Digital Signal Processing

Frequency Domain Characteristics of LTI System

Question 1
Marks : +2 | -2
Pass Ratio : 100%
What is the output sequence of the system with impulse response h(n)=(1/2)nu(n) when the input of the system is the complex exponential sequence x(n)=Aejnπ/2?
\\(Ae^{j(\\frac{nπ}{2}-26.6°)}\\)
\\(\\frac{2}{\\sqrt{5}} Ae^{j(\\frac{nπ}{2}-26.6°)}\\)
\\(\\frac{2}{\\sqrt{5}} Ae^{j({nπ}{2}+26.6°)}\\)
\\(Ae^{j(\\frac{nπ}{2}+26.6°)}\\)
Explanation:
First we evaluate the Fourier transform of the impulse response of the system h(n)
Question 2
Marks : +2 | -2
Pass Ratio : 100%
If an LTI system is described by the difference equation y(n)=ay(n-1)+bx(n), 0 < a < 1, then what is the parameter ‘b’ so that the maximum value of |H(ω)| is unity?
a
1-a
1+a
none of the mentioned
Explanation:
We know that,
Question 3
Marks : +2 | -2
Pass Ratio : 100%
What is the magnitude of H(ω) for the three point moving average system whose output is given by y(n)=\\(\\frac{1}{3}[x(n+1)+x(n)+x(n-1)]\\)?
\\(\\frac{1}{3}|1-2cosω|\\)
\\(\\frac{1}{3}|1+2cosω|\\)
|1-2cosω|
|1+2cosω|
Explanation:
For a three point moving average system, we can define the output of the system as
Question 4
Marks : +2 | -2
Pass Ratio : 100%
If the system gives an output y(n)=H(ω)x(n) with x(n) = Aejωnas input signal, then x(n) is said to be Eigen function of the system.
True
False
Explanation:
An Eigen function of a system is an input signal that produces an output that differs from the input by a constant multiplicative factor known as Eigen value of the system.
Question 5
Marks : +2 | -2
Pass Ratio : 100%
If the Eigen function of an LTI system is x(n)= Aejnπ and the impulse response of the system is h(n)=(1/2)nu(n), then what is the Eigen value of the system?
3/2
-3/2
-2/3
2/3
Explanation:
First we evaluate the Fourier transform of the impulse response of the system h(n)
Question 6
Marks : +2 | -2
Pass Ratio : 100%
What is the response of the system with impulse response h(n)=(1/2)nu(n) and the input signal x(n)=10-5sinπn/2+20cosπn?
20-\\(\\frac{10}{\\sqrt{5}} sin(π/2n-26.60)+ \\frac{40}{3}cosπn\\)
20-\\(\\frac{10}{\\sqrt{5}} sin(π/2n-26.60)+ 40cosπn\\)
20-\\(\\frac{10}{\\sqrt{5}} sin(π/2n+26.60)+ \\frac{40}{3cosπn}\\)
None of the mentioned
Explanation:
The frequency response of the system is
Question 7
Marks : +2 | -2
Pass Ratio : 100%
What is the magnitude of the frequency response of the system described by the difference equation y(n)=ay(n-1)+bx(n), 0<a<1?
\\(\\frac{|b|}{\\sqrt{1+2acosω+a^2}}\\)
\\(\\frac{|b|}{1-2acosω+a^2}\\)
\\(\\frac{|b|}{1+2acosω+a^2}\\)
\\(\\frac{|b|}{\\sqrt{1-2acosω+a^2}}\\)
Explanation:
Given y(n)=ay(n-1)+bx(n)
Question 8
Marks : +2 | -2
Pass Ratio : 100%
If x(n)=Aejωn is the input of an LTI system and h(n) is the response of the system, then what is the output y(n) of the system?
H(-ω)x(n)
-H(ω)x(n)
H(ω)x(n)
None of the mentioned
Explanation:
If x(n)= Aejωn is the input and h(n) is the response o the system, then we know that
Question 9
Marks : +2 | -2
Pass Ratio : 100%
If h(n) is the real valued impulse response sequence of an LTI system, then what is the imaginary part of Fourier transform of the impulse response?
–\\(\\sum_{k=-∞}^∞ h(k) sin⁡ωk\\)
\\(\\sum_{k=-∞}^∞ h(k) sin⁡ωk\\)
–\\(\\sum_{k=-∞}^∞ h(k) cos⁡ωk\\)
\\(\\sum_{k=-∞}^∞ h(k) cos⁡ωk\\)
Explanation:
From the definition of H(ω), we have
Question 10
Marks : +2 | -2
Pass Ratio : 100%
If h(n) is the real valued impulse response sequence of an LTI system, then what is the phase of H(ω) in terms of HR(ω) and HI(ω)?
\\(tan^{-1}\\frac{H_R (ω)}{H_I (ω)}\\)
–\\(tan^{-1}\\frac{H_R (ω)}{H_I (ω)}\\)
\\(tan^{-1}\\frac{H_I (ω)}{H_R (ω)}\\)
–\\(tan^{-1}\\frac{H_I (ω)}{H_R (ω)}\\)
Explanation:
If h(n) is the real valued impulse response sequence of an LTI system, then H(ω) can be represented as HR(ω)+j HI(ω).