1.2 Rational z-Transform 1.2 Rational z-Transform 1.2 Rational z-Transform At a root z=5 of the numerator polynomial, Consider:G()=zwn凸ΠH(e-s) .A physical interpretation of the concepts of G()=0,and as a result,these values of z poles and zeros can be given by plotting the are known as the zeros of G(z) .Note G(z)has Mfinite zeros and N finite log-magnitude 20logG()as shown on next slide for .At a root 2=乙，of the denominator poles polynomial,G(z)=0,and as a result,these .If N>Mthere are additional N-Mzeros at z G()= 1-2.4z+2.88z-2 values of z are known as the poles of G(z) =0 (the origin in the z-plane) 1-0.8z+0.64z2 If N<Mthere are additional M-N poles atz =0 2.Region of Convergence of a 2.Region of Convergence of a 1.2 Rational z-Transform Rational z-Transform Rational z-Transform Two poles at 30.4±j0.6928 ROC of a z-transform is an important concept. Moreover.if the ROC of a z-transform includes the unit circle,the DTFT of the Without the knowledge of the ROC.there is no sequence is obtained by simply evaluating the unique relationship between a sequence and its z-transform on the unit circle. z-transform. .There is a relationship between the ROC of the Hence.the z-transform must always be z-transform of the impulse response of a causal specified with its ROC. LTI discrete-time system and its BIBO Two zeros at stability. 2-1.2±12 2 1025 1. 2 Rational z-Transform At a root of the numerator polynomial, , and as a result, these values of z are known as the zeros of G(z) At a root of the denominator polynomial, , and as a result, these values of z are known as the poles of G(z) l z  () 0 G l  () 0 G zl  l z   26 1. 2 Rational z-Transform Consider: Note G(z) has M finite zeros and N finite poles If N > M there are additional NˉM zeros at z = 0 (the origin in the z-plane) If N < M there are additional MˉN poles at z = 0     ( ) 0 1 0 1 ( ) M N M l l N l l p z Gz z d z          27 1. 2 Rational z-Transform A physical interpretation of the concepts of poles and zeros can be given by plotting the log-magnitude 20log10|G(z)| as shown on next slide for 1 2 1 2 1 2.4 2.88 ( ) 1 0.8 0.64 z z G z z z          28 1. 2 Rational z-Transform Two poles at z=0.4fj0.6928 Two zeros at z=1.2fj1.2 29 2. Region of Convergence of a Rational z-Transform ROC of a z-transform is an important concept. Without the knowledge of the ROC, there is no unique relationship between a sequence and its z-transform. Hence, the z-transform must always be specified with its ROC. 30 2. Region of Convergence of a Rational z-Transform Moreover, if the ROC of a z-transform includes the unit circle, the DTFT of the sequence is obtained by simply evaluating the z-transform on the unit circle. There is a relationship between the ROC of the z-transform of the impulse response of a causal LTI discrete-time system and its BIBO stability.