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