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274 Transform Analysis of Linear Time-lnvariant Systems Chap. 5 
greater. Furthermore, if the number of poles and zeros of H(z) were not restricled, 
the number of choices for H(z) would be unlimited. To see this, assume that H(z) has 
a factor of lhe form 
i.e., 
z-1 - a* 
1 az-1 ' 
(5.88) 
Factors of this form are referred to as all-pass factors, since they have unity magnitude 
on lhe unit circle; lhey are discussed in more detail in Section 5.5. It is easily verified 
that 
C(z) H(z)H*(l/z*) H1 (z)Hj(l/ z*); (5.89) 
i.e., all-pass factors cancel in C(z) and therefore would not be identifiable from the 
pote-zero plot of C(z). Consequently, if the number of poles and zeros of H(z) is 
unspecified, then, given C(z), any choice for H(z) can be cascaded with an arbitrary 
number of all-pass factors with poles inside the unit circle (i.e., lal < 1). 
5.5 ALL-PASS SYSTEMS 
As indicated in the discussion of Example 5.12, a stable system function of the form 
Hap(Z) z-
1 
- a* 
1 az-1 (5.90) 
has a frequency-response magnitude that is independent of w. This can be seen by 
writing Hap(eiw) in the form 
(5.91) 
. 1 a•eiw 
= e-JW . • 
1 - ae-Jw 
ln Eq. (5.91), the term e-iw has unity magnitude, and the remaining numerator and 
denominator factors are complex conjugates of each other and therefore have the sarne 
magnitude. Consequently, iHap(eiw)I = l. A system for which the frequency-response 
magnitude is a constant is called an a/1-pass system, since the system passes all of the 
frequency components of its input with constant gain or attenuation. The most general 
form for the system function of an all-pass system with a real-valued impulse response 
is a product of factors like Eq. (5.90), with complex poles being paired with their con-
jugates; i.e., 
M, -1 d Me ( -1 *)( -1 ) 
II z - k II z - ek z - ek Hap(z) = A ---- 1 , 1 - d z-1 (1 - e z-1)(1 - e*z- ) k=I k k=l k k (5.92) 
where Ais a positive constant and the dk's are the real poles, and the ek's the complex 
poles, of Hap( z). For causal and stable all-pass systems, idki < 1 and lekl < 1. ln terms 
Sec. 5.5 AII-Pass Systems 
jm 
Unit z-plane 
2 'lfte 
Figure 5.21 Typical pole-zero plot for 
an all-pass system. 
275 
of our general notation for system functions, all-pass systems have M N = 2Mc + M, 
poles and zeros. Figure 5.21 shows a typical pole-zero plot for an all-pass system. ln 
this case M, = 2 and Me = 1. Note that each pole of Hap(z) is paired with a conjugate 
reciprocai zero. 
The frequency response for a general all-pass system can be expressed in terms of 
the frequency responses of first-order all-pass systems like that specified in Eq. (5.90). 
For a causal all-pass system, each of these terms consists of a single pole inside the 
unit circle and a zero at the conjuga te reciprocai location. The magnitude response for 
such a term is, as we have shown, unity. Thus, the log magnitude in dB is zero. With a 
expressed in polar formas a = rei0 , the phase function for Eq. (5.90) ís 
< . . = -w - 2 arctan ------ . [
e-im re-i0 ] [ rsin(w-0) ] 
1 re1°e-Jw 1 - r cos(w 0) (5.93) 
Likewise, the phase of a second-order all-pass system with poles at z = rei0 and z 
re- i0 is 
<[ . . . . = -2w - 2 arctan [ 
(e-iw - re-i0)(e-im rei6) ] [ r sin(w 0) ] 
(1 - re1°e-1w)(l re-1°e-1w) 1 - r cos(w 0) 
[ 
r sin(w + 0) ] 
-2 arctan ------ . 
1 - r cos( w + 0) 
Example 5.13 First- and Second..Order AII-Pass Systems 
(5.94) 
Figure 5.22 shows plots of the log magnitude, phase, and group delay for two first-
order all-pass systems, one with a real pole at z = 0.9 (0 = O, r = 0.9) and another 
with a pole at z = -0.9 (0 = :rc, r 0.9). For both systems, the radii of the poles are 
r = 0.9. Likewise, Figure 5.23 shows the sarne functions for a second-order all-pass 
system with poles at z = 0.9eirr/4 and z 0.9e~irr/4 _