# Fourier Transforms Properties MCQ’s

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

1. If x(n) is a real sequence, then what is the value of XI(ω)?
a) $$\sum_{n=-∞}^∞ x(n)sin⁡(ωn)$$
b) –$$\sum_{n=-∞}^∞ x(n)sin⁡(ωn)$$
c) $$\sum_{n=-∞}^∞ x(n)cos⁡(ωn)$$
d) –$$\sum_{n=-∞}^∞ x(n)cos⁡(ωn)$$

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 XR(ω)?
a) $$\sum_{n=0}^∞$$xR (n)cosωn-xI (n)sinωn
b) $$\sum_{n=0}^∞$$xR (n)cosωn+xI (n)sinωn
c) $$\sum_{n=-∞}^∞$$xR (n)cosωn+xI (n)sinωn
d) $$\sum_{n=-∞}^∞$$xR (n)cosωn-xI (n)sinωn

3. If x(n) is a real signal, then x(n)=$$\frac{1}{π}\int_0^π$$[XR(ω) cosωn- XI(ω) sinωn] dω.
a) True
b) False

4. What is the Fourier transform of the signal x(n)=a|n|, |a|<1?
a) $$\frac{1+a^2}{1-2acosω+a^2}$$
b) $$\frac{1-a^2}{1-2acosω+a^2}$$
c) $$\frac{2a}{1-2acosω+a^2}$$
d) None of the mentioned

5. What is the value of XI(ω) given $$\frac{1}{1-ae^{-jω}}$$, |a|<1?
a) $$\frac{asinω}{1-2acosω+a^2}$$
b) $$\frac{1+acosω}{1-2acosω+a^2}$$
c) $$\frac{1-acosω}{1-2acosω+a^2}$$
d) $$\frac{-asinω}{1-2acosω+a^2}$$

6. 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)

7. 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) $$\frac{1}{2π} \int_0^{2π}$$[XR(ω) sinωn+ XI(ω) cosωn] dω
b) $$\int_0^{2π}$$[XR(ω) sinωn+ XI(ω) cosωn] dω
c) $$\frac{1}{2π} \int_0^{2π}$$[XR(ω) sinωn – XI(ω) cosωn] dω
d) None of the mentioned

8. If x(n) is a real and odd sequence, then what is the expression for x(n)?
a) $$\frac{1}{π} \int_0^π$$[XI(ω) sinωn] dω
b) –$$\frac{1}{π} \int_0^π$$[XI(ω) sinωn] dω
c) $$\frac{1}{π} \int_0^π$$[XI(ω) cosωn] dω
d) –$$\frac{1}{π} \int_0^π$$[XI(ω) cosωn] dω

9. What is the value of |X(ω)| given X(ω)=1/(1-ae-jω), |a|<1?
a) $$\frac{1}{\sqrt{1-2acosω+a^2}}$$
b) $$\frac{1}{\sqrt{1+2acosω+a^2}}$$
c) $$\frac{1}{1-2acosω+a^2}$$
d) $$\frac{1}{1+2acosω+a^2}$$

10. What is the energy density spectrum of the signal x(n)=anu(n), |a|<1?
a) $$\frac{1}{1+2acosω+a^2}$$
b) $$\frac{1}{1-2acosω+a^2}$$
c) $$\frac{1}{1-2acosω-a^2}$$
d) $$\frac{1}{1+2acosω-a^2}$$

11. 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)

12. What is the value of XR(ω) given X(ω)=$$\frac{1}{1-ae^{-jω}}$$,|a|<1?
a) $$\frac{asinω}{1-2acosω+a^2}$$
b) $$\frac{1+acosω}{1-2acosω+a^2}$$
c) $$\frac{1-acosω}{1-2acosω+a^2}$$
d) $$\frac{-asinω}{1-2acosω+a^2}$$

13. If x(n)=A, -M<n<M,; x(n)=0, elsewhere. Then what is the Fourier transform of the signal?
a) A$$\frac{sin⁡(M-\frac{1}{2})ω}{sin⁡(\frac{ω}{2})}$$
b) A2$$\frac{sin⁡(M+\frac{1}{2})ω}{sin⁡(\frac{ω}{2})}$$
c) A$$\frac{sin⁡(M+\frac{1}{2})ω}{sin⁡(\frac{ω}{2})}$$
d) $$\frac{sin⁡(M-\frac{1}{2})ω}{sin⁡(\frac{ω}{2})}$$

14. 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}