CSEITQUESTIONS.BLOGSPOT.IN CSEITQUESTIONS.BLOGSPOT.IN CSEITQUESTIONS.BLOGSPOT.IN IT6502 DIGITAL SIGNAL PROCESSING UNIT 1 SIGNAL AND PROCESSING PART A 1. Calculate the minimum sampling frequency required for x(t) = 0.5 sin 50πt+ 0.25 sin 25πt, so as to avoid aliasing. 2. State any two properties of Auto correlation function. 3. State sampling theorem. 4. Distinguish between power and energy signal with an example. 5. Define and express the transfer function of Nth order LTI system. 6. Compare linear convolution and circular convolution. 7. State low pass sampling theorem. 8. What is correlation?What are its types? jωn 9. Find the energy and power of x(n) = Ae u(n). 10. Determine which of the following sequences is periodic, and compute their fundamental period. (a) j7πn Ae (b) sin(3n) 11. Is the system y (n) = ln [x (n)] is linear and time invariant? n 12. Determine Z transform of x(n) = 5 u(n) n 13. Find the signal energy of (1/2) u(n) 14. Determine whether the following sinusoids is periodic, if periodic then compute their fundamental period. (a) cos 0.01πn (b) sin(62πn/10) x (n) 15. Check whether the system y(n) = e is linear. n 16. Determine Z transform of x(n) = a u(n) 17. Define the auto correlation and cross correlation. 18. List any four properties of Z-Transform. 19. What are the properties of convolution? 20. List the properties of discrete time sinusoidal signals? PART B n n 1. (i) Find the convolutionx(n) * h(n) , wherex(n) = a u(n) h(n) = β u(n) (ii) Find the Z-transform of the following sequences : n x(n) = (0.5) u(n) + u(n −1) x(n) = δ(n − 5) . 2. (i) State and explain sampling theorems. (ii) Find the Z-transform auto correlation function. 3. (i) Suppose a LTI system with input x(n ) and output y(n ) is characterized by its unit sample n response h(n ) = (0.8) u(n ). Find the response y(n ) of such a system to the input signalx(n ) = u(n ). (ii) A causal system is represented by the following differenceEquation 1 CSEITQUESTIONS.BLOGSPOT.IN CSEITQUESTIONS.BLOGSPOT.IN CSEITQUESTIONS.BLOGSPOT.IN Compute the system function H (z )and find the unit sample response of the system in analytical form. n 4. (i) Compute the normalized autocorrelation of the signal x(n ) = a u(n ),0 < a <1 (ii) Determine the impulse response for the cascade of two LTI system having impulse responses n n h1(n) = (0.5) u(n) and h2(n) = (0.2) u(n) , 5. (i) Find the inverse Z-Transform of using (1) Residue method and (2) Convolution method. (ii) State and prove circular convolution. 6. Determine the causual signal x(n)for the following Z-transform 3 2 (i) X(z) = (z +z) / ((z-0.5) (z-0.25)) (ii) X(z) = (1+z ) / (1-z +0.5z ) -1 -1 -2 7. (i) Find inverse Z – transfer of X(Z) = ifROC : |Z| > 1, (2) ROC : |Z| < 0.5, (3) ROC : . . 0.5 < |Z| <1 (iii) Derive expressions to relate Z – transfer and DFT 8. (i)Determine the transfer function, and impulse response of the system y(n) – y(n – 1) + y(n – 2) = x(n) + x(n – 1). (ii) Find the convolution sum of 1 2 2,0,1 1 0 andh(n) = δ(n) – δ(n – 1) + δ(n – 2) – δ(n – 3). 9. (i) Find the Z transform of x(n) n = 2 u(n – 2) 2 x(n) = n u(n) (ii) State and explainscaling, linearity and time delay properties of Z transform 10. (i)Derive the equation for convolution sum and summarize the steps involved in computing convolution. (ii) State, prove and explain the sampling theorem 11. Determine the casual signal x(n) for the following Z‐transform 2 CSEITQUESTIONS.BLOGSPOT.IN CSEITQUESTIONS.BLOGSPOT.IN CSEITQUESTIONS.BLOGSPOT.IN 12. (i) Explain the different types of digital signal representation with examples (ii) What is Nyquist rate? Explain its significance while sampling analog signals (ii) List the properties of ROC. 13. Check whether the following systems are linear nonlinear, time variant or invariant, causal non‐causal, stable and unstable 1. y(n) = cos[x(n)] 2.y(n)=x(‐n+2) 3. y(n)=x(2n) 4.y(n)=x(n)cosω(n) 14. Find the convolution of the signal x(n) = {1,2,-3,4} and h(n) = {-5,-6,7,8,9} using tabulation method Compute the normalized autocorrelation of the signal. UNIT 2 FREQUENCY TRANSFORMS PART A 1. Write down DFT pair of equations. 