Find the transient and steady-state response for the input

Question

Q1. Design a causal 4^th order FIR band-pass filter with linear phase for cutoff frequencies at 2000 Hz and 4000 Hz and sampling frequency of 16kHz.

a) Use a rectangular window. Give precise numerical values for the filter coefficients b_i and a_i.

b) We give the values of the following 5-point Kaiser windows. Choose one of the three following Kaiser windows that is likely to produce the desired 4th order filter with the best cutoff and most reduced ripple effects.

Kaiser A .choice: 0.9403 0.9849 1.0000 0.9849 0.9403 (beta=0.5)

Kaiser B .choice: 0.8244 0.9545 1.0000 0.9545 0.8244 (beta=0.9)

Kaiser C .choice: 0.4387 0.8348 1.0000 0.8348 0.4387 (beta=2)

Indicate (circle above) clearly the choice and determine the coefficients b_i and a_i of the filter.

Q2. For the following causal system

H(z) = z-1/(z-0.5)²

Find the transient and steady-state response for the input x(n)=u(n) using the inverse z-transform and the residue theorem.