IEEE TRANSACTIONS ON SIGNAL PROCESSING. VOL. 39, NO. 9.SEPTEMBER 1991 1973 The Theory of Bandpass Sampling Rodney G. Vaughan, Member, IEEE, Neil L. Scott, and D. Rod White Abstruct-The sampling of bandpass signals is discussed with respect to band position, noise considerations, and parameter sensitivity. For first-order sampling, the acceptable and unac- ceptable sample rates are presented, with specific discussion of the practical rates which are nonminimum. The minimum sam- pling rate is pathological in that any imperfection in the imple- 0 fs fl fc fu mentation will cause aliasing. The susceptibility of the accept- able sampling rates to aliasing decreases with increasing guard- Fig. 1, The bandpass situation as an analog signal spectrum. The sampling rate is expressed as A Hz and the band is located at ( f r , A,). Only the band size from the minimum sampling rate. positive frequencies are shown. In applying bandpass sampling to relocate signals to a base- band position, the signal-to-noise ratio is not preserved owing to the out-of-band noise being aliased. The degradation in sig- nal-to-noise ratio is quantified in terms of the position of the bandpass signal. For the construction of a bandpass signal from second-order samples, the cost of implementing the interpolant (dynamic range and length) depends on Kohlenberg's sampling factor k, the relative delay between the uniform sampling streams. An elaboration on the disallowed discrete values of k shows that some allowed values are better than others for implementation. Optimum values of k correspond to the quadrature sampling values with the band being either integer or half-integer posi- tioned. I. INTRODUCTION (b) Fig. 2. Examples of (a) integer band positioning for c = 3, and (b) half- T HE subject of bandpass sampling is an aspect of dig- ital signal processing which has relevance to a variety of disciplines including, for example, optics [ 11, radar [2], integer band positioning for c = 3. sonar [3], communications [4], biomedical signals [ 5 ] , bandwidths from the origin, i.e., f L = c ( f u - f L ) , c = 0 , power measurement [6], and general instrumentation, +1, +2, * * ,. and c = 0 is the low-pass case. Fig. 2(a) such as sampling oscilloscopes. indicates integer band positioning, for c = 3. Historically, Cauchy seems to have been the first to hy- Another special case is half-integer positioning, when pothesize the bandpass sampling requirement [7]. Nyquist fr. = ( ( 2 c + 1)/2)(fu - f L ) , Fig. 2(b) shows an example [8] and Gabor [9] also alluded to the bandpass case. Koh- of half-integer positioning for c = 2. These cases of band lenberg [ 101 introduced second-order sampling and pro- positioning are useful for illustrating aspects of aliasing, vided the basic interpolation formula, which was then re- and are particularly important in the application of sam- ported in the texts by Goldman [ 111 and Middleton [ 121. pling to the effective relocation of signals between band- The bandpass situation is depicted in Fig. 1, where the pass and low-pass positions. sampling rate is expressed asfy Hz and the bandpass sig- The classical bandpass theorem for uniform sampling nal is located between f L Hz and f u Hz. The signal band- states that the signal can be reconstructed if the sampling width B is less than f u - f L , and the positive frequency rate is at leastff""") = 2f u / n , where n is the largest integer band can be expressed as the interval ( f L , f u ) . within f u / B , denoted by n = I,[ f,/B]. The bund position refers to the fractional number of This minimum rate for uniform sampling is of interest bandwidths from the origin at which the lower band edge from a theoretical viewpoint, but for practical applica- resides. A special case is integer bund positioning, which tions this rate is often presented in a misleading way. Fig. holds when the band is located at an integral number of 3 from Feldman and Bennett [ 161, also reproduced in sev- eral texts (e.g., [13]-[15]), illustrates the minimum uni- Manuscript received July 5, 1989; revised June 24, 1990. form sampling rate for bandpass signals. The theoretical R. G. Vaughan and N . L. Scott are with Industrial Development, DSIR, minimum rate f, = 2 B is seen to apply only for integer Lower Hutt, New Zealand. band positioning. The skew lines represent the minimum D. R. White is with Physical Sciences, DSIR, Lower Hutt, New Zea- land. uniform rate required when the passband is not located in IEEE Log Number 9101475. an integer position. The vertical lines, which represent a 1053-587X/91/0900-1973$01.OO 0 1991 IEEE T 1 bandwidths be- to sample at above the theoretical minimum rate or. ject is discussed and results are depicted graphically. and in general it 2fu (the low-pass case rate) are not. Gregg The interpolant has the properties that S(0) = 1 . Gaskell [ I ] gives an expression (see (16)) for the valid rates. NO. (5) Shannon's theorem) where the equality in f . is zero. some authors mistakenly inter.(t>= cos [27r(f L + B)t . between -& and +&. hold. but otherwise there is little for which the interpolant blows up. S ( p / B ) [ 171 states without elaboration that rates between f I""") and = 0 f o r p # 0. sitivity is discussed in Section 111. the interpolation is given by '0 B ZB 38 48 5B 6B 7B f" Fig. This illustrates the need Geometrically. length machines is affected by the choice of k . but implementation difficulty using finite word- the theoretical minimum but still well below the frequen.r7rBkI SOW = (3) 2 n Bt sin r 7r Bk S. Gregg's 1171 +f 6+ k) S (-t + + k)] (I) version of Fig.f u ) . ternatively stated. the signal is band limited to less than B and the signal occu. Three special cases are noteworthy: tioned. tween the two-sided spectrum. 3 to mean that no aliasing will kBr.( r + l)nBk] (4) 27rBt sin ( r + 1)aBk and r is an integer given by the band position such that if there is no signal component at the frequencies f L o r i . 