1.Impulse Invariance Method 1.Impulse Invariance Method 1.Impulse Invariance Method Due to sampling the mapping is many-to-one Assume that H(s)has the form of .H(z)converges ife<1 or a>0,indicating The strips of length 2 /T are all mapped onto H(s)=4 that H (s)is stable the unit circle The corresponding signal in time-domain is Generalizing to higher order(N)analog .Only if h(r)is a band-limited signal,no alias transfer functions will occur h(1)=ST-(H(s)=Ae-u(r) H.=克4 Hence,this method is not suitable for ·By sampling h(i) s+ar highpass and bandstop filters design h(n)=h(nT)=Ae-"u(nT) h,0-∑Aen'ut0 H)=∑n=em 1-e2 →He)=月 1-er2可 1.Impulse Invariance Method 1.Impulse Invariance Method 2.Bilinear Transform Method Example 。Magnitude Response Definition- First Order Butterworth Filter Designed Using the To avoid aliasing,the mapping from s-plane Impulse Invariant Method (T=1) to z-plane should be one-to-one,i.e.,a single H.(s)= 8+1 +h.(t)=et)+H(z）= point in the s-plane should be mapped to a l-e2可 unique point in the z-plane and vice versa zero at z=0 1)The entire jn-axis should be mapped onto pole at z=1/e the unit circle 2)The entire left-half s-plane should be mapped inside the unit circle13 1. Impulse Invariance Method Due to sampling the mapping is many-to-one The strips of length 2±/T are all mapped onto the unit circle Only if ha(t) is a band-limited signal, no alias will occur Hence, this method is not suitable for highpass highpass and bandstop filters design 14 1. Impulse Invariance Method Assume that Ha(s) has the form of The corresponding signal in time-domain is By sampling ha(t) ( ) a A H s s   1 ( ) ST { ( )} ( ) t a a h t H s Ae u t  () ( ) ( ) nT a h n h nT Ae u nT  1 0 () () 1 n nT n T n n A H z hnz A e z e z        15 1. Impulse Invariance Method H(z) converges if or ¢>0, indicating that Ha(s) is stable Generalizing to higher order (N) analog transfer functions 1 T e    0 1 ( ) N k a k k A H s s    1 () () k N t a k k h t Ae ut   1 1 ( ) 1 N k T k A H z e z   16 1. Impulse Invariance Method Example First Order Butterworth Filter Designed Using the Impulse Invariant Method (T=1) zero at z=0 pole at z=1/e 1 ( ) 1 H s a s  () () t a h t e ut 1 1 1 ( ) 1 H z e z 17 1. Impulse Invariance Method Magnitude Response 18 2. Bilinear Transform Method Definition – To avoid aliasing, the mapping from s-plane to z-plane should be one-to-one, i.e., a single point in the s-plane should be mapped to a unique point in the z-plane and vice versa 1) The entire j-axis should be mapped onto the unit circle 2) The entire left-half s-plane should be mapped inside the unit circle