Frequency Domain Characteristics of LTI System MCQ Quiz – Objective Question with Answer for Frequency Domain Characteristics of LTI System

1. 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?

A. H(-ω)x(n)
B. -H(ω)x(n)
C. H(ω)x(n)
D. None of the mentioned

Answer: C

If x(n)= Aejωn is the input and h(n) is the response o the system, then we know that

y(n)=\(\sum_{k=-∞}^∞ h(k)x(n-k)\)

=>y(n)=\(\sum_{k=-∞}^∞ h(k)Ae^{jω(n-k)}\)

= A \([\sum_{k=-∞}^∞ h(k) e^{-jωk}] e^{jωn}\)

= A. H(ω). ejωn

= H(ω)x(n)

 

2. 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 the Eigenfunction of the system.
A. True
B. False

Answer: A

An Eigenfunction of a system is an input signal that produces an output that differs from the input by a constant multiplicative factor known as the Eigenvalue of the system.

 

3. 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?

A. \(Ae^{j(\frac{nπ}{2}-26.6°)}\)

B. \(\frac{2}{\sqrt{5}} Ae^{j(\frac{nπ}{2}-26.6°)}\)

C. \(\frac{2}{\sqrt{5}} Ae^{j({nπ}{2}+26.6°)}\)

D. \(Ae^{j(\frac{nπ}{2}+26.6°)}\)

Answer: B

First, we evaluate the Fourier transform of the impulse response of the system h(n)

H(ω)=\(\sum_{n=-∞}^∞ h(n) e^{-jωn} = \frac{1}{1-1/2 e^{-jω}}\)

At ω=π/2, the above equation yields,

H(π/2)=\(\frac{1}{1+j 1/2}=\frac{2}{\sqrt{5}} e^{-j26.6°}\)

We know that if the input signal is a complex exponential signal, then y(n)=x(n) . H(ω)

=>y(n)=\(\frac{2}{\sqrt{5}} Ae^{j(\frac{nπ}{2}-26.6°)}\)

4. If the Eigenfunction 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 Eigenvalue of the system?

A. 3/2
B. -3/2
C. -2/3
D. 2/3

Answer: D
First, we evaluate the Fourier transform of the impulse response of the system h(n)

H(ω)=\(\sum_{n=-∞}^∞ h(n) e^{-jωn} = \frac{1}{1-\frac{1}{2} e^{-jω}}\)

At ω=π, the above equation yields,

H(π)=\(\frac{1}{1+\frac{1}{2}}\)=2/3

If the input signal is a complex exponential signal, then the input is known as the Eigenfunction and H(ω) is called the Eigenvalue of the system. So, the Eigenvalue of the system mentioned above is 2/3.

 

5. If h(n) is the real-valued impulse response sequence of an LTI system, then what is the imaginary part of the Fourier transform of the impulse response?

A. –\(\sum_{k=-∞}^∞ h(k) sin⁡ωk\)

B. \(\sum_{k=-∞}^∞ h(k) sin⁡ωk\)

C. –\(\sum_{k=-∞}^∞ h(k) cos⁡ωk\)

D. \(\sum_{k=-∞}^∞ h(k) cos⁡ωk\)

Answer: A

From the definition of H(ω), we have

H(ω)=\(\sum_{k=-∞}^∞h(k) e^{-jωk}\)

=\(\sum_{k=-∞}^∞h(k) cos⁡ωk-j\sum_{k=-∞}^∞h(k) sin⁡ωk\)

= HR(ω)+j HI(ω)

=> HI(ω)=-\(\sum_{k=-∞}^∞h(k) sin⁡ωk\)

 

6. 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)?

A. \(tan^{-1}\frac{H_R (ω)}{H_I (ω)}\)

B. –\(tan^{-1}\frac{H_R (ω)}{H_I (ω)}\)

C. \(tan^{-1}\frac{H_I (ω)}{H_R (ω)}\)

D. –\(tan^{-1}\frac{H_I (ω)}{H_R (ω)}\)

Answer: C

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).

=> tanθ=\(\frac{H_I (ω)}{H_R (ω)}\) => Phase of H(ω)=\(tan^{-1}\frac{H_I (ω)}{H_R (ω)}\)

 

7. What is the magnitude of H(ω) for the three point moving average system whose output is given by
y(n)=\(\frac{1}{3}[x(n+1)+x(n)+x(n-1)]\)?

A. \(\frac{1}{3}|1-2cosω|\)

B. \(\frac{1}{3}|1+2cosω|\)

C. |1-2cosω|

D. |1+2cosω|

Answer: B

For a three point moving average system, we can define the output of the system as

y(n)=\(\frac{1}{3}[x(n+1)+x(n)+x(n-1)]=>h(n)=\{\frac{1}{3}, \frac{1}{3}, \frac{1}{3}\}\)

it follows that H(ω)=\(\frac{1}{3}(e^{jω}+1+e^{-jω})=\frac{1}{3}(1+2cosω)\)

=>| H(ω)|=\(\frac{1}{3}\)|1+2cosω|

8. 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?

A. 20-\(\frac{10}{\sqrt{5}} sin(π/2n-26.60)+ \frac{40}{3}cosπn\)

B. 20-\(\frac{10}{\sqrt{5}} sin(π/2n-26.60)+ 40cosπn\)

C. 20-\(\frac{10}{\sqrt{5}} sin(π/2n+26.60)+ \frac{40}{3cosπn}\)

D. None of the mentioned

Answer: A

The frequency response of the system is

H(ω)=\(\sum_{n=-∞}^∞ h(n) e^{-jωn} = \frac{1}{1-\frac{1}{2} e^{-jω}}\)

For first term, ω=0=>H(0)=2

For second term, ω=π/2=>H(π/2)=\(\frac{1}{1+j\frac{1}{2}} = \frac{2}{\sqrt{5}} e^{-j26.6°}\)

For third term, ω=π=> H(π)=\(\frac{1}{1+\frac{1}{2}}\) = 2/3

Hence the response of the system to x(n) is

y(n)=20-\(\frac{10}{\sqrt{5}} sin(π/2n-26.6^0)+ \frac{40}{3}cosπn\)

 

9. 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?

A. \(\frac{|b|}{\sqrt{1+2acosω+a^2}}\)

B. \(\frac{|b|}{1-2acosω+a^2}\)

C. \(\frac{|b|}{1+2acosω+a^2}\)

D. \(\frac{|b|}{\sqrt{1-2acosω+a^2}}\)

Answer: D

Given y(n)=ay(n-1)+bx(n)

=>H(ω)=\(\frac{|b|}{1-ae^{-jω}}\)

By calculating the magnitude of the above equation we get

|H(ω)|=\(\frac{|b|}{\sqrt{1-2acosω+a^2}}\)

 

10. 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. a
B. 1-a
C. 1+a
D. none of the mentioned

Answer: B

We know that,

|H(ω)|=\(\frac{|b|}{\sqrt{1-2acosω+a^2}}\)
Since the parameter ‘a’ is positive, the denominator of |H(ω)| becomes minimum at ω=0. So, |H(ω)| attains its maximum value at ω=0. At this frequency we have,

\(\frac{|b|}{1-a}\) = 1 => b=±(1-A..

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