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

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

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

This is the end of the preview. Sign up to access the rest of the document.

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&gt; 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 &lt; Wn , a nd (ii) - 40u - 40logw n for W &gt; · 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 ( &lt; 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 &quot;&quot;&quot; -: ::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&lt;~ OF ~~~~~~§§~~~~rf~;:~~i~=li--li--r-i!Tlll-r'rww~--'As~ptotel i I II I' '&quot; ~ ..g 1i i&quot;f - 30 - 60 i Ii --+b....::.5~p....A~~-'I~\I~t\~f.t:=_&quot;_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...
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