IMPORTANT QUESTIONS - EC6502 Principles of Digital Signal Processing-1

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EC6502 Principles of  Digital Signal Processing

Question Bank

UNIT I

Part A

1. Write the formula for DFT and IDFT.
2. Name any two properties of DFT.
3. State and prove Parseval theorem.
4. State and prove time-shifting property of DFT.
5. State and prove circular convolution.
6. What are ‘twiddle factor’ of the DFT?(May-2012)
7. State the relationship between DTFT and DFT.(Nov-2011,May-2014)
8. Determine the value of W¹⁶ for 64-point DFT.
9. Give the number of complex addition and complex multiplication required for the direct computation of N-point DFT.
10. What is zero padding? What are its uses?(Nov-2011,Nov-2013)
11. Define circular convolution.
12. Distinguish between Circular convolution and linear convolution.
13. Write briefly about Overlap-save method.
14. Write briefly about Overlap-add method
15. State the difference between Overlap-save method and Overlap-add method.
16. What is the need for FFT?
17. What is the main advantage of FFT?(May-2011)
18. Calculate the number of complex multiplication and complex addition needed in the calculation of DFT using FFT algorithm with 32-point sequence.(Nov-2010,Nov-2013)
19. What is radix-2 FFT?
20. What is DIT FFT algorithm?
21. What is DIF FFT algorithm?
22. What are the differences and similarities between DIT and DIF algorithm?
23. Draw the basic butterfly diagram of radix 2 DIT FFT.
24. Draw the basic butterfly diagram of radix 2 DIF FFT.
25. What is meant by ‘in-place’ in DIT and DIF algorithm?(Nov-2014)
26. Give the computation efficiency of FFT over DFT.
27. Explain how you would use the FFT algorithm to compute the IDFT.
28. What is bit reversal?(May-2014,May-2011)
29. How many stages of decimations are required in the case of 64pt Radix-2 DIT-FFT algorithm?(May-2012)
30. 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}(Nov-2010)

31. Compute the 4-point DFT of the following sequences,

                                                                i.      x[n] = 2ⁿ

                                                              ii.      x[n] = 2^-n

                                                            iii.      x[n] = sin(nπ/2)

                                                            iv.      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}

32. Find IDFT of X(k) = {1, 0, 1, 0}.

Ans.: x[n] = {0.5, 0, 0.5, 0}

33. 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}

34. 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

35. 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

36. 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}

37. 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}

38. 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

39. 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}

40. 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}(Nov-2014,Nov-11,Nov-2013,Nov-2010)

Ans.: y[n] = {1, 1, 1, 4, 6, 4, 1, 4, 8, 5, 1}

3.      Draw the signal flow graph for 8/16-point DFT using (a) DIT algorithm and DIF algorithm.(May-2014)

4.       Compute the IDFT for the sequence X(k) = 2^-kwhere k = 0 to 7 using DIF FFT algorithm.(Nov-14)

5.      Find the DFT of a sequence x[n] = {1, 2, 3, 4, 4, 3, 2, 1} using DIT algorithm.(Nov-2010,Nov-2013)

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, 0, 0} 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, 0, 0, 0} using DIF algorithm.(Nov-2011)

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 with impulse response, h[n] = {1, 2, 3} to input sequence x[n] = {1, 2, 2, 1}. Use circular convolution in frequency domain.

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}(Nov-2013,May-2012)

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.  State any six properties of DFT(Nov-2014)

20.  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}
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