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Unformatted text preview: , one can quickly observe t he effect of particular
form of compensation on Mp a nd wp. 7.4 (a) F ig. 7 .13 ( a)vector representation of complex numbers (b) vector representation offactors of H{s).
the polezero locations of a system transfer function and its frequency response (or
filtering characteristics). A system transfer function can be expressed as
(7.28a)
where Zl, Z 2, . .. , Z n are the zeros of H (s) and the characteristic roots .A 1, .A2, . .. ,
.An are the poles of H (s). Now t he value of t he t ransfer function H (s) a t some
frequency s = p is
(7.28b)
This equation consists of factors of the form p  Zi a nd p  .Ai. T he factor p  z is a
complex number represented by a vector drawn from point z t o the point p in the
complex plane, as illustrated in Fig. 7.13a. T he length of this line segment is Ip  z I,
the magnitude of p  z. T he angle of this directed line segment (with horizontal
axis) is L(p  z). To compute H (s) a t s = p, we draw line segments from all poles
and zeros of H (s) t o t he point p , as shown in Fig. 7.13b. T he vector connecting
a zero Zi t o the point p is p  Zi. Let the length of this vector be r i, a nd let its
angle with the horizontal axis b e <Pi. T hen p  Zi = r ieN)i. Similarly, t he vector
connecting a pole .Ai t o the point p is p  .Ai = d iejOi, where d i a nd (}i are the length
and the angle (with t he horizontal axis), respectively, of t he vector p  .Ai. Now
from Eq. (7.28b) it follows t hat Filter Design by Placement o f Poles and Zeros OF H(s) I n t his section we explore t he s trong dependence of frequency response on t he
location of poles and zeros of H(s). T his dependence points t o a simple intuitive
procedure t o filter design. A systematic filter design procedure to meet given specifications is discussed later in Secs. 7.5, 7.6, a nd 7.7. 7.41 Re .... Dependence of Frequency Response on poles and Zeros o f H (s) F requency response of a system is basically t he information a bout t he filtering
capability of t he system. We now examine t he close connection t hat exists between t We c an find s imilar c ontours for c onstant a ( the closedloop phase response). ( rlej1), ) he j ¢2) . .. ( rnej¢n)
H (8 )I.=p = bn ( dlej01 ) (d2ej02) . .. ( dne j01 ) = b r lr2··· r n e j [(¢, + ¢2+··+¢n)(O,+02+ '+On)]
n Therefore d 1d2·· · d n q r2··· r n I H(s)ls=p=b nd d 1 2 ··· d n p roduct of the distances of zeros to p
=bn~~~~~~~
p roduct of t he distances of poles to p (7.29a) 498 7 Frequency Response a nd Analog Filters a nd l id. = s um of zero angles t o p  sum of pole angles t o p (7.29b) Using this procedure, we c an determine H (s) for any value o f s. To compute t he
frequency response H (jw), we use s = jw (a point on the imaginary axis), connect
all poles a nd z eros t o t he p oint j w, a nd d etermine IH(jw)1 a nd L H(jw) from Eqs.
(7.29). We r epeat t his procedure for all values of w from 0 t o 0 0 t o o btain t he
frequency response. jmo...
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This note was uploaded on 04/14/2013 for the course ENG 350 taught by Professor Bayliss during the Spring '13 term at Northwestern.
 Spring '13
 Bayliss
 Signal Processing, The Land

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