1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

9.3 • DIGITAL SIGNAL F ILTERS 383

and

(

z -pei")

z = 1 + pz-^1 if 0 < p < 1 and <f> = 7r,

and

(

-z-z -P) = l -pz-,^1

in the denominator of H(z). They will contribute to the term

1 + a1z-^1 + a2z-^2 + ... + ap_ 1 z-P+l + apz- P

zP + a1zP- l + a2zP-^2 + ... + ap_1z + ap

= zP (9-43)

(iii) Attenuating Factors

To attenuate the signals cos(IJn) and sin(IJn), use factors of the form

if 0 < p < 1 and 0 < 8 < 7r,

and

z -pe'"

( ) = 1 + pz-^1 if 0 < p < 1 and 8 = 7r.

z

The factor

(z -P) = 1 - pz- 1

z

is a special case that attenuates low-frequency signals. These factors will con-

tribute to the term (9-42).

(iv) Combination of Factors

The transfer function H(z) could have a zero or pole at the origin, but this has

no net effect on the output signal. The other zeros and poles determine the

nature of the filter. A conjugate pair of zeros e±ie of H(z) on the unit circle

will "zero-out" the signals cos(IJn) and sin(IJn). If 0 < p ~ 1, the conjugate pair

of zeros pe±i^9 of H (z) will attenuate the signals cos(IJn) and sin(IJn), and the

conjugate pair of poles pe±•4> of H(z) will amplify the signals cos(IJn) and sin(IJn).

It is useful to plot the location of the zeros and poles and note their magnitude

and argument. As a general rule, zeros are used to attenuate signals and poles

are used to amplify signals. The primary goal of filter design is to construct

H(z) so that the amplitude response A(IJ) has a desired shape. The following

examples have been chosen to illustrate these concepts. Books on digital signal

filter design will explain the process in detail.