2. Calculate % saving in computing through radix –2, DFT algorithm of DFTcoefficients. Assume N = 512. 3. State and prove Parseval's theorem. 4. Compute the DFT of the four point sequence x(n ) = {0,1,2,3}. 5. What is the relation between DFT and Z-Transform? 6. What is phase factor or twiddle factor? 7. List the uses of FFT in linear filtering? n 8. Find the DTFT of x(n)=-b .u(-n-1). 9. Compute the IDFT of Y(k)={1,0,1,0}. 10. How DFT is differ from DTFT. 11. Find DFT of sequence x(n) = {1, 1, -2, -2} 12. What are the computational saving (both complex multiplication and complex addition) in using N – point FFT algorithm. 13. What do you mean by in – place computation? 14. Differentiate between DIT and DIF FFT algorithm. 15. What is meant by radix 4 FFT? 16. List any four properties of DFT. 3 CSEITQUESTIONS.BLOGSPOT.IN CSEITQUESTIONS.BLOGSPOT.IN CSEITQUESTIONS.BLOGSPOT.IN 17. Compute DFT of x(n) = {1, -1, 1, -1} K 18. Find the value of WN when N = 8 and K = 2 and also k = 3 19. How many multiplication and additions are required to compute N point DFT using Radix 2 FFT? 20. Give transform pair equation of DCT? PART B 1. (i) Explain, how linear convolution of two finite sequences are obtained via DFT. (ii) Compute the DFT of the following sequences : x = [1,0,−1,0] (2) x = [ j,0, j,1] when j = −1 . 2. Draw the flow chart for N = 8 using tadix-2, DIF algorithm for finding DFT coefficients. 3. By means of the DFT and IDFT, determine the response at the FIR filter with the impulse response h(n ) = [1,2,3] and the input sequencex(n ) = [1,2,2,1]. 4. Compute the DFT of the following sequence x(n ) using the decimation in time FFT algorithmx(n ) = [1,−1,-1,-1,1,1,1,-1]. 5. (i) Evaluate the 8-point for the following sequences using DIT-FFT algorithm (ii) Calculate the percentage of saving in calculations in a 1024-point radix -2 FFT, when compared to direct DFT. 6. Determine the response of LTI system when the input sequence x (n ) = {−1, 1, 2, 1, −1 } by radix 2 DIT FFT. The impulse response of the system is h(n) = {−1, 1, −1, 1}. 7. (i)Find 8-point DFT for the following sequence using direct method {1,1,1,1,1,1,0,0} (ii)state any six properties of DFT. 8. Compute 8 point DFT of the following sequence reusing radix – 2 DIT FFT algorithm x(n) = {1,-1,-1,-1,1,1,1,-1} 9. (i)Discuss the properties of DFT. (ii)Discuss the use of FFT algorithm in linear filtering and correlation 10. Find DFT for {1,1,2,0,1,2,0,1} using FFT DIT butterfly algorithm and plot the spectrum. 11. Compute the eight point DFT of the given sequence x(n) = { ½, ½, ½, ½, 0, 0, 0, 0} using radix – 2 DIT - DFT algorithm. 12. (i)Find 8-point DFT for the following sequence using direct method {1,1,1,1,1,1,0,0} (ii) State any six properties of DFT. 13. a) Draw the flow chart for N = 8 using tadix-2, DIF algorithm for finding DFT coefficients. b) Determine the following system for static, linear, time variance, causal. i) y(n) = x(n+2);ii) y(n) = x(n2); ) y(n) = x(-n); 14. Determine the response of LTI system when the input sequence x (n ) = {−1, 1, 2, 1, −1 } by radix 2 DIT FFT. The impulse response of the system is h(n) = {−1, 1, −1, 1}. 