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 Y(e0)=∑xmle Jon 1=-0 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 gn], its z-transform G(z) is defined as G(z)=∑g[n] 1=-00 where 2=re(z)+j Im(zEC 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(reJu) noyon ∑gne 1=-00 The above can be interpreted as the dtFt of the modified sequence gin]r") For r=l(i.e,==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 ∑g[r] 1=-0 For a given sequence, the set r of values of 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(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 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