1.Definition 1.Definition 1.Definition The transfer function has Mzeros on the unit Example 2 Note:Poles farthest from z=0 have a circle at z=e,K=0.1.2.....M-1 A causal LTI IIR digital filter is described magnitude 0.74 ROC:>0.74 .There are poles atz=0 and a single pole at by a constant coefficient difference equation. 2=1 given by The pole at z=1 y(m）-x(n-1)-12x(n-2)+x(n-3) exactly cancels the +1.3)(n-1)-1.04y(n-2)+0.222y(n-3) zero at z=1.The Its transfer function is therefore given by ROC is the entire z-plane except z z1-12z2+2 H(e)=Y(e) X(②)1-13z1+1.0422-0.22223 2.Frequency Response from Transfer 2.Frequency Response from Transfer 2.Frequency Response from Transfer Function Function Function .If the ROC of the transfer function H(z) It is convenient to visualize the contributions includes the unit circle,then the frequency .For a stable rational transfer function in the form of the zero factor (-and the pole response H(e)of the LTI digital filter can be obtained simply as follows: He)=凸nΠe- factor (z)from the factored form of do I(2-A) the frequency response.The magnitude H(e)=H(2) function is given by The factored form of the frequency response For a real coefficient transfer function H(z), is given by it can be shown that He)=凸emwΠ(- Πae- H(=H(e)H'() do I(em- 2aΠe-a =H(e)H(e)=H(2)H(z) 0 aΠaem-人7 1. Definition The transfer function has M zeros on the unit circle at There are poles at z = 0 and a single pole at z = 1 2 / , 0,1, 2, , 1 j kM ze K M    The pole at z = 1 exactly cancels the zero at z = 1. The ROC is the entire z-plane except z = 0 -1 -0.5 0 0.5 1 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 7 Real Part Imaginary Part M=8 8 1. Definition Example 2 A causal LTI IIR digital filter is described by a constant coefficient difference equation, given by Its transfer function is therefore given by ( ) ( 1) 1.2 ( 2) ( 3) 1.3 ( 1) 1.04 ( 2) 0.222 ( 3) yn xn xn xn yn yn yn           1 23 12 3 ( ) 1.2 ( ) ( ) 1 1.3 1.04 0.222 Yz z z z H z Xz z z z           9 1. Definition Note: Poles farthest from z = 0 have a magnitude 0.74 ROC: z  0.74 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 Real Part Imaginary Part 10 2. Frequency Response from Transfer Function If the ROC of the transfer function H(z) includes the unit circle, then the frequency response of the LTI digital filter can be obtained simply as follows: For a real coefficient transfer function H(z) , it can be shown that ( ) () j j z e He Hz   ( ) j H e 2 * 1 () ()() ( ) ( ) () ( ) j j jj j j z e He He H e He He HzHz       11 2. Frequency Response from Transfer Function For a stable rational transfer function in the form The factored form of the frequency response is given by     0 ( ) 1 0 1 ( ) M N M k k N k k p z Hz z d z               ( ) 1 0 0 1 ( ) M j k k j j NM N j k k p e He e d e           12 2. Frequency Response from Transfer Function It is convenient to visualize the contributions of the zero factor zero factor and the pole factor from the factored form of the frequency response. The magnitude function is given by z k  ( ) 1 0 0 1 1 0 0 1 ( ) M j k k j j NM N j k k M j k k N j k k p e He e d e p e d e                     z  k