4 CSEITQUESTIONS.BLOGSPOT.IN CSEITQUESTIONS.BLOGSPOT.IN CSEITQUESTIONS.BLOGSPOT.IN UNIT 3 IIR FILTER DESIGN PART A 1. What are the limitations of Impulse invariant method of designing digitalfilters? 2. Draw the ideal gain Vs frequency characteristics of :HPF and BPF. 3. What is meant by warping? 4. What are the limitations of impulse invariance method? 5. Sketch the various tolerance limits to approximate an ideal lowpassandhighpass filter. 6. What is the importance of poles in filter design? 7. Compare bilinear and impulse invariant transformation. 8. What is aliasing? 9. Define Bilinear transformation with expressions. 10. Mention the properties of Butterworth filter. 11. What are the characteristics of Chebyshev filter? 12. Write the transformation equation to convert low pass filter into band stop filter. 13. Define Phase Delay and Group Delay. 14. Why IIR filters do not have linear phase? 15. Use the backward difference for the derivative and convert the analog filter to digital filter given 2 H(s)=1/(s +16) 16. State the relationship between the analog and digital frequencies when converting an analog filter using bilinear transformation. 17. Explain the advantage and drawback of Bilinear transformation. 18. Compare the Butterworth and Chebyshev Type-1 filters. 19. What is the main drawback of impulse invariant mapping? 20. Compare the digital and analog filter. PART B 1. Design digital low pass filter using Bilinear transformation, Given that Assume sampling frequency of 100 rad/sec. 2. Design FIR filter using impulse invariance technique. Given that and implement the resulting digital filter by adder, multipliers and delays Assume sampling period T = 1 sec. 3. (i) Find the H (z ) corresponding to the impulse invariance design using a sample rate of 1/T samples/sec for an analog filter H (s) specified as follows : (ii) Design a digital low pass filter using the bilinear transform to satisfy the following characteristics 5 CSEITQUESTIONS.BLOGSPOT.IN CSEITQUESTIONS.BLOGSPOT.IN CSEITQUESTIONS.BLOGSPOT.IN (1) Monotonic stop band and pass band (2) − 3 dBcutoff frequency of 0.5 πrad (3) magnitude down at least −15 dB at 0.75πrad. 4. D esign an II R filter using impulse invariance t echnique for the given A ssume T = 1 sec. Realize this filter using dire ct form I a nd direct form II. 5. The specification of the d esired lowpass filter is D esign a Butterworth digital filter using bilinear transformati on. 6. The specification of the d esired low pass filter is D esign a Chebyshev digital filter usi ng impulse invariant tra nsformation. 7. D esign an IIR digital low pass butter worth filter to meet the following requirements: Pass band ripple (peak to peak): ≤ 0.5dB, Pass b and edge: 1.2kHz, Stop band attenu ation: ≥ 40 dB, Stop ban d edge: 2.0 kH z, Sampling rate: 8.0 kH z. Use bilinear transfor mation techn ique. 8. (i) Discuss the limitation of designin g an IIR filte r using impulse invaria nt method (ii)convert the analog filter with system transfer f unction using bilinear t transformation n 2 Ha(S)=(S+0.3) / ((S+0.3) +16) 9. The specification of the d sired low pass filter is 0.8 ≤ │H(ω)│ ≤ 1.0; 0 ≤ ω ≤ 0.2 │H(ω)│ ≤ 0.2; 0.32 ≤ ω ≤ D ensign butter worth digital l filter using g impulse invariant Transformation. 