1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

382 CHAPTER. 9 • Z-TR.ANSFOR.MS AND APPLICATIONS

The fundamental theorem of algebra implies that the numerator has Q roots
(called zeros) and the denominator has P roots (called poles). The zeros { Zq} ~=l

may be chosen in conjugate pairs on the unit circle and lzql = 1 for q = 1, 2, ... , Q.

For stability, all the poles {wv}:=l must lie inside t he unit circle and lw,,I < 1

for p = 1, 2, ... , P. Furthermore, the poles are chosen to be real numbers and/or

in conjugate pairs. This will guarantee that the recursion coefficients are all real

numbers. UR filters may be all pole or zero-pole and stability is a concern; FIR

filters and all zero-filters are always stable.

9 .3 .4 Desig n of Filte rs

In practice, recursion formula (9-31) is used to calculate the output signal. How-

ever, digital filter design is based on the preceding t heory. One starts by selecting

the location of zeros and poles corresponding to filter design requirements and

constructing the transfer function H (z) = ~~~l · Since the coefficients in H(z}

are real, all zeros and poles having an imaginary component must occur in con-

j ugate pairs. Then the recursion coefficients are identified in (9-34) and used

in (9-·31} to write the recursive filter. Both the numerator and denominator of

H (z) can be factored into quadratic factors with real coefficients and possibly

one or two linear factors with real coefficients. The following principles are used

to construct H(z).

(i) Zeroing Out Factors

To filter out the signals cos( lln) and sin( lln), use factors of the form

z - ei^8 z - e-iQ..

(-- )( ) = 1 - (e'^8 + e- '^8 )z-^1 + z-^2 if 0 < B < -ir,

z z

and

z -ei"

(--}= l+z-^1 ifB= n ,

z

in the numerator of H(z). They will contribute to t he term

bo + b,z-^1 + ~z-^2 + ... + bq-1z-Q+1 + bqz- Q =

zQbo + b1zQ- i + ~zQ-z + ... + bQ-1z + bQ

zQ

(ii) Boosting Up Factors

To amplify the signals cos(Bn) and sin(Bn), use factors of the form

(9-42)

if 0 < p < 1 and 0 < ¢> < -ir,