# 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 X_{I}(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)=x_{R}(n)+jx_{I}(n) is a complex sequence whose Fourier transform is given as X(ω)=X_{R}(ω)+jX_{I}(ω), then what is the value of X_{R}(ω)?

a) \(\sum_{n=0}^∞\)x_{R} (n)cosωn-x_{I} (n)sinωn

b) \(\sum_{n=0}^∞\)x_{R} (n)cosωn+x_{I} (n)sinωn

c) \(\sum_{n=-∞}^∞\)x_{R} (n)cosωn+x_{I} (n)sinωn

d) \(\sum_{n=-∞}^∞\)x_{R} (n)cosωn-x_{I} (n)sinωn

3. If x(n) is a real signal, then x(n)=\(\frac{1}{π}\int_0^π\)[X_{R}(ω) cosωn- X_{I}(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 X_{I}(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)=x_{R}(n)+jx_{I}(n) is a complex sequence whose Fourier transform is given as X(ω)=X_{R}(ω)+jX_{I}(ω), then what is the value of x_{I}(n)?

a) \(\frac{1}{2π} \int_0^{2π}\)[X_{R}(ω) sinωn+ X_{I}(ω) cosωn] dω

b) \(\int_0^{2π}\)[X_{R}(ω) sinωn+ X_{I}(ω) cosωn] dω

c) \(\frac{1}{2π} \int_0^{2π}\)[X_{R}(ω) sinωn – X_{I}(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) \(\frac{1}{π} \int_0^π\)[X_{I}(ω) sinωn] dω

b) –\(\frac{1}{π} \int_0^π\)[X_{I}(ω) sinωn] dω

c) \(\frac{1}{π} \int_0^π\)[X_{I}(ω) cosωn] dω

d) –\(\frac{1}{π} \int_0^π\)[X_{I}(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)=a^{n}u(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) e^{jωk}. X(-ω)

b) e^{jωk}. X(ω)

c) e^{-jωk}. X(-ω)

d) e^{-jωk}. X(ω)

12. What is the value of X_{R}(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) A^{2}\(\frac{sin(M+\frac{1}{2})ω}{sin(\frac{ω}{2})}\)

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 x_{1}(n)=x_{2}(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}

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