10. Determine the system function H(z) of the chebyshevs low pass digital filter with the specifications =1dB ripple in the pass band 0 =1dB ripple in the stop band 0 Using bilinear transformation (assume T=1sec) 11. Obtain the direct form I, direct form ii ,cascade, parallel form realization for the system y(n)= -0.1y(n-1)+0.2y(n-2)+3x(n)+3.6x(n-1)+0.6 x(n-2) 12. Apply Bilinear Transformation to H(s) =2/(S+2) (S+3) with T=0.1 sec. 6 CSEITQUESTIONS.BLOGSPOT.IN CSEITQUESTIONS.BLOGSPOT.IN CSEITQUESTIONS.BLOGSPOT.IN UNIT 4 FINITE IMPULSE RESPONSE DIGITAL FILTERS PART – A 1 2 3 4 5 6 7 8 9 10 11 12 Compare FIR filters and FIR filters with regard to stability and complexity List out the conditions for the FIR filter to be linear phase. What is Gibb’s phenomenon or Gibb’s oscillation? Write the equations for rectangular window and hamming window. Write the equations for blackmanwindow.andhanning window. Distinguish between FIR and IIR filters. Compare the digital and analog filter. What are the desirable properties of windowing technique? Write the equation of Bartlett window. Draw the Direct form I structure of the FIR filter. Write the steps involved in FIR filter design. Draw the direct form implementation of the FIR system having difference equation y(n) = x(n) – 2x(n-1) + 3x(n-2) – 10x(n-6) 13 14 15 16 17 18 19 20 Obtain direct cascade realization of the system H(Z) = (1+5Z +6Z )(1+Z ) What are advantages and disadvantages of FIR filter? What is the reason that FIR filter is always stable? What is the necessary and sufficient condition for linear phase characteristic in FIR filter? State the properties of FIR filter? What are called symmetric and antisymmetric FIR filters? State the condition for a digital filter to be causal and stable? Write the procedure for designing FIR filter using windows. 21 Write the procedure for designing FIR filter using frequency sampling method. PART - B -1 -2 -1 1. Design the first 15 coefficients of FIR filters of magnitude specification is given below : 2. Draw THREE different FIR structures for the H(z) given below: -1 -2 -1 H(Z) = (1+5Z +6Z )(1+Z ) 3. Design and obtain the coefficients of a 15 tap linear phase FIR low pass filter using Hamming window to meet the given frequency response. 7 CSEITQUESTIONS.BLOGSPOT.IN CSEITQUESTIONS.BLOGSPOT.IN CSEITQUESTIONS.BLOGSPOT.IN 4. Determine the coefficients of a linear phase FIR filter of length M = 15 which has a symmetric unit s ample response and a frequency response that satisfies the conditions 5. Design an FI R filter using handing window with the following specification 6. (I) Design a single toroth filter to reject frequencies in the range 1 to 2 red/sec using rectangular window with N =7 . (ii) Compare Hamming window and Kaiser Window. 7. Prove that an FIR filter has linear phase it the unit sample response satisfies the condition h (n) = h (N-1-n).also discuss symmetric and anti symmetric cases of FIR filter when N is even. 8. (I) Explain briefly how the zeros in FIR filter is located. (ii) Using a rectangular window technique, design a low pass filter with pass band gain of unity cut off frequency of 1000Hz and working at a sampling frequency of 5 kHz. The length of the impulse response should be 7. 9. Design an FI R low pass digital filter using the frequency sampling method for the following specifications Cut off frequency = 1500Hz Sampling frequency = 15000Hz Order of the filter N = 10 Filter Length require d L = N+1 = 11 10. Design a FIR low pass filter having the following specifications using Henning window Assume N = 7 11. (I) Realize the following FIR system using minim um number of multipliers -1 -2 -3 -4 (i)H(Z) = 1 + 2Z + 0.5Z - 0.5Z - 0 .5Z (4) -1 -2 -3 -4 -5 -6 (ii) H(Z) = 1 + 2Z + 3Z + 4Z + 3 Z + 2Z + Z (4) (iii) Design a digital FIR band pass filter with lower cut off frequency 2000Hz and upper cut off frequency 3200 Hz using Hamming window of length N = 7. Sampling rate is 10000Hz. (8) 12. (i) Consider a n FIR lattice filter with coefficients k1 = 1/2 ;k2 = 1/3 ; k3 = 1/4. Determine the FI R filter coefficients for the direct form structure (8) 8 CSEITQUESTIONS.BLOGSPOT.IN CSEITQUESTIONS.BLOGSPOT.IN CSEITQUESTIONS.BLOGSPOT.IN (ii) Using a rectangular window technique, design a low pass filter with pass band gain of unity cut off frequency of 1000Hz and working at a sampling frequency of 5 kHz. The length of the impulse response should be 7. (8) UNIT – V FINITE WORD LENGTH EFFECTS PART – A 1. What is truncation? 