorials. we can parametrize this circle in terms of cosines and sines. Furthermore. For example. starting with the 2-D orthogonal components of the unit circle. . for purely real x. Also. this allows for the definition of the trigonometric functions for complex arguments z: where i 2 = −1. respectively. This exhibits a deep relationship between the complex sine and cosine functions and their real (sin. trigonometric functions become essential in the geometric interpretation of complex analysis. It is also sometimes useful to express the complex sine and cosine functions in terms of the real and imaginary parts of their arguments. In this way. The sine and cosine defined by this are entire functions. and as above. of the complex exponential function when its argument is purely imaginary: Euler's formula illustrated with the three dimensional helix. parametrized by e ix. cosh) counterparts. if one considers the unit circle in the complex plane.Trigonometric functions 55 Relationship to exponential function and complex numbers It can be shown from the series definitions[10] that the sine and cosine functions are the imaginary and real parts. This identity is called Euler's formula. cos) and hyperbolic real (sinh. sine and cosine (using θ = t ). with the above identity. the relationship between the complex exponential and the trigonometric functions becomes more apparent. ) It turns out that this differential equation can be used not only to define the sine and cosine functions but also to prove the trigonometric identities for the sine and cosine functions. Hue varies with argument. (See linear differential equation.Trigonometric functions Complex graphs In the following graphs. The tangent function is the unique solution of the nonlinear differential equation satisfying the initial condition y(0) = 0. measured from the positive real axis. together they form a basis of V. and the range values are indicated at each point by color. (more) 56 Trigonometric functions in the complex plane Definitions via differential equations Both the sine and cosine functions satisfy the differential equation That is to say. each is the additive inverse of its own second derivative. or angle.[11] . the observation that sine and cosine satisfies y′′ = −y means that they are eigenfunctions of the second-derivative operator. • the sine function is the unique solution satisfying the initial condition • the cosine function is the unique solution satisfying the initial condition and . the domain is the complex plane pictured. Brightness indicates the size (absolute value) of the range value. with black being zero. Further. This method of defining the sine and cosine functions is essentially equivalent to using Euler's formula. Within the 2-dimensional function space V consisting of all solutions of this equation. Since the sine and cosine functions are linearly independent. There is a very interesting visual proof that the tangent function satisfies this differential equation. the sum formulas reduce to simpler equations known as the double-angle formulae. much like the logarithm function. using arguments that date to Ptolemy. the Pythagorean identity is written where sin2 x + cos2 x is standard notation for (sin x)2 + (cos x)2. In symbolic form. These identities can also be used to derive the product-to-sum identities that were used in antiquity to transform the product of two numbers into a sum of numbers and greatly speed operations. then This means that the second derivative of a sine in degrees does not satisfy the differential equation but rather The cosine's second derivative behaves similarly. Other key relationships are the sum and difference formulas. which give the sine and cosine of the sum and difference of two angles in terms of sines and cosines of the angles themselves. the square of the sine plus the square of the cosine is 1. This means that these sines and cosines are different functions. Here. These can be derived geometrically. k is a constant that represents a mapping between units. and that the fourth derivative of sine will be sine again only if the argument is in radians. Identities Many identities interrelate the trigonometric functions. In particular. . Among the most frequently used is the Pythagorean identity. When the two angles are equal. If an argument to sine or cosine in radians is scaled by frequency.Trigonometric functions 57 The significance of radians Radians specify an angle by measuring the length around the path of the unit circle and constitute a special argument to the sine and cosine functions. This is easy to see by studying a right triangle of hypotenuse 1 and applying the Pythagorean theorem. If x is in degrees. then the derivatives will scale by amplitude. One can also produce them algebraically using Euler's formula. which states that for any angle. only sines and cosines that map radians to ratios satisfy the differential equations that classically describe them. using the periodicity and symmetries of the trigonometric functions. Taylor and Laurent series) with range reduction and a table lookup—they first look up the closest angle in a small table. Below is the list of the derivatives and integrals of the six basic trigonometric functions. The first step in computing any trigonometric function is range reduction—reducing the given angle to a "reduced angle" inside a small range of angles. are also possible.Trigonometric functions 58 Calculus For integrals and derivatives of trigonometric functions. Other derivations. however. Computation The computation of trigonometric functions is a complicated subject. The number C is a constant of integration. and then use the . Such tables have been available for as long as trigonometric functions have been described (see History below). and typically for higher or variable precisions. Modern computers use a variety of techniques. which can today be avoided by most people because of the widespread availability of computers and scientific calculators that provide built-in trigonometric functions for any angle. This section. we say that there exists exactly one pair of real functions — and — such that for all real numbers and . and such derivations can be extended to the complex numbers. for example. and Padé approximation. best uniform approximation. As an example. one can define the trigonometric functions using functional equations based on properties like the sum and difference formulas. describes details of their computation in three important contexts: the historical use of trigonometric tables. see the relevant sections of Differentiation of trigonometric functions. is to combine a polynomial or rational approximation (such as Chebyshev approximation. people typically evaluated trigonometric functions by interpolating from a detailed table of their values. especially on higher-end processors with floating point units. and a few "important" angles where simple exact values are easily found. Lists of integrals and List of integrals of trigonometric functions. Prior to computers. and were typically generated by repeated application of the half-angle and angle-addition identities starting from a known value (such as sin(π/2) = 1). starting from other functional equations. Symbolically. the following equations hold: with the added condition that . say 0 to π/2. Definitions using functional equations In mathematical analysis. one can prove that only two real functions satisfy those conditions. the modern techniques used by computers. calculated to many significant figures. this derivation can be used to define trigonometry in Galois fields.[12] One common method. Taking as given these formulas and the Pythagorean identity. we can choose tangent of an angle of Therefore: radians (45°) can then be found using the Pythagorean theorem: radians (45°). For example. By dividing it into two. which uses only addition. All its angles are π/3 radians (60 degrees). Function sin cos tan cot sec csc .[13] Devices that lack hardware multipliers often use an algorithm called CORDIC (as well as related techniques). For very high precision calculations. Then the . For this triangle. and table lookup. This yields: Special values in trigonometric functions There are some commonly used special values in trigonometric functions.[14] Finally. for some simple angles.Trigonometric functions polynomial to compute the correction. bitshift. the sine. the next largest side =(√3)/2 and the hypotenuse = 1. Consider a right triangle where the two other angles are equal. These methods are commonly implemented in hardware floating-point units for performance reasons. The values of sine. which itself approximates the trigonometric function by the (complex) elliptic integral. as shown in the following table. we obtain a right triangle with π/6 radians (30 degrees) and π/3 radians (60 degrees) angles. we start with an equilateral triangle of side length 1. the values can be easily computed by hand using the Pythagorean theorem. trigonometric functions can be approximated by the arithmetic-geometric mean. subtraction. cosine and tangent of any integer multiple of radians (3°) can be found exactly by hand. the shortest side = 1/2. as in the following examples. when series expansion convergence becomes too slow. cosine and 59 To determine the trigonometric functions for angles of π/3 radians (60 degrees) and π/6 radians (30 degrees). and therefore are both length of side b and the length of side a are equal. The arcsine. Just like the sine and cosine. Using the complex logarithm. the inverse functions could be confused with the multiplicative inverses of the functions. for example. though "arcsec" can be confused with "arcsecond". and hence not injective. When this notation is used. These functions may also be defined by proving that they are antiderivatives of other functions. For example. can be written as the following integral: Analogous formulas for the other functions can be found at Inverse trigonometric functions. one can generalize all these functions to complex arguments: .Trigonometric functions 60 Inverse functions The trigonometric functions are periodic. the inverse trigonometric functions can also be defined in terms of infinite series. etc. the notations sin−1 and cos−1 are often used for arcsin and arccos. The notation using the "arc-" prefix avoids such confusion. The principal inverses are usually defined as: Function Definition Value Field For inverse trigonometric functions. In the following. so strictly they do not have an inverse function. Therefore to define an inverse function we must restrict their domains so that the trigonometric function is bijective. the functions on the left are defined by the equation on the right. these are not proved identities. a figure formed with a trigonometry-based function. B and C: or. In this formula the angle at C is opposite to the side c. mainly because of the following two results. equivalently. This is a common situation occurring in triangulation. . and c and angles opposite those sides A. The law of cosines can be used to determine a side of a triangle if two sides and the angle between them are known. a technique to determine unknown distances by measuring two angles and an accessible enclosed distance. are of crucial importance in trigonometry. It can also be used to find the cosines of an angle (and consequently the angles themselves) if the lengths of all the sides are known. b.Trigonometric functions 61 Properties and applications The trigonometric functions. It can be proven by dividing the triangle into two right ones and using the above definition of sine. This theorem can be proven by dividing the triangle into two right ones and using the Pythagorean theorem. The law of sines is useful for computing the lengths of the unknown sides in a triangle if two angles and one side are known. as the name suggests. or equivalently. Law of sines The law of sines states that for an arbitrary triangle with sides a. Law of cosines The law of cosines (also known as the cosine formula) is an extension of the Pythagorean theorem: A Lissajous curve. where R is the triangle's circumradius. for the lengths and corresponding opposite angles. but the patterns of sums and differences. .Trigonometric functions 62 Law of tangents The following all form the law of tangents[15] The explanation of the formulae in words would be cumbersome. Law of cotangents If (the radius of the inscribed circle for the triangle) and (the semi-perimeter for the triangle). then the following all form the law of cotangents[16] It follows that In words the theorem is: the cotangent of a half-angle equals the ratio of the semi-perimeter minus the opposite side to the said angle. to the inradius for the triangle. are apparent in the theorem. The oscillation about the sawtooth is called the Gibbs phenomenon For example. the pendular motion of a mass hanging by a string. . such as the movement of a mass attached to a spring and. All basis functions have nodes at the nodes of the sawtooth.Trigonometric functions 63 Other useful properties Periodic functions The trigonometric functions are also important in physics. which models many natural phenomena.