Digital Signal Processing

Frequency Analysis of Discrete Time Signal

Question 1
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 2
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 3
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 4
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 5
Marks : +2 | -2
Pass Ratio : 100%
If x(n) is a stable sequence so that X(z) converges on to a unit circle, then the complex cepstrum signal is defined as ____________
X(ln X(z))
ln X(z)
X-1(ln X(z))
None of the mentioned
Explanation:
Let us consider a sequence x(n) having a z-transform X(z). We assume that x(n) is a stable sequence so that X(z) converges on to the unit circle. The complex cepstrum of the signal x(n) is defined as the sequence cx(n), which is the inverse z-transform of Cx(z), where Cx(z)=ln X(z)
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%
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 8
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 9
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 10
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.