Inverse z-Transform Chapter 6B Part B ◆Inverse z.Transform ●●● ●●● ●●● z-Transform Properties Inverse z-Transform z-Transform 09 and ZT Properties 3 1.Inverse z-Transform 1.1 General Expression 1.1 General Expression ●Recall that,,for=red“,thez-transform G(2) ◆By making a change of variable z=re“,the 1.1 General Expression given by previous equation can be converted into a 1.2 Inverse z-Transform by Partial-Fraction Expansion ()=G(re)=g(n)r"e contour integral given by 1.3 Partial-Fraction Using MATLAB gm=2[G2]=2 G(=)2"d is merely the DTFT of the modified sequence 1.4 Inverse z-Transform via Long Division g(n)rn where c is a counterclockwise contour of 1.5 Inverse z-Transform Using MATLAB integration defined by=r Accordingly,the inverse DTFT is thus given But the integral remains unchanged when c is replaced with any contour c'encircling the point z=0 in the ROC of G(z) 4 5Chapter 6B z-Transform Part B Inverse z-Transform and ZT Properties 3 Inverse z-Transform Inverse z Inverse z-Transform z-Transform Properties Transform Properties 4 1. Inverse z-Transform 1.1 General Expression 1.2 Inverse z-Transform by Partial Transform by Partial-Fraction Expansion 1.3 Partial 1.3 Partial-Fraction Using MATLAB 1.4 Inverse z-Transform via Long Division 1.5 Inverse z-Transform Using MATLAB 5 1.1 General Expression Recall that, for z= rej¹, the z-transform G(z) given by is merely the DTFT of the modified sequence g(n)r-n Accordingly, the inverse DTFT is thus given by - () ( ) () j n jn n G z G re g n r e       1 () ( ) 2 n j jn g n r G re e d        6 1.1 General Expression By making a change of variable z= rej¹ , the previous equation can be converted into a contour integral given by where c is a counterclockwise contour of integration defined by |z|= r But the integral remains unchanged when c is replaced with any contour c’ encircling the point z=0 in the ROC of G(z) 1 1 1 ( ) [ ( )] ( ) 2 n c g n G z G z z dz  j