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 WN is given as:
a) ej2πN
b) e -j2πN
c) e-j2π/N
d) ej2π/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) 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
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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 N2 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
WNk+N/2= -WNk
The matrix W4 may be expressed as

8. If X(k) is the N point DFT of a sequence whose Fourier series coefficients is given by ck, then which of the following is true?
a) X(k)=Nck
b) X(k)=ck/N
c) X(k)=N/ck
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 W4100=Wx200, 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.
To practice all areas of Digital Signal Processing, here is complete set of 1000+ Multiple Choice Questions and Answers.