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 ( )