transformation from analog filter to digital filter H(S)HZ), viz. mapping from s plane to Z plane, must satisfy the same frequency response( map imaginary axis to the unit circle causality and stabil ity is preserved map poles from left half plane to the inside circle) S plane→ Z plane Imaginary axIs → the unit circle Ha(s) poles in left half plane >inside the unit circle
H (s) a H(z) imaginary axis →the unit circle poles in left half plane →inside the unit circle S plane → Z plane transformation from analog filter to digital filter: H(s)→H(z)，viz. mapping from S plane to Z plane, must satisfy: the same frequency response（map imaginary axis to the unit circle）； causality and stability is preserved（map poles from left half plane to the inside circle）
7.1.1 filter design by impulse invariance according to hn]=Tchinldi H(s)→h(t)-hmmm,hn→>H(z) conversion formula H(s)=∑ k TA H(z)=H2(s)1 I-eskld
7.1.1 filter design by impulse invariance ( ) ( ) [ ] ( ) [ ] [ ] [ ] [ ] H s h t h n H z h n T h nT h n Td hc n Td c c d c d → ⎯⎯⎯⎯⎯→ → = = according to ： − = − − − − − = − = = − = − 1 0 1 1 1 1 0 1 ( ) ( ) ( ) 1 N k s T d k e z T s s c N k k k c e z T A H z H s s s A H s k d s k Td d k conversion formula：
Relationship between poles causal and stable): S et relation between frequencies Q=97,-x<O<丌,-0<g2< s plane Z plane 3π/Td
-3π/Td π/Td 3π/Td jΩ -π/Td = Td ,− ,− relation between frequencies: S plane Z plane Relationship between poles（causal and stable): k Td s k s → e
relation between frequency response: when aliasing is small, the frequency response is the same H(e10)=〉H2(g+j H 2丌 strongpoint: linear frequency mapping shortcoming: aliasing in frequency response restriction in application: cannot used in high-pass and bandstop filter
relation between frequency response: when aliasing is small, the frequency response is the same. Td d k c j T H e H j jk = =− = + ) 2 ( ) ( strongpoint：linear frequency mapping; shortcoming：aliasing in frequency response。 restriction in application：can not used in high-pass and bandstop filter
Design steps: Equivalent analog specificat ion-p-32pl, s-s>, digital specificat ion I digital specificat ion =0n/7a,9s=0s/7 i> prototype analog specihcat ion (2)design Hc(s) (3) H()=H(s) SkId about td independent of T do not influence aliasing arbitrary value, generally, take 1(attention (1) and (3) have the same value
(1)[equivalent analog specificat ion ,digital specificat ion ] ' , ' ⎯⎯⎯⎯⎯⎯→ p = p T S = S T digital specificat ion prototype analog specificat ion / , / ⎯⎯⎯⎯⎯⎯⎯→ p = p Td S =S Td (2)design H (s) c Design steps： about Td: independent of T; do not influence aliasing; arbitrary value，generally, take 1（attention（1）and（3）have the same value）. 1 1 1 ( ) ( ) − − − − = e z T s s c s k Td d k （3） H z H s