A filter described by the equation: y(n) = x(n) + x(n-1) + 0.9 y(n-1) - 0.81 y(n-2)
(a) Find the transfer function H(z) for the filter and find the poles and zeros of the filter.
(b) Plot the poles and zeros of the filter using zplane(b,a) and tell whether or not this filter is stable.
(c) Plot the magnitude and phase of the frequency response of the filter. Annotate the plots to indicate the magnitude and phase response at points ω=0.33π and ω=π.
(d) Generate 200 samples of the signal x(n) = sin(0.33π n) + 5 cos(nπ) and process them through the filter. Plot both the filter's input x(n) and output y(n) on the same graph.
(e) How are the amplitudes of the two sinusoids affected by the filter?
(f) Determine the equation for the steady-state output y_{ss}(n) of the filter whose input is x(n).