1974 IEEE TRANSACTIONS ON SIGNAL PROCESSING.e. pass sampling at above the minimum rate are reviewed. the first term of the existing information is clarified and the cases for band.fL)t . integer band positioning ( r odd) and for these conditions The theoretical minimum uniform sampling rate fF = there are exactly r bandwidths between -fL and +fL.( r + l ) n B k ] . SEPTEMBER 1991 to be applied independent of the band position. > 4B. S(0 = so(0 + sI(0 (2) Note that f. This is analogous to the anomaly of the low-pass case (cf. Each sequence is of rate B (i. Kohlenberg's [ 101 second-order sampling theorem 'An alternative definition for r in ( 5 ) is to place the equality with the allows the minimum rate. (7) occur as long asf. The situation now corre- The discussion includes sensitivity of sampling rates and sponds to either integer band positioning ( r even) or half- offers a practical guide to bandpass uniform sampling. In Section 11. and S( p / B + k ) = 0 . 1 . The right-hand side of ( 5 ) can be expressed in terms of pies the interval (0.cos [27rfLt . 9. 1. For this alternative definition. This sen- cies of the signal being sampled. The theoretical minimum sampling rate is pathological in the sense that any engineering imperfections in an im- plementation will cause aliasing. 3. and a swap of the term that drops out of 'The low-pass case is exceptional in that the component atfi = 0 can be the interpolant when the equality holds.. right-hand term instead of the left-hand term. this sub.. 1 I . and the restrictions kBr # 0. al.Ir <. Minimum sampling frequency for band of width B [ 161 discontinuity.* 2 B can be applied only when the band is integer posi. These rates correspond to the region above the function In the general case when the equality of ( 5 ) does not in Fig. in the form of an average rate. I Kida and Kuroda [ 181 discuss the the center frequency f c .e. disallowed signal components for higher order sampling. . 2.rnBk] . The When the equality in ( 5 ) holds. a point elucidated in Section 11. guard-bands need to be included. 1.id). * . 3. * * on k are removed. 2fL 2h + .. = 2 B is a valid uniform sampling rate only where cos [27r(rb . This causes a swap in the above meaning of odd and even r . r recovered because i t contains trivial phase information cannot be zero and the low-pass case corresponds to r = 1 . So(t). i. i. L 2fu can B B hold only if there is no signal component atL. VOL 39. Nevertheless. sampling every 1 / B s). The tech- nique is second order in that two interleaved uniform sam- ple sequences are applied. has extrema which are larger than unity. Using the samples to construct a signal f ( t ) at ( f L . is tions. k is otherwise unre- information regarding bandpass sampling at rates above stricted. h)t . in general. valid. interpolant. there are at least r . 3 displays more clearly the discontinuity in which p is the sample number index and the interpolant in the function by using dotted lines for the vertical sec. traverse disallowed values of the sampling frequency. kB(r + 1 ) = 0.cos [2n(rB .e. the value of k may not take on the discrete values pret figures such as Fig. 2 .. and the sequence separation is denoted by k s. e. contributions from the aliased spectra. Section IV describes and quantifies Quadrature sampling is important because. where n is the integer given by Similarly. using an image rejecting mixer). which is a form of large wedge to the left of the figure. The low- plied the special case of uniform sampling combined with pass casefs 2 2fu. 4 is a corrected and extended version of Fig. the in-phase and quadrature components are periment to demonstrate the effect. * . 3 .Q(t) sin o. in which the signal-to-noise ratio is ideally pre- and the simplified interpolant served. occurs at the tips to derive the required interpolation weights used for time of the wedges.t (11) timates the effect of applying “acceptable” sampling rates with practical. (8) ever a band-pass signal is reproduced at a baseband po- 2nBt sin nBk sition by sampling. Their innovation was to “decimate for implementation. Rice and Wu [ 1 9 ] ap. combined into the baseband. ing filter. This es- f (t) = Z(t) cos a c t . as discussed in Section whence I and by Kida and Kuroda [ 181. rad/s. requires integer for bandpass sampling owing to (postfilter) thermal noise band positioning. with its well-known sinc ( 2 n B t ) in. which. In form interpolator. The areas inside the wedges are the allowed zones for Grace and Pitt [ 3 ] drew attention to this special case of sampling without aliasing. This is in contrast to (9) analog complex mixing (i. the aliasing noise sampled explicitly from the bandpass signal. Even with an ideal antialias- form rate sampling. Each wedge across quadrature sampling in that k = 1/ 2 B = 1 /4 fc + m / 2fc the figure corresponds to a successive value of n .5 f S s . Fig. the allowed sampling rates are repre- quadrature sampling using the m = 0 case and also used sented by a vertical line within the allowed areas and decimation before the interpolation filter. tous presence in less critical applications. The shaded area represents Kohlenberg’s sampling scheme. They The theoretical minimum sampling rate f. The conditions for acceptable uniform sampling rates can be written [ l ] requires sampling at times 2fu . . as the name the signal-to-noise ratio degradation. rectangular bandpass fil- where Z(t) and Q ( t ) are quadrature components of the cen. Equations ( 1 6 ) and ( 17) are depicted graphically in Fig. given by n = 1 . (17) The equality conditions of (16) hold only if the band is confined to the interval (fL. and these rates are clearly dangerous points aligning the samples.1 P actp =pa. despite its ubiqui- S.COS TBk ( 2 ~ B- SLP(t) = . Jackson and above the band position given on the abscissa. since any engineering imperfection by two” the samples before applying the Hilbert trans. This is seen resulting from using typical bandpass filters is presented by denoting the bandpass signal in the usual way: in terms of a signal-to-(aliasing) distortion ratio.2. the signal-to-noise ratio (SNR) is not preserved terpolant.. 11. uniform sampling rates that result in aliasing.h n n . normalized by B .