June 12, 2021

Fourier Transforms Properties MCQ’s

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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}

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