z-Transform From our earlier discussion on the uniform convergence of the dtft, it follows that the series G(re)=∑g[ e on converges ifigIn]r is absolutely summable. i.e. if g n<∞ 1=-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 g[n] is an annular region of the z plane R。-<z< Rg where o≤Rg<R+≤o 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(z)denote an analytic(or holomorphic) function over an annular region Q centered at z O 8 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(z)can be expressed as the bilateral series f()=∑an(x-= 1=-00 where f(=)(z-=0)(n 2丌j 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"un] and its ROC NowX(=)=∑am]z=∑a"z n=-0 0 The above power series converges to X(z)= for az<1 1-az ROC is the annular region z> al 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