华南理工大学电子与信息学院：《数字信号与处理》（英文版）Lecture 8 z-Transform

z-Transform The dtft provides a frequency-domain representation of discrete-time signals and lti discrete-time systems Because of the convergence condition, in many cases. the dtft of a sequence may not exist as a result. it is not possible to make use of such frequency-domain characterization in these cases Copyright C 2001, S K. Mitra

Copyright © 2001, S. K. Mitra 1 z-Transform • The DTFT provides a frequency-domain representation of discrete-time signals and LTI discrete-time systems • Because of the convergence condition, in many cases, the DTFT of a sequence may not exist • As a result, it is not possible to make use of such frequency-domain characterization in these cases

z-Transform a generalization of the dtft defined by (e)=∑x on nle n=-00 leads to the z-transform z-transform may exist for many sequences for which the dtft does not exist Moreover, use of z-transform techniques permits simple algebraic manipulations Copyright C 2001, S K. Mitra

Copyright © 2001, S. K. Mitra 2 z-Transform • A generalization of the DTFT defined by leads to the z-transform • z-transform may exist for many sequences for which the DTFT does not exist • Moreover, use of z-transform techniques permits simple algebraic manipulations =   =−  −  n j j n X (e ) x[n]e

z-Transform Consequently z-transform has become an important tool in the analysis and design of digital filters For a given sequence gln], its z-transform G(z)is defined as G()=∑gn]zn n=-0 Where z=Re(二)+jlm(二)∈ C is a complex variable Copyright C 2001, S K. Mitra

Copyright © 2001, S. K. Mitra 3 z-Transform • Consequently, z-transform has become an important tool in the analysis and design of digital filters • For a given sequence g[n], its z-transform G(z) is defined as where is a complex variable   =− − = n n G(z) g[n]z z z j z = +  Re( ) Im( )

z-Transform If we let z=re/o. then the z-transform reduces to G(re0)=∑g[ n]"e The above can be interpreted as the dtFt of the modified sequence gn]r Forr=l(i.e,z=1), z-transform reduces to its dtFT, provided the latter exists 4 Copyright C 2001, S K. Mitra

Copyright © 2001, S. K. Mitra 4 z-Transform • If we let , then the z-transform reduces to • The above can be interpreted as the DTFT of the modified sequence • For r = 1 (i.e., |z| = 1), z-transform reduces to its DTFT, provided the latter exists  = j z r e =   =−  − −  n j n j n G(r e ) g[n]r e { [ ] } n g n r −

z-Transform The contour z=l is a circle in the z-plane of unity radius and is called the unit circle Like the dtft. there are conditions on the convergence of the infinite series ∑8团m]zn n=-00 For a given sequence, the set r of values of z for which its z-transform converges is called the region of convergence ROC Copyright C 2001, S K. Mitra

Copyright © 2001, S. K. Mitra 5 z-Transform • The contour |z| = 1 is a circle in the z-plane of unity radius and is called the unit circle • Like the DTFT, there are conditions on the convergence of the infinite series • For a given sequence, the set R of values of z for which its z-transform converges is called the region of convergence (ROC)   =− − n n g[n]z

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 z￾plane: 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 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 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 aza 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