[17] Under rather general conditions. but all but the fundamental have additional nodes. The superposition of several terms in the expansion of a sawtooth wave are shown underneath. The sine and cosine functions are one-dimensional projections of uniform circular motion. the expansion of the periodic function ƒ(t) takes the form: Click on the image to see an animation of the additive synthesis of a square wave with an increasing number of harmonics Superimposed sinusoidal wave basis functions (bottom) form a sawtooth wave (top) when added. a periodic function ƒ(x) can be expressed as a sum of sine waves or cosine waves in a Fourier series. The characteristic wave patterns of periodic functions are useful for modeling recurring phenomena such as sound or light waves. the basis functions have wavelengths λ/k (k = integer) shorter than the wavelength λ of the sawtooth itself (except for k = 1).[18] Denoting the sine or cosine basis functions by φk. for example. Trigonometric functions also prove to be useful in the study of general periodic functions. are used to describe simple harmonic motion. for small angles. the square wave can be written as the Fourier series In the animation of a square wave at top right it can be seen that just a few terms already produce a fairly good approximation. The sine and the cosine functions. [19] All six trigonometric functions in current use were known in Islamic mathematics by the 9th century. and Rheticus' student Valentinus Otho Madhava of Sangamagrama (c. see Euler's_formula#Using_power_series [11] Needham. 149.[23] Finally. p. page 149 Stanley.com: Tangent def 5b. [12] Kantabutra. as well as the near-modern abbreviations sin. [9] Remmert. Brent. [14] "R. com/ browse/ tangent) See Boyer (1991). Ziegler. Springer-Verlag. 5c (http:/ / dictionary. Visual Complex Analysis. Proofs from THE BOOK (http:/ / www. used in solving triangles.[20] al-Khwārizmī produced tables of sines. Notes [1] [2] [3] [4] [5] [6] [7] [8] Dictionary.. p. and cosec. 1145/ 321941. under the mistaken impression that jb stood for the word jaib. abbreviated to jiva. See Maor (1998) See Ahlfors. Martin. but are now seldom used.). jya-ardha. and methods like Gal's accurate tables. and Payne and Hanek reduction algorithms can be used. reference. Extract of page 327 (http:/ / books. this transliteration was mis-translated in the 12th century into Latin as sinus. Abramowitz. 1400) made early strides in the analysis of trigonometric functions in terms of infinite series. p. Nasir al-Din al-Tusi. whereas secant stems from Latin secans — "cutting" — since the line cuts the circle. vowels not being written in Arabic. Next. the haversine (haversin(θ) = versin(θ) / 2 = sin2(θ/2)). com/ books?id=CC0dQxtYb6kC& pg=PA327) [10] For a demonstration. Theory of complex functions (http:/ / books. "Fast Multiple-Precision Evaluation of Elementary Functions".Trigonometric functions 64 History While the early study of trigonometry can be traced to antiquity. cot. They were studied by authors including Omar Khayyám. and 'tan' is by the 16th century French mathematician Albert Girard. acm. 321944). Stanley. 242 (1976). Many more relations between these functions are listed in the article about trigonometric identities. the word sine derives from the Sanskrit word for half the chord. Reinhold (1991). also defining them as infinite series and presenting "Euler's formula".[22] Leonhard Euler's Introductio in analysin infinitorum (1748) was mostly responsible for establishing the analytic treatment of trigonometric functions in Europe. tang. Cody and Waite reduction. Rheticus. cosines and tangents. via translation from Sanskrit to Arabic and then from Arabic to Latin. since the line touches the circle of unit radius.. The functions sine and cosine can be traced to the jyā and koti-jyā functions used in Gupta period Indian astronomy (Aryabhatiya." (http:/ / doi. as was the law of sines. Surya Siddhanta). Springer. ACM 23. as does sinus in Latin. the versine (versin(θ) = 1 − cos(θ) = 2 sin2(θ/2)) (which appeared in the earliest tables [2] ). google.. com/ mathematics/ book/ 978-3-642-00855-9) (Second ed. P. Vol I.[24] The word tangent comes from Latin tangens meaning "touching". springer. doing that while maintaining precision is nontrivial. In a paper published in 1682. Enumerative Combinatorics. Jamshīd al-Kāshī (14th century). Etymologically. the trigonometric functions as they are in use today were developed in the medieval period.[21] The first published use of the abbreviations 'sin'. written jb. The chord function was discovered by Hipparchus of Nicaea (180–125 BC) and Ptolemy of Roman Egypt (90–165 AD).[2] A few functions were common historically. ISBN 0198534469. Weisstein. Leibniz proved that sin x is not an algebraic function of x. Ulugh Beg (14th century). J. 'cos'. ISBN 0-387-97195-5. Günter M. such as the chord (crd(θ) = 2 sin(θ/2)). ISBN 9773540678657. Bhāskara II. This was transliterated in Arabic as jiba. com/ books?id=CC0dQxtYb6kC).. [13] However. Vol I Aigner. pages 43–44. (2000). . . English usage converted the Latin word sinus to sine. Enumerative Combinatorics. org/ 10... cos. . . 327.. which means "bosom" or "bay" or "fold" in Arabic. Regiomontanus (1464). google. sec. the exsecant (exsec(θ) = sec(θ) − 1) and the excosecant (excsc(θ) = exsec(π/2 − θ) = csc(θ) − 1). [17] Stanley J Farlow (1993). . 2002): ISBN 0-691-09541-8. MacTutor History of Mathematics archive.st-andrews. Pan Reference Books. 1976. ISBN 0-19-853446-9. Press. Elements of the History of Mathematics.. (2000).mcs.dcs.pupress. uk/ ~history/ Biographies/ Madhava. Fourier Analysis and its Applications (Reprint of Wadsworth & Brooks/Cole 1992 ed.F. p. "Tangent" (http://mathworld. (1991). (1998).usfca. sa/ PHYS/ alshukri/ PHYS215/ Islamic_astronomy. dcs. J. London. John Wiley & Sons. Islamic Astronomy (http:/ / faculty.html) from MathWorld. chapter 3. Stegun.html). Scientific American.). School of Mathematics and Statistics University of St Andrews. MacTutor History of Mathematics archive. Gerald B Folland (2009). 65 References • Abramowitz. • Lars Ahlfors. . Handbook of Mathematical Functions with Formulas. and E. "On hardware for computing exponential and trigonometric functions. • Maor. accessed 21 January 2006. New York. • O'Connor. Partial differential equations for scientists and engineers (http:/ / books.wolfram. [18] See for example.F. [23] See Maor (1998). Carl B.J.ac. [22] Nicolás Bourbaki (1994).uk/ ~history/Mathematicians/Madhava. "Preface" (http://www. regarding the etymology. 77 ff. • O'Connor. Robertson. • Weisstein. [16] The Universal Encyclopaedia of Mathematics. "Trigonometric functions" (http://www-gap.com/Tangent. . Princeton Univ.st-and.uk/history/Projects/ Pearce/Chapters/Ch9_3. (2002). and Mathematical Tables. (1991). 1960. ISBN 0-691-00659-8. The Crest of the Peacock: Non-European Roots of Mathematics. Eli. (2000). ISBN 0821847902. ISBN 0471543977. (1996).edu/vca/PDF/vca-preface. An accurate elementary mathematical library for the IEEE floating point standard. html). . • Gal. McGraw-Hill Book Company. com/ books?id=DLUYeSb49eAC& pg=PA82) (Reprint of Wiley 1982 ed. New York.). Shmuel and Bachelis.edu/books/maor/).html).. MacTutor History of Mathematics archive.html). 82. 1966. • Pearce. "Convergence and completeness" (http:/ / books. Robertson. google. edu. . p. Retrieved 2010-37-13. page 530. "Madhava of Sangamagrama" (http:/ / www-gap.uk/~history/ HistTopics/Trigonometric_functions.st-and. [19] Boyer. [20] Owen Gingerich (1986).edu/vca/). ISBN 0-486-61272-4. Carl B. Boris. Dover. • Kantabutra. Reprint edition (February 25. George G. Trigonometric Delights (http://www. English version George Allen and Unwin. Graphs. ACM Transaction on Mathematical Software (1991).. 1960. Penguin Books. 210. Inc. Complex Analysis: an introduction to the theory of analytic functions of one complex variable. (1964). "Madhava of Sangamagramma" (http://www-history. 328–339 (1996). ISBN 0-471-54397-7. com/ books?id=idAomhpwI8MC& pg=PA77).Trigonometric functions [15] The Universal Encyclopaedia of Mathematics. edu/ ~djoyce/ trig/ ).).. "Madhava of Sangamagramma" (http://www-groups. [21] J J O'Connor and E F Robertson. A History of Mathematics (Second ed.. Springer. J.dcs. second edition. • Needham.. John Wiley & Sons. . Translated from the German version Meyers Rechenduden. htm). Inc.usfca.. Translated from the German version Meyers Rechenduden. Ian G. Milton and Irene A. 74. pp.ac.. Eric W. google. st-and. A History of Mathematics. 1964.ac. clarku. Tristan. 254.J. 1976. and E. Computers 45 (3). Oxford University Press. Vitit. page 529. Scotland. (1999). • Joseph. 2nd edition." IEEE Trans. [24] "Clark University" (http:/ / www. Pan Reference Books. English version George Allen and Unwin. ac. kfupm. 1964. 2nd ed.princeton. ISBN 048667620X. Courier Dover Publications. • Boyer. American Mathematical Society.pdf)" to Visual Complex Analysis (http://www. Retrieved 2007-09-08. p. and chaos theory. The idea is to enhance the real numbers by introducing a non-real number i whose square is −1. Italian mathematician Gerolamo Cardano is the first known to have conceived complex numbers. Complex numbers are a solution to this problem. Some authors write a+ib instead of a+bi. (http://www. For example. Re is the real axis.com/library/module_viewer. has no real solution x. Definition A complex number is an expression of the form where a and b are real numbers and i is a mathematical symbol which is called the imaginary unit. so x2 + 1 cannot be zero. applied mathematics. since the square of x is 0 or positive.Trigonometric functions 66 External links • Visionlearning Module on Wave Mathematics (http://www. For example. respectively.visionlearning. . Complex numbers are used in many scientific fields. electromagnetism. so complex numbers are written as a + bj or a + jb. Im is the imaginary axis. representing the complex plane. In some disciplines (in particular. trigonometric and hyperbolic functions • Dave's draggable diagram. so that x = i and x = −i are the two solutions to the preceding equation.[1] Introduction and definition Complex numbers have been introduced to allow for solutions of certain equations that have no real solution: the equation A complex number can be visually represented as a pair of numbers forming a vector on a diagram called an Argand diagram. he called them "fictitious".clarku. quantum physics.trixon.5 + 2i is a complex number. including engineering. electrical engineering. The real number a of the complex number z = a + bi is called the real part of z and the real number b is the imaginary part.edu/~djoyce/trig/) (Requires java browser plugin) Complex number A complex number is a number consisting of a real part and an imaginary part.se/): Visualization of the unit circle. the imaginary unit i is instead written as j. where i is a symbol for current). and i is the square root of –1.[2] They are denoted Re(z) or ℜ(z) and Im(z) or ℑ(z). Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part. php?mid=131&l=&c3=) • GonioLab (http://glab. In this way the complex numbers contain the ordinary real numbers while extending them in order to solve problems that would be impossible with only real numbers. −3. during his attempts to find solutions to cubic equations in the 16th century. those of the form 0 + bi. Complex numbers whose real part is zero. These are emphasised in a complex number's polar form diagram.Complex number A real number a can usually be regarded as a complex number with an imaginary part of zero. The rules for addition. subtraction. Work on the problem of general polynomials ultimately led to the fundamental theorem of algebra. Complex numbers thus form an algebraically closed field. the angles they make with the x axis). though his understanding was rudimentary. when b is negative. that is to say. a + 0i. It is common to write a for a + 0i and bi for 0 + bi. or algebraic form. The set of all complex numbers is denoted by C or . a situation known as casus irreducibilis. are called imaginary numbers. rectangular. Viewed in this way the multiplication of a complex number by i corresponds to rotating a complex number anticlockwise through 90° about the origin. and division of complex numbers were developed by the Italian mathematician Rafael Bombelli. The defining characteristic of a position vector is that it has magnitude Figure 1: A complex number plotted as a point (red) and position vector (blue) on an Argand and direction. even when the final solutions are real numbers. However the sets are defined differently and have slightly different operations defined. .e. and imaginary part as vertical (see Figure 1). is the rectangular expression and it turns out notably that the operations of addition and of the point. Moreover. who extended this abstraction to the theory of quaternions. 67 The complex plane A complex number can be viewed as a point or position vector in a two-dimensional Cartesian coordinate system called the complex plane or Argand diagram (see Pedoe 1988 and Solomentsev 2001). named after Jean-Robert Argand. for example 3 − 4i instead of 3 + (−4)i. Many mathematicians contributed to the full development of complex numbers. These two values used to identify a given complex number are therefore called its Cartesian. a solution exists to every polynomial equation of degree one or higher.