13.1 Analog/Digital Lowpass Butterworth Filter

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1 CHAPTER 3 IIR FILTER DESIGN 3. Analog/Digial Lowpass Buerworh Filer This docuen designs a lowpass digial IIR filer of he Buerworh ype. A bilinear ransforaion is perfored o creae a digial filer fro he analog design. You specify: T, he sapling rae fp, he passband edge frequency fs, he sopband edge frequency G, he iniu sopband aenuaion Mahcad calculaes he required filer order and consrucs he ransfer funcion in he s-plane. This docuen prewarps he analog filer frequencies, and subsequenly gives he digial ransfer funcion. Filer oupu is displayed for soe siple inpu sequences in wo ways: by direc convoluion wih he runcaed ipulse response, and by sequence generaion using a difference equaion. Bacground Digial filers are ofen designed using analog echniques, and hen convering he analog ransfer funcion ino digial ers. This approach was adoped because analog filer design is a well-undersood and refined science, and in soe cases easier o ipleen. Bilinear Transforaion There are a nuber of ehods for convering funcions fro he coninuous-ie Laplace Transfor doain o he discree-ie z-ransfor doain. One of he os universal eploys a bilinear ransforaion beween he s-plane and he z-plane: s= 2 T z + z where T is he sapling rae.

2 Frequency and Phase Warping The coninuous-ie frequency axis is apped ono he uni circle in he z-plane. Since he ransforaion is, necessarily, nonlinear, here will be soe "warping" of he frequencies as hey conver fro he analog o digial doain. I is no so criical ha soe frequencies are unevenly disribued, excep in he case of he design frequencies a he band edges. I is possible o "prewarp" hese frequencies so he digial filer sill ees design consrains, using he relaion = ω s 2 T an ω z 2 where z varies beween and 2. The bilinear ransforaion also warps he phase of he filer, paricularly a he edges of he frequency range. The phase characerisic of all filers designed by his echnique is nonlinear. Mahcad Ipleenaion The filer design depends on four paraeers: he sapling inerval, passband and sopband edges, and he desired aenuaion in he sop band. Lowpass Filer Specificaions sapling inerval (sec): iplies a bandlii (Hz) of: passband edge (Hz): sopband edge (Hz): aenuaion for frequencies above he sopband edge: T. s f ax = 5 Hz 2 T f p 9 Hz f s Hz G.2 To prepare for analog filer design, prewarp he passband and sopband frequencies, and noralize he frequency range so ha he passband edge is a (see Pars and Burrus).

3 Buerworh Filer Design Prewarp frequencies for he analog filer: ωπ π 2 π f p = Hz ωσ σ 2 π f s = Hz u p 2 = T an ωπ π T Hz u s 2 = 2 T an ωσ σ T Hz 2 Frequency noralizaion: u u s =.8 u p The docuen hen calculaes he approxiae order for a Buerworh-ype filer ha will ee he specificaions, and sores he poles of he ransfer funcion in he array p. log 2 N ceil = G 5 2 log(u) If he filer does no ee specs, redefine N here for a higher order filer : N= 5 N p sin π ( 2 + ) + j cos π ( 2 + ) 2 N 2 N The analog ransfer funcion is: H s (s) s p x.5, H s ( j x) x Fig. 3. Chec he noralized frequency response of he analog filer

4 Digial Filer Transforaion Find he corresponding digial ransfer funcion using a bilinear ransforaion. q an ωπ π T 2 H(z) H s q z z + A plo of he digial frequency filer response shows ha we have correcly e our sopband and passband requireens. f Hz,. T 2 T H(exp( j 2 π f T)) f (Hz) f p (Hz) f s (Hz) The N zeros of he ransfer funcion are all a z = -. The N poles are given by: q+ p a q p Fig. 3.2 Digial frequency response in herz Ia I j 4 π f T e a Re j 4 π f T e

5 Transfor Expansion in Parial Fracions The previous plo shows he locaion of he poles of Hz inside he uni circle. These poles are in conjugae pairs, wih one real pole if he filer lengh is odd. The denoinaor can be wrien as a produc of quadraic facors of he for z2 + bz + c. If N is odd, here is one linear facor of he for z - blin. The coefficiens b and c are given by: odd od ( N, 2) > ax if odd, floor, N N 2 2 ax b 2 Rea 2 c a b ax ifodd, a.599 b = N 2, 2 Rea ax c ax if odd,, a ax c = When he ransfer funcion is expressed as a raional funcion in z, he nueraor is M ( z+ ) N where M H s (q) = j 25 Filer Ipulse Response To generae he ipulse response for he digial filer, expand H in parial fracions. The coefficiens for his expansion can be convenienly generaed in Mahcad by carrying ou a conour inegral which finds he residue a each pole. The plo of he ipulse response below shows he firs 5 ers, calculaed fro he previously generaed poles and parial fracion coefficiens. r. K 2 π Hr exp( j θ) + a r exp ( j θ) dθ 2 π H expanded in parial fracions is: H(z) =M + K z a The coefficiens K are:.4+.79i K =

6 For ore inforaion on parial fracion expansion, see Secion. z-transfor and Inverse Transfor The beginning of he infinie ipulse response sequence is given by: 8 h K a h M h h Taing real values since iaginary pars are negligible (approxiaion errors). Fig. 3.4 Infinie ipulse response of filer Design Chec To calculae he filer oupu for a sine-wave inpu, convolve he inpu signals wih he firs 5 ers of he ipulse response. Two inpu signals wih differen frequencies are exained. The low-frequency inpu x is no aenuaed, whereas he higher-frequency inpu x2 is, as we would expec. Also, noice he phase shifing, as discussed in he Bacground secion. 8 Define low and high frequency inpus and heir oupu convoluion. Noe: Frequencies above he band edge should no be used in his calculaion: hey will no be adequaely sapled.

7 ω 2 π T.9 f p x () sin ω y ω 2 2 π T. f s x( ) h x2 () sin ω 2 y2 x2( ) h Inpu signals Oupus x() y x2() y