z-Transform From our earlier discussion on the uniform convergence of the DTFT, it follows that the series G(re0)=∑g noyon nre 1=-00 converges ifigin]r"is absolutely summable, 1.e < =-0 Copyright C 2001, S K. Mitra
Copyright © 2001, S. K. Mitra 6 z-Transform • From our earlier discussion on the uniform convergence of the DTFT, it follows that the series converges if is absolutely summable, i.e., if = =− − − n j n j n G(r e ) g[n]r e { [ ] } n g n r − =− − n n g[n]r
z-Transform In general, the roc of a z-transform of a sequence gIn is an annular region of the z ane <z<R+ Where0≤Ro<Ro+≤∞ ROC Copyright C 2001, S K. Mitra
Copyright © 2001, S. K. Mitra 7 z-Transform • In general, the ROC of a z-transform of a sequence g[n] is an annular region of the zplane: where − + Rg z Rg 0 Rg − Rg + ROC Rg − Rg +
Cauchy-Laurent Series The z-transform is a form of the Cauchy Laurent series and is an analytic function at every point in the roc Let f(a) denote an analytic(or holomorphic function over an annular region Q centered at z Im(=) Re(=) Copyright C 2001, S K Mitra
Copyright © 2001, S. K. Mitra 8 • The z-transform is a form of the CauchyLaurent series and is an analytic function at every point in the ROC • Let f (z) denote an analytic (or holomorphic) function over an annular region centered at Cauchy-Laurent Series o z Re( )z Im( )z o z
Cauchy-Laurent Series Then f()can be expressed as the bilateral series n=-00 where 2丌 ∮(=)(=-=0)m r being a closed and counterclockwise integration contour contained in Q Copyright C 2001, S K. Mitra
Copyright © 2001, S. K. Mitra 9 • Then f (z) can be expressed as the bilateral series being a closed and counterclockwise integration contour contained in Cauchy-Laurent Series ( 1) ( ) ( ) 1 ( )( ) 2 n n o n n n o f z z z f z z z dz j =− − + = − = − where
z-Transform Example -Determine the z-transform X(z) of the causal sequence xn]=a"[n] and its ROC NoWX(z)=∑am]zn=∑a" n=0 The above power series converges to X(=)= 1-0-1, for az<1 ROC is the annular region >a Copyright C 2001, S K Mitra
Copyright © 2001, S. K. Mitra 10 z-Transform • Example - Determine the z-transform X(z) of the causal sequence and its ROC • Now • The above power series converges to • ROC is the annular region |z| > || x[n] [n] n = = = = − =− − 0 ( ) [ ] n n n n n n X z n z z , for 1 1 1 ( ) 1 1 − = − − z z X z