Echo Measurement

The questions below are due on Thursday April 24, 2025; 02:00:00 PM.
 
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Assume that a single echo interferes with a speaker's voice that is being recorded by a microphone as illustrated in the following figure.

We can represent this recording situation as a linear, time-invariant system, with the speaker's voice as the input and the recorded microphone signal as the output. Assume that the impulse response of this system is

h(t) = \delta(t\!-\!T_1)+\epsilon\,\delta(t\!-\!T_2)

where T_1 represents the delay of the direct path from speaker to microphone, T_2 represents that delay through the echo path, and \epsilon represents the amplitude of the echo.

Part a.

On the axes below, sketch the magnitude and angle of the frequency response of this system in the absence of an echo (i.e., when \epsilon = 0). Label all important values.

Part b.

Now consider the case where there is an echo. The following plots show that magnitude and angle of the frequency response of this system for |\omega|\lt1500,rad/s, for some values of T_1, T_2, and \epsilon.

Determine values of T_1, T_2, and \epsilon that are consistent with the graphs above.

Please upload a single pdf file that contains your answers to all parts of this problem:

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