[3] A more abstract formalism for the complex numbers was further developed by the Irish mathematician William Rowan Hamilton. This conundrum led Italian mathematician Gerolamo Cardano to conceive of complex numbers in around 1545. The numbers are conventionally plotted using the real part as the horizontal component. where any polynomial equation has a root. it is common to write a − bi instead of a + (−b)i. multiplication. for instance comparison operations are not defined for complex numbers. that is to say. History in brief Main section: History The solution of a general cubic equation in radicals (without trigonometric functions) may require intermediate calculations containing the square roots of negative numbers. multiplication take on a very natural geometric character when complex numbers are viewed as position vectors: addition corresponds to vector addition while multiplication corresponds to multiplying their magnitudes and adding their arguments (i. which shows that with complex numbers. Geometrically. Inversive geometry. can also be expressed in terms of complex numbers. It is denoted or .Complex number 68 Elementary operations Conjugation The complex conjugate of the complex number z = x + yi is defined to be x − yi. a complex number is real if and only if it equals its conjugate. Conjugation distributes over the standard arithmetic operations: The reciprocal of a nonzero complex number z = x + yi is given by This formula can be used to compute the multiplicative inverse of a complex number if it is given in rectangular coordinates. conjugating twice gives the original complex number: . The real and imaginary parts of a complex number can be extracted using the conjugate: Geometric representation of and its conjugate in the complex plane Moreover. is the "reflection" of z about the real axis. In particular. a branch of geometry studying more general reflections than ones about a line. . the square of the imaginary unit is −1: The preceding definition of multiplication of general complex numbers is the natural way of extending this fundamental property of the imaginary unit. subtraction is defined by Using the visualization of complex numbers in the complex plane. X is the point such that the triangles with vertices 0. That is to say: Addition of two complex numbers can be done geometrically by constructing a parallelogram. Indeed. are congruent. which is described above. A. Equivalently. Similarly. and X. interpreted as points of the complex plane. B. and real division: Division can be defined in this way because of the following observation: . A and B.Complex number 69 Addition and subtraction Complex numbers are added by adding the real and imaginary parts of the summands. treating i as a variable. The division of two complex numbers is defined in terms of complex multiplication. A. Multiplication and division The multiplication of two complex numbers is defined by the following formula: In particular. B. is the point X obtained by building a parallelogram three of whose vertices are 0. the addition has the following geometric interpretation: the sum of two complex numbers A and B. the formula follows from this (distributive law) (commutative law of addition—the order of the summands can be changed) (commutative law of multiplication—the order of the factors can be changed) (fundamental property of the imaginary unit). and is written as with the modulus. by Pythagoras' theorem. the argument can be found from the rectangular form : [6] . In general. and the angle of the line through P and O.Complex number 70 As shown earlier. y = 0). Polar form Absolute value and argument Another way of encoding points in the complex plane other than using the x. If z is a real number (i. This idea leads to the polar form of complex numbers.. then r = |x|. 0) (origin). Square root The square roots of a + bi (with b ≠ 0) are . is the complex conjugate of the denominator .[4] [5] Here is called the modulus of a + bi. and the square root with non-negative real part is called the principal square root. or are polar expressions of the point. As . r is the distance of the point P representing the complex number z to the origin. The real part c and the imaginary part d of the denominator must not both be zero for division to be defined. The argument or phase of z is the angle of the radius OP with the positive real axis.and y-coordinates is to use the distance of a point P to O. where and where sgn is the signum function. This can be seen by squaring to obtain a + bi. the point whose coordinates are (0. The absolute value (or modulus or magnitude) of a complex number z = x+yi is Figure 2: The argument φ and modulus r locate a point on an Argand diagram.e. division and exponentiation are simpler than the ones using Cartesian coordinates. Together. which gives back i 2 = −1. Values in the range are obtained by adding if the value is negative. The value of φ equals the result of atan2: . The polar angle for the complex number 0 is undefined. the principal value in the interval is chosen. The picture at the right illustrates the multiplication of . the polar form. For example. division and exponentiation in polar form The relevance of representing complex numbers in polar form stems from the fact that the formulas for multiplication. this is sometimes abbreviated to In angle notation. r and φ give another way of representing complex numbers.Complex number 71 The value of φ must always be expressed in radians. often used in electronics to represent a phasor with amplitude r and phase φ it is written as[7] Multiplication. the arg function is sometimes considered as multivalued. the absolute values are multiplied and the arguments are added to yield the polar form of the product. multiplying by i corresponds to a quarter-rotation counter-clockwise. Normally. but arbitrary choice of the angle 0 is common. The red triangle is rotated to match the vertex of the blue one and stretched by √5. In other words. as the combination of modulus and argument fully specify the position of a point on the plane. the length of the hypotenuse of the blue triangle. Hence. as given above. Given two complex numbers z1 = r1(cos φ1 + isin φ1) and z2 =r2(cos φ2 + isin φ2) the formula for multiplication is Multiplication of 2+i (blue triangle) and 3+i (red triangle). It can change by any multiple of 2π and still give the same angle. Recovering the original rectangular co-ordinates from the polar form is done by the formula called trigonometric form Using Euler's formula this can be written as Using the cis function. On the other hand. Briefly. and complex Lie algebra. formulas like this—known as Machin-like formulas—are used for high-precision approximations of π. C is not an ordered field. for example the law of commutativity of addition and multiplication for any two complex numbers z1 and z2: These two laws and the other requirements on a field can be proven by the formulas given above. every nonzero complex number has a reciprocal complex number. Formulas such as (which holds for positive real numbers). complex matrix. the formula 72 holds. . in any ordered field. the thing's name is usually modified to reflect that fact. As the arctan function can be approximated highly efficiently. Second. the nth root of z is considered as a multivalued function (in z). for any complex number a. Thus. Here is the usual (positive) nth root of the positive real number r.Complex number Since the real and imaginary part of 5+5i are equal. the argument of that number is 45 degrees. its negative −a is also a complex number. When the underlying field for a mathematical topic or construct is the field of complex numbers. do in general not hold for complex numbers. respectively. any two complex numbers can be added and multiplied to yield another complex number. While the nth root of a positive real number r is chosen to be the positive real number c satisfying cn = x there is no natural way of distinguishing one particular complex nth root of a complex number. Moreover. division is given by This also implies de Moivre's formula for exponentiation of complex numbers with integer exponents: The n-th roots of z are given by for any integer k satisfying 0 ≤ k ≤ n − 1. In fact. using the fact that the real numbers themselves form a field. it is also the sum of the angles at the origin of the red and blue triangle are arctan(1/3) and arctan(1/2). or π/4 (in radian). as opposed to a usual function f. Unlike the reals. that is to say. this means that the following facts hold: first. and third. for which f(z) is a uniquely defined number. Properties Field structure The set C of complex numbers is a field. For example: complex analysis. the square of any element is necessarily positive. Similarly. it is not possible to define a relation z1 < z2 that is compatible with the addition and multiplication. complex polynomial. so i2 = −1 precludes the existence of an ordering on C. Therefore. these operations satisfy a number of laws. is nonzero. an. This is the statement of the fundamental theorem of algebra. .. any complex matrix has at least one (complex) eigenvalue. Because of this fact.Complex number 73 Solutions of polynomial equations Given any complex numbers (called coefficients) a0.. so these two fields are isomorphic. The following description of C as a topological field (that is. since √2 is not a rational number) nor the real numbers R (the polynomial x2 + a does not have a real solution for a > 0. Another consequence of this algebraic characterization is that C contains many proper subfields which are isomorphic to C (the same is true of R. • If x and y are distinct elements of P. since C can be distinguished from R because the nonzero complex numbers are connected. the properties of nearness and continuity. Any field F with these properties can be endowed with a topology by taking the sets B(x. For example. only. Second. C has a nontrivial involutive automorphism such that xx∗ is in P for any nonzero x in C. theorems that hold "for any algebraically closed field". . Because of this fact. (namely the complex conjugation). then either x−y or y−x is in P. its transcendence degree over Q. This means that 1 + 1 + .. C is isomorphic to the field of complex Puiseux series. where x ranges over the field and p ranges over P. This property does not hold for the field of rational numbers Q (the polynomial x2 − 2 does not have a rational root. or a proof combining Galois theory and the fact that any real polynomial of odd degree has at least one root. That is to say. + 1 ≠ 0 for any number of summands (all of which equal one).. Characterization as a topological field The preceding characterization of C describes the algebraic aspects of C. which matter in areas such as analysis and topology. With this topology F is isomorphic as a topological field to C. then S + P = x + P for some x in C. Also. which allows to specify notions such as convergence) does take into account the topological properties. The only connected locally compact topological fields are R and C. C contains a subset P (namely the set of positive real numbers) of nonzero elements satisfying the following three conditions: • P is closed under addition. it has characteristic 0. a1. Third. C is called an algebraically closed field. or topological ones such as the winding number. either by analytic methods such as Liouville's theorem. It can be shown that any field having these properties is isomorphic (as a field) to C.. p) = {y | P − (y − x)(y −x) ∈ P} as a base. provided that at least one of the higher coefficients... are not dealt with. For example. the prime field of C is the cardinality of the continuum. This gives another characterization of C as a topological field. However. fixing P and . which contains many sub fields isomorphic to itself). Algebraic characterization The field C has the following three properties: first. since the square of x is positive for any real number x). specifying an isomorphism requires the axiom of choice. Moreover. multiplication and taking inverses. the algebraic closure of Qp also satisfies these three properties. while the nonzero real numbers are not. apply to C. it is algebraically closed (see above). a field that is equipped with a topology. • If S is any nonempty subset of P. an. There are various proofs of this theorem.. the equation has at least one complex solution z. Though this representation of complex numbers with matricies is the most common.. the square of the absolute value of a complex number expressed as a matrix is equal to the determinant of that matrix: The conjugate corresponds to the transpose of the matrix. correspond to the ones yielded by this abstract algebraic approach—the two definitions of the field C are said to be isomorphic (as fields). The usual addition and multiplication of polynomials endows the set R[X] of all such polynomials with a ring structure. . This extension field contains two square roots of −1. and the sum and product of complex numbers corresponds to the sum and product of such matrices. the distributive law is required to hold for any three elements x. Matrix representation of complex numbers Complex numbers can also be represented by 2×2 matrices that have the following form: Here the entries a and b are real numbers.b) of real numbers. the above formulas for addition and multiplication read It is then just a matter of notation to express (a. elements of the extension field can be written as ordered pairs (a. this also shows that C is an algebraic closure of R. More rigorously. y and z of a field. The sum and product of two such matrices is again of this form. The quotient ring R[X]/(X2+1) can be shown to be a field. See the article on 2 × 2 real matrices for other representations of complex numbers. This characterization relies on the notion of fields and polynomials. Though this low-level construction does accurately