SRI VENKATESWARA COLLEGE OF ENGINEERING
Department of Electronics and Communication Engineering
EC2302 Digital Signal Processing
Question Bank
UNIT I
Part A
1. Define DFT and IDFT.
2. Name any two properties of DFT.
3. State and prove Parsevals theorem.
4. State and prove time-shifting property of DFT.
5. State and prove circular convolution.
6. What are ‘twiddle factors’?
7. State the relationship between DTFT and DFT.
8. State the relationship between DFT and z-transform.
9. Determine the value of W16 for 64-point DFT.
10. Give the number of complex addition and complex multiplication required for the direct computation of N-point DFT.
11. What is zero padding? What are its uses?
12. Define periodic convolution.
13. Distinguish between Circular convolution and linear convolution.
14. Write briefly about Overlap-save method.
15. Write briefly about Overlap-add method
16. State the difference between Overlap-save method and Overlap-add method.
17. Calculate the number of complex multiplication and complex addition needed in the calculation of DFT using FFT algorithm with 32-point sequence.
18. What is radix-2 FFT?
19. What is DIT FFT algorithm?
20. What is DIF FFT algorithm?
21. What are the differences and similarities between DIT and DIF algorithm?
22. Draw the basic butterfly diagram of radix 2 DIT FFT.
23. Draw the basic butterfly diagram of radix 2 DIF FFT.
24. What is meant by ‘in-place’ in DIT and DIF algorithm?
25. Give the computation efficiency of FFT over DFT.
26. Explain how the FFT algorithm can be used to compute the IDFT.
27. Given two sequences of length N=4 defined by x[n] = {1, 2, 2, 1} and
h[n] = {2, 1, 1, 2}, Determine the periodic convolution.
Ans.: y[n] = {9, 10, 9, 8}
28. Compute the 4-point DFT of the following sequences,
- x[n] = 2n
- x[n] = 2-n
- x[n] = sin(nπ/2)
- x[n] = cos(nπ/2)
Ans.: (a) X(k) = {15, -3+j6, -5, -3-j6}
(b) X(k) = {15/8, (3/4)-j(3/8), (5/8), (3/4)-j(3/8)}
(c) X(k) = {0, -j2, 0, j2}
(d) X(k) = {1, 1-j√2, 1, 1+j√2}
29. Find IDFT of X(k) = {1, 0, 1, 0}.
Ans.: x[n] = {0.5, 0, 0.5, 0}
30. Find the IDFT of the sequence X(k) = {10, -2+j2, -2, -2-j2} using DIT and DIF algorithm.
Ans.: x[n] = {1, 2, 3, 4}
31. The first five samples of 8-point DFT of a real valued sequence are {28, -4+j9.565, -4+j4, -4+j1.656, -4}. Determine the remaining three samples.
Ans.: X(5) = -4-j1.656, X(6) = -4-j4, X(7) = -4-j9.565
32. For the 8-sample sequence x[n] = {1, 2, 3, 5, 5, 3, 2, 1}, the first five DFT coefficients are {22, -7.536-j3.121, 1+j, -0.465-j1.121, 0}. Determine the remaining three DFT coefficients.
Ans.: X(5)= -0.465+j1.121, X(6)= 1-j, X(7)= -7.536+j3.121
33. Consider the finite sequence x[n] = {1, 2, 2, 1}. The 5-point DFT of x[n] is denoted by X(k). Plot the sequence whose DFT is Y(k) = e-j4πk/5 X(k).
Ans.: y[n] = {1, 0, 1, 2, 2}
34. If the DFT of the sequence x[n] = {1, 2, 1, 1, 2, -1} is X(k). Plot the sequence whose DFT is Y(k) = e-jπk X(k).
Ans.: y[n] = {1, 2, -1, 1, 2, 1}
35. Consider the 8-point decimation-in-frequency (DIF) flow graph. What is the gain of the “signal path” that goes from x[5] to X(3)?
Ans.: (X(3)/x[5]) = 0.707+j0.707
36. Compute 4-point DFT of a sequence x[n] = {0, 1, 2, 3} using DIF and DIT algorithm.
Ans.: X(k) = {6, -2+j2, -2, -2-j2}
37. Consider the 8-point decimation-in-time (DIT) flow graph. What is the gain of the “signal path” that goes from x[3] to X(2)?
Ans.: (X(2)/x[3]) = j
Part B
1. An input sequence x[n] = {2, 1, 0, 1, 2} is applied to DSP system having an impulse sequence h[n] = {5,3,2,1}. Determine the output sequence produced by (a) Linear convolution and (b) Verify the same through circular convolution.
