Digital Signal Processing

Frequency Analysis of Discrete Time Signal

Question 1
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\\)
Question 2
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 3
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 4
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 5
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.
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%
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%
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 9
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,
Question 10
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