390 CHAPTER, 9 • Z-TR.ANSFOR.MS AND APPLICATIONS
Imz'-~~n~~~no..._~,___;:::,~n 9
4 2Figure 9.13 Amplitude response A(8) a.nd zero-pole plot for the combination filter
y{n) = x[n) + x[n -l ] + x(n -2] + x[n -3] + ~J2y[n - l j - ~y{n - 2).Remark 9.18
The zeros e±if and ei" of H(z) determine which signals are zeroed out and the
arguments of the poles ~e±lf of H(z) point to frequencies that are boosted up
by the filter. •Remark 9. 19
The flat portion of the graph in the interval [O, %1 makes this filter more practical
for boosting low frequencies than the filter in Example 9.22(a). •Remark 9.20
The higher frequencies in the interval rn, 7r] are attenuated more than they are
in Example 9.23{b). The situation is illustrated in Figure 9.13. •
A signal processing engineer uses complex analysis to construct filters with
the desired amplitude and phase response characteristics. Finite impulse re-
sponse {FIR) filters have only zeros, whereas infinite impulse response {IIR)
filters have poles and may have zeros as well. The area of filter design involves
many types, such as low pass, high pass, all pass, band pass, and band stop.
Special forms of such filters include, but are not limited to, Bessel, Butterworth,
Chebyshev, Gaussian, moving average, single pole, and Remez. More informa-
tion about filter design can he found in books on digital signal processing.-------~EXERCISES FOR SECTION 9.3
- Use direct substitution and trigonometric identit ies to show the following:
(a) y[nl = x{n]+x[n-l]+x[n-2] will "zerc>out" x[n) = cos(2;n) and x[n] =
sin(^2 ; n).