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Unformatted text preview: Ve216 Lecture Notes Dianguang Ma Spring 2009 Chapter 7 (Part I) Representing Signals by Using DiscreteTime Complex Exponentials: The zTransform 7.1 Introduction The ztransform is a generalization of the DTFT. It provides a broader characterization of discretetime LTI systems and their interaction with signals than is possible with the DTFT. The primary roles of the ztransform in engineering practice are the study of system characteristics and the derivation of computational structures for implementing discretetime systems on computers. The unilateral ztransform is also used to solve difference equations subject to initial conditions. 7.2 The zTransform Consider applying a complex exponential input of the form x[n]= z n to an LTI system with impulse response h[n]. The system output is given by [ ] [ ]* [ ] [ ] [ ] [ ] [ ] ( ) ( ) [ ] : transfer function k n k n k k k n k k y n h n x n h k x n k h k z z h k z H z z H z h k z =  = =  = = = = = = @ 7.2 The zTransform Given the simplicity of describing the action of the system on inputs of the form z n , we now seek a representation of arbitrary signals as a weighted superposition of eigenfunction z n . Recall that an eigenfunction is a signal that passes through the system without being modified except for multiplication by a scalar. Hence, we identify z n as an eigenfunction of the LTI system and H(z) as the corresponding eigenvalue. n z z H n y ) ( ] [ = 7.2 The zTransform The transfer function H(z) is the ztransform of h[n], or the DTFT of h[n]rn . ( 29 { } n n n j n n n j n n r n h DTFT e r n h re n h z n h z H  =   =  = = = = = ] [ ] [ ) ]( [ ] [ ) ( We may recover h[n] by the inverse ztransform of H(z).   = = = dz z z H j d re re H n h d e re H r n h n n j j n j j n 1 2 2 ) ( 2 1 ) )( ( 2 1 ] [ ) ( 2 1 ] [ 7.2 The zTransform For an arbitrary signal x[n], the ztransform of x[n] is  = = n n z n x z X ] [ ) ( And the inverse ztransform of X(z) is We express this relationship with the notation  = dz z z X j n x n 1 ) ( 2 1 ] [ ) ( ] [ z X n x z 7.2 The zTransform =   = = = = j e z j n n n z X e X r n x DTFT z n x z X ) ( ) ( } ] [ { ] [ ) ( The ztransform of x[n] can be interpreted as the DTFT of x[n] after multiplication by a real exponential signal rn . Hence, a necessary condition for convergence of the z transform is the absolute summability of x[n]rn . That is, we must have < =  =  = n n n n r n x r n x ] [ ] [ 7.2 The zTransform The range of r for which the ztransform converges is termed the region of convergence (ROC)....
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 Spring '09
 DianguangMa

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