250+ TOP MCQs on Efficient Computation of DFT FFT Algorithms – 2 & Answers

Digital Signal Processing Questions & Answers for freshers focuses on “Efficient Computation of DFT FFT Algorithms”.

1. 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 decimation-in-time algorithm.
A. True
B. False
Answer: A
Clarification: Let us consider the computation of the N=2^v point DFT by the divide and conquer approach. We select M=N/2 and L=2. This selection results in a split of 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), respectively, that is
f₁(n)=x(2n)
f₂(n)=x(2n+1), n=0,1,2…N/2-1
Thus f₁(n) and f₂(n) are obtained by decimating x(n) by a factor of 2, and hence the resulting FFT algorithm is called a decimation-in-time algorithm.

2. 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)-W_N^k F₂(k)
C. F₁(k)+W_N^k F₂(k)
D. None of the mentioned
Answer: C
Clarification: From the question, it is given that
f₁(n)=x(2n)
f₂(n)=x(2n+1), n=0,1,2…N/2-1
X(k)=(sum_{n=0}^{N-1} x(n) W_N^{kn}), k=0,1,2..N-1
=(sum_{n ,even} x(n) W_N^{kn}+sum_{n ,odd} x(n) W_N^{kn})
=(sum_{m=0}^{(frac{N}{2})-1} x(2m)W_N^{2km}+sum_{m=0}^{(frac{N}{2})-1} x(2m+1) W_N ^{k(2m+1)})
=(sum_{m=0}^{(frac{N}{2})-1} f_1(m) W_{N/2}^{km} + W_N^k sum_{m=0}^{(N/2)-1} f_2(m) W_{(frac{N}{2})}^{km})
X(k)=F₁(k)+ W_N^k F₂(k).

3. If X(k) is the N/2 point DFT of the sequence x(n), then what is the value of X(k+N/2)?
A. F₁(k)+F₂(k)
B. F₁(k)-W_N^k F₂(k)
C. F₁(k)+W_N^k F₂(k)
D. None of the mentioned
Answer: B
Clarification: We know that, X(k) = F₁(k)+W_N^k F₂(k)
We know that F₁(k) and F₂(k) are periodic, with period N/2, we have F₁(k+N/2) = F₁(k) and F₂(k+N/2)= F₂(k). In addition, the factor W_N^k+N/2 = -W_N^k.
Thus we get, X(k+N/2)= F₁(k)- W_N^k F₂(k).

4. How many complex multiplications are required to compute X(k)?
A. N(N+1)
B. N(N-1)/2
C. N2/2
D. N(N+1)/2
Answer: D
Clarification: We observe that the direct computation of F₁(k) requires (N/2)² complex multiplications. The same applies to the computation of F₂(k). Furthermore, there are N/2 additional complex multiplications required to compute W_N^k. Hence it requires N(N+1)/2 complex multiplications to compute X(k).

5. The total number of complex multiplications required to compute N point DFT by radix-2 FFT is?
A. (N/2)log₂N
B. Nlog₂N
C. (N/2)logN
D. None of the mentioned
Answer: A
Clarification: The decimation of the data sequence should be repeated again and again until the resulting sequences are reduced to one point sequences. For N=2^v, this decimation can be performed v=log₂N times. Thus the total number of complex multiplications is reduced to (N/2)log₂N.

6. The total number of complex additions required to compute N point DFT by radix-2 FFT is?
A. (N/2)log₂N
B. Nlog₂N
C. (N/2)logN
D. None of the mentioned
Answer: B
Clarification: The decimation of the data sequence should be repeated again and again until the resulting sequences are reduced to one point sequences. For N=2^v, this decimation can be performed v=log₂N times. Thus the total number of complex additions is reduced to Nlog₂N.

7. The following butterfly diagram is used in the computation of __________

A. Decimation-in-time FFT
B. Decimation-in-frequency FFT
C. All of the mentioned
D. None of the mentioned
Answer: A
Clarification: The above given diagram is the basic butterfly computation in the decimation-in-time FFT algorithm.

8. For a decimation-in-time FFT algorithm, which of the following is true?
A. Both input and output are in order
B. Both input and output are shuffled
C. Input is shuffled and output is in order
D. Input is in order and output is shuffled
Answer: C
Clarification: In decimation-in-time FFT algorithm, the input is taken in bit reversal order and the output is obtained in the order.

9. The following butterfly diagram is used in the computation of __________

A. Decimation-in-time FFT
B. Decimation-in-frequency FFT
C. All of the mentioned
D. None of the mentioned
Answer: B
Clarification: The above given diagram is the basic butterfly computation in the decimation-in-frequency FFT algorithm.

10. For a decimation-in-time FFT algorithm, which of the following is true?
A. Both input and output are in order
B. Both input and output are shuffled
C. Input is shuffled and output is in order
D. Input is in order and output is shuffled
Answer: D
Clarification: In decimation-in-frequency FFT algorithm, the input is taken in order and the output is obtained in the bit reversal order.

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