Complete the following:
a. Show that the amplitude gain of any linear-phase filter with weights
in the form of (5.3) may be expressed in the following form:
[Hint: Use the weight vector in (5.3) centered at zero, noting that
the time shift has no effect on the amplitude spectrum.] Compare
this result with (2.32), and note that here we have the magnitude of
a continuous Fourier series for a function of frequency (not time).
b. Express the amplitude gain at half the sampling rate as a simple, nontrigonometric sum. Use this expression to compute the
power gains of the two filters in Exercise 5.1 at half the sampling
rate. If you have worked Exercise 5.4, use this result to verify the
ends of the two dB plots.