Page 357 - Schaum's Outline of Theory and Problems of Signals and Systems

FOURIER ANALYSIS OF DISCRETE-TIME SIGNALS AND SYSTEMS [CHAP. 6

0.414
Fig. 6-33 Simulation of an RC filter by the bilinear transformation method.

6.48. Let h[n] denote the impulse response of a desired IIR filter with frequency response
H(R) and let h,[n] denote the impulse response of an FIR filter of length N with
frequency response H,(R). Show that when

ho[n] = (:["I OsnsN-1 (6.198)
otherwise
the mean-square error e2 defined by

is minimized.
By definition (6.27)
ffi m
H(R) = C h[n]e-J'" and H,(R)= z ho[n]e-JRn
n = -m n- -m
Let

where e[n] = h[n] - h,[n]. By Parseval's theorem (6.66) we have

The last two terms in Eq. (6.201) are two positive constants. Thus, E' is minimized when

that is,

Note that Eq. (6.198) can be expressed as

where w[n] is known as a rectangular window function given by
OsnsN-1
w[n] =
otherwise