Digital Signal Processing

Frequency Analysis of Discrete Time Signal

Question 1
Marks : +2 | -2
Pass Ratio : 100%
If cx(n) is the complex cepstrum sequence obtained from the inverse Fourier transform of ln X(ω), then what is the expression for cθ(n)?
\\(\\frac{1}{2π} \\int_0^π \\theta(ω) e^{jωn} dω\\)
\\(\\frac{1}{2π} \\int_{-π}^π \\theta(ω) e^{-jωn} dω\\)
\\(\\frac{1}{2π} \\int_0^π \\theta(ω) e^{jωn} dω\\)
\\(\\frac{1}{2π} \\int_{-π}^π \\theta(ω) e^{jωn} dω\\)
Explanation:
We know that,
Question 2
Marks : +2 | -2
Pass Ratio : 100%
Which of the following represents the phase associated with the frequency component of discrete-time Fourier series(DTFS)?
ej2Ï€kn/N
e-j2Ï€kn/N
ej2Ï€knN
none of the mentioned
Explanation:
We know that,
Question 3
Marks : +2 | -2
Pass Ratio : 100%
The Fourier series for the signal x(n)=cos√2πn exists.
True
False
Explanation:
For ω0=√2π, we have f0=1/√2. Since f0 is not a rational number, the signal is not periodic. Consequently, this signal cannot be expanded in a Fourier series.
Question 4
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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 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 6
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 7
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What is the period of the Fourier transform X(ω) of the signal x(n)?
Ï€
1
Non-periodic
2Ï€
Explanation:
Let X(ω) be the Fourier transform of a discrete time signal x(n) which is given as
Question 8
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 9
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 10
Marks : +2 | -2
Pass Ratio : 100%
What is the average power of the discrete time periodic signal x(n) with period N?
\\(\\frac{1}{N} \\sum_{n=0}^{N}|x(n)|\\)
\\(\\frac{1}{N} \\sum_{n=0}^{N-1}|x(n)|\\)
\\(\\frac{1}{N} \\sum_{n=0}^{N}|x(n)|^2\\)
\\(\\frac{1}{N} \\sum_{n=0}^{N-1}|x(n)|^2 \\)
Explanation:
Let us consider a discrete time periodic signal x(n) with period N.