*2. where m = 2 / r ( = 2 for integer band positioning).i. The sampling frequency. UNIFORMSAMPLING To extract Z(t) directly from f ( t ) .VAUGHAN et al. Waters and Jarrett [4] implemented a sampling system..: THE THEORY OF BANDPASS SAMPLING 1975 i) The low-pass case is f L = 0 and r = 0 for which the Matthewson [ 2 ] also discussed implementation of quad- interpolant reduces to rature sampling. with a simple ex- indicates. f (ti) = -Q($>. the noise from all the aliased bands is ii) The special case of k = 1/ 2 B corresponds to uni. cor- used the fact that Z(t) and Q(t) are a Hilbert transform pair responding to integer band positioning. instead of ideal. k. such as sam- nBt pling oscilloscopes.h): a frequency component at f = f L orf = f u will be aliased. Also.(t) = ___ cos 2nfct.e. tp = - 2fc’ p = 0 . to obtain Q ( t ) (13) 1 s n 5 lgk]. is on the The signals Z(t) and Q ( t ) are band limited to the interval ordinate. The degradation in signal-to-noise ratio does not sin n B t seem to have been discussed to date. as mentioned above. (15) 4. i. B. ter angular frequency U. corresponds to the the band being integer positioned. When- COS t nBk) .. and the abscissa represents the band position (0. will move the sampling rate into the disallowed area.e. ters. B / 2 ) so that each stream requires sampling at a rate f u / B . This has clear im- iii) The quadrature sampling case has plications for communications and radar receivers where bandpass sampling is applied to relocate the signal effec- tively to a low(er) pass position. The spectrum of a sampled signal is periodic. = 2 B . because any sampling rate variation will not only cause aliasing. Sampling at nonminimum rates is equivalent to aug. 4 2 % B 3 1 2 0 1 0 1 2 3 4 5 6 I - fu B Fig. and this range is divided into values above and below a menting the signal band with a guard-band. f. If the operating point is the vertical midpoint of the wedge. . =f. The giving the lower and upper guard-bands as total guard-band is given by B G T = f. This case would Fig. borders. The lower order and the guard-bands become 1 I . Brown [20] has pointed out that for symmetric double wedge is then given by sideband signals. In fact. showing the nth wedge have an allowed operating “region” in Fig. The case also represents a pathological condi- The allowable range of sampling frequencies is tion. any nominal operating point (see Fig. 5 ) allowed operating point away from the tips of the wedges of Fig. NO. 4 of straight with the guard-bands and the sampling frequency toler- lines bisecting the disallowed areas between the parallel ance. SEPTEMBER 1991 1 5 6 4 5 3 fs . to show the practical areas of nonminimum sampling rates. and the band is located at (fL. 5 is an expansion of Fig. where creases with increasing n.). but will also cause the signal to become folded incorrectly with a likely loss of otherwise retriev- able information. the asymmetry de- and its location (ft.1 To find the relation between the guard-bands and the BGL = Afsu Yy- sampling frequency. f. 4. The allowed and disallowed (shaded areas) uniform sampling rates versus the band position. This is found from the overall bandwidth (signal band- width plus total guard-band) Note that symmetric guard-bands imply asymmetric sam- W=B + BGT (18) pling frequency tolerances. 4.f . 3 . 1916 IEEE TRANSACTIONS ON SIGNAL PROCESSING. which is labelled on the right-hand ordinate of Fig. then the sampling rate is f. However. 39.2 B Hz. 4. allowing a theoretical mini- mum sampling rate of 1/B for a band-limited signal where the total bandwidth (both sidebands) is B .U + ALL (23) tween positions where the spectrum will be aliased. B is the bandwidth. respectively. The right-hand ordinate is the total guard-band (sum of upper and lower guard-bands). 4 can be interpreted as having a guard-band be. ) . Ah = Af. the order of the wedge n is required. with B G T = B G L + B G u . n’ . The figure is an extension of Fig. VOL.+ BGu (20) (26) and B G L and B G u are the lower and upper guard-bands. 9. is the sampling rate. the spectra can “fold over” each other without loss of information. .5-kHz guard.n) = 0 ($) (30) pling rate and the guard-band size.1 i.[10703/30] = 356 and the sampling rate has cision requirements and increase the guard-band. This demonstrates the tradeoff between the precision required of the sam- AA - 2B I n(n . for this example. indicated to give a feel for practical requirements. . Af. locus approaches a vertical line at f. above (i./B . If a of the 429th wedge. 6 as the line I / ( n ( n .[10702. This sensitivity corresponds to the vertical locus within the 428th wedge. wedge (samef.B . infinitesimally to the left of) the tip The f . fL f" fi Fig.. corresponding to a zero allowable line gives the relative precision required of the sampling sampling rate range. the creasing separation of the band-pass signal and the origin. while remaining above the tip of the 428th bands so that the overall band is (10. and the line is -0(l/n2).B) . and parallel to the f./B = n as Ah ap- i.356 = 72 separate ranges of dis. 10. above the tip An example illustrates the use of the equations.703 MHz)..282 kHz. . / B de- required to avoid aliasing clearly increases with the in. (For the above-mentioned points.1) B ('r ./ B ./B .5)/428 = 50. where n = 1. but of limits of 2(10703)/356 = 60. is on the lower edge of the nth rate when operating with f.7. the line trates the pitfall of arbitrarily increasing the sampling rate would be an order of magnitude in A h / 2 B higher than from its theoretical minimum value for band-pass sam. 5 . i. 4.e.n = l line. With Note. then the rela- tive precision of the sampling rate becomes. n' . various nth wedges. 6 which steepens as f . 