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

27rt 27rfst also hajw is not generally bandlimited

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Unformatted text preview: nal a liasing, we m ust u se [see E q. (8.17c)] 1 T<- ( 12.58) - 2Fh LH • 00_ E xample 1 2.5 U sing t he b ilinear transformation, synthesize Ha(S}=~ L Ha(joo) S +We . ..............~............... F rom Eq. (12.57), we obtain -1112 H[z] = --;-;;--_W--;.,e_ _ (~: ~ ~) +We F ig. 1 2.12 Bilinear t ransformation m ethod of design: response. WeT(z + I} (2 + w eT}z - (2 - weT) We should use Eq. (12.58) to select a suitable value for T . However, t o facilitate comparison w ith t he impulse invariance method, we choose here t he s ame value for T as t hat in E xample 12.4: T = l;"c' T he s ubstitution of w eT = 7r / 10 in t he above equation yields H[z] = 0.1357 ( T = 7r X 1 0- 6 F rom this we o btain 0.024(1 + cos w T} [ 1 - 0.9518 cos w T ] 1/2 T= LH[eiWT] = t an- 1 sin w T _ t an-1 sin w T (12.59b) 1 + cos w T cos w T - 0.7285 F igure 12.12 shows IHI a nd L H as computed from Eqs. (12.59). Compare these with t he filter characteristics obtained from t he impulse invariant m ethod (Fig. 12.11). • C omputer E xample C 12.4 Using M ATLAB, find t he b ilinear transformed digital filter t o realize t he first-order analog B utterworth filter in E xample 12.5. 5 T he analog filter transfer function is 105 /(s + 10 ) a nd t he s ampling interval T = 6 1 O-67r. Hence, t he s ampling frequency F s= 10 /7r. A s uitable MATLAB function t o solve this problem is ' bilinear'. T he i nput d ata a re t he coefficients of t he n umerator a nd t he d enominator polynomials o f Ha(s} [entered as (n + 1}-element vectors num a nd den] a nd t he s ampling frequency F s Hz. MATLAB returns b a nd a, t he n umerator a nd t he d enominator polynomial coefficients of t he desired digital filter H[z]. iwT H[e iWT ] = 0.1357(e + I} eJwT - 0.7284 _ 0.1357( cos w T + 1 + j sin wT} - cos w T - 0.7284 + j sin w T (a) amplitude response (b) phase o z+ 1 ) z - 0.7284 Hence IH[eiwTl\ = (b) 7r X 1 0- 6 (12.59a) F s=10'6/pi; n um=[O 1 0'5];den=[1 1 0'5]; [ b,aj=bilinear(num,den,Fs) MATLAB r eturns b=O. 1358 0 .1358 a nd a =l - 0.7285. Therefore a nd H[z] (12.56a) T 0.1358(z + I} z - 0.7285 which agrees with t he answer found i n E xample 12.5. To plot t he a mplitude a nd t he p hase response, we can use t he l ast 8 functions in Example C12.1. 0 Similarly, starting with the power series for esT yields the transformation s = ~(z - 1) = (12.56b) These are strikingly simple transformations, which work reasonably well for lowpass and bandpass filters with low resonant frequencies. They cannot be used for highpass and bandstop filters, however, and they are inferior to bilinear transformation. The transformation in Eq. (12.56b) also has a stability problem. Frequency Prewarping Inherent in Bilinear Transformation F igure 1 2.12 s hows t hat IH[ejwTJ\ ~ IHa(jw)1 f or s mall w. F orlarge v alues o f w, t he e rror i ncreases. M oreover, IH [ejwTJ\ = 0 a t w = 7r IT. I n f act, i t a ppears a s i f t he 744 12 Frequency Response a nd D igital Filters 12.6 Recursive F ilter design: T...
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