describe the structure of the complex numbers.. an are real numbers. Moreover. This ring is called polynomial ring. In this notation. namely (the cosets of) X and −X. the imaginary unit. respectively. subtraction. say. Equivalently. complex numbers have been defined by introducing i. the above formulas for addition etc.Complex number 74 Formal construction Formal development Above. Together with the above-mentioned fact that C is algebraically closed. A field is a set endowed with an addition. For example. . b) as a + ib. the following equivalent definition reveals the algebraic nature of C more immediately.. Moreover. many other representations arise from matrices other than that square to the negative of the identity matrix. which means that each element of the extension field can be uniquely written as a linear combination in these two elements. b) of real numbers. rational numbers. A polynomial p(X) with real coefficients is an expression of the form where the a0. as a symbol. multiplication and division operations which behave as is familiar from. The geometric description of the multiplication of complex numbers can also be phrased in terms of rotation matrices by using this correspondence between complex numbers and such matrices. The set R of real numbers does form a field. the set C of complex numbers can be defined as the set R2 of ordered pairs (a. (The cosets of) 1 and X form a basis of R[X]/(X2+1) as a real vector space. A sequence of complex numbers is said to converge if and only if its real and imaginary parts do. complex functions have four-dimensional graphs and may usefully be illustrated by color coding a three-dimensional graph to suggest four dimensions. endowed with the metric is a complete metric space. as well as hyperbolic functions such as sinh also carry over to complex arguments without change. and ln the (real) natural logarithm. unique only up to a multiple of 2π. Often. there is an infinitude of complex solutions z of the equation for any complex number w ≠ 0. The principal value of log is often taken by restricting the .Complex number 75 Complex analysis The study of functions of a complex variable is known as complex analysis and has enormous practical use in applied mathematics as well as in other branches of mathematics. It can be shown that any such solution z—called complex logarithm of a—satisfies where arg is the argument defined above. or by animating the complex function's dynamic transformation of the complex plane. As arg is a multivalued function. Like in real analysis. Euler's identity states: for any real number φ. This is equivalent to the (ε. where the absolute value of real numbers is replaced by the one of complex numbers. C. From a more abstract point of view. log is also multivalued. Complex exponential and related functions Color wheel graph of sin(1/z). this notion of convergence is used to construct a number of elementary functions: the exponential function exp(z). which notably includes the triangle inequality for any two complex numbers z1 and z2. Black parts inside refer to numbers having large absolute values. is defined as the infinite series and the series defining the real trigonmetric functions sine and cosine. the most natural proofs for statements in real analysis or even number theory employ techniques from complex analysis (see prime number theorem for an example). also written ez. Unlike real functions which are commonly represented as two-dimensional graphs. The notions of convergent series and continuous functions in (real) analysis have natural analogs in complex analysis. δ)-definition of limits. in particular Unlike in the situation of real numbers. Signal analysis Complex numbers are used in signal analysis and other fields for a convenient description for periodically varying signals. it is a nonminimum phase system. Complex exponentiation zω is defined as 76 Consequently. Applications Some applications of complex numbers are: Control theory In control theory. For example. it is especially important whether the poles and zeros are in the left or right half planes. This map is holomorphic if and only if b = 0. The root locus. Nyquist plot. and Nichols plot techniques all make use of the complex plane. The system's poles and zeros are then analyzed in the complex plane. it will have marginal stability. In the root locus method. for some natural number n. capacitors.Complex number imaginary part to the interval (−π. but does not satisfy the Cauchy-Riemann equations. For given real functions representing actual physical quantities. For example. functions that can locally be written as f(z)/(z − z0)n with a holomorphic function f(z). often in terms of sines and cosines. have real part greater than or less than zero. In electrical engineering. still share some of the features of holomorphic functions. corresponding complex functions are considered of which the real parts are the original quantities.e. The second summand real-differentiable. If a system has poles that are • in the right half plane. Holomorphic functions A function f : C → C is called holomorphic if it satisfies the Cauchy-Riemann equations. • on the imaginary axis. they are in general multi-valued. Other functions have essential singularities. If Fourier analysis is employed to write a given real-valued signal as a sum of periodic functions. Meromorphic functions. frequency-dependent resistances . For ω = 1 / n. the absolute value |z| of the corresponding z is the amplitude and the argument arg(z) the phase.π]. these periodic functions are often written as complex valued functions of the form where ω represents the angular frequency and the complex number z encodes the phase and amplitude as explained above. it will be unstable. this recovers the non-unicity of n-th roots mentioned above. • all in the left half plane. it will be stable. such as sin(1/z) at z = 0. The treatment of resistors. the Fourier transform is used to analyze varying voltages and currents. is Complex analysis shows some features not apparent in real analysis. and inductors can then be unified by introducing imaginary. i. systems are often transformed from the time domain to the frequency domain using the Laplace transform. If a system has zeros in the right half plane. any R-linear map C → C can be written in the form with complex coefficients a and b. For a sine wave of a given frequency. any two holomorphic functions f and g that agree on an arbitrarily small open subset of C necessarily agree everywhere. complex functions are used to describe potential flow in two dimensions. still images. 77 Improper integrals In applied fields. restore. which utilize digital versions of Fourier analysis (and wavelet analysis) to transmit. by means of complex-valued functions. Quantum mechanics The complex number field is relevant in the mathematical formulations of quantum mechanics. Likewise. complex numbers are often used to compute certain real-valued improper integrals. (Electrical engineers and some physicists use the letter j for the imaginary unit since i is typically reserved for varying currents and may come into conflict with i. Fractals Certain fractals are plotted in the complex plane. compress. Relativity In special and general relativity. This use is also extended into digital signal processing and digital image processing.Complex number for the latter two and combining all three in a single complex number called the impedance. where complex Hilbert spaces provide the context for one such formulation that is convenient and perhaps most standard. in difference equations. the complex roots r of the characteristic equation of the difference equation system are used. (This is no longer standard in classical relativity. to attempt to solve the system in terms of base functions of the form f(t) = r t. and otherwise process digital audio signals.) Complex numbers are essential to spinors. but is used in an essential way in quantum field theory. see methods of contour integration. Fluid dynamics In fluid dynamics. Dynamic equations In differential equations. Several methods exist to do this. the Mandelbrot set and Julia sets. e. Algebraic number theory .g. it is common to first find all complex roots r of the characteristic equation of a linear differential equation or equation system and then attempt to solve the system in terms of base functions of the form f(t) = ert. which are a generalization of the tensors used in relativity. some formulas for the metric on spacetime become simpler if one takes the time variable to be imaginary. The original foundation formulas of quantum mechanics – the Schrödinger equation and Heisenberg's matrix mechanics – make use of complex numbers. and video signals.) This approach is called phasor calculus. any nonconstant polynomial equation (in complex coefficients) has a solution in C. The roots of such equations are called algebraic numbers–they are a principal object of study in algebraic number theory. although negative quantities were not conceived of in Hellenistic mathematics and Heron merely replaced it by its positive. Tartaglia's cubic formula gives the solution to the equation x3 − x = 0 as . often integers or rationals. where x and y are integers. As an example. in which analytic methods can be used. algebraic methods can be used to study geometric questions and vice versa. This is done by encoding number-theoretic information in complex-valued functions. With algebraic methods. the volume of an impossible frustum of a pyramid to arrive at the term in his calculations. In this way. even if one was only interested in real solutions. b. that is to say integers satisfying Construction of a regular polygon using straightedge and compass. (which implies that the triangle having sidelengths a. History The earliest fleeting reference to square roots of negative numbers can perhaps be said to occur in the work of the Greek mathematician Heron of Alexandria in the 1st century AD. For example. the algebraic closure of Q. sometimes required the manipulation of square roots of negative numbers. A fortiori. the Riemann zeta-function ζ(s) is related to the distribution of prime numbers. apparently in error. more specifically applying the machinery of field theory to the number field containing roots of unity. and c is a right triangle). by taking advantage of the fact that they can be regarded as complex numbers. c). where in his Stereometrica he considers. it can be shown that it is not possible to construct a regular nonagon using only compass and straightedge–a purely geometric problem. Another example are Pythagorean triples (a. Gerolamo Cardano). the same is true if the equation has rational coefficients. which also contains all algebraic numbers. b. Compared to Q. It was soon realized that these formulas. They can be studied by considering Gaussian integers. C has the advantage of being easily understandable in geometric terms.Complex number 78 As mentioned above. that is. numbers of the form x + iy.[8] The impetus to study complex numbers proper first arose in the 16th century when algebraic solutions for the roots of cubic and quartic polynomials were discovered by Italian mathematicians (see Niccolo Fontana Tartaglia. Analytic number theory Analytic number theory studies numbers. as later mathematicians showed rigorously. two of which are complex and when the three cube roots of -1 of which two are complex. For instance.. This difficulty eventually led to the convention of using the special symbol i in place of guard against this mistake .) A further source of confusion was that the equation inconsistent with the algebraic identity incorrect use of this identity (and the related identity seemed to be capriciously . which is valid for non-negative real numbers a and b. Even so Euler considered it natural to introduce students to complex numbers much earlier than we do today.Complex number 79 The three cube roots of −1.. In the 18th century complex numbers gained wider use. The bedeviled Euler. Elements of Algebra [10]. are substituted into this expression the three real roots. he introduces these numbers almost at once and then uses them in a natural way throughout. b positive and the other negative. mais qu’il n’y a quelquefois aucune quantité qui corresponde à celle qu’on imagine. ) in the case when both a and b are negative even to and which was also used in complex number calculations with one of a. but sometimes there exists no quantity that matches that which we imagine. 1 and −1. result. although he was at pains to stress their imaginary nature [9] [. in 1730 Abraham de Moivre noted that the complicated identities relating trigonometric functions of an integer multiple of an angle to powers of trigonometric functions of that angle could be simply re-expressed by the following well-known formula which bears his name.. ([.. Of course this particular equation can be solved at sight but it does illustrate that when general formulas are used to solve cubic equations with real roots then. In his elementary algebra text book. de Moivre's formula: .] sometimes only imaginary.] quelquefois seulement imaginaires c’est-à-dire que l’on peut toujours en imaginer autant que j'ai dit en chaque équation. the use of complex numbers is unavoidable. 