Question

Consider the DT-LTI system with impulse response: h[n]=(n−1)(u[n−1]−u[n−5]).

Derive on paper the response of the system (y[n]paper ) to the input signal x[n]=u[n−2]−u[n−6].

Answer 1

To find the response of the system to the input signal, we convolve the impulse response with the input signal.

Step-by-step explanation:

To find the response of the system to the input signal, we convolve the impulse response with the input signal. Let's simplify the input signal:

x[n]=u[n−2]−u[n−6]

We can express this as:

x[n]=u[n−2]−u[n−6] = u[n−2]−u[n−5]+u[n−5]−u[n−6]

Now we apply the convolution property:

y[n] = x[n] * h[n] = (u[n−2]−u[n−5]) * [(n−1)(u[n−1]−u[n−5])]

By distributing and rearranging terms:

y[n] = (n−1)(u[n−1]−u[n−2])−(n−1)(u[n−5]−u[n−6])

This is the response of the system to the input signal x[n]=u[n−2]−u[n−6].