This set of Digital Signal Processing Multiple Choice Questions & Answers (MCQs) focuses on “Frequency Domain Sampling DFT”.

1. If x(n) is a finite duration sequence of length L, then the discrete Fourier transform X(k) of x(n) is given as:

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Explanation: If x(n) is a finite duration sequence of length L, then the Fourier transform of x(n) is given as

If we sample X(ω) at equally spaced frequencies ω=2πk/N, k=0,1,2…N-1 where N>L, the resultant samples are

2. If X(k) discrete Fourier transform of x(n), then the inverse discrete Fourier transform of X(k) is:

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Explanation: If X(k) discrete Fourier transform of x(n), then the inverse discrete Fourier transform of X(k) is given as

3. A finite duration sequence of length L is given as x(n) =1 for 0≤n≤L-1=0 otherwise , then what is the N point DFT of this sequence for N=L?

a) X(k) =L for k=0, 1,2….L-1

b) X(k) =L for k=0

=0 for k=1,2….L-1

c) X(k) =L for k=0

=1 for k=1,2….L-1

d) None of the mentioned

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Explanation: The Fourier transform of this sequence is

If N=L, then X(k)= L for k=0

=0 for k=1,2….L-1

4. The Nth rot of unity W_{N} is given as:

a) e^{j2πN}

b) e ^{-j2πN}

c) e^{-j2π/N}

d) e^{j2π/N}

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Explanation: We know that the Discrete Fourier transform of a signal x(n) is given as

Thus we get Nth rot of unity WN= e

^{-j2π/N}

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

a) N^{2} complex multiplications and N(N-1) complex additions

b) N^{2} complex additions and N(N-1) complex multiplications

c) N^{2} complex multiplications and N(N+1) complex additions

d) N^{2} complex additions and N(N+1) complex multiplications

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Explanation: The formula for calculating N point DFT is given as

From the formula given at every step of computing we are performing N complex multiplications and N-1 complex additions. So, in a total to perform N-point DFT we perform N

^{2}complex multiplications and N(N-1) complex additions.

6. Which of the following is true?

d) None of the mentioned

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Explanation: If XN represents the N point DFT of the sequence xN in the matrix form, then we know that

7. What is the DFT of the four point sequence x(n)={0,1,2,3}?

a) {6,-2+2j-2,-2-2j}

b) {6,-2-2j,2,-2+2j}

c) {6,-2+2j,-2,-2-2j}

d) {6,-2-2j,-2,-2+2j}

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Explanation: The first step is to determine the matrix W4. By exploiting the periodicity property of W4 and the symmetry property

W

_{N}

^{k+N/2}= -W

_{N}

^{k}

The matrix W

_{4}may be expressed as

8. If X(k) is the N point DFT of a sequence whose Fourier series coefficients is given by c_{k}, then which of the following is true?

a) X(k)=Nc_{k}

b) X(k)=c_{k}/N

c) X(k)=N/c_{k}

d) None of the mentioned

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9. What is the DFT of the four point sequence x(n)={0,1,2,3}?

a) {6,-2+2j-2,-2-2j}

b) {6,-2-2j,2,-2+2j}

c) {6,-2-2j,-2,-2+2j}

d) {6,-2+2j,-2,-2-2j}

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Answer: Given x(n)={0,1,2,3}

We know that the 4-point DFT of the above given sequence is given by the expression

In this case N=4

=>X(0)=6,X(1)=-2+2j,X(2)=-2,X(3)=-2-2j.

10. If W_{4}^{100}=W_{x}^{200}, then what is the value of x?

a) 2

b) 4

c) 8

d) 16

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Explanation: We know that according to the periodicity and symmetry property,

100/4=200/x=>x=8.

**Sanfoundry Global Education & Learning Series – Digital Signal Processing.**

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