A three-point moving-average filter is of the form: y[n] = (x [n 1] + x[n] +

A three-point moving-average filter  is of the form:

y[n] = β(αx [n − 1] + x[n] + αx[n + 1])

where α and β are constants, and x[n] is the input and y[n] is the output of the filter.

(a) Determine the transfer function H(z) = Y(z)/X(z) of the filter and from it find the frequency response H(e^jω) of the filter in  terms of α and β.

(b) Let α = 0.5, find then β so that the dc gain of the filter is unity, and the filter has a zero phase. For the given and obtained  values of α and β, sketch H(e^jω) and find the poles and zeros of H(z) and plot them in the Z-plane.

(c) Suppose we let v[n] = y[n – 1] be the output of a second filter.  Is this filter causal? Find its transfer function G(z) = V(z)/X(z). Use MATLAB to compute the unwrapped phases of G(z) and to  plot the poles and zeros of G(z) and H(z) and explain the relation between G(z) and H(z).