# Fourier Transforms Properties

This set of Digital Signal Processing Multiple Choice Questions & Answers (MCQs) focuses on “Properties of Fourier Transform for Discrete Time Signals”.

1. If x(n)=xR(n)+jxI(n) is a complex sequence whose Fourier transform is given as X(ω)=XR(ω)+jXI(ω), then what is the value of XR(ω)?
a) ∑∞n=0xR (n)cosωn-xI (n)sinωn
b) ∑∞n=0xR (n)cosωn+xI (n)sinωn
c) ∑∞n=−∞xR (n)cosωn+xI (n)sinωn
d) ∑∞n=−∞xR (n)cosωn-xI (n)sinωn

2. If x(n)=xR(n)+jxI(n) is a complex sequence whose Fourier transform is given as X(ω)=XR(ω)+jXI(ω), then what is the value of xI(n)?
a) 12π∫2π0[XR(ω) sinωn+ XI(ω) cosωn] dω
b) ∫2π0[XR(ω) sinωn+ XI(ω) cosωn] dω
c) 12π∫2π0[XR(ω) sinωn – XI(ω) cosωn] dω
d) None of the mentioned

3. If x(n) is a real sequence, then what is the value of XI(ω)?
a) ∑∞n=−∞x(n)sin(ωn)
b) –∑∞n=−∞x(n)sin(ωn)
c) ∑∞n=−∞x(n)cos(ωn)
d) –∑∞n=−∞x(n)cos(ωn)

4. Which of the following relations are true if x(n) is real?
a) X(ω)=X(-ω)
b) X(ω)=-X(-ω)
c) X*(ω)=X(ω)
d) X*(ω)=X(-ω)

5. If x(n) is a real signal, then x(n)=1π∫π0[XR(ω) cosωn- XI(ω) sinωn] dω.
a) True
b) False

6. If x(n) is a real and odd sequence, then what is the expression for x(n)?
a) 1π∫π0[XI(ω) sinωn] dω
b) –1π∫π0[XI(ω) sinωn] dω
c) 1π∫π0[XI(ω) cosωn] dω
d) –1π∫π0[XI(ω) cosωn] dω

7. What is the value of XR(ω) given X(ω)=11−ae−jω,|a|<1?
a) asinω1−2acosω+a2
b) 1+acosω1−2acosω+a2
c) 1−acosω1−2acosω+a2
d) −asinω1−2acosω+a2

8. What is the value of XI(ω) given 11−ae−jω, |a|<1?
a) asinω1−2acosω+a2
b) 1+acosω1−2acosω+a2
c) 1−acosω1−2acosω+a2
d) −asinω1−2acosω+a2

9. What is the value of |X(ω)| given X(ω)=1/(1-ae-jω), |a|<1?
a) 11−2acosω+a2√
b) 11+2acosω+a2√
c) 11−2acosω+a2
d) 11+2acosω+a2

10. If x(n)=A, -M<n<M,; x(n)=0, elsewhere. Then what is the Fourier transform of the signal?
a) Asin(M−12)ωsin(ω2)
b) A2sin(M+12)ωsin(ω2)
c) Asin(M+12)ωsin(ω2)
d) sin(M−12)ωsin(ω2)

11. What is the Fourier transform of the signal x(n)=a|n|, |a|<1?
a) 1+a21−2acosω+a2
b) 1−a21−2acosω+a2
c) 2a1−2acosω+a2
d) None of the mentioned

12. If X(ω) is the Fourier transform of the signal x(n), then what is the Fourier transform of the signal x(n-k)?
a) ejωk. X(-ω)
b) ejωk. X(ω)
c) e-jωk. X(-ω)
d) e-jωk. X(ω)

13. What is the convolution of the sequences of x1(n)=x2(n)={1,1,1}?
a) {1,2,3,2,1}
b) {1,2,3,2,1}
c) {1,1,1,1,1}
d) {1,1,1,1,1}

14. What is the energy density spectrum of the signal x(n)=anu(n), |a|<1?
a) 11+2acosω+a2
b) 11−2acosω+a2
c) 11−2acosω−a2
d) 11+2acosω−a2