0. as it was noticed that formal manipulation of complex expressions could be used to simplify calculations involving trigonometric functions. Rafael Bombelli was the first to explicitly address these seemingly paradoxical solutions of cubic equations and developed the rules for complex arithmetic trying to resolve these issues. that is one can imagine as many as I said in each equation. The term "imaginary" for these quantities was coined by René Descartes in 1637. Otto Hölder. and absolute value. it is not the only one. With respect to the basis 1. and the modulus. Hermann Schwarz. Cauchy (1828) called 2 2 the direction the reduced form (l'expression . Gauss had earlier published an essentially topological proof of the theorem in 1797 but expressed his doubts at the time about "the true metaphysics of the square root of -1". the one mentioned in the section on matrix representation of complex numbers above. It can be carried further to higher dimensions. Any matrix has the property that its square is the negative of the identity matrix: J2 = -I. Then . The English mathematician G. largely establishing modern notation and terminology. Henri Poincaré. Karl Weierstrass and many others. However. often used for . Gauss used i for number for a + bi. yielding the quaternions H and octonions O which (as a real vector space) are of dimension 4 and 8. this matrix is i. In 1806 Jean-Robert Argand independently issued a pamphlet on complex numbers and provided a rigorous proof of the fundamental theorem of algebra. commencing around 1825 in Cauchy's case. x·y ≠ y·x for two quaternions. the multiplication of octonions fails (in addition to not being commutative) to be associative: (x·y)·z ≠ x·(y·z). although it had been anticipated as early as 1685 in Wallis's De Algebra tractatus. for modulus. It was not until 1831 that he overcame these doubts and published his treatise on complex numbers as points in the plane.. Argand called factor.e. is due to Hankel (1867).Complex number In 1748 Leonhard Euler went further and obtained Euler's formula of complex analysis: 80 by formally manipulating complex power series and observed that this formula could be used to reduce any trigonometric identity to much simpler exponential identities. By Hurwitz's theorem they are the only ones. is due to Weierstrass. all of these are normed division algebras over R. The expression direction coefficient. However. respectively. The Cayley-Dickson construction is closely related to the regular representation of C. with increasing dimension. H. and called a + b the norm. the sedenions fail to have this structure. i. thought of as an R-algebra (an R-vector space with a multiplication). Later classical writers on the general theory include Richard Dedekind. the algebraic properties familiar from real and complex numbers vanish: the quaternions are only a skew field. This means the following: the R-linear map for some fixed complex number w can be represented by a 2×2 matrix (once a basis has been chosen). The next step in the Cayley-Dickson construction. Hardy remarked that Gauss was the first mathematician to use complex numbers in 'a really confident and scientific way' although mathematicians such as Niels Henrik Abel and Carl Gustav Jacob Jacobi were necessarily using them routinely before Gauss published his 1831 treatise.[11] Augustin Louis Cauchy and Bernhard Riemann together brought the fundamental ideas of complex analysis to a high state of completion. i. Generalizations and related notions The process of extending the field R of reals to C is known as Cayley-Dickson construction. introduced the term complex réduite) and apparently introduced the term argument. Wessel's memoir appeared in the Proceedings of the Copenhagen Academy but went largely unnoticed. Felix Klein. with respect to the basis 1. The idea of a complex number as a point in the complex plane (above) was first described by Caspar Wessel in 1799.e. While this is a linear representation of C in the 2 × 2 real matrices. i. The common terms used in the theory are chiefly due to the founders. ISBN 978-1556080104 . p. p.de/c/c024140. org/ kb/ thread. Notes [1] Burton (1995. Richard N. §9. (1986). com/ books?id=rFhiJqkrALIC& pg=PA14) [7] Nilsson. WT.htm). 66. Susan A. Prentice Hall. Kapil D. An Imaginary Tale: The Story of (http:/ / mathforum. This field is called p-adic complex numbers by analogy. ISBN 9780691127989. .springer. (2005). This is generalized by the notion of a linear complex structure.7. McGraw-Hill.. Wright. G. google. math. 338. BP (2007). Irene A. ISBN 978-0070006577 • Conway. Nation. com/ books?id=g5j-cT-vg_wC& pg=PA66) [3] Katz (2004. uk/ C. with respect to the usual absolute value metric. which are elements of the ring R[x]/(x2 − 1) (as opposed to R[x]/(x2 + 1)). The fields R and Qp and their finite field extensions. Ltd. Encyclopaedia of Mathematics. Michiel. which are thereby analogous to R. . E. Riedel. Miltonn. htm) [5] Cooke. 14. Handbook of mathematical functions with formulas. Springer. Richard D. Geometry: A comprehensive course. by Ostrowski's theorem. (2001).Complex number 81 is also isomorphic to the field C. John B. com/ books?id=rFhiJqkrALIC) (2nd ed. Vetterling. ISBN 0618825150. sfu. Other choices of metrics on Q lead to the fields Qp of p-adic numbers (for any prime number p). the field of rational numbers. p.). New York: John Wiley & Sons. The algebraic closure of of Qp still carry a norm. mat. Paul J. . H. jspa?forumID=149& threadID=383188& messageID=1181284). College Algebra and Trigonometry (http:/ / books. . References Mathematical references • Ahlfors.. org/ ebooks/ 26400). ISBN 0-486-65812-0 • Press. google. Foundations of Discrete Mathematics.S. ISBN 0-470-25952-3. in Hazewinkel.nrbook. ca/ ~cbm/ aands/ page_17. ISBN 0-486-61272-4. Dan (1988). PHI Learning Pvt. 17 (http:/ / www. Barker. Cengage Learning. SA. New York: Cambridge University Press. . 189 (fourth edition). com/ books?id=lUcTsYopfhkC). and gives an alternative complex structure on R2. For example this notion contains the split-complex numbers.com/empanel/index. Springer. ISBN 0199219869. Vernon C. page 338 (http:/ / books.). com/ books?id=sxmM8RFL99wC) (8th ed. Sangwin/ euler/ [11] Hardy. including C. google. C. ISBN 978-0-470-21152-6 • Pedoe. Chapter P.26. Extract of chapter 1.. ISBN 978-0-521-88068-8 • Solomentsev. Courier Dover Publications.. H. Retrieved 20 April 2011. OUP Oxford.). Classical algebra: its nature. H. google.. John Wiley and Sons. p. (2007). google. James William. p. ISBN 0-387-90328-3 • Joshi. An Introduction to the Theory of Numbers. .). google. (1989). com/ books?id=sxmM8RFL99wC& pg=PA338) [8] Nahin. Numerical Recipes: The Art of Scientific Computing (3rd ed. [9] Descartes. com/ ?id=g5j-cT-vg_wC) (6 ed. Dover Publications. bham. ISBN 0486600688. "Section 5. google.D. E. Princeton University Press. Lars (1979). p. p. com/ books?id=MtU8uP7XMvoC). and uses (http:/ / books.). page 14 (http:/ / books.4) [4] Abramowitz. [10] http:/ / web. graphs.. Roger (2008). "Complex number" (http://eom. Section 3. ISBN 81-203-2641-5. René (1954) [1637]. and O. In this ring. (2007). Chapter 9. Flannery.5 Complex Arithmetic" (http:// apps.. (2008). M. The completion turns out to be algebraically closed. Functions of One Complex Variable I. There are no other nontrivial ways of completing Q than R and Qp. WH. p. 66 (http:/ / books. (1964). the equation a2 = 1 has four solutions. Retrieved 20 April 2011. google. J. Teukolsky.html?pg=225). Hypercomplex numbers also generalize R. are local fields. The field R is the completion of Q. 294) [2] Aufmann. ISBN 0-131-98925-1. google. and mathematical tables (http:/ / books.1. but (unlike C) are not complete with respect to it. Dover. p. (2000) [1938]. com/ books?id=lUcTsYopfhkC& pg=PA59) [6] Kasana. Complex Variables: Theory And Applications (http:/ / books. 59. ac. La Géométrie (http:/ / www. Extract: page 59 (http:/ / books. Complex analysis (3rd ed. Stegun. origins. gutenberg. Electric circuits (http:/ / books. 17. . (http://www. External links • Imaginary Numbers (http://www. Further reading • The Road to Reality: A Complete Guide to the Laws of the Universe. ISBN 978-0-07-009465-9 • Katz. Springer.co. An Imaginary Tale: The Story of (hardcover ed.uk/programmes/b00tt6b2) on In Our Time at the BBC.). . Alfred A. ISBN 0-19-853447-7 (hardcover. Julia sets. ISBN 0-679-45443-8. • Unknown Quantity: A Real and Imaginary History of Algebra.dimensions-math.. Ebbinghaus .-D. Brief Version. Paul J.). Numbers (hardcover ed. • H. (1995). (1998).org/convergence/1/?pa=content& sa=viewDocument&nodeId=640&bodyId=1038) at Convergence. Chapters 4-7 in particular deal extensively (and enthusiastically) with complex numbers.Complex number 82 Historical references • Burton. The History of Mathematics (3rd ed. History of complex numbers and complex analysis with compelling and useful visual interpretations. ISBN 0-309-09657-X (hardcover 2006).. Clarendon Press. Addison-Wesley.maa. by Tristan Needham. A very readable history with emphasis on solving polynomial equations and the structures of modern algebra. Chapter 6 discusses transformations of the complex plane. • Euler's work on Complex Roots of Polynomials (http://mathdl. Victor J.). by Roger Penrose. (2004).htm) Chapter 5 presents an introduction to complex arithmetic and stereographic projection.org/Dim_regarder_E. • John and Betty's Journey Through Complex Numbers (http://mathforum. and the Mandelbrot set. New York: McGraw-Hill. ISBN 0-691-02795-1 A gentle introduction to the history of complex numbers and the beginnings of complex analysis. Knopf. ISBN 978-0-321-16193-2 • Nahin. • Visual Complex Analysis. Princeton University Press. ISBN 0-387-97497-0 An advanced perspective on the historical development of the concept of number. by John Derbyshire. (1991). David M. Joseph Henry Press. A History of Mathematics. 2005. 1997).bbc.org/johnandbetty/) • Dimensions: a math film. MAA Mathematical Sciences Digital Library. the channels are combined ('mixed-down') to two channels for stereo or more for surround sound. a microphone — typically a sensitive one — is placed at some distance from the sound source. especially for directional mics which exhibit bass boost from the proximity effect. just out of sight. These are the relative level (or loudness) difference between the two channels Δ L. The "interaural" signals (binaural ILD and ITD) at the ears are not the stereo microphone signals which are coming from the loudspeakers. Choice of technique depends on a number of factors. or having a microphone nearby is not appropriate. including room reverberation. This can be a concern. • Situational circumstances: Sometimes a microphone should not be visible. • Processing: If the signal is destined to be heavily processed. Multi-track recording Often each instrument or vocalist is miked separately. in situations where ambient noise is useful (hall reverberation. Loudspeaker signals are different from the sound . These signals are normally not mixed. a microphone is placed relatively close to an instrument or sound source. or voice sources. • Type of sound-source: Acoustic instruments produce a very different sound than electric instruments. audience reaction). and can give a sound that is more natural. In this way there is always a certain distance between the actor and the microphone. it can be a desired outcome. film. or when trying to avoid feedback in an amplified performance. natural mix of the sound source or sources. The artists need not perform in the same place at the same time.Microphone practice 83 Microphone practice There exist a number of well-developed microphone techniques used for miking musical. • In ambient or distant micing. At a later stage. Close micing often affects the frequency response of the microphone. especially in amplified performances. The goal of this technique is to get a broader. Basic techniques There are several classes of microphone placement for recording and amplification. • The use of a windshield as well as a pop shield. and are called "interchannel" signals (Δ L and Δ t). Microphones may also be used to record the overall effect. where audio feedback can be a significant problem. This permits greater control over the final sound. Generally effects such as reverberation are added to each recorded channel. and is commonly used when attempting to record a number of separate instruments while keeping the signals separate. and different levels sent to left and right final channels to position the artist in the stereo sound-stage. but recording two channels (stereo recording) is simpler and cheaper. and individual tracks (or sections of tracks) can be re-recorded to correct errors. stereo or multi-channel. along with ambient sound. with one or more microphones recording to separate channels (tracks). • In close micing. including reverberation from the room or hall. In scenes for a movie the microphone may be held above the pictureframe. a different type of input may be required. Stereo recording techniques There are two features of sound that the human brain uses to place objects in the stereo sound-field between the loudspeakers. designed to reduce vocal plosives. or just the effect of the performance room. which are again different from the human voice. including: • The collection of extraneous noise. or "mixed down". This serves to reduce extraneous noise. • Choice of a signal type: Mono. and the time delay difference in arrival times for the same sound in each channel Δ t. Alternatively. Microphone practice arriving at the ear.