6./B .2f. the line interpolating the various bands. For practical situa- tions. (smaller n ) . / B . Afs/2B = BGT/B * l / n ' ..017 kHz.n = 1 as a function of wedge.e. an allowable range of Ah = 152 Hz. This illus.e. Moving to lower order wedges 25-kHz signal is allocated symmetric 2./B allowed sampling rates have been traversed.5/25] = 428. For example. a modest increase in the sampling rate permits a . The relative precision required of the sampling oscil- The theoretical minimum sampling rate (for the signal lator is depicted in Fig. then the difference between the minimum allowed set of sampling rates and the maximum minimum and maximum allowed sampling rates is available guard-band.l)(f.n = 1 line represents operating with the of the (n + 1)th wedge. a total of 428 . related to the inverse square of the separation of the BGL = Ah 4 band from the origin. increasing n.) For the example with n = 428.n = 1 line. After intersecting the f. an operating point within a given wedge corresponds to a The relative precision of the minimum sampling rate positive slope locus in Fig.. and Afsu and AfsL represent the allowable upward and downward range of the sampling frequency from its nominal value.1)). for n' = n + 10. This without guard-bands). represent guard-bands against aliasing of the signal occupying bandwidth B . If an operating point is within the nth proaches zero at the tip of the wedge.: THE THEORY OF BANDPASS SAMPLING 1977 &l BGL+ BGU f.n') runs above thef. the inclusion of larger guard-bands by operating in the retical minimum rate to a rate which includes the guard. The positions of a typical crystal and RC oscillator are 2(10702.n = 1. and isf4min)= n. Parameters in the nth wedge from Fig. the rel- n' ative frequency precision required of the sampling rate is BGu = 4 1/(428 * 427) = 5 ppm./2B = l/n'(n' .e. = 60. from (24) so that the relative precision required offs is and (25). If the relative Ah = n-1 n (29) guard-band is now held at a constant size. The guard-band size decreases with 2(f. Varying pling. creases. wedge. f. / B ) will relax the sampling oscillator pre- then n' = 1. BGLand B. that in moving from the theo. increasing n .VAUGHAN et al.n = 1 line in Fig.130 kHz to 2(10700)/355 course requires higher sampling rates. ( t ) . P 4 (32) where F(f ) is the analog spectrum. and are I Ro: fL If IrB .) (38) 111. * denotes convolu- tion. caused by the sampling stream being delayed by k seconds. is the same for FA and FE. The RI: el = nBk(r + 1) (40) interest in second-order sampling has been not only from and the theoretical viewpoint of maintaining a constant mini- mum average sampling rate independent of the band po. The phase shifts are. it must be borne in mind that so far the discussion has been limited to Ro: FA = + F(-fb) F(f) (35) avoiding aliasing in the sampling process. Exact F~ = F(f) + eJZeoF(-fb) (36) (re)construction of the bandpass analog waveform re. whence S(t) = S . these operating points are within encompass the band whereas R I can encompass the band. 9. 1 . illustrates the geometric aliasing relations. z Equations ( 3 1 ) and ( 3 2 ) show that the sampled spectrum is a series of analog spectra located B Hz apart and scaled by B ./B .~ analog spectrum. a point addressed RI: FA = F ( f ) + F(-f. Ro: fb =f+ 2fL . Ro cannot ing tof. Fig. From here on. the required range ( 3 8 ) and hereon. Note that from the choice of definition for r. 3 . 39. i. 7. which is an elaboration of Linden’s [21] Fig. The line is an interpolation of the operating points correspond. 4 . the nth wedge at a position above the tip of the (n + 1)th wedge. SEPTEMBER 1991 1 For second-order sampling.f L (33) 1 0 1 10 100 1000 lL and B Fig. VOL. The scaling factor B in ( 3 1) and ( 3 2 ) is omitted in (35)- struction using various k .e. specifically. (34) pling rates. The relative precision of the set of minimum allowed uniform sam- RI: rB . in the aliased spectra. realizable precision requirement. ( t ) terms of the interpolant (see (2)) are ap- plied. (number of bits in the coefficients) and length (number of The aliased contribution (the term containing F( -f ’)) coefficients) for realizing the interpolant as a finite im. and quires exact implementation of the sampling parameters used in the interpolant computation.rB (41) sition.( r + l ) B . but also from its practical application in the sim- plified form of quadrature sampling a bandpass signal. there is a progressive phase change of 27r Bk imposed on F( f ) at successive spectral locations. (42) This section concerns aspects of bandpass signal con. For FE. In terms of Fig. INTERPOLATION IN SECOND-ORDER SAMPLING where Kohlenberg’s second-order sampling result gives a the. R. The application is the generation of a band-pass. is constant throughout each region. the U explicit frequency dependence of FA and FB is dropped. the spectra resulting from each of the two interleaved sampling sequences applied to the bandpass signal are [21] 10. 4. = f + 2fL . and the sampling rate sensitivity. except for a phase shift which pulse response filter. FB = F ( f ) + e-j2”F(-f. NO. pB) (31) FB(f) = F ( f ) * c Be-JZTBkp6(f P . and p is a spectral position index.’ FA(f> = F(f) * B 6(f . are In addressing sampling rate variation. integer or half-integer band po- sitioning. These regions correspond to where the So(t) and S .n = 1. There are two regions in the band in which the aliasing arises from different aliases of the negative frequencies from the 10. 6 .: f .1978 IEEE TRANSACTIONS ON SIGNAL PROCESSING. different for each of the Ro and each of the R I from second-order samples. signal however.f L s f If L + B. as shown in the example The spectra resulting from the two sampling sequences above.. Ro: Bo = nBkr (39) oretical minimum average sampling rate which would correspond to the straight line fs = 2 B on Fig.) (37) within the following section. e.. The bandpass signal construction is (cf. F(f) is the analog spectrum. = 27r(rB . Considerable insight into the inter. ( 1 ) ) As already noted for integer and half-integer band po- sitioning.J m (47) + ~ sincef.) + cos ( a u t . m = 0. (eJ2'lFA . The aliased bands are progressively phase shifted relative to the analog band as indicated by the notation y' = e-J2Bi1.e. = ( r 1 / 2 ) B . 2m + 1 k = which are unique except when k = 0. e. f 2 .t . 7. + I . as evident in (10). for the narrow-band situation. 