Digital Signal Processing

Frequency Analysis of Discrete Time Signal

Question 1
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 2
Marks : +2 | -2
Pass Ratio : 100%
What is the Fourier series representation of a signal x(n) whose period is N?
\\(\\sum_{k=0}^{N+1}c_k e^{j2Ï€kn/N}\\)
\\(\\sum_{k=0}^{N-1}c_k e^{j2Ï€kn/N}\\)
\\(\\sum_{k=0}^Nc_k e^{j2Ï€kn/N}\\)
\\(\\sum_{k=0}^{N-1}c_k e^{-j2Ï€kn/N}\\)
Explanation:
Here, the frequency F0 of a continuous time signal is divided into 2Ï€/N intervals.
Question 3
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 4
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 5
Marks : +2 | -2
Pass Ratio : 100%
What is the equation for average power of discrete time periodic signal x(n) with period N in terms of Fourier series coefficient ck?
\\(\\sum_{k=0}^{N-1}|c_k|\\)
\\(\\sum_{k=0}^{N-1}|c_k|^2\\)
\\(\\sum_{k=0}^N|c_k|^2\\)
\\(\\sum_{k=0}^N|c_k|\\)
Explanation:
We know that Px=\\(\\frac{1}{N} \\sum_{n=0}^{N-1}|x(n)|^2\\)
Question 6
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 7
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 8
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 9
Marks : +2 | -2
Pass Ratio : 100%
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 10
Marks : +2 | -2
Pass Ratio : 100%
What is the Fourier series representation of a signal x(n) whose period is N?
\\(\\sum_{k=0}^{\\infty}|c_k|^2\\)
\\(\\sum_{k=-\\infty}^{\\infty}|c_k|\\)
\\(\\sum_{k=-\\infty}^0|c_k|^2\\)
\\(\\sum_{k=-\\infty}^{\\infty}|c_k|^2\\)
Explanation:
The average power of a periodic signal x(t) is given as \\(\\frac{1}{T_p}\\int_{t_0}^{t_0+T_p}|x(t)|^2 dt\\)