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).
Q.
(a) Find the spectrum of this waveform: x = [ exp( -[0:1:49]/10), exp(-[50:-1:1]/10)]. Subplot only the magnitude by using the spectrum program from the class notes.
(b) Using this information from the 100-point DFT, re-create the waveform by summing the sinusoidal contributions from all the DFT coefficients as done in lecture. Show both the waveform x and its recreation xhat on another subplot. (The waveforms might merge into one figure so the mean square error is effectively zero in this case.)
(d) Using MATLAB, find the index n0 at which the spectral magnitude falls below 0.05 of its maximum (which is a point beyond which there is little energy left in the higher frequencies.)
(e) Using only the first n0 DFT coefficients, generate the waveform and plot in Matlab by using the symmetry properties of the transform. Plot xhat over the actual signal and annotate the plot to indicate the value of the calculated mse. Title and label the axes. Submit plot and code.