7.) general interpolant form ( 2 ) predominantly resembles a So@)= (48) modulated (byf.t . polant can be gained by expressing it in terms of the fre. although the relation is not simple.) -cos (WLI . 2 / B . Using the angular frequencies w.VAUGHAN et a l .e . FA+ FE) (6"'FA I Fa) In R. In Fig. so that these values of k indeed is used for brevity. e.(FA + FE). correspond to quadrature samples. 0. When k de- l . the phase factor notation (53) e .fL) which since the interpolant becomes larger.e. The cos (w.) sinc function.~ 2 ~ B k r.-.. (e . becomes zero. and RI in the analog spec. and F A ( f )and F B ( f )are the spectra from each of the two interleaved (by k s) uniform sample sequences. : arg (y"") and the length (number of filter taps) required for the in- = -281: terpolant contribute to the cost of an implementation. 2B(r 1) + whence the sampling sequences coincide. i.For integer and half- integer band positioning. Both the dynamic is the common border of R. range (number of bits required for the filter coefficients) trum. -cos (w.. : THE THEORY OF BANDPASS SAMPLING 1919 Flfl = k 1 (0' or?. Further sim- 1 plification in the signal construction occurs if e-'"' = Ro: F(f) = @8" . >> B . wL = 2 n f L . In this case.. = the bandpass signal becomes more difficult to implement 2af. 2 f L / B = r and R. the filter length is also which shows the composition of the interpolant in the clearly defined for a given coefficient size as long as 5. 7.) The dynamic range is a minimum for quadrature sampling SI(t) = (49) when the interpolant assumes its maximum value at t = w B t sin 0. m RI phase factor O = e x p I-jZflBkI Fig. .e. . construction of quencies in Fig.. in which case 1 F(f) = e-~201 . i 2 0 0 ~ ~. wB = 2 n B and w . (50) so the disallowed values f o r k are This occurs when Ro: m k = -. for quadrature sampling. i. and noting that Ro: arg (y")= -200. so the dynamic range and w B t sin Bo the length are related. R .. .FB) (44) F(f) = . e (45) j2nkB(r + 1) = b ' ( 2 m + 1)a (51) rB' i. the region Ro becomes zero. The geometric relations for aliasing in second-order bandpass sampling and general band position. same form as the low-pass case of (8). 0.F ~ ) (43) -1. As k approaches its disallowed values. 1 / B . For general values of A r and eo. coefficients.2fL (57) The low-pass case illustrates the increase in interpolant B length with a poor choice of k . the magnitude of the first term should lating F. For constructing a bandpass signal whose band is not integer or half-integer positioned. 8(a). clearly not viable (i. one quadrature sampling location which is at k = 1 / 2 B .. is optimum in this ’ I2 case. such as those due to 4r+)B 2 ( 1 ) quantization.e. The units of a i are watts/hertz and particularly clear for the low-pass case. the quadrature sampling values nal construction may arise because of the need to include “guard bits” in the sampled signal and the interpolant m -~ - 1 . For a given dynamic range and coef. m = 0 to 10. The uni- sulting uncertainty in the constructed signal is of interest.. m = 0.1980 IEEE TRANSACTIONS ON SIGNAL PROCESSING. f l .3.5. The variance of a component (10) is independent of k ) . for minimum filter length. the alter- example where r = 9 and A r = 0 (half-integer band po- native to applying the interpolant of ( 5 4 ) directly is to sitioning). where there is only the noise power in the constructed signal is found by in. NO. . how the errors in the sampled data magnify borne in mind that half-integer band positions alias to be during the construction of an analog bandpass signal. for integer or S. 8(b). The specific aim is to get an idea of of integer band positioning ( r even and A r = 0) and is the cost of a poor choice of k . Since the quad- rature sampling locations are at the minima. will increase. . it is only possible to apply this in the special case frequency domain. The The required resampling is a shift of one of the sample asymptotes correspond to the disallowed values if k . VOL. 5 2 . . 8(a) for the In a nonuniformly sampled low-pass system. The minimum noise power is a 2 B / 2 . small where a i Aand aiBare the variances of the components of perturbations (relative to l / r ) of k from the quadrature FA and FE. 39. the quadrature sampling points are at k = Letting a’ = a i A= ait?. Consequently. This is resents a best case. is plotted against k normalized by B in Fig. This is evident in Fig. from A r < 1 . 8(a) in F can then be written in terms of its magnitude: where any choice of quadrature sampling gives the opti- mum k with respect to sensitivity of the signal construc- tion to uncertainty in the sampled data. in the form the choice of definition of r . Nb spectrally inverted at the low-pass position. For the half-integer Further expense in the implementation of bandpass sig- band positioning case. 8 . viz. it should be choices of k . so ( 5 5 ) rep. uniform sampling where k = 1 / 2 B (corresponding to quadrature sampling since fc = B / 2 ) . caused for example values do not result in a large penalty in the uncertainty by k becoming very small. and in Fig. both the dynamic range and the length of the inter- polant increase. tegrating over the band of F .e. A r = r . and from (8). m / 10B and k = m / 11B for Fig. (59) In terms of an implementation. This is because errors in the measurement or k = l + 9 + 1 20B’ 20B ’ generation process of FA and FE. as already and is estimated here using a sensitivity analysis in the noted.. viz. are located at the function minima of Fig. . k = 1 / 2 B is a disallowed value) Progress can be made by assuming that the variations in Fig. configuring the system such that the signal is in- teger positioned and applying uniform sampling is the Ar (- = log2 sin2 eo + sin e) el . and these do not occur at the func- and integrating. a. The interpolant is simplified and is of corresponding (same analog frequency) components in the same for each of the quadrature sampling values (NB: FA and FB are uncorrelated. sequences using the sinc 2 n Bt interpolant on alternate k = m / l O B . The generate uniform samples and interpolate with sinc 2 n Bt. SEPTEMBER 1991 viates from the values corresponding to quadrature sam. For correlated variations.p(t) = -Cot TBk ~ nBt + sin2nBt ~ (54) half-integer band positioning and quadrature sampling. the noise power in F is tion minima. . In Fig. ( 5 9 ) indicates for poor simplest arrangement. In an implementation. For baseband interpolation of a sampled bandpass signal. the low-pass interpolant can be written. occurs for A r = 0 and el = sin2 a B t 2nBt n/4 + mn. for Fig. caused by constructing an analog bandpass signal. or at least kept small with respect to the second term. number of bits of resolution irrecoverably lost in calcu- ficient word size. 9. 