Suppose an IIR system is represented by a difference equation y[n] = a y[n − 1] + x[n], where x[n] is the input and y[n] is the output.

(a) If the input is x[n] = u[n] and it is known that the steady-state response is y[n] = 2, what would be a for that to be possible (in steady state x[n] = 1 and y[n] = y[n − 1] = 2 since n → ∞).

(b) Writing the system input as x[n] = u[n] = δ[n] + δ[n − 1] +δ[n − 2] + ⋯then according to the linearity and time-invari-ance the output should be

y[n] = h[n] + h[n − 1] + h[n − 2] + ···

Use the value for a found above, that the initial condition is zero, i.e., y[ − 1] = 0, and that the input is x[n] = u[n], to find the values of the impulse response h[n] for n ≥ 0 using the above equation. The system is causal.

(c) Use the function filter to compute the impulse response h[n] and compare it with the one obtained above.