[1] A stereo effect is achieved through differences in sound pressure level between two microphones. almost holographic soundstage. At a distance of about 50 cm (0. especially if employed close to the sound source(s). who called it "Stereosonic". See the article "Binaural recording for earphones". XY Stereo When the microphones are bidirectional and placed facing +-45° with respect to the sound source. 84 Various methods of stereo recording X-Y technique: intensity stereophony Here there are two directional microphones at the same place. At 70 cm distance it is about equivalent to the pickup angle of the near-coincident ORTF setup. A further refinement of the Blumlein Pair was developed by EMI in 1958. If the distance is increased between the microphones it effectively decreases the pickup angle. They added a little in-phase crosstalk above 700 Hz to better align the mid and treble phantom sources with the bass ones. so capturing time-of-arrival stereo information as well as some level (amplitude) difference information. Due to the lack of differences in time-of-arrival and phase ambiguities. [2] Blumlein Stereo A-B technique: time-of-arrival stereophony This uses two parallel omnidirectional microphones some distance apart. and typically placed at 90° or more to each other. . the sonic characteristic of X-Y recordings is generally less "spacey" and has less depth compared to recordings employing an AB setup. The sonic image produced by this configuration is considered by many authorities to create a realistic.5 m) the time delay for a signal reaching first one and then the other microphone from the side is approximately 1. the X-Y-setup is called a Blumlein Pair.5 ms (1 to 2 ms). This makes it especially useful for film-based projects. There is some controversy as to whether MS micing technique can create translation issues when used with matrix encoded cinema surround formats such as Dolby SR LtRt. often mounted on a bar to define the separation. or even on the top of a handheld digital recorder. the microphones would ideally be in exactly the same place. This only causes problems with sound from above or below the height of the microphones. Right = Mid − Side (the polarity-reversed side-signal). This can be an important factor in choosing which technique to use. matching a zoom lens. The equipment for the techniques also varies from the bulky to the small and convenient. Since M/S setups can give a variable soundstage width. out-of-phase parts of the signal will cancel. there may be some loss of treble when played back in mono. the stereo width can be manipulated after the recording has taken place. if the Mid and Side signals are recorded (rather than the matrixed Left and Right). which may cause the unwanted reduction or loss of some parts of the signal. This configuration produces a completely mono-compatible signal and. so they are often separated vertically. The left and right channels are produced through a simple matrix: Left = Mid + Side. One mic is physically inverted over the other. they are the least compatible with mono. • In the X-Y techniques. Mid-Side Stereo • Since the A-B techniques use phase differences to give the stereo image.Microphone practice 85 M/S technique: Mid/Side stereophony This coincident technique employs a bidirectional microphone facing sideways and a cardioid (generally a variety of cardioid. A-B techniques generally use two separate microphone units. so they share the same distance. • The M/S technique is ideal for mono compatibility. although Alan Blumlein described the usage of an omnidirectional transducer in his original patent) at an angle of 90° facing the sound source. Choosing a technique If a stereo signal is to be reproduced in mono. X-Y microphone capsules can be mounted in one unit. which relies on phase relationships between left and right channels of the stereo recording in order to decode surround information. . since summing Left+Right just gives the Mid signal back. which is not possible – if they are slightly separated left to right. they are often used in small 'pencil microphones' to mount on video cameras. Transverse waves occur when a disturbance sends waves perpendicular (at right angles) to the original wave.Intensity Stereo (http://www. google. ISBN 0240519612. of oscillations or vibrations around almost fixed locations. instead. For example. a cork on rippling water will bob up and down. electromagnetic radiation (including visible light. Focal Press. "The Stereophonic Zoom" (http:/ / www. uk/ books?id=w8kXMVKOsY0C& pg=PA170) (2 ed. When air molecules collide with their neighbors. The Microphone Book (http:/ / books. John (2004). which will cause a cascade of collisions between neighbouring molecules. Waves are described by a wave equation which sets out how the disturbance proceeds over time. a wave is a disturbance that travels through space and time. X-rays. with little or no associated mass transport. they also bounce away from them (restoring force). sound waves propagate via air molecules bumping into their neighbors. This keeps the molecules from continuing to travel in the direction of the wave. This type of wave consists of periodic oscillations in electrical and magnetic fields. The mathematical form of this equation varies depending on the type of wave. The deformation reverses itself owing to restoring forces resulting from its deformation.). A main distinction can be made between transverse and longitudinal waves. co. rycote. Waves travel and the wave motion transfers energy from one point to another.Microphone practice 86 References [1] Michael Williams.com/images/uploads/ The_Stereophonic_Zoom. 170. often with no permanent displacement of the particles of the medium—that is.com/ Visualization-Blumlein-E.Blumlein Eight/Eight 90° . . Longitudinal waves occur when a disturbance sends waves in the same direction as the original wave. For example. and radio waves). Surface waves in water One type of wave is a mechanical wave. p. staying in about the same place while the wave itself moves onwards. They consist. [2] Eargle.pdf) • Visualization XY Stereo System . com/ images/ uploads/ The_Stereophonic_Zoom. This transfers some energy to these neighbors. External links • Michael Williams. which propagates through a medium in which the substance of this medium is deformed. pdf) (PDF).sengpielaudio. usually accompanied by the transfer of energy. . ultraviolet radiation. The Stereophonic Zoom (http://www. Another type of wave can travel through a vacuum. infrared radiation. gamma rays. e. .htm) Wave In physics.rycote. Rycote Microphone Windshields Ltd.g. where the concept of medium does not apply. based on the mechanical origin of acoustic waves. its position within the vibration cycle) is different for adjacent points in space because the vibration reaches these points at different times. wave theory represents a particular branch of physics that is concerned with the properties of wave processes independently from their physical origin. if all the parts were independent. Concepts such as mass. and sound waves generated by radio receivers. in Hunt 1992. may be generalized to all waves.[1] For example. shock waves etc. An attempt to define the necessary and sufficient characteristics that qualify a phenomenon to be called a wave results in a fuzzy border line. 132). p. a moving disturbance in space–time can exist if and only if the medium involved is neither infinitely stiff nor infinitely pliable. There are water waves on the ocean surface. telephone handsets and living creatures (as voices). within some specific contexts (for example. however. all-encompassing definition for the term wave is not straightforward.Wave 87 General features A single. the phase of a vibration (that is. light waves emitted by the Sun. is still a researched area in the study of sound. If all the parts making up a medium were rigidly bound. For such reasons. and so on. wave processes revealed from the study of waves other than sound waves can be significant to the understanding of sound phenomena. The term wave is often intuitively understood as referring to a transport of spatial disturbances that are generally not accompanied by a motion of the medium occupying this space as a whole. This principle was first introduced in Young's study of light and. momentum. the energy of a vibration is moving away from the source in the form of a disturbance within the surrounding medium (Hall 1980. or elasticity. then there would not be any transmission of the vibration and again. It may appear that the description of waves is closely related to their physical origin for each specific instance of a wave process. This is impossible because it would violate general relativity. However. in the case of air: vortices. where energy is moving in both directions equally. with no delay in the transmission of the vibration and therefore no wave motion. no wave motion. Other properties.. On the other hand. dispersion etc. .. this notion is problematic for a standing wave (for example. then they would all vibrate as one. although they are usually described in an origin-specific manner. become therefore crucial in describing acoustic (as distinct from optic) wave processes. radio waves broadcast by radio stations. For example. a vibration is not necessarily a wave. inertia. acoustics is distinguished from optics in that sound waves are related to a mechanical rather than an electromagnetic wave transfer caused by vibration. scattering of sound by sound). they reveal a characteristic that is relevant to all waves regardless of origin: within a wave. In a wave. A relevant example is Thomas Young's principle of interference (Young. or for electromagnetic / light waves in a vacuum. However. Although the above statements are meaningless in the case of waves that do not require a medium. microwaves used in microwave ovens. For example. A vibration can be defined as a back-and-forth motion around a reference value. in the case of solids: Rayleigh waves. 1802. Similarly. 8). p. radiation pressure. This difference in origin introduces certain wave characteristics particular to the properties of the medium involved. a wave on a string). [6] . Consider the string to have a single spatial dimension. E.Wave 88 Mathematical description of one-dimensional waves Wave equation Consider a traveling transverse wave (which may be a pulse) on a string (the medium). let the positive direction be to the right. the wave shaped like the function F will move in the positive x-direction at velocity v (and G will propagate at the same speed in the negative x-direction). this periodicity of F implies a periodicity in time as well: F(x − v(t + T)) = F(x − vt) provided vT = λ. not nonlinear). that is. Consider this wave as traveling • in the direction in space. That is. F(x + λ − vt) = F(x − vt).. so an observation of the wave at a fixed location x finds the wave undulating periodically in time with period T = λ/v. This wave can also be represented by the partial differential equation General solutions are based upon Duhamel's principle. can be measured between any two corresponding points on a waveform (waveform (waveform or.[4] Wave forms The form or shape of F in d'Alembert's formula involves the argument x − vt. the periodicity of F in space means that a snapshot of the wave at a given time t finds the wave varying periodically in space with period λ (the Sine.g. or shape This wave can then be described by the two-dimensional functions Wavelength λ.[5] In the case of a periodic function F with period λ. wavelength of the wave). Constant values of this argument correspond to constant values of F. In a similar fashion. and these constant values occur if x increases at the same rate that vt increases. square. triangle and sawtooth waveforms. by d'Alembert's formula: [3] traveling to the right) traveling to the left) representing two component waveforms and traveling through the medium in opposite directions. and the negative direction be to the left. • with constant amplitude • with constant velocity .[2] • with constant waveform. where is • independent of wavelength (no dispersion) • independent of amplitude (linear media. more generally. the pulse shape changes in a manner often described using an envelope equation. and the green dots propagate with the group velocity. To understand them.w. where velocity is the amplitude envelope of the wave. or may be modulated so as to vary with time and/or position. For simplification. makes clear that this expression describes a vibration of wavelength traveling in the x-direction with a constant phase velocity . the modulated wave can be written in the form:[7] [8] [9] Illustration of the envelope (the slowly varying red curve) of an amplitude modulated wave. The outline of the variation in amplitude is called the envelope of the wave. the phase velocity and the group velocity. Mathematically.[10] [11] Phase velocity and group velocity There are two velocities that are associated with waves. is the wavenumber and is the phase.Wave 89 Amplitude and modulation The amplitude of a wave may be constant (in which case the wave is a c. Frequency dispersion in groups of gravity waves on the surface of deep water. The most basic wave (a form of plane wave) may be expressed in the form: This shows a wave with the Group velocity and Phase velocity going in different directions. examination is restricted to one dimension.