8(a) and (b). the noise power ’ The first term decays as 1/ n Bt whereas the second term relative to its minimum value can be expressed as the decays as 1 / 2 n Bt. . Assuming integer band denotes the extent of the Ro portion of the band and 0 I positioning. and both k = samples. . 8(b). form sampling case is given by k = 1 / 2 B and. be minimized. functions are symmetric about k = 0 and k = 1 / 2 B .. i. The effect of the choice of k on the re. where pling. it is unwise to use an arbitrary choice of quadrature sampling since the construction process could well give rise to an error 1 I . 8(a). 8(b) for r = 10 and A r = 0. may in turn create errors in the recon- structed signal..and using ( 4 3 ) and (44)in (55) 1/ 2 1 B . 3 / 2 1 B . - 1s ul 10 where el. the first quadrature sampling value system.. bands. . NOISEI N BANDPASS SAMPLED SIGNALS (neither integer nor half-integer band positioning).. in which tem requires finding this minimum. such as 455 kHz. which is the band noise. in Noise considerations are important not only where the general. using guard-bands) with r = 356 and B = 30 kHz.e. expressed here for R. and an alias suppression of 40 dB. and ratio will be poorer than that from an equivalent analog so the configuration of the optimum general sampling sys- system (i. The analog signal- The accurate construction of the bandpass signal from to-noise ratio is thus SIN. i.(r + l)B. 3 ns.. the periodicity of the spectrum means that wide- is close to the first minimum of the function. The graphs represent a cost of a poor choice correspondingly simpler. in-band noise power density of Np.. A k -0 .e. In this case of general band positioning. is thus thenlthe constructed spectrum of interest F ( f ) and its al- ias F( -f') in terms of the wanted spectrum is. An estimate of the equivalent number of bits of resolution irrecov- i. (b) r = 10. For lower intermediate frequencies in communications. The rel- ative difference between the constructed spectrum and the wanted spectrum can be expressed as a result which is also readily obtained from time domain considerations. : VAUGHAN et al.. the resulting signal-to-noise amounts to the same sampling sequence interleaving). If the sample sequences are interleaved by k.5 IV. but also in the measurement of noise it- self.: THE THEORY OF BANDPASS SAMPLING 25 20 - -fn. such as in communi- cation receivers. only. the re- quired accuracy of the sample sequence interleaving is OI 02 0. is all combined into each of thefs/2 optimum quadrature sampling requires the minimum k . The best general value of k occurs at the In applying bandpass sampling to relocate a bandpass first function minimum (or the last minimum. The average sample rate is ( A ) = 2 B .7 MHz. This disadvantage. The signal-to-noise ratio for its samples also depends on the accuracy of the sampling the sampled signal becomes degraded by at least the noise rate and k . A r = 0 (half-integer band positioning). A r = 0. 8. (a) r = 9. The minima do not. and integer positioned. = -nk. of k in the second-order bandpass sampling process. an ideal image-rejecting mixer). etc. The minima correspond to cases of quadrature sampling. out-of-band noise power density No. For the previous example (in Section 11. correspond to quadrature sampling. which signal to a low-pass position.e. as a function of k . For large values of r pertinent to commu- nications or radar. as the signal-to-noise ratio is preserved. k = 1 /4fc. the requirement is erably lost in interpolating a bandpass signal using uncertain baseband rate samples. andfL = 10. signal-to-noise ratio is important. make Consider a system with a bandpass signal of spectral it worthwhile arranging the band to be integer or half- power density S . such as the thermal noise introduced by the global minimum. For small A k 05 which approximates how the degree to which the alias is 00 suppressed depends on the difference of k and k. A k - 7 ns ( r = 15). viz. In a sampling in the signal. and and the interpolant implementation features k = k. As r increases. such as in noise power measurements. aliased from the bands between dc and the passband. relevant hardware. the demands on A k become con- siderable. However... + A k . well as the need here for a complicated interpolant.5 Ob 07 01 09 1 00 03 0-4 - k 10 Ak=-*- B 357 ~ 3 0 ( 1 0 ) ~ (b) Fig. the timing error A k becomes increas- ingly critical. .e. NO. giving a sampled spectra SNR of about 21 dB (ENBW) of the analog signal.50 In the case where the noise spectrum is not uniform. sumption in (67) that the signal power at the filter input i. or even normally be the dominant effect on the signal distortion higher levels.. Fig 9 Averaged sampled spectra of a 10 9-kHz carrier amplitude modu- noting N E A as the equivalent noise spectral density lated by a 200-Hz sinusoid in the presence of white noise The sampling (ENSD) and B E . respectively. This is because of the as- The minimum degradation given by (64) is 10 log 32. In practice. The test signal was a 10. and the spectra calculated by The limits are chosen so that the wanted signal band oc- FFT on a LeCroy oscilloscope. giving a Nyquist bandwidth of 625 Hz. . The average spectra is shown in Fig. for example.) has similar. and (b) 1250 Hz. ratio. then the degradation of the signal-to-noise w 9 -20 ratio in decibels is simply e _J -30 a f -40 DsNR z 10 log II. is 20 kHz. Nevertheless. 