2. What is product quantization error? 3. What is meant by fixed point arithmetic? Give example 4. Explain the meaning of limit cycle oscillator 5. What is overflow oscillations? 6. What are the advantages of floating point arithmetic? 7. Compare truncation with rounding errors. 8. What is dead – band of a filter? 9. What do you understand by input quantization error? 10. State the methods used to prevent overflow? 11. Compare fixed point and floating point arithmetic? 12. What are the two types of quantization employed in a digital system? 13. What is rounding and what is the range of rounding? 14. What is quantization step size? 15. Define Noise transfer function? 16. What are limit cycles? 17. What is meant by block floating point representation? What are its advantages? 18. What are the three-quantization errors to finite word length registers in digital filters? 19. What is coefficient quantization error? What is its effect? 20. Why rounding is preferred to truncation in realizing digital filter? 21. State the need for scaling in filter implementation. 22. What is product round off noise? 9 CSEITQUESTIONS.BLOGSPOT.IN CSEITQUESTIONS.BLOGSPOT.IN CSEITQUESTIONS.BLOGSPOT.IN PART – B 1. Discuss in detail the errors resulting from rounding and truncation? 2. Explain the limit cycle oscillations due to product round off and overflow errors? (Nov2010) 3. Explain the characteristics of limit cycle oscillations with respect to the system described by the difference equation y(n)=0.95y(n-1)+x(n). x(n)=0; y(n-1) =13. Determine the dead band of the system . 4. The output of A/D converter is applied to digital filter with t he system function Find t e output noise power from the digital filter when the input signal is quantized to have 8 bits. 5. (i)Explain the effects of c o-efficient quantization in FIR filters? (ii)Distinguish between fixed point and floating point arithmetic 6. With respect to finite word length effects in digital filters, with examples discuss about (i) Over flow limit cycle oscillation (ii) Signal scaling 7. Consider a second order IIR filter with Find the effect on quantization on pole locations of the given system function i n direct form and in cascade form. Assume b = 3 bits. 8. W hat is called quantization noise? Derive the expression for quantization noise power. 9. How to prevent limit cycle oscillations? Explain. 10. (i) Compare the truncation and rounding errors using fixed point and floating point representation. (ii) Represent the following numbers in floating point format with five bits for mantissa and three bits for exponent. (a) 710 (b) 0.2510 (c) - 710 (d) - 0.2510 11. (I) Explain the characteristics of limit cycle oscillation with respect to the system described by the difference equation : y(n) = 0.95 y(n-1) + x (n) ; x(n)= 0 And y (n-1) = 13. (ii) Explain the effects of coefficient quantization in FI R filters. 10 CSEITQUESTIONS.BLOGSPOT.IN CSEITQUESTIONS.BLOGSPOT.IN CSEITQUESTIONS.BLOGSPOT.IN 12. Explain the characteristics of a limit cycle oscillation with respect to the system described b y the Equation Determine the dead band of the filter x (n) = (3/4) δ (n). 11
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