[12] . The red dot moves with the phase velocity. The fast varying blue curve is the carrier wave. in cases where the group velocity varies with wavelength. Otherwise. one must consider several types of waveform. Rewriting the argument. this equation can be simplified as: showing that the envelope moves with the group velocity and retains its shape. . or continuous wave). which can be related to the usual sine and cosine forms using Euler's formula. If the group [10] (see below) is wavelength-independent. which is being modulated. the group velocity is showing that the velocity of a localized particle in quantum mechanics is its group velocity. Consequently. say φ(k1). and N is the amplitude of the wave. For example. For example. a common choice for Ao is a Gaussian wave packet:[13] where σ determines the spread of k1-values about k. the shape of the wave packet broadens with time. The exponential function inside the integral for ψ oscillates rapidly with its argument. and the particle becomes less localized.Wave The other type of wave to be considered is one with localized structure described by an envelope. this rate of variation is:[12] where the evaluation is made at k1 = k because A(k1) is centered there.[12] Because the group velocity varies with k. and where it varies rapidly. contributing little to ψ.[12] However. (This observation is the basis for the method of stationary phase for evaluation of such integrals. the velocity of the constituent waves of the wave packet travel at a rate that varies with their wavelength. an exception occurs at the location where the argument φ of the exponential varies slowly. This result shows that the position x where the phase changes slowly.[14] ) The condition for φ to vary slowly is that its rate of change with k1 be small. where . and they cannot maintain the same interference pattern as the wave propagates. Sinusoidal waves Mathematically. so some move faster than others. the most basic wave is the (spatially) one-dimensional sine wave (or harmonic wave or sinusoid) with an amplitude described by the equation: Sinusoidal waves correspond to simple harmonic motion. which may be expressed mathematically as. the position where ψ is appreciable. for example: 90 where now A(k1) (the integral is the inverse fourier transform of A(k1)) is a function exhibiting a sharp peak in a region of wave vectors Δk surrounding the point k1 = k. interfere destructively. In exponential form: with Ao the magnitude of A. for that wave situation. in quantum mechanics the energy of a particle represented as a wave packet is E = ħω = (ħk)2/(2m). the exponentials cancel each other out. moves with time at a speed called the group velocity: The group velocity therefore depends upon the dispersion relation connecting ω and k.[15] In other words. g. volts/meter). the response to complex waves cannot be determined from a sine-wave decomposition.[24] When a medium is nonlinear.[20] [21] The sinusoid is defined for all times and distances.[18] Due to the Kramers–Kronig relations.. A wavenumber meter).[22] [23] In particular. so the sine wave propagating in a dispersive medium is attenuated in certain frequency ranges that depend upon the medium. It is related to the frequency or period by The wavelength of a sinusoidal waveform traveling at constant speed is given by:[16] where is called the phase speed (magnitude of the phase velocity) of the wave and is the wave's frequency. . and electromagnetic waves (a form of transverse vacuum wave) express the amplitude in terms of its electric field (e. generally is . making it easy to analyze. or nearly so. in an ocean wave approaching shore. an arbitrary wave shape can be decomposed into an infinite set of sinusoidal waves by the use of Fourier analysis.[17] Although arbitrary wave shapes will propagate unchanged in lossless linear time-invariant systems. These are related by: In other words. pascals). the simple case of a single sinusoidal wave can be applied to more general cases. the frequency and period of a wave are reciprocals. a linear medium with dispersion also exhibits loss. sound waves) use units of pressure (e.g. The analysis of the wave can be based upon comparison of the local wavelength with the local water depth. the incoming wave undulates with a varying local wavelength that depends in part on the depth of the sea floor compared to the wave height..g. can be associated with the wavelength by the relation The period is the time for one complete cycle of an oscillation of a wave. 91 • • • • • The units of the amplitude depend on the type of wave.[19] The sine function is periodic. meters). maximum distance from the highest point of the disturbance in the medium (the crest) to the equilibrium point during one wave cycle. so the calculation of arbitrary wave behavior can be found by adding up responses to individual sinusoidal waves using the superposition principle to find the solution for a general waveform.g. is the space coordinate is the time coordinate is the wavenumber is the angular frequency is the phase. longitudinal mechanical waves (e. in the presence of dispersion the sine wave is the unique shape that will propagate unchanged but for phase and amplitude. the spatial frequency of the wave in radians per unit distance (typically per measured in meters. Transverse mechanical waves (e. this is the maximum vertical distance between the baseline and the wave. The wavelength is the distance between two sequential crests or troughs (or other equivalent points). In the illustration to the right.. a wave on a string) have an amplitude expressed as a distance (e.. so the sine wave or sinusoid has a wavelength in space and a period in time. Fortunately.g. As a result.. Wavelength can be a useful concept even if the wave is not periodic in space. many media are linear. The angular frequency represents the frequency in radians per second. For example.Wave • is the maximum amplitude of the wave. whereas in physical situations we usually deal with waves that exist for a limited span in space and duration in time. The frequency is the number of periods per unit time (per second) and is typically measured in hertz. Wave 92 Standing waves A standing wave, also known as a stationary wave, is a wave that remains in a constant position. This phenomenon can occur because the medium is moving in the opposite direction to the wave, or it can arise in a stationary medium as a result of interference between two waves traveling in opposite directions. Standing wave in stationary medium. The red dots represent the wave nodes The sum of two counter-propagating waves (of equal amplitude and frequency) creates a standing wave. Standing waves commonly arise when a boundary blocks further propagation of the wave, thus causing wave reflection, and therefore introducing a counter-propagating wave. For example when a violin string is displaced, transverse waves propagate out to where the string is held in place at the bridge and the nut, where the waves are reflected back. At the bridge and nut, the two opposed waves are in antiphase and cancel each other, producing a node. Halfway between two nodes there is an antinode, where the two counter-propagating waves enhance each other maximally. There is no net propagation of energy over time. One-dimensional standing waves; the fundamental mode and the first 6 overtones. A two-dimensional standing wave on a disk; this is the fundamental mode. A standing wave on a disk with two nodal lines crossing at the center; this is an overtone. Physical properties Waves exhibit common behaviors under a number of standard situations, e.g., Transmission and media Waves normally move in in a straight line (i.e. rectilinearly) through a transmission medium. Such media can be classified into one or more of the following categories: • A bounded medium if it is finite in extent, otherwise an unbounded medium • A linear medium if the amplitudes of different waves at any particular point in the medium can be added • A uniform medium or homogeneous medium if its physical properties are unchanged at different locations in space • An anisotropic medium if one or more of its physical properties differ in one or more directions Light beam exhibiting reflection, refraction, transmission and dispersion when encountering a prism • An isotropic medium if its physical properties are the same in all directions Wave 93 Reflection When a wave strikes a reflective surface, it changes direction, such that the angle made by the incident wave and line normal to the surface equals the angle made by the reflected wave and the same normal line. Interference Waves that encounter each other combine through superposition to create a new wave called an interference pattern. Important interference patterns occur for waves that are in phase. Refraction Refraction is the phenomenon of a wave changing its speed. Mathematically, this means that the size of the phase velocity changes. Typically, refraction occurs when a wave passes from one medium into another. The amount by which a wave is refracted by a material is given by the refractive index of the material. The directions of incidence and refraction are related to the refractive indices of the two materials by Snell's law. Diffraction A wave exhibits diffraction when it encounters an obstacle that bends the wave or when it spreads after emerging from an opening. Diffraction effects are more pronounced when the size of the obstacle or opening is comparable to the wavelength of the wave. Sinusoidal traveling plane wave entering a region of lower wave velocity at an angle, illustrating the decrease in wavelength and change of direction (refraction) that results. Polarization A wave is polarized if it oscillates in one direction or plane. A wave can be polarized by the use of a polarizing filter. The polarization of a transverse wave describes the direction of oscillation in the plane perpendicular to the direction of travel. Longitudinal waves such as sound waves do not exhibit polarization. For these waves the direction of oscillation is along the direction of travel. Wave 94 Dispersion A wave undergoes dispersion when either the phase velocity or the group velocity depends on the wave frequency. Dispersion is most easily seen by letting white light pass through a prism, the result of which is to produce the spectrum of colours of the rainbow. Isaac Newton performed experiments with light and prisms, presenting his findings in the Opticks (1704) that white light consists of several colours and that these colours cannot be decomposed any further.[25] Mechanical waves Waves on strings The speed of a wave traveling along a vibrating string ( v ) is directly proportional to the square root of the tension of the string ( T ) over the linear mass density ( μ ): where the linear density μ is the mass per unit length of the string. Acoustic waves Acoustic or sound waves travel at speed given by or the square root of the adiabatic bulk modulus divided by the ambient fluid density (see speed of sound). Water waves • Ripples on the surface of a pond are actually a combination of transverse and longitudinal waves; therefore, the points on the surface follow orbital paths. • Sound—a mechanical wave that propagates through gases, liquids, solids and plasmas; • Inertial waves, which occur in rotating fluids and are restored by the Coriolis effect; • Ocean surface waves, which are perturbations that propagate through water. Shock waves giving rise to various types of radiation such as radio waves. infrared. Quantum mechanics also describes particle properties that other waves. de Broglie waves Louis de Broglie postulated that all particles with momentum have a wavelength A propagating wave packet. in vacuum. infrared. uv) An electromagnetic wave consists of two waves that are oscillations of the electric and magnetic fields. which can be modeled as kinematic waves[26] • Metachronal wave refers to the appearance of a traveling wave produced by coordinated sequential actions. In the 19th century.Wave 95 Other • Waves of traffic. the electric and magnetic fields satisfy the wave equation both with speed equal to that of the speed of light. Electromagnetic waves (radio. such as light and sound. Electromagnetic waves can have different frequencies (and thus wavelengths). visible light. and so forth. James Clerk Maxwell showed that. . the envelope of the wave packet moves at a different [27] speed than the constituent waves. have on the atomic scale and below. propagation of different densities of motor vehicles. From this emerged the idea that light is an electromagnetic wave. visible. ultraviolet and X-rays. in general. An electromagnetic wave travels in a direction that is at right angles to the oscillation direction of both fields. microwaves. micro. that is. Quantum mechanical waves The Schrödinger equation describes the wave-like behavior of particles in quantum mechanics. Solutions of this equation are wave functions which can be used to describe the probability density of a particle. [33] Given the Gaussian: the Fourier transform is: The Gaussian in space therefore is made up of waves: that is. de Broglie proposed a superposition of different wavelengths ranging around a central value in a wave packet. It is well known from the theory of Fourier analysis. and the local wavelength deviates on either side of the main wavelength value. and the more localized the envelope. In a wave packet. where the central wavelength is related to the central wave vector k0 as λ0 = 2π / k0. The parameter σ decides the spatial spread of the Gaussian along the x-axis. while the Fourier transform shows a spread in wave vector k determined by 1/σ. The Fourier transform of a Gaussian is itself a Gaussian. the larger the spread in required wavelengths.