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

15a the log amplitude function 20u is p lotted as a

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Unformatted text preview: agnitude 0 T he log a mplitude is given by -2 Q. log amplitude = - 20 log /1 + 2 j( -4 (0- O.Ola O.OSa O .la F ig. 7 .5 O.Sa Sa a SOa IOa For W «':: W n> For -6 W » Wn + (~:) 21 (7.20) t he log amplitude becomes IOOa log amplitude Errors in asymptotic approximation of a first-order pole. L HUW)=-L(l+j:) (:J =-tan-l(~) , ~ - 20 log 1 = 0 (7.21) t he log amplitude is log amplitude ~ -201og 1( - :J 21 = - 401og (:J (7.22a) and, for W » W = - 40logw - 40logw n «':: a, (7.22b) = - 40u - 40logw n Let us investigate the asymptotic behavior of this function. For (7.22c) T he two asymptotes are (i) zero for W < Wn , a nd (ii) - 40u - 40logw n for W > · T he second asymptote is a s traight line with a slope of - 40 d B/decade (or - 12dB/octave) when plotted against t he log W scale. I t begins a t W = Wn [see Eq. (7.22b)]. T he a symptotes are depicted in Fig. 7.6a. T he e xact log amplitude is given by [see Eq. (7.20)] Wn a, -tan-l(~) ~ - 90 0 T he actual p lot along with the asymptotes is depicted in Fig. 7.4b. In this case, we use a three-line segment asymptotic plot for greater accuracy. The asymptotes are (i) a phase angle of 00 for W : :; a /lO, (ii) a phase angle of - 90 0 for W ~ lOa, a nd a s traight line with a slope - 45°/decade connecting these two asymptote~ (from W = a /IO t o lOa) crossing the W axis a t W = a /10. I t can be seen from FIg. 7.4b t hat t he asymptotes are very close t o t he curve a nd t he maximum error is 5.7 0 • A p lot of the e rror as a function of W is i llustrated in Fig. 7.5b. T he actual .plot can be obtained b y adding the error to the asymptotic plot. T he phase functIOn for a pole a t - a is shown in Fig. 7.4b. The phase for a zero at - a (shown d otted in Fig. log amplitude = - 20 log 1{[ (:J 2]2 + 4(2 (:J 2 }t (7.23) Clearly, t he log amplitude in this case involves a parameter ( . For each value of ( , we have a different plot. For complex conjugate poles,t ( < 1. Hence, we must sketch a family of curves for a n umber of values of ( in t he r ange 0 t o 1. This is i llustrated in Fig. 7.6a. T he e rror between the actual plot and the asymptotes t For ( ~ 1, t he t wo p oles in t he s econd-order factor a re no longer complex b ut r eal, a nd e ach o f t hese two real poles c an b e d ealt w ith as s eparate f irst-order factors. 484 7 Frequency Response a nd Analog Filters 7.2 Bode P lots 485 20 10 i:I:I """ -: ::r:: 0 b Il .9 5 - --- ..-.-- ?:l -10 - 20 10.5 I I q.707 ; -5------:---.t=lt--·--·t-_t_--I-+~H_T----_t_--,!___+-1:-r--j ---r-f-I . I -6 O.loon 0.2con 0.500n oo=oon 200n 500n 100ln 90 , -------T----,---.-4-,,-,-.. .-----~--_.---r--r_r_r<~ OF ~~~~~~§§~~~~rf~;:~~i~=li--li--r-i!Tlll-r'rww~--'As~ptotel i I II I' '" ~ ..g 1i i"f - 30 - 60 i Ii --+b....::.5~p....A~~-'I~\I~t\~f.t:=_"_c::=_=----_..L--_=_=_;;t;~~;-._=rl_-+ll~~IDI-'- _ C=O.l --------t~ I I -90 I ! I : : i ii I 3 0- I ! -30 I -·120 60 - ------+------+---'---t - 60 I --150 F ig. 7.7 Errors in the asymptotic approximation of a second-order pole. - 180 0.2oon F ig. 7 .6 0.500n OOn 2 con 500n Amplitude and phase response of a second-order pole. (7.24) is shown in Fig. 7.7. T he...
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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.

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