1- Suppose that you are recording sound in a room with an echo. so your microphone picks up both the original signal and a delayed, attenuated version of the signal. You decide to model this process as an LTI system with impulse response h(t)=delta (t)+beta delta (t-alpha ) a)(5p) Determine the frequency response, H(w) , of the system. b)(20p) For alpha =2 and beta =1 , find and plot the magnitude response and phase response of the system.

Question

1- Suppose that you are recording sound in a room with an echo. so your microphone picks up both the original signal and a delayed, attenuated version of the signal. You decide to model this process as an LTI system with impulse response h(t)=delta (t)+beta delta (t-alpha ) a)(5p) Determine the frequency response, H(w) , of the system. b)(20p) For alpha =2 and beta =1 , find and plot the magnitude response and phase response of the system.

Answer 1

a) To determine the frequency response,

, of the system, we need to take the Fourier transform of the impulse response,

.The impulse response is given by:

Taking the Fourier transform of

, we get:

Using the properties of the Fourier transform, we have:

and

Substituting these values, we get:

b) For

and

, the magnitude response and phase response of the system can be found by substituting these values into the expression for

.Magnitude response:

Phase response:

Substituting

and

, we get:

To plot the magnitude response and phase response, we can use a graphing tool or software. The magnitude response will show the variation of the magnitude of

with respect to the frequency, while the phase response will show the variation of the phase of

with respect to the frequency.

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