2.Bilinear Transform Method 2.Bilinear Transform Method 2.Bilinear Transform Method 2)Employ impulse invariance method to s'-plane with One-to-one mapping from s to s' mapping 2=47 n'=m 1 s-plane z-plane mapping 77 Derivation of the bilinear transform: 1)One-to-one mapping froms tos'which compresses the entire s-plane into the strip -T<Im(s)<T s-plane s'plane -17 9 2.Bilinear Transform Method 2.Bilinear Transform Method 2.Bilinear Transform Method The normalized frequency now The desired transformation from s to z(via s) The bilinear transform:s= 21-21 corresponds toΩ"T T1+2 @=2 tan- @=2tan- 2T) 2 子m The s-plane transfer function H(s)gives a .Thus,the entire j-axis is compressed to the ●As we know e-e 1-e2 plane transfer function interval (-for in a one-to-one /tanx= ete-Ite2 G()=H(s儿2- manner ·Solving z gives: T14 ·Hence The mapping is highly nonlinear 2 0）_21-em .However,for small ='T it is =j片ta2)厂T1+e网 21-2 /-到 approximately linear 。Lets=jQ andz=e“,we can arrive ats= The bilinear transform T1+2-1 23 2419 2. Bilinear Transform Method Derivation of the bilinear transform: Derivation of the bilinear transform: 1) One-to-one mapping from s to s’ which compresses the entire s-plane into the strip ˉ±/T < Im (s’) < ±/T  j 0 j z Im 1 Re z s-plane z-plane mapping 20 2. Bilinear Transform Method 2) Employ impulse invariance method to s’-plane with z=es’T  j 0 s-plane s’-plane  j 0  /T  /T mapping 21 2. Bilinear Transform Method 2 1 ' tan 2 T T     One-to-one mapping from s to s’ 0  /T  /T ' 22 2. Bilinear Transform Method The normalized frequency ¹ now corresponds to ’T Thus, the entire j-axis is compressed to the interval (ˉ±,±) for¹ in a one-to-one manner The mapping is highly nonlinear However, for small ¹=’T it is approximately linear 1 2 tan 2 T      23 2. Bilinear Transform Method The desired transformation from s to z (via s’) As we know Hence Let s=j and z=ej¹ ,we can arrive at 1 2 tan 2 T      2 tan T 2      2 2 1 tan 1 jx jx jx jx jx jx ee e j x ee e   2 21 tan 2 1 j j e j j T Te        1 1 2 1 1 z s T z The bilinear transform  24 2. Bilinear Transform Method The bilinear transform: The bilinear transform: The s-plane transfer function Ha(s) gives a z￾plane transfer function Solving z gives: 1 1 2 1 1 () () z a s T z Gz H s  1 1 2 2 T T z s s         1 1 2 1 1 z s T z