[32] or from the Heisenberg uncertainty principle (in the case of quantum mechanics) that a narrow range of wavelengths is necessary to produce a localized wave packet. the electrons in a CRT display have a de Broglie wavelength of about 10−13 m. the wavelength of the particle is not precise. This hypothesis was at the basis of quantum mechanics.[28] a waveform often used in quantum mechanics to describe the wave function of a particle. .[30] For example.Wave 96 where h is Planck's constant. this wavelength is called the de Broglie wavelength. Nowadays. a wave like this with definite wavelength is not localized in space. In representing the wave function of a localized particle. the larger the extent in k. a Gaussian wavefunction ψ might take the form:[31] at some initial time t = 0. and so cannot represent a particle localized in space. a number of waves of wavelengths λ such that kλ = 2 π. That is. and hence in λ = 2π/k.[29] Gaussian wave packets also are used to analyze water waves. and p is the magnitude of the momentum of the particle. the wave packet is often taken to have a Gaussian shape and is called a Gaussian wave packet. To localize a particle. For example. the smaller the extent in space. A wave representing such a particle traveling in the k-direction is expressed by the wave function: where the wavelength is determined by the wave vector k as: and the momentum by: However. p. com/ ?id=o7sXm2GSr9IC& pg=PA77). google. ISBN 0080439306. . Ostrovsky & Alexander I. Erasmo Recami. Oscillations and waves (http:/ / books. . 242. Such nonuniform traveling waves are common in many physical problems. Geometric wave equations. Cambridge University Press. . google. google. Wiley. 60–61.3 Pulse envelope in nondispersive media" (http:/ / books. google. (http:/ / books. In Hugo E. pp. com/ ?id=geYKPFoLgoMC& pg=PA365). the phase term kx is typically replaced by the integral of k(x)dx. google. Quantum many-particle systems (http:/ / books. ISBN 0521631343. com/ ?id=WdPGzHG3DN0C& pg=PA128). . Michael Struwe (2000). "Waves and their properties" (http:/ / books.). Quantum Mechanics (http:/ / books.).). Fitts (1999). [15] Donald D. . Wiley. . Slawinski (2003). com/ ?id=xxbXgL967PwC& pg=PA329). American Mathematical Society Bookstore. ISBN 0470108851. 365. p. Ihsan Shatah. pp. p. Cambridge University Press. Courier Dover. com/ gp/ product/ 0801873258). ISBN 0763759937. [6] Alexander McPherson (2009). "Wave equations" (http:/ / books. Localized Waves. pp. Modulated waves: theory and application (http:/ / www. pp. . . google. WKB method In a nonuniform medium. Elsevier. "§13. Davide Janner (2008). Michel Zamboni-Rached. google. pp. 50–52. 329. 339 ff.). Quantum Mechanics: An introduction (http:/ / books. 121. 37 ff. Foundations for guided-wave optics. [13] See. Springer. Eq. [4] Jalal M. [12] Albert Messiah (1999). . [3] Karl F Graaf (1991). 9. Dover. 13–14. in which the wavenumber k can depend on the location as well as the frequency. Johns Hopkins University Press. google. Cassidy. Henri Orland (1998). [9] Mark Lundstrom (2000). p. ISBN 0471756873. Seismic waves and rays in elastic media. google. Fundamentals of carrier transport (http:/ / books. . p. ISBN 0521658411. . com/ ?id=6TCm8Xy-sLUC& pg=PA242). Floyd James Rutherford (2002). Cuvillier Verlag.7. [17] Paul R Pinet (2009). for example. "Localization and Wannier wave packets in photonic crystals" (http:/ / books. 131 ff. 128 ff. Wave motion in elastic solids (http:/ / books. cit. . ISBN 0801873258. Westview Press. pp. com/ ?id=s7bp6ezoRhcC& pg=PA134). [5] Louis Lyons (1998). gravitational waves are disturbances in the curvature of spacetime. 77. com/ ?id=mx5CfeeEkm0C& pg=PA121) (Reprint in Advanced Book Classics ed. com/ ?id=6kOoT_AX2CwC& pg=PA9). com/ ?id=7qCMUfwoQcAC&pg=PA61) (2nd ed. Cambridge University Press. [8] Fritz Kurt Kneubühl (1997). [7] Christian Jirauschek (2005). google. ISBN 052143601X.). google. Springer. Birkhäuser. . ISBN 3540674586. Gerald James Holton. google. Not to be confused with gravity waves. . FEW-cycle Laser Dynamics and Carrier-envelope Phase Detection (http:/ / books. 15 ff. Introduction to Macromolecular Crystallography (2 ed. com/ ?id=5cZFRwLuhdQC& printsec=frontcover) (Reprint of Oxford 1975 ed. Animation showing the effect of a cross-polarized gravitational wave on a ring of test particles References [1] Lev A. ISBN 9780486409245. ISBN 354062001X. op. predicted by Einstein's theory of general relativity. . 33. google. Hernández-Figueroa. "The linear wave equation" (http:/ / books. Understanding physics (http:/ / books. 363. com/ ?id=mwssSDXzkNcC& pg=PA52& dq=intitle:quantum+ inauthor:messiah+ "group+ velocity"+ "center+ of+ the+ wave+ packet") (Reprint of two-volume Wiley 1958 ed. Allan Bromley (2007). . [11] Stefano Longhi. pp. ISBN 0470185902. com/ ?id=8t4DiXKIvRgC& pg=PA15). ISBN 3865374190. [10] Chin-Lin Chen (2006). .Wave 97 Gravitational waves Researchers believe that gravitational waves also travel through space. [14] John W. Principles of quantum mechanics: as applied to chemistry and chemical physics (http:/ / books. [16] David C. Potapov (2002). google. com/ ?id=rpQo7f9F1xUC& pg=PA340). ISBN 0821827499. com/ ?id=FTdDMtpkSkIC& pg=PA33). according to the WKB method. although gravitational waves have never been directly detected. Wiley-Interscience. amazon. ISBN 0738200522. 2(a) in Walter Greiner. google. . including the mechanics of the cochlea and waves on hanging ropes. google. ISBN 9780486667454. Negele. All you wanted to know about mathematics but were afraid to ask (http:/ / books. p. ISBN 0387987568. com/ ?id=LxzWPskhns0C& pg=PA363). pp. com/ ?id=zsasG2axbSoC& pg=PA37). p. D. [2] Michael A. p. pp. O. google. Marton & Claire Marton.36 and associated discussion in Otfried Madelung (1996). .42. google.1 The Principle of Superposition" (http:/ / books. or compounded of these. image.. p. . [32] Siegmund Brandt. acoustic wave propagation and scattering. com/ ?id=Kf8fyvRd280C& pg=PA32).). google. S. [30] Chiang C. Springer. Fitts (1999). . ISBN 0521598273.1287/opre.1098/rspa. [19] See Eq. Introduction to solid-state theory (http:/ / books. The physics and chemistry of the interstellar medium (http:/ / books. and Eq. p. and Greated. • Hunt. [21] Seth Stein. 57. Cambridge University Press. Refractions. "Prop VII Theor V" (http:/ / gallica. L. A. G.. Quantum Mechanics (http:/ / books. 61) …the individual waves move more slowly than the packet and therefore pass back through the packet as it advances" [28] Ming Chiang Li (1980). . . ISBN 0865420785. 208. The versatile soliton (http:/ / books. The picture book of quantum mechanics (http:/ / books. Introductory physics series). com/ ?id=g5q6tZRwUu4C& pg=PA271).229. com/ ?id=UfSk45nCVKMC& pg=PA134). D. com/ ?id=TC4MCYBSJJcC& pg=PA106). pp. 31. google. Cambridge University Press. com/ ?id=oRSHWmaiZwUC& pg=PA208& dq=sine+ wave+ medium+ + linear+ time-invariant). p. G. Baltimore: The Johns Hopkins University Press.. Michael E. "Thus. p. Opticks: Or. .html#pub17). William R.. p. p. google.). Isaac (1704). google. Richards (1956). com/ ?id=YWd7zHU0U7UC& pg=PA57). Advances in Electronics and Electron Physics.. Belmont. [27] A. Wysession (2003). 32. . com/ ?id=WHMNEL-9lqkC& pg=PA47) (2nd ed. In L. Allan Bromley (2007). Electromagnetic Radiation: Variational Methods. com/ ?id=8t4DiXKIvRgC& pg=PA17). ISBN 3540293043. [20] Aleksandr Tikhonovich Filippov (2000). google. ISBN 0801858704. google. V. C. Principles of quantum mechanics (http:/ / books.Wave [18] Mischa Schwartz. "Wave packet solutions" (http:/ / books. ISBN 0521826349. A treatise of the Reflections. Lighthill. 5. Serway and John W. Quantum Mechanics (http:/ / books. Operations Research 4 (1): 42–51. . Cantrell (2000). google. London. doi:10. [23] Kimball A. Jewett (2005). 3. . 16. • Hall. 59 ff. Wysession (2003).0088. [25] Newton. com/ ?id=7qCMUfwoQcAC& pg=PA60) (2 ed. And: P. google. . Modulated Waves. (1992). Whitham (1955). 271. 6.I. 106. 1.5 in F Mainardi (1996). 119 ff.. ISBN 0521620058. com/ ?id=QKsiFdOvcwsC& pg=PA677).. New York: Schirmer Books. (http:/ / books. (1971). p. . 47. Springer. "Electron Interference" (http:/ / books. M. ISBN 0817636358. Tielens (2005). World Scientific. Wiley-Blackwell. "§14.). Julian Seymour Schwinger (2006). (1987).. Potapov.4.). Theory and Applications. Eq. [31] Walter Greiner. California: Wadsworth Publishing Company. ISBN 9810242719.). and Seymour Stein (1995). ISBN 053449143X. The applied dynamics of ocean surface waves (http:/ / books. 60. com/ ?id=yK_J-3_p8_oC& pg=PA261) (3rd ed. Herbert Überall. Hans Dieter Dahmen (2001). In Ardéshir Guran. . ISBN 0387951415. Vibrations and Waves (M. 118. an arbitrary function f(r.aip. (1999). bnf. E. p. An introduction to seismology. google. . 261 ff. Courier Dover Publications. google. M. ISBN 9780780347151. Quantum Mechanics for Applied Physics and Engineering (Reprint of Academic Press 1981 ed. Bibcode 1955RSPSA.). com/ ?id=VM4GFlzHg34C& pg=PA23) (3rd ed. The Musician’s Guide to Acoustics. ISBN 0865420785. google. Series A 229: 281–345. Cambridge University Press. 53. com/ ?id=7qCMUfwoQcAC& pg=PA60) (2nd ed. Principles of physics (4th ed. Mei (1989). com/ ?id=wMnvg681JXMC& pg=PA119). com/ ?id=3SOwc6npkIwC& pg=PA59). New York: Acoustical Society of America Press. op. and earth structure (http:/ / books. J.1955.Donald D.. p.. OCLC 163810889. Bostrom.P. ISBN 3540674586. Inflexions and Colours of Light. Origins in Acoustics (http://asa. 677. T. Fromhold (1991). p. google. Cambridge University Press. . Springer. ISBN 0534007589. Waveguides and Accelerators (http:/ / books. 17. A theory of traffic flow on long crowded roads". • French. p." [26] M. earthquakes. . Acoustic Interactions with Submerged Elastic Structures: Nondestructive testing. google. . 98 Sources • Campbell. Electronic basis of the strength of materials (http:/ / books. pagination). are either the Colours of homogeneal Lights. Academic Press. fr/ ark:/ 12148/ bpt6k3362k. ISBN 0120146533. com/ ?id=1DZz341Pp50C& pg=PA433). Springer. World Scientific. John Wiley and Sons. com/ ?id=Kf8fyvRd280C& pg=PA31). A. p. G. Cengage Learning. Nelson Thornes. [33] Cyrus D. Proceedings of the Royal Society of London. Modern mathematical methods for physicists and engineers (http:/ / books. google. [22] Seth Stein. Allan Bromley (2007). 23. [29] See for example Walter Greiner. f128. "Shockwaves on the highway". 134. cit. Springer. p. • Ostrovsky. Springer. B.. com/ ?id=x_h2rai2pYwC& pg=PA16). 60. "All the Colours in the Universe which are made by Light.1. doi:10. ISBN 3540674586. google. Also Two treatises of the Species and Magnitude of Curvilinear Figures. F. ISBN 354060443X.org/publications. Michael E. "On kinematic waves. Milton. and John Joseph Gilman (2003).). ISBN 9971507897.. Bennett. t) can be synthesized by a proper superposition of the functions exp[i (k·r−ωt)]…" [24] Raymond A. ISBN 0-393-09936-9. A. I.10 and discussion in A. ISBN 0486667413. . . 433. D. "Transient waves in linear viscoelastic media" (http:/ / books. pp. ISBN 0521658411.. p. Musical Acoustics: An Introduction. . . p. (1980). Leroy. A. Communication Systems and Techniques (http:/ / books.T. "(p. google.281L. p. II. D. lecture notes and exams. Velocities of waves Phase velocity • Group velocity • Front velocity • Signal velocity . htm-simulation) • MIT OpenCourseWare 8.phy. 99 External links • Interactive Visual Representation of Waves (http://resonanceswavesandfields.hk/wiki/englishhtm/Interference.htm) • Simulation of interference of water waves (http://www.phy.hk/wiki/englishhtm/Lwave.com/html_books/lm/ch19/ch19. Los Angeles.edu/courses/physics/ 8-03-physics-iii-vibrations-and-waves-fall-2004/) Free.phy.scienceaid.org/papers/diss.htm) • Simulation of longitudinal traveling wave (http://www.html) • Simulation of diffraction of water wave passing through a gap (http://www. assignments.com/2007/08/ true-waves.co.acousticslab.03: Vibrations and Waves (http://ocw. uk/feschools/) • chapter from an online textbook (http://www.htm).htm) • Sounds Amazing—AS and A-Level learning resource for sound and waves (http://www.phy.cbu. (2001) (http://www. Perceptual and Physical Properties of Amplitude Fluctuation and their Musical Significance.hk/wiki/englishhtm/ Diffraction.phy.html) • Simulation of waves on a string (http://www.acoustics. Doctoral Dissertation.lightandmatter.uk/physics/waves/ properties.htm) • Simulation of transverse traveling wave (http://www.Wave • Vassilakis.hk/wiki/englishhtm/TwaveA.html) • Science Aid: Wave properties—Concise guide aimed at teens (http://www.blogspot.mit. independent study course with video lectures.N.ac.htm) • Simulation of stationary wave on a string (http://www. P.hk/wiki/englishhtm/StatWave.net/applets/mechanical-waves) • of longitudinal and transverse mechanical wave (http://www.edu/~jvarrian/applets/waves1/lontra_g. University of California.salford.physics-lab. Furrykef. Tim32. HenningThielemann. Omegatron. FHannes. Furrykef. Quadell. Nuclearwombat. Darkwind. Bongomatic. Arabic Pilot. TodKarlson. Andykass.164. Efryevt. Teorth. Oking83. David Gerard. Keenan Pepper. Bkell. Qwfp. Goldencako. Adashiel. EverettColdwell. Kohtala. Funandtrvl. Kramer. Knight1993. Haddock420. Teapeat. Movses. Pak21. BigJohnHenry. Bemoeial.wikipedia. Stevenj. NightFalcon90909. Tbackstr. AllTheThings. 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