Digital Signal Processing

Properties of Fourier Transform for Discrete Time Signals

Question 1
Marks : +2 | -2
Pass Ratio : 100%
What is the value of |X(ω)| given X(ω)=1/(1-ae-jω), |a|<1?
\\(\\frac{1}{\\sqrt{1-2acosω+a^2}}\\)
\\(\\frac{1}{\\sqrt{1+2acosω+a^2}}\\)
\\(\\frac{1}{1-2acosω+a^2}\\)
\\(\\frac{1}{1+2acosω+a^2}\\)
Explanation:
For the given X(ω)=1/(1-ae-jω), |a|<1 we obtain
Question 2
Marks : +2 | -2
Pass Ratio : 100%
What is the value of XR(ω) given X(ω)=\\(\\frac{1}{1-ae^{-jω}}\\),|a|<1?
\\(\\frac{asinω}{1-2acosω+a^2}\\)
\\(\\frac{1+acosω}{1-2acosω+a^2}\\)
\\(\\frac{1-acosω}{1-2acosω+a^2}\\)
\\(\\frac{-asinω}{1-2acosω+a^2}\\)
Explanation:
Given, X(ω)=\\(\\frac{1}{1-ae^{-jω}}\\), |a|<1
Question 3
Marks : +2 | -2
Pass Ratio : 100%
If x(n) is a real signal, then x(n)=\\(\\frac{1}{π}\\int_0^π\\)[XR(ω) cosωn- XI(ω) sinωn] dω.
True
False
Explanation:
We know that if x(n) is a real signal, then xI(n)=0 and xR(n)=x(n)
Question 4
Marks : +2 | -2
Pass Ratio : 100%
Which of the following relations are true if x(n) is real?
X(ω)=X(-ω)
X(ω)=-X(-ω)
X*(ω)=X(ω)
X*(ω)=X(-ω)
Explanation:
We know that, if x(n) is a real sequence
Question 5
Marks : +2 | -2
Pass Ratio : 100%
If x(n) is a real sequence, then what is the value of XI(ω)?
\\(\\sum_{n=-∞}^∞ x(n)sin⁡(ωn)\\)
–\\(\\sum_{n=-∞}^∞ x(n)sin⁡(ωn)\\)
\\(\\sum_{n=-∞}^∞ x(n)cos⁡(ωn)\\)
–\\(\\sum_{n=-∞}^∞ x(n)cos⁡(ωn)\\)
Explanation:
If the signal x(n) is real, then xI(n)=0
Question 6
Marks : +2 | -2
Pass Ratio : 100%
If x(n)=A, -M<n<M,; x(n)=0, elsewhere. Then what is the Fourier transform of the signal?
A\\(\\frac{sin⁡(M-\\frac{1}{2})ω}{sin⁡(\\frac{ω}{2})}\\)
A2\\(\\frac{sin⁡(M+\\frac{1}{2})ω}{sin⁡(\\frac{ω}{2})}\\)
A\\(\\frac{sin⁡(M+\\frac{1}{2})ω}{sin⁡(\\frac{ω}{2})}\\)
\\(\\frac{sin⁡(M-\\frac{1}{2})ω}{sin⁡(\\frac{ω}{2})}\\)
Explanation:
Clearly, x(n)=x(-n). Thus the signal x(n) is real and even signal. So, we know that
Question 7
Marks : +2 | -2
Pass Ratio : 100%
If x(n) is a real and odd sequence, then what is the expression for x(n)?
\\(\\frac{1}{π} \\int_0^π\\)[XI(ω) sinωn] dω
–\\(\\frac{1}{Ï€} \\int_0^Ï€\\)[XI(ω) sinωn] dω
\\(\\frac{1}{π} \\int_0^π\\)[XI(ω) cosωn] dω
–\\(\\frac{1}{Ï€} \\int_0^Ï€\\)[XI(ω) cosωn] dω
Explanation:
If x(n) is real and odd then, x(n)cosωn is odd and x(n) sinωn is even. Consequently
Question 8
Marks : +2 | -2
Pass Ratio : 100%
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)?
\\(\\frac{1}{2π} \\int_0^{2π}\\)[XR(ω) sinωn+ XI(ω) cosωn] dω
\\(\\int_0^{2π}\\)[XR(ω) sinωn+ XI(ω) cosωn] dω
\\(\\frac{1}{2Ï€} \\int_0^{2Ï€}\\)[XR(ω) sinωn – XI(ω) cosωn] dω
None of the mentioned
Explanation:
We know that the inverse transform or the synthesis equation of a signal x(n) is given as
Question 9
Marks : +2 | -2
Pass Ratio : 100%
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)?
\\(\\sum_{n=0}^∞\\)xR (n)cosωn-xI (n)sinωn
\\(\\sum_{n=0}^∞\\)xR (n)cosωn+xI (n)sinωn
\\(\\sum_{n=-∞}^∞\\)xR (n)cosωn+xI (n)sinωn
\\(\\sum_{n=-∞}^∞\\)xR (n)cosωn-xI (n)sinωn
Explanation:
We know that X(ω)=\\(\\sum_{n=-∞}^∞\\) x(n)e-jωn
Question 10
Marks : +2 | -2
Pass Ratio : 100%
What is the value of XI(ω) given \\(\\frac{1}{1-ae^{-jω}}\\), |a|<1?
\\(\\frac{asinω}{1-2acosω+a^2}\\)
\\(\\frac{1+acosω}{1-2acosω+a^2}\\)
\\(\\frac{1-acosω}{1-2acosω+a^2}\\)
\\(\\frac{-asinω}{1-2acosω+a^2}\\)
Explanation:
Given, X(ω)=\\(\\frac{1}{1-ae^{-jω}}\\), |a|<1