Signal Processing and Linear Systems-B.P.Lathi copy

Thus one unit along t he u scale is t he same as one

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: a ctual plot can be obtained by adding t he e rror t o t he a symptotic p lot. . . For second-order zeros (complex conjugate zeros), t he plots are mirror Images (about t he O-dB line) of t he plots depicted in Fig. 7.6a. N~te t he reso~ance phenomenon o f t he complex conjugate poles. This phenomenon IS barely noticeable for ( > 0.707 b ut becomes pronounced as ( --> O. Phase T he p hase function for second-order poles, as apparent in Eq. (7.19b), is For W «Wn , L H(jw) "" 0 LH(jw)::= - 180° Hence, t he phase --> - 180° as W --> 0 0. As in t he case of amplitude, we also have a family of phase plots for various values of ( , as illustrated in Fig. 7.6b. A convenient asymptote for t he p hase of complex conjugate poles is a s tep function t hat is 0° for W < W n a nd - 180° for W > W n . A n error plot for such a n a symptote is shown in Fig. 7.7 for various values o f ( . T he e xact phase is t he a symptotic value plus t he error. 7 4 86 F requency R esponse a nd A nalog F ilters 7.2 F or c omplex c onjugate z eros, t he a mplitude a nd p hase p lots a re m irror i mages o f t hose for c omplex c onjugate p oles. W e s hall d emonstrate t he a pplication o f t hese t echniques w ith t wo e xamples. • B ode P lots 4 87 50 As 45 "" ::t: 1 .'(\ : X1 i : ", r tlll " - - . --1' -: : Ex'\Ct 1,*', ' i ll E xample 7 .3 S ketch t he B ode plots for t he t ransfer function I I \ I I', ImptoUl'P~I:I ' " ' I, ',~; ' 1/11" I , ' I ' I' " : !:,' , 40 I r 1 : i 00 b I) .£ H 8 _ 208(8 + 100) ( ) - (8 + 2)(8 + 10) 0 35 N F irst, we w rite t he t ransfer function in normalized form 2 0x100 H (8) = 8(1+1~O) '2><lO (1 + ~) (1 + fa) 8(1+1~O) = 100..,..--'-,...,...""-'-"...,...,(1 + ~) (1 + fa) 30 25 f---fl----f-H-IH-H+--+-+-+- P hase P lots We draw t he a symptotes corresponding to each of t he four factors: I ~-t+HfI " i ii I ' I ", 1 Here, t he c onstant t erm is 100; t hat is, 40 d B (20 log 100 = 40). T his t erm c an b e a dded t o t he p lot by s imply relabeling t he h orizontal axis (from where t he a symptotes begin) as t he 40 d B line (see Fig. 7.8a). Such a s tep implies shifting t he h orizontal axis upward by 40 dB. This is precisely w hat is desired. I n a ddition, we have two first-order poles a t - 2 a nd - 10, o ne zero a t t he origin, and one zero a t - 100. S tep 1 . F or each of these terms, we d raw a n a symptotic plot as follows (see Fig. 7.8a): (i) For t he z ero a t t he origin draw a s traight line with a slope of 20 d B/decade passing t hrough w = 1. (ii) For t he p ole a t - 2, d raw a s traight line with a slope of - 20 d B/decade (for w > 2) b eginning a t t he c orner frequency w = 2. (iii) For t he p ole a t - 10, d raw a straight line with a slope of - 20 d B/decade b eginning a t t he c orner frequency w = 10. (iv) For t he z ero a t - 100, d raw a s traight line with a slope of 20 d B/decade b eginning a t t he c orner frequency w = 100...
View Full Document

This note was uploaded on 04/14/2013 for the course ENG 350 taught by Professor Bayliss during the Spring '13 term at Northwestern.

Ask a homework question - tutors are online