Digital Signal Processing

Frequency Analysis of Discrete Time Signal

Question 1
Marks : +2 | -2
Pass Ratio : 100%
What is the energy density spectrum Sxx(ω) of the signal x(n)=anu(n), |a|<1?
\\(\\frac{1}{1+2acosω+a^2}\\)
\\(\\frac{1}{1+2asinω+a^2}\\)
\\(\\frac{1}{1-2asinω+a^2}\\)
\\(\\frac{1}{1-2acosω+a^2}\\)
Explanation:
Since |a|<1, the sequence x(n) is absolutely summable, as can be verified by applying the geometric summation formula.
Question 2
Marks : +2 | -2
Pass Ratio : 100%
What is the Fourier transform of the signal x(n) which is defined as shown in the graph below?
Ae-j(ω/2)(L)\\(\\frac{sin⁡(\\frac{ωL}{2})}{sin⁡(\\frac{ω}{2})}\\)
Aej(ω/2)(L-1)\\(\\frac{sin⁡(\\frac{ωL}{2})}{sin⁡(\\frac{ω}{2})}\\)
Ae-j(ω/2)(L-1)\\(\\frac{sin⁡(\\frac{ωL}{2})}{sin⁡(\\frac{ω}{2})}\\)
None of the mentioned
Explanation:
The Fourier transform of this signal is
Question 3
Marks : +2 | -2
Pass Ratio : 100%
What is the Fourier transform of the signal x(n)=u(n)?
\\(\\frac{1}{2sin⁡(ω/2)} e^{j(ω+π)}\\)
\\(\\frac{1}{2sin⁡(ω/2)} e^{j(ω-π)}\\)
\\(\\frac{1}{2sin⁡(ω/2)} e^{j(ω+π)/2}\\)
\\(\\frac{1}{2sin⁡(ω/2)} e^{j(ω-π)/2}\\)
Explanation:
Given x(n)=u(n)
Question 4
Marks : +2 | -2
Pass Ratio : 100%
What is the Fourier transform X(ω) of a finite energy discrete time signal x(n)?
\\(\\sum_{n=-∞}^∞x(n)e^{-jωn}\\)
\\(\\sum_{n=0}^∞x(n)e^{-jωn}\\)
\\(\\sum_{n=0}^{N-1}x(n)e^{-jωn}\\)
None of the mentioned
Explanation:
If we consider a signal x(n) which is discrete in nature and has finite energy, then the Fourier transform of that signal is given as
Question 5
Marks : +2 | -2
Pass Ratio : 100%
What are the Fourier series coefficients for the signal x(n)=cosπn/3?
c1=c2=c3=c4=0,c1=c5=1/2
c0=c1=c2=c3=c4=c5=0
c0=c1=c2=c3=c4=c5=1/2
none of the mentioned
Explanation:
In this case, f0=1/6 and hence x(n) is periodic with fundamental period N=6.
Question 6
Marks : +2 | -2
Pass Ratio : 100%
What is the expression for Fourier series coefficient ck in terms of the discrete signal x(n)?
\\(\\frac{1}{N} \\sum_{n=0}^{N-1}x(n)e^{j2Ï€kn/N}\\)
\\(N\\sum_{n=0}^{N-1}x(n)e^{-j2Ï€kn/N}\\)
\\(\\frac{1}{N} \\sum_{n=0}^{N+1}x(n)e^{-j2Ï€kn/N}\\)
\\(\\frac{1}{N} \\sum_{n=0}^{N-1}x(n)e^{-j2Ï€kn/N}\\)
Explanation:
We know that, the Fourier series representation of a discrete signal x(n) is given as
Question 7
Marks : +2 | -2
Pass Ratio : 100%
Which of the following condition is to be satisfied for the Fourier transform of a sequence to be equal as the Z-transform of the same sequence?
|z|=1
|z|<1
|z|>1
Can never be equal
Explanation:
Let us consider the signal to be x(n)
Question 8
Marks : +2 | -2
Pass Ratio : 100%
For a signal x(n) to exhibit even symmetry, it should satisfy the condition |X(-ω)|=| X(ω)|.
True
False
Explanation:
We know that, if a signal x(n) is real, then
Question 9
Marks : +2 | -2
Pass Ratio : 100%
The sequence x(n)=\\(\\frac{sin⁡ ω_c n}{πn}\\) does not have both z-transform and Fourier transform.
True
False
Explanation:
The given x(n) do not have Z-transform. But the sequence have finite energy. So, the given sequence x(n) has a Fourier transform.
Question 10
Marks : +2 | -2
Pass Ratio : 100%
Which of the following relation is true if the signal x(n) is real?
X*(ω)=X(ω)
X*(ω)=X(-ω)
X*(ω)=-X(ω)
None of the mentioned
Explanation:
We know that,