Ans.: y[n] = {10, 11, 7, 9, 14, 8, 5, 2}
2. Convolve the following sequence using (a) Overlap-save method and (b) Overlap-add method,
x[n] = {1, -1, 2, 1, 2, -1, 1, 3, 1} and h[n] = {1, 2, 1}
Ans.: y[n] = {1, 1, 1, 4, 6, 4, 1, 4, 8, 5, 1}
3. Draw the signal flow graph for 16-point DFT using DIT algorithm and DIF algorithm.
4. Compute the IDFT for the sequence X(k) = 2-k where k = 0 to 7 using DIF FFT algorithm.
5. Find the DFT of a sequence x[n] = {1, 2, 3, 4, 4, 3, 2, 1} using DIT algorithm.
Ans.: X(k) = {20, -5.828-j2.414, 0, 0.172-j0.414, 0, 0.172+j0.414, 0, -5.828-j2.414}
6. Find the DFT of a sequence x[n] = {1, 1, 1, 1, 1, 1} using DIF algorithm.
Ans.: X(k) = {6, -0.707-j0.707, 1-j, 0.707+j0.293, 0, 0.707-j0.293, 1+j, -0.707+j0.707}
7. Find the DFT of a sequence x[n] = {1, 1, 1, 1, 1} using DIF algorithm.
Ans.: X(k) = {5, -j2.414, 1, -j0.414, 1, j0.414, 1,j2.414}
8. Find the IDFT of a sequence X(k) = {5, 0, 1-j, 0, 1, 0, 1+j, 0} using DIT algorithm.
Ans.: x[n] = {1, 0.75, 0.5, 0.25, 1, 0.75, 0.5, 0.25}
9. Consider two sequence x[n] = cos(nπ/2) and h[n] = 2n. Determine the output sequence y[n] by circular convolution using concentric circle method. Take N=4.
Ans.: y[n] = {-3, -6, 3, 6}
10. Determine the output sequence y[n] of FIR filter using DFT , IDFT methods with impulse response, h[n] = {1, 2, 3} to input sequence x[n] = {1, 2, 2, 1}.
Ans.: y[n] = {1, 4, 9, 11, 8, 3}
11. Find the Linear convolution through circular convolution of x1[n] and x2[n].
x1[n] = δ[n] + δ[n-1] + δ[n-2]
x2[n] = 2δ[n] - δ[n-1] + 2δ[n-2]
Ans.: x3[n] = 2δ[n] + δ[n-1] + 3δ[n-2] + δ[n-3] + 2δ[n-4]
12. Given two sequence x1[n] = {1, 2, 3, 1} and x2[n] = {4, 3, 2, 2}. Find x3[n] such that (i) X3(k) = X1(k).X2(k), (ii) using concentric circle method and (iii) Matrix method.
Ans.: X3(k) = {17, 19, 22, 19}
13. Find the output y[n] of a filter whose impulse response is h[n] = {1, 1, 1} and input signal is x[n] = {3, -1, 0, 1, 3, 2, 0, 1, 2, 1} using (i) Overlap save method and (ii) Overlap add method.
Ans.: y[n] = (3, 2, 2, 0, 4, 6, 5, 3, 3, 4, 3, 1)
14. Find the output y[n] of a filter whose impulse response is h[n] = {1, 2} and input signal is x[n] = {1, 2, -1, 2, 3, -2, -3, -1, 1, 1, 2, -1} using (i) Overlap save method and (ii) Overlap add method.
Ans.: y[n] = (1, 4, 3, 0, 7, 4, -7, -7, -1, 3, 4, 3, -2)
15. Determine the output of a linear FIR filter whose impulse response h[n] = {1, -3, 5} and input signal x[n] = {-1, 4, 7, 3, -2, 9, 10, 12, -5, 8} using (i) Overlap-save method and (ii) Overlap-add method.
Ans.: y[n] = (-1, 7, -10, 2, 24, 30, -27, 27, 9, 83, -49, 40)
16. Determine the DFT of the given data sequence x[n] = {2, 1, 4, 6, 5, 8, 3, 9} by DIT algorithm.
Ans.: X(k) = {38, -5.828+j6.11, j6,-0.412+j8.1, -10, -0.412-j8.1, -j6, -5.828-j6.11}
17. Determine the DFT of the given data sequence x[n] = {-1, 2, -3, 4, 9, -20, 12, 6} by DIT and DIF algorithm.
Ans.: X(k) = {9, 6.968+j0.86, -1+j28, -26.968-j29.14, 25, 26.968+j29.14, -1-j28,
6.968-j0.86}
18. Calculate IDFT for the given coefficients X(k) = {38, -5.828+j6.07, j6, -0.172+j8.07, -10, -0.172-j8.07, -j6, -5.828-j6.07} using DIT and DIF algorithm.
Ans.: x[n] = {2, 1, 4, 6, 5, 8, 3, 9}
19. Compute IDFT of the sequence X(k) = {7, -0.707-j0.707, -j, 0.707-j0.707, 1, 0.707 +j0.707, j, -0.707+j0.707} using DIT and DIF algorithm.
Ans.: x[n] = {1, 1, 1, 1, 1, 1, 1, 0}