10 [23] shows the signal distortion ratio for two To get a feel for the filter requirements. aliasing filter is necessary. then the increase in the noise power is at 2 5 P -30 -40 least 15 and 25 dB. giving a sampled spectra SNR of about 38 dB. The signal-to-noise ratio is lectingf.7 MHz). 0 For the example with B = 30 kHz located at the integer B -10 position n = r / 2 = 30(fu . of guard-bands will normally ensure this will not be the and assumes that B E . then of the mean-square aliasing error and the total signal power passed through the filter and can be interpreted as NEABEA = NEsB. 9. and an infinitesimally small sampling aperture. is then at least som “)I2 do A simple experiment using a noise source and a sam- pling oscilloscope demonstrates the effect. However. In practice. Np >> No and the signal-to-noise ratio is estab- lished before sampling.50 Equivalently. trading off the filter requirements against sampling rate 9(b). (65) a signal-to-noise ratio. while in practice it is a case at the edges of the filter.1982 IEEE TRANSACTIONS ON SIGNAL PROCESSING. if Np = No. De.95-kHz carrier amplitude-modulated by a 200-Hz sine wave. a 0 100 200 300 400 500 600 700 800 conservative estimate of the noise power density in each FREQUENCY (Hz) of the bands in the sampled spectrum can be obtained by (b) equating the noise power before and after sampling. The resultant signal was where H(o)is the filter transfer function andf. the presence considering only the aliased noise from the noise source. or n = 356(fu -20 . ideal filters. windowed using a minimum 3-term metic center frequency of the upper and lower 3-dB points. Fig. even when nications cases. The average of 10 power cupies a Nyquist bandwidth. It is here defined for the bandpass The degradation in a signal-to-noise ratio is NEs/NEA case [23] as which. the out-of-band ing the antenna. filter types against the sampling rate normalized to the 1 . etc. as the equivalent noise bandwidth rate is (a) 125 kHz. this requires se- spectra is shown in Fig. On the other hand. Note that this analysis assumes a flat analog (a) noise spectrum. The signal was then sampled at 1250 Hz sitioned. tive signal-to-noise ratio [23]. the potential for signal-to-noise degradation power level in the channel of interest. A Bruel and Kjaer 0-20 kHz noise source sig- nal was added to the test signal. In radio commu- in bandpass sampling systems is considerable. It is the ratio B). the antialiasing filter will power (adjacent band signals. but also the receiver front end. and is only one of many factors contributing to the effec- The observed degradation is thus about 17 dB. In most antialiasing filter. adjacent little greater. in decibels. than that in the band of interest. so an anti. 39. the filter H(o)not only represents the the analog spectrum contains only thermal noise. 10.455 kHz). SEPTEMBER 1991 Often. (64) . channel power levels may be tens of decibels above the In general. go -10 ing n >> 1. about 15 dB. Recall that this minimum is given by is constant across the spectrum. and NEs as the ENSD of the sampled signal with 2 B the sampling rate (it is known [22] that the ENBW of a sampled system cannot exceed tortion ratio (SDR) [24] is a useful metric. VOL. is the arith- sampled at 125 kHz. includ- communications systems. with the signal-to-noise ratio now being -21 dB. such that the band is integer or half-integer po- about 38 dB. the effective noise temperature is increased FREQUENCY (Hz) by at least n. and assum. the signal dis. 9(a). Blackman-Harris function. and n = This signal distortion ratio offers only a rough guide for 20 000/625 32. Modulation. Dec. it is wise to configure the system such that the band is integer or half-integer positioned. 15. The advantages are a simplified interpolant and the poten- tial to apply quadrature sampling with the flexibility to Fig. 10. 1978. Ints. D. the curves which are valid. guard-band size. 1432-1436.. vol. D. however. vol. vol. This matter is discussed in [23].e. This degradation is unavoid- able and a simple estimate for the minimum degradation is available in terms of the band position. H.” J .” J . Ltd. The sampling represents the optimum sample sequence inter- envelope edges for each curve represent Q = 1 and Q = 200 1231. i. “Certain topics in telegraph transmission theory. Stockton. Grace and S. there are only discrete points on bandpass signal from its samples. the signal becomes degraded owing to aliasing of the noise (introduced after any band- pass antialiasing filter) between dc and at least the original bandpass spectral position. pp. 1953. New York: Van Nostrand.” AIEE Trans. . 1979. pp.. S . E. quadrature sampling spacing should not be applied 1). “Digital processing of bandpass over the 3-dB bandwidth only. J. ” IEEE Trans. M. Quadrature for second-order and eighth-order Buttenvorth and Chebychev filters. R. 1960. J. Linear Systems.” GECJ.. J . no. New York: Wiley. Elecrron. 7 . Fourier Transforms. 1978. S . Noise and Spectral Analysis.. In the sampling process. Shanmugam. Information Theory (Prentice-Hall Electrical Engineer- be taken when increasing the sampling rate from its min. The envelopes enclose the For cases where the band is not integer of half-integer curves for filters with Q values from 1 to 200. 1986. 1949. B. 490-595. Clarke and J. IO. sample at above the theoretical minimum rate. Pitt. For practical bandpass sampling. Kohlenberg. and Optics. and displays the least 3-dB bandwidth of the filter Because of the selection sensitivity to error propagation in the construction of a of fs mentioned above. Care must S . Middleton. For constructing an analog bandpass signal from its second-order samples. M. Digital and Analogue Communications Sysrems. vol. R e s . 1953. Waters and B . i. herent detection. An Introducrion io Siatisiical Communication Theory. Feldman and W . In applying Fig. AES. no. 1965. 1928. K. 9 . Aerosp. owing to possible error propagation from im- r = 36. Inst. r = 2fL/B (for integer band positioning) by Q = ( r + Here.” J . but again. D. “Bandwidth and transmission performance. the curves of Fig. . 