DFT Properties

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

1. If x(n) and X(k) are an N-point DFT pair, then x(n+N)=x(n).
a) True
b) False

2. If x(n) and X(k) are an N-point DFT pair, then X(k+N)=?
a) X(-k)
b) -X(k)
c) X(k)
d) None of the mentioned

3. If X1(k) and X2(k) are the N-point DFTs of X1(n) and x2(n) respectively, then what is the N-point DFT of x(n)=ax1(n)+bx2(n)?
a) X1(ak)+X2(bk)
b) aX1(k)+bX2(k)
c) eakX1(k)+ebkX2(k)
d) None of the mentioned

4. If x(n) is a complex valued sequence given by x(n)=xR(n)+jxI(n), then what is the DFT of xR(n)?
a) ∑Nn=0xR(n)cos2πknN+xI(n)sin2πknN
b) ∑Nn=0xR(n)cos2πknN−xI(n)sin2πknN
c) ∑N−1n=0xR(n)cos2πknN−xI(n)sin2πknN
d) ∑N−1n=0xR(n)cos2πknN+xI(n)sin2πknN

5. If x(n) is a real sequence and X(k) is its N-point DFT, then which of the following is true?
a) X(N-k)=X(-k)
b) X(N-k)=X*(k)
c) X(-k)=X*(k)
d) All of the mentioned

6. If x(n) is real and even, then what is the DFT of x(n)?
a) ∑N−1n=0x(n)sin2πknN
b) ∑N−1n=0x(n)cos2πknN
c) -j∑N−1n=0x(n)sin2πknN
d) None of the mentioned

7. If x(n) is real and odd, then what is the IDFT of the given sequence?
a) j1N∑N−1k=0x(k)sin2πknN
b) 1N∑N−1k=0x(k)cos2πknN
c) −j1N∑N−1k=0x(k)sin2πknN
d) None of the mentioned

8. If X1(n), x2(n) and x3(m) are three sequences each of length N whose DFTs are given as X1(k), X2(k) and X3(k) respectively and X3(k)=X1(k).X2(k), then what is the expression for x3(m)?
a) ∑N−1n=0x1(n)x2(m+n)
b) ∑N−1n=0x1(n)x2(m−n)
c) ∑N−1n=0x1(n)x2(m−n)N
d) ∑N−1n=0x1(n)x2(m+n)N

9. What is the circular convolution of the sequences X1(n)={2,1,2,1} and x2(n)={1,2,3,4}?
a) {14,14,16,16}
b) {16,16,14,14}
c) {2,3,6,4}
d) {14,16,14,16}

10. What is the circular convolution of the sequences X1(n)={2,1,2,1} and x2(n)={1,2,3,4}, find using the DFT and IDFT concepts?
a) {16,16,14,14}
b) {14,16,14,16}
c) {14,14,16,16}
d) None of the mentioned

11. If X(k) is the N-point DFT of a sequence x(n), then circular time shift property is that N-point DFT of x((n-l))N is X(k)e-j2πkl/N.
a) True
b) False

12. If X(k) is the N-point DFT of a sequence x(n), then what is the DFT of x*(n)?
a) X(N-k)
b) X*(k)
c) X*(N-k)
d) None of the mentioned