1. The effect of round-off errors due to the multiplications performed in the DFT with fixed-point arithmetic is known as Quantization error.

A. True

B. False

2. What is the model that has been adopted for characterizing a round-off error in multiplication?

A. Multiplicative white noise model

B. Subtractive white noise model

C. Additive white noise model

D. None of the mentioned

3. How many quantization errors are present in one complex-valued multiplication?

A. One

B. Two

C. Three

D. Four

4. What is the total number of quantization errors in the computation of single point DFT of a sequence of length N?

A. 2N

B. 4N

C. 8N

D. 12N

5. What is the range in which the quantization errors due to rounding off are uniformly distributed as random variables if Δ=2-b?

A. (0,Δ)

B. (-Δ,0)

C. (-Δ/2,Δ/2)

D. None of the mentioned

6. The 4N quantization errors are mutually uncorrelated.

A. True

B. False

7. The 4N quantization errors are correlated with the sequence {x(n)}.

A. True

B. False

8. How is the variance of the quantization error related to the size of the DFT?

A. Equal

B. Inversely proportional

C. Square proportional

D. Proportional

9. Every fourfold increase in the size N of the DFT requires an additional bit in computational precision to offset the additional quantization errors.

A. True

B. False

10. What is the variance of the output DFT coefficients |X(k)|?

A. \(\frac{1}{N}\)

B. \(\frac{1}{2N}\)

C. \(\frac{1}{3N}\)

D. \(\frac{1}{4N}\)

11. What is the signal-to-noise ratio?

A. σX2.σq2

B. σX2/σq2

C. σX2+σq2

D. σX2-σq2

12. How many number of bits are required to compute the DFT of a 1024 point sequence with an SNR of 30db?

A. 15

B. 10

C. 5

D. 20

13. How many number of butterflies are required per output point in the FFT algorithm?

A. N

B. N+1

C. 2N

D. N-1

14. What is the value of the variance of quantization error in FFT algorithm, compared to that of direct computation?

A. Greater

B. Less

C. Equal

D. Cannot be compared

15. How many number of bits are required to compute the FFT of a 1024 point sequence with an SNR of 30db?

A. 11

B. 10

C. 5

D. 20

16. Which of the following is true regarding the number of computations required to compute an N-point DFT?

A. N2 complex multiplications and N(N-1) complex additions

B. N2 complex additions and N(N-1) complex multiplications

C. N2 complex multiplications and N(N+1) complex additions

D. N2 complex additions and N(N+1) complex multiplications

17. Which of the following is true regarding the number of computations required to compute DFT at any one value of ‘k’?

A. 4N-2 real multiplications and 4N real additions

B. 4N real multiplications and 4N-4 real additions

C. 4N-2 real multiplications and 4N+2 real additions

D. 4N real multiplications and 4N-2 real additions

18. WN^{k+N/2}=?

A. WN^{k}

B. -WN^{k}

C. WN^{-k}

D. None of the mentioned

19. What is the real part of the N point DFT XR(k) of a complex valued sequence x(n)?

A. \(\sum_{n=0}^{N-1} [x_R (n) cos\frac{2πkn}{N} – x_I (n) sin\frac{2πkn}{N}]\)

B. \(\sum_{n=0}^{N-1} [x_R (n) sin\frac{2πkn}{N} + x_I (n) cos\frac{2πkn}{N}]\)

C. \(\sum_{n=0}^{N-1} [x_R (n) cos\frac{2πkn}{N} + x_I (n) sin\frac{2πkn}{N}]\)

D. None of the mentioned

20. The computation of XR(k) for a complex-valued x(n) of N points requires ________

A. 2N2 evaluations of trigonometric functions

B. 4N2 real multiplications

C. 4N(N-1) real additions

D. All of the mentioned

21. Divide-and-conquer approach is based on the decomposition of an N-point DFT into successively smaller DFTs. This basic approach leads to FFT algorithms.

A. True

B. False

22. If the arrangement is of the form in which the first row consists of the first M elements of x(n), the second row consists of the next M elements of x(n), and so on, then which of the following mapping represents the above arrangement?

A. n=l+mL

B. n=Ml+m

C. n=ML+l

D. none of the mentioned

23. If N=LM, then what is the value of WNmqL?

A. WMmq

B. WLmq

C. WNmq

D. None of the mentioned

24. How many complex multiplications are performed in computing the N-point DFT of a sequence using the divide-and-conquer method if N=LM

A. N(L+M+2)

B. N(L+M-2)

C. N(L+M-1)

D. N(L+M+1)

25. How many complex additions are performed in computing the N-point DFT of a sequence using the divide-and-conquer method if N=LM?

A. N(L+M+2)

B. N(L+M-2)

C. N(L+M-1)

D. N(L+M+1)

26. Which is the correct order of the following steps to be done in one of the algorithms of the divide and conquer method?

A. Store the signal column-wise

B. Compute the M-point DFT of each row

C. Multiply the resulting array by the phase factors WNlq.

D. All of the above

27. If we store the signal row-wise then the result must be read column-wise.

A. True

B. False

28. If we store the signal row-wise and compute the L point DFT at each column, the resulting array must be multiplied by which of the following factors?

A. WN^{lq}

B. WN^{pq}

C. WN^{lq}

D. WN^{pm}

29. If we split the N point data sequence into two N/2 point data sequences f1(n) and f2(n) corresponding to the even-numbered and odd-numbered samples of x(n), then such an FFT algorithm is known as a decimation-in-time algorithm.

A. True

B. False

30. If we split the N point data sequence into two N/2 point data sequences f1(n) and f2(n) corresponding to the even-numbered and odd-numbered samples of x(n) and F1(k) and F2(k) are the N/2 point DFTs of f1(k) and f2(k) respectively, then what is the N/2 point DFT X(k) of x(n)

A. F1(k)+F2(k)

B. F1(k)-WNk F2(k)

C. F1(k)+WNk F2(k)

D. None of the mentioned