1.1 Definition of z-Transform 1.1 Definition of z-Transform 1.1 Definition of z-Transform From our earlier discussion on the uniform In general,the ROC R of a z-transform of a An example of ROC ,m母 convergence of the DTFT,it follows that the sequence g(n)is an annular region of the z- Z-plane series Gre")=g(nr"e plane: R.<R converges if g(n)"is absolutely summable, where0≤R-<R,≤o ·ReH i.e.,if Elscrk Note:The z-transform is a form of a Laurent series and is an analytic function at every point in the ROC. The gray region unit cirele represents the ROC 1.1 Definition of z-Transform 1.1 Definition of z-Transform 1.1 Definition of z-Transform Comments 3 If R.<R-,then the ROC is a null space 1 The complex variable z is called the complex and the ZT does not exist. Therefore the discrete-time Fourier transform frequency given by z=re",where r is the G(")may be viewed as a special case of the ④The function=l(or=ed“)is a circle of unit attenuation and is the real frequency. z-transform G(z). radius in the z-plane and is called the unit 2 Since the ROC is defined in terms of the circle.If the ROC contains the unit circle. 5 If g(n)=h(n)is the impulse response of some magnitude r.the shape of the ROC is an then we can evaluate G(z)on the unit circle. system,its z-transform G()=H(z)is called as annulus.Note that R.may be equal to 0 System Function or Transfer Function of this and/or Rcould possibly be infinity. G()l=G(e)=g(me system. 113 1. 1 Definition of 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 g n z ( ) () j n jn n G re g n r e      - ( ) n n gnr     14 1. 1 Definition of z-Transform In general, the ROC of a z-transform of a sequence g(n) is an annular region of the z￾plane: where Note: The z-transform is a form of a Laurent series and is an analytic function at every point in the ROC. g g R zR     0 g g  R R   15 g R  g R  1. 1 Definition of z-Transform An example of ROC unit circle 1 z-plane Re{z} jIm{z} 0 The gray region represents the ROC 16 1. 1 Definition of z-Transform Comments ķ The complex variable z is called the complex frequency given by z= rej¹, where r is the attenuation and ¹ is the real frequency. real frequency. ĸ Since the ROC is defined in terms of the magnitude r, the shape of the ROC is an annulus. Note that may be equal to 0 and/or could possibly be infinity. g R  g R  17 1. 1 Definition of z-Transform Ĺ If , then the ROC is a null space and the ZT does not exist. ĺ The function r=1 (or z= ej¹) is a circle of unit radius in the z-plane and is called the unit circle. If the ROC contains the unit circle contains the unit circle, then we can evaluate G(z) on the unit circle. g g R R    - - ( )| ( ) ( ) j j jn z e n Gz Ge g n e       18 1. 1 Definition of z-Transform Therefore the discrete-time Fourier transform G(ej¹) may be viewed as a special case of the z-transform G(z). Ļ If g(n)=h(n) is the impulse response of some system, its z-transform G(z)=H(z) is called as System Function or Transfer Function of this system.