Digital Signal Processing Questions and Answers – Frequency Domain Sampling DFT

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 ____________
a) \(\sum_{n=0}^{N-1}x(n)e^{-j2πkn/N}\)(L<N)(k=0,1,2…N-1)
b) \(\sum_{n=0}^{N-1}x(n)e^{j2πkn/N}\)(L<N)(k=0,1,2…N-1)
c) \(\sum_{n=0}^{N-1}x(n)e^{j2πkn/N}\)(L>N)(k=0,1,2…N-1)
d) \(\sum_{n=0}^{N-1}x(n)e^{-j2πkn/N}\)(L>N)(k=0,1,2…N-1)
View Answer

Answer: a
Explanation: If x(n) is a finite duration sequence of length L, then the Fourier transform of x(n) is given as
X(ω)=\(\sum_{n=0}^{L-1} x(n)e^{-jωn}\)
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
X(k)=\(\sum_{n=0}^{N-1}x(n)e^{-j2πkn/N}\)

2. If X(k) discrete Fourier transform of x(n), then the inverse discrete Fourier transform of X(k) is?
a) \(\frac{1}{N} \sum_{k=0}^{N-1}X(k)e^{-j2πkn/N}\)
b) \(\sum_{k=0}^{N-1}X(k)e^{-j2πkn/N}\)
c) \(\sum_{k=0}^{N-1}X(k)e^{j2πkn/N}\)
d) \(\frac{1}{N} \sum_{k=0}^{N-1}X(k)e^{j2πkn/N}\)
View Answer

Answer: d
Explanation: If X(k) discrete Fourier transform of x(n), then the inverse discrete Fourier transform of X(k) is given as
x(n)=\(\frac{1}{N} \sum_{k=0}^{N-1}X(k)e^{j2πkn/N}\)

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
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c)

X(k) = L for k=0
=1 for k=1,2....L-1

d) None of the mentioned
View Answer

Answer: b
Explanation: The Fourier transform of this sequence is
X(ω)=\(\sum_{n=0}^{L-1} x(n)e^{-jωn}=\sum_{n=0}^{L-1}e^{-jωn}\)
The discrete Fourier transform is given as X(k)=\(\sum_{n=0}^{N-1}e^{-j2πkn/N}\)
If N=L, then X(k)= L for k=0
=0 for k=1,2….L-1
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4. The Nth rot of unity WN is given as ______________
a) ej2πN
b) e-j2πN
c) e-j2π/N
d) ej2π/N
View Answer

Answer: c
Explanation: We know that the Discrete Fourier transform of a signal x(n) is given as X(k)=\(\sum_{n=0}^{N-1}x(n)e^{-j2πkn/N}=\sum_{n=0}^{N-1}x(n)W_N^{kn}\)
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
View Answer

Answer: a
Explanation: The formula for calculating N point DFT is given as
X(k)=\(\sum_{n=0}^{N-1}x(n)e^{-j2πkn/N}\)
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.
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6. Which of the following is true?
a) WN*=\(\frac{1}{N} W_N^{-1}\)
b) WN-1=\(\frac{1}{N} W_N*\)
c) WN-1=WN*
d) None of the mentioned
View Answer

Answer: b
Explanation: If XN represents the N point DFT of the sequence xN in the matrix form, then we know that
XN=WN.xN
By pre-multiplying both sides by WN-1, we get
xN=WN-1.XN
But we know that the inverse DFT of XN is defined as
xN=\(\frac{1}{N} W_N*X_N\)
Thus by comparing the above two equations we get
WN-1 = \(\frac{1}{N} W_N*\)

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}
View Answer

Answer: c
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
W4=\(\begin{bmatrix}W_4^0& W_4^0& W_4^0& W_4^1\\W_4^0& W_4^0& W_4^2& W_4^3\\W_4^0& W_4^2& W_4^0& W_4^3\\W_4^4& W_4^6& W_4^6& W_4^9\end{bmatrix}=\begin{bmatrix}W_4^0& W_4^0& W_4^0& W_4^1\\W_4^0& W_4^0& W_4^2& W_4^3\\W_4^0& W_4^2& W_4^0& W_4^3\\W_4^0& W_4^2& W_4^2& W_4^1\end{bmatrix}\)

=\(\begin{bmatrix}1&1&1&1\\1&-j&-1&j\\1&-1&1&-1\\1&j&-1&-j\end{bmatrix}\)
Then X4=W4.x4=\(\begin{bmatrix}6\\ -2+2j\\ -2\\-2-2j\end{bmatrix}\)
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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
View Answer

Answer: a
Explanation: The Fourier series coefficients are given by the expression
ck=\(\frac{1}{N} \sum_{n=0}^{N-1}x(n)e^{-j2πkn/N}=\frac{1}{N}X(k)\) => X(k)= Nck

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}
View Answer

Answer: d
Explanation: Given x(n)={0,1,2,3}
We know that the 4-point DFT of the above given sequence is given by the expression
X(k)=\(\sum_{n=0}^{N-1}x(n)e^{-j2πkn/N}\)
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
View Answer

Answer: c
Explanation: We know that according to the periodicity and symmetry property,
100/4=200/x=>x=8.

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Manish Bhojasia, a technology veteran with 20+ years @ Cisco & Wipro, is Founder and CTO at Sanfoundry. He lives in Bangalore, and focuses on development of Linux Kernel, SAN Technologies, Advanced C, Data Structures & Alogrithms. Stay connected with him at LinkedIn.

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