Consider a discrete-time LTI system with impulse response of the form: h[n] = [n

Question

Consider a discrete-time LTI system with impulse response of the form:

h[n] = [n] + [n D]

where D is a positive integer and 0 < < 1. If the input to this system is a signal x, the output is the the signal

(x h)[n] = x[n] + x[n D].

So the system output consists of x itself, together with an echo, which is scaled down by a factor of .

Suppose the sampling frequncy of our audio file is Fs Hz and we want an inter-echo delay of seconds. Find a formula for D in terms of Fs and .Then find the value of D if the sampling frequency is Fs = 8192 Hz and the desired delay is = 0.75 seconds.

Explanation / Answer

Sampling frequency = Fs Hz.

i.e. Fs samples per second

1 second = Fs samples

T seconds = Fs * T samples

Therefore D = Fs * T

If Fs = 8192 and T = 0.75

then D = Fs*T = 8192*0.75 = 6144

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