0. = 50 ksamples/s) with k = 1/4fc is usually close to the optimum. R.e. this defi. corresponding tolerance to variations in the sampling rate. 1977. New For sampling an analog bandpass signal in order to pro. Panter. New York: rate is equivalent to the introduction of guard-bands or a McGraw-Hill. Phys. Matthewson.e. 4. the denominator cor. positioned. contribute to the aliasing distortion.” Alia Frequenza. it F. 1982. 93. vol. 24. Haykin. London: Constable and Co. 1978.. aliasing will occur. Jarrett. 28. Nyquist. C. Jackson and P. and the various allowed nonminimum C. 10. F. 1968. vol. P. I . “Principles and theory of watt- should be borne in mind that the signal distortion ratio is meters operating on the basis of regularly sampled pairs. “Quadrature sampling of high frequency For such a definition.. York: Wiley. W . D.f. vol. 12. a filter equivalent to an eighth-order Chebychev with Q = 16. the simplest system is to con- figure the band to be integer positioned and apply uniform (which is also quadrature) sampling. nition does not appreciably shift the curves of Fig. XLVII. Gabor./&& = 2 (i.VAUGHAN et a1 THE THEORY OF BANDPASS SAMPLING 1983 cess digitally the signal. Strictly speak. Del Re. Phys. Elec. “Bandpass signal sampling and co- change by less than a couple of decibels. only an approximation to the effective signal-to-noise ra. D. Operating at a nonminimum sampling P. 4. Soc. “Bandpass signal filtering and reconstruction through minimum-sampling-rate digital processing.. D. “Exact interpolation of band-limited functions. Black.” Bell Syst. vol. ing. Sept. there is a need to Appl. imum value because there are bands of rates for which New York: McGraw-Hill. W. Tech. Sysi. The relations between the sampling rate tolerance. CONCLUSION 1946. no.. Eng. New tortion power” ratio if the integration in the numerator is York: Wiley. “Theory of communication. Amer. Sri. signals. The signal distortion ratio (SDR) as defined for the bandpass case use several values for the sequence spacing. Acoust. A. a signal distortion ratio of 37 dB requires a sam. 10 waveforms.18. 1953. Gaskell. ing Series). responds to the signal power only rather than total power. Bennett. 47. leaving in the sense that it requires the shortest filter length (least number of taps) for interpolant implementation using a given coefficient word size. H. J. only interleaved portions of the filter “stop band” no. Nov. R. the optimum sampling sequence interleaving eter Q = fc/B3dBis related to the band position parameter for bandpass signal construction is not trivial to find.. although the minimum quadrature spac- pling rate off. tio caused by the aliasing. vol. The param.. in general. S . the New York: Wiley. The definition of the signal distortion ratio in (67) can REFERENCES perhaps be more easily related to a signal power to “dis. In the example with a 25-kHz band at 455 kHz. V . 1982. Analog and Digital Communications Systems. 44.” J . sampling rates give practical design choices. Modulation Theory. Goldman. Communication Sjistems. Gregg. perfect samples. “Relations between the possibility of resto.E. Australia). currently Project Leader in Communications Technology at DSIR. Sysr. New Zealand. [23] N. “A discussion of sampling theorem. L. Instrum. Japan. 1219-1226. Phys- [201 J. and [ Z l ] D. “On uniform sampling of amplitude modulated sig.. A. His recent work has been 47. Electron. 69. I . L. Signal Processing rind its Appli- cations (ISSPA’90) (Gold Coast. and M.” Pror. Signal Processing und i f s Ap- plications (ISSPA ’90)(Gold Coast. His current research areas include anten. vol. “The signal distortion ratio of filters due to aliasing. M . Rice and K. New York: Wiley (Series in Telecommunications). 6. NO. pp. He is curacy Johnson noise thermometry. electronic and software system design. Hamilton. White.” IEEE Trans. interactive graph- 1983. 9. 736-739. Truns. degrees in electrical engineering from Canterbury University. G. Neil L. rangement of sampling points. ment of Scientific and Industrial Research. Jr. “The noise bandwidth of sampled data systems. and high-ac- nas. Brown. New Zea- land. Aug.D. vol. 1988 (1st ed. Syst. pp.” Proc. New Zealand. 1986. perature measurement and control.” IEEE Trans.E. and the Ph. Electron. His main activities are in the fields of tem- industrial application. no. time series analysis. 1989. Linden. Standards Section of the Department of Scientific puter-controlled instmments for laboratory and and Industrial Research in Lower Hutt. instrumentation.and com. L. on sampling theory. He worked in the New Zealand Post Office on D. Int. G.E. Scott received the B. 1 . “The signal distortion ratio of filters due to aliasing.. Aerosp. ics and Engineering Laboratory. AES-18. degree from Aalborg University. VOL. 1-10. no. New Zealand.. 38. (Elec- ration of bandpass-type band-limited waves by interpolation and ar. 39. Aerosp. Symp. Mens.. He then moved to the Depart- pp. “Quadrature sampling with high dynamic tricity on hydropower station remote control and range. 2nd ed. I t i t . Vaughan (M’83) received the B. Rod White received the M. and digital signal processing. multiple path propagation. Australia). 1990. Gagliardi. Kuroda. symp. IRE. Scott and R . and the application of DSP in [22] D. where he has nals. 1959. followed by research in digital mobile communications. Since then he has been with the Temperature entific and Industrial Research. arrays. Kida and T. Vaughan.” in Proc. Scott and R. New Zealand. Nov. no. Denmark. and M..1984 IEEE TRANSACTIONS ON SIGNAL PROCESSING. Iniroducfion to Communications Engineering. 1982. 1241 R. 4. in 1980. Vaughan. July worked on image processing. W u . trical) degrees from the University of Canterbury. W . Commun. Aug. 4 . Waikato.” Electron. [23] N. 1036-1043. degree in phys- ics with first class honors from the University of tall traffic analysis before joining the Physics and Engineering Laboratory of the Department of Sci. vol. pp. digital signal processing. There he developed microprocessor. vol.Sc. AES-19. Rodney G. ics. R.” IEEE communications.E. SEPTEMBER 1991 [I81 T. 1990. H. pt. vol. 1978). He worked for six years at New Zealand Elec- [I91 D.