Digital Signal Processing Multiple Choice Questions 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)
Answer: A
Clarification: 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})
Answer: D
Clarification: 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
C.
X(k) = L for k=0 =1 for k=1,2....L-1
D. None of the mentioned
Answer: B
Clarification: 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
4. The Nth rot of unity WN is given as ______________
A. ej2πN
B. e-j2πN
C. e-j2π/N
D. ej2π/N
Answer: C
Clarification: 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
Answer: A
Clarification: 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.
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
Answer: B
Clarification: 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}
Answer: C
Clarification: 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_4^9end{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_4^1end{bmatrix})
=(begin{bmatrix}1&1&1&1\1&-j&-1&j\1&-1&1&-1\1&j&-1&-jend{bmatrix})
Then X4=W4.x4=(begin{bmatrix}6\ -2+2j\ -2\-2-2jend{bmatrix})
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
Answer: A
Clarification: 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}
Answer: D
Clarification: 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
Answer: C
Clarification: We know that according to the periodicity and symmetry property,
100/4=200/x=>x=8.