For a LTID system transfer function of Hz z 01z 12 with i

For a LTID system transfer function of H[z] = z + 0.1/z + 1.2 with input x[n] = (0.2)^n u[n] and output y[u] Find the zero-state response.

Solution

First, it\'s important to realize that the response of an LTI system to a scaled and shifted impulse

x[n]=a(nk)x[n]=a(nk)

is simply a scaled and shifted version of its impulse response:

y[n]=ah[nk]

Furthermore, the response to a sum of signals equals the sum of the responses to the individual components of the input signal. Consequently, the response to the given input signal

x[n]=2[n]3[n2]

is

y[n]=2h[n]3h[n2] (1)

Now you just have to express the impulse response h[n] in terms of -impulses. By noting that

u[n]u[n3]=[n]+[n1]+[n2]

you get

h[n]=[n]+[n1]+4[n2] (2)

Plugging (2) into (1) gives the desired result.