【正文】 的低通或帶通濾波器 )， 而且高頻衰減越快 ， 混疊效應(yīng)越小 。 至于高通和帶阻濾波器 ， 由于它們在高頻部分不衰減 ， 因此將完全混淆在低頻響應(yīng)中 。 如果要對高通和帶阻濾波器采用脈沖響應(yīng)不變法 ， 就必須先對高通和帶阻濾波器加一保護(hù)濾波器 ， 濾掉高于折疊頻率以上的頻率 ， 然后再使用脈沖響應(yīng)不變法轉(zhuǎn)換為數(shù)字濾波器 。 當(dāng)然這樣會進(jìn)一步增加設(shè)計復(fù)雜性和濾波器的階數(shù) 。 Bilinear Transformation The technique discussed in this subsection avoids the problem of aliasing by using the bilinear transformation , an algebraic transformation between the variables s and z that maps the entire axis in the splane to one revolution of the unit circle in the zplane. With denoting the continuoustime system function and H (z) the discretetime system function, the bilinear transformation corresponds to replacing s by that is, ? ?cHs1121 , ( 7 . 2 0 )1dzsTz???????????? ?1121 , ( 7 . 2 1 )1c dzH z HTz?????????????????To develop the properties of the algebraic transformation specified in Eq.(), we solve for z to obtain ? ?? ?1 / 2 , ( 7 .2 2 )1 / 2ddTszTs???and , substituting into Eq.() ,we obtain 1 / 2 / 2 , ( 7 . 2 3 )1 / 2 / 2ddT j TzT j T??? ? ??? ? ?sj?? ? ?If then , from Eq.(), it follows that ∣ z∣ 1 for any values of . Similarly , if , then ∣ z∣ 1 for all . Next , to show that the axis of the splane maps onto the unit circle, we substitute into Eq.(), obtaining From Eq.() , it is clear that ∣ z∣ =1 for all values of s on the axis . That is , the axis maps onto the unit circle , so Eq.() takes the form To derive a relationship between the variables and , it is useful to return to Eq.() and substitute . We obtain 0??0??? ?j?sj? ? ?1 / 2 , ( 7 .2 4 )1 / 2ddjTzjT?????j?j?1 / 2 , ( 7 . 2 5 )1 / 2j ddjTejT? ???????jze ??21 , ( 7 . 2 6 )1jjdesTe????????????? or, equivalently, Equating real and imaginary parts on both sides of Eq.() leads to the relations and or ? ?? ? ? ?/2/22 s in / 222 ta n / 2 , 7 . 2 7 )2 c o s / 2jjddej jsjT e T??????????? ? ? ? ?????? ?2 ta n / 2 , ( 7 . 2 8 )dT???? ?2 a r c ta n / 2 , ( 7 .2 9 )dT? ??0? ?these properties of the bilinear transformation as a mapping from the splane to the zplane are summarized in Figures and . from Eq.() and Figure , we see that the range of frequencies maps to while the range maps to . The bilinear transformation avoids the problem of aliasing encountered with the use of impulse invariance, because it maps the entire. 0 ???? 0 ????0? ? ? ? ? 0??? ? ?imaginary axis of the splane onto the unit circle in the zplane. The price paid for this, however, is the nonlinear pression of the frequency axis depicted in Figure Consequently, the design of discretetime filters using the bilinear transformation is useful only when this pression can be tolerated or pensated for, as in the case of filters that approximate ideal piecewiseconstant magnituderesponse characteristics. This is illustrated in Figure . where we show how a continuoustime frequency response and tolerance scheme maps to a corresponding discretetime frequency response and tolerance scheme through the frequency warping of Eqs.() and (). Typical frequencyselective continuoustime approximations are Butterworth , Chebyshev, and elliptic filters. As discussed in Appendix B a Butterworth continuoustime filter is monotonic in the passband and in the stopband . A type I Chebyshev filter has an equiripple characteristic in the passband and monotonically in the stopband. A type II Chebyshev filter is monotonic in the passband and equiripple in the stopband. An elliptic filter is equiripple in both the passband and the stopband. Although the bilinear transform can be used effectively in mapping a piecewiseconstant magnituderesponse characteristic from the splane to the zplane , the distortion in the frequency axis also manifests itself as a warping of the phase response of the filter. For example, Figure shows the result of applying the bilinear transformation to an ideal linear phase factor . If we substitute Eq.() for s and evaluate the result on the unit circle , the phase angle is . In , the solid curve shows the function , which is obtained by using the small angle approximation . sae?? ? ? ?2 / ta n / 2dT???? ? ? ?2 / ta n / 2dT???? ?/ 2 ta n / 2??? Mapping of the splane using the bilinear transform. Mapping of the continuoustime frequency axis onto the discretetime frequency axis by bilinear transformation. Figure Frequency warping inherent in the bilinear transformation of a continuoustime lowpass filter into a discretetime lowpass filter. To achieve the desired discretetime cutoff frequencies , the continuoustime cutoff frequencies must be prewarped as indicated. Figure lllustration of the effect of the bilinear transformation on a linear phase characteristic. Examples of Bilinear Transformation Design Example Bilinear Transformation of a Butterworth Filter Consider the discretetime filter specifications of Example , in which we illustrated the impulse invariance technique for the design of a discretetime filter. The specifications on the discretetime filter are In carrying out the design using the bilinear transformation , the critical frequencies of the discretetime filter must be prewarped to the corresponding continuoustime frequencies using Eq.() ? ?? ?0. 89 12 5 1 , 0 0. 2 , ( 7. 30 )0. 17 78 3 , 0. 3 , ( 7. 30 )jjH