上海交通大学：《数字信号处理 Digital Signal Processing（B）》教学资源_Handouts_ch11 IIR

Digital Signal Processing 主讲：张君 上洛文大

Digital Signal Processing 主讲：张君

IIR Filter Design Method Digital Signal Processing--IIR Digital Filter Design bilinear digital filter transformation analog filter specifications 2=g(o) specifications analog filter design method bilinear digital filter transformation analog filter H( s-fe) Ha(s) Butterworth Chebyshev Elliptic 上游充通大粤

Digital Signal Processing—— IIR Digital Filter Design IIR Filter Design Method digital filter specifications analog filter specifications analog filter Ha(s) digital filter H(z) analog filter design method bilinear transformation bilinear transformation Ω=g(ω) s=f(z) Butterworth Chebyshev Elliptic

Other Design Method Digital Signal Processing--IIR Digital Filter Design =2aey-H,ea min 上游充通大

Digital Signal Processing—— IIR Digital Filter Design Other Design Method [ ( ) ( ) ] min 2 1 ∑= − M i j d j i i H e H e ω ω ε =

Filter Design by Impulse Invariance G U Digital Signal Processing--IIR Digital Filter Design h(n)=ha(nT) Ha(s)=L[ha(t)] H(Z)=Z[h(n)] a(=之 4 i=1 s-Si h,()=之Aeu(t), u(t) i=l hn)）=ha(nT)=∑Aern)）=∑A(er)y”un) i=l i=l 上游充通大

Digital Signal Processing—— IIR Digital Filter Design Filter Design by Impulse Invariance h(n)=ha(nT) Ha(s)=L[ha(t)] H(z)=Z[h(n)] ∑= − = N i i i s s A Ha s 1 ( ) ∑= = N i s t a i h t Ae u t u t i 1 ( ) ( ), ( ) ∑ ∑ = = = = = N i N i s T n i s nT i h n ha nT Ae u n A e u n i i 1 1 ( ) ( ) ( ) ( ) ( )

Filter Design by Impulse Invariance Digital Signal Processing--IIR Digital Filter Design H(e)=之Aer:"=∑4∑(e:'y° n=0i=1 i= n=0 1-(erz1) (ex2)=0, 1-ez1 k-→0 e-1-。 ao=立+w 上游充通大￥

Digital Signal Processing—— IIR Digital Filter Design Filter Design by Impulse Invariance ∑∑ ∑ ∑ ∞ = = ∞ = − = − = = 0 1 0 1 1 ( ) ( ) n N i n s T n N i i s nT n i H z Ae z A e z i i − →∞ − − − s T k s T k e z e z i i 1 1 1 1 ( ) ( ) 0, 1 =∞ = − k s T k e z i ∑ = − − = N i s T i e z A H z i 1 1 1 ( ) ∑ ∞ =−∞       = + m a a m T H s j T H s 1 2π ( ) ˆ

Filter Design by Impulse Invariance Digital Signal Processing--IIR Digital Filter Design (s)=[[h (t)>(t-nT)le-"dt n=-00 =∑h.u)δt-nI)ed =∑h,(nT)ew7 H(a)=∑h(n)zn n=-00 ST s plane ~z plane z=e 上游充通大粤

Digital Signal Processing—— IIR Digital Filter Design Filter Design by Impulse Invariance ∑ ∑ ∫ ∫ ∑ ∞ =−∞ − ∞ =−∞ ∞ −∞ − ∞ −∞ − ∞ =−∞ = = − = − n nsT n st st n h nT e h t t nT e dt H s h t t nT e dt ( ) ( ) ( ) ( ) [ ( ) ( )] ˆ a a a a δ δ ∑ ∞ =−∞ − = n n H(z) h(n)z sT s plane ~z plane z = e

Filter Design by Impulse Invariance Digital Signal Processing--IIR Digital Filter Design 视 jIm() 3π π Re(z) Z plane S plane ⊙：一兀~亚洛充五大￥

Digital Signal Processing—— IIR Digital Filter Design Filter Design by Impulse Invariance jΩ 0 σ T −π T 3π T − 3π T π j Im(z) Re(z) 0 ω S plane Z plane ω : −π ~ π

Filter Design by Impulse Invariance Digital Signal Processing--IIR Digital Filter Design He=月.o-7立.(+m 2o+2 21 (s+o1)2+2 -o1±j21 e sin T 1-2z-ea!cos T+2e-2iT 上浒充通大￥

Digital Signal Processing—— IIR Digital Filter Design Filter Design by Impulse Invariance ( )       = ∑ + ∞ =−∞ T m H j j T H e a m j π ω ω 1 2 ( ) ∑ ∞ =−∞ =       = = + m z e m T H s j T H z ST H s 1 2π ( ) ˆ a a 1 1 1 1 2 2 1 1 1 1 1 1 1 2 2 1 , ( ) sin 1 2 cos T T T j s ze T ze T ze σ σ σ σ σ − − − − Ω − ±Ω + +Ω Ω − Ω+

Example (Filter Design by Impulse Invariance) Digital Signal Processing--IIR Digital Filter Design 2 1 1 H(S)= (s+1)(s+3)s+1S+3 1 1 H(z)= 1-z-e-T 1-2-e-3T 2-(e-r -e-37) =1-z'(ef+e3r)+e7z2 2 (2+1)(2+3)(3-2)+j42 (e-T-e-T)e-jo H()=H()1-(e 上游充重大￥

Digital Signal Processing—— IIR Digital Filter Design Example (Filter Design by Impulse Invariance) 3 1 1 1 ( 1)( 3) 2 ( ) + − + = + + = s s s s H s − Ω + Ω = Ω + Ω + Ω = = Ω = (3 ) 4 2 ( 1)( 3) 2 ( ) ( ) 2 j j j Ha j H s s j T T z e z e H z 1 1 3 1 1 1 1 ( ) − − − − − − − = 1 3 4 2 1 3 1 ( ) ( ) − − − − − − − − − + + − = z e e e z z e e T T T T T ω ω ω ω ω 3 4 2 3 1 ( ) ( ) ( ) ( ) T T j T j T T j z e j e e e e e e e e H e H z j − − − − − − − − = − + + − = =

Design Procedure Digital Signal Processing--IIR Digital Filter Design Given the digital lowpass filter wp,ws,Rp and As,we want to determine H(z)by designing an equivalent analog filter and mapping it into the desired digital filter. 1.Choose T and determine the analog frequencies: Qp=wp/T,s-ws/T 2.Design an analog filter Ha(s)using the specifications with one of the three prototypes of the previous section. Z。 3.Using partial fraction expansion,expand Ha(s)into 4.Now transform analog poles (pk}into digital poles {epkr}to obtain the digital filter e-21e&. R 上浒充通大￥

Design Procedure Digital Signal Processing—— IIR Digital Filter Design Given the digital lowpass filter wp,ws,Rp and As, we want to determine H(z) by designing an equivalent analog filter and mapping it into the desired digital filter. Design Procedure: 1. Choose T and determine the analog frequencies: Ωp=wp/T, Ωs=ws/T 2. Design an analog filter Ha(s) using the specifications with one of the three prototypes of the previous section. 3. Using partial fraction expansion, expand Ha(s) into 4. Now transform analog poles {pk} into digital poles {epkT} to obtain the digital filter ∑ = − = N k k k a s p R H s 1 ( ) ∑ = − − = N k p T k e z R H z 1 k 1 1 ( )