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Unformatted text preview: MIT OpenCourseWare http://ocw.mit.edu HST.582J / 6.555J / 16.456J Biomedical Signal and Image Processing Spring 2007 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms . HarvardMIT Division of Health Sciences and Technology HST.582J: Biomedical Signal and Image Processing, Spring 2007 Course Director: Dr. Julie Greenberg HST582J/6.555J/16.456J Biomedical Signal and Image Processing Spring 2007 Chapter 6 ZTRANSFORMS c Bertrand Delgutte and Julie Greenberg, 1999 Introduction Chapters 2 and 3 considered the design and analysis of digital filters in the time and frequency domains and illustrated the use of the discretetime Fourier transform to determine the fre quency response of linear, timeinvariant systems. The limitation of that approach is that the DTFT only exists if the signal/unitsample response is absolutely summable or contains finite energy. The Ztransform is a generalization of the DTFT and applies to signals/unitsample responses that do not meet either of these criteria. Therefore, the Ztransform is a useful tool for investigating issues related to stability, including analysis of feedback systems. 6.1 Definition and properties 6.1.1 Definition The Ztransform can be used to characterize the response of linear, timeinvariant filters to complex exponential signals. Specifically, consider the response of a filter with unitsample response h [ n ] to the complex exponential z n , where z is an arbitrary complex number: ∞ y [ n ] = h [ n ] ∗ z n = h [ m ] z n − m m = −∞ This can be written as: ∞ y [ n ] = z n h [ m ] z − m = x [ n ] H ( z ) m = −∞ with ∞ − n H ( z ) = h [ n ] z (6.1) n = −∞ Thus, the output is equal to the input multiplied by the complex constant H ( z ). The complex n exponential signal z is said to be an eigenfunction of the linear, timeinvariant system h [ n ]. When considered as a function of z , the eigenvalue H ( z ) is the Ztransform of h [ n ]. Its argument is the complex variable z which defines a plane, with the real part and the imaginary part as orthogonal coordinates. For LTI systems described by a unitsample response h [ n ], H ( z ) is also referred to as the system function . The system function is a generalization of the frequency response, since the DTFT is a special case of the Ztransform, that is, H ( f ) = H ( z )  j 2 πf z = e Cite as: Julie Greenberg, and Bertrand Delgutte. Course materials for HST.582J / 6.555J / 16.456J, Biomedical Signal and Image Processing, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]. 6.1.2 Ztransforms of filters defined by a difference equation Consider the class of digital filters described by linear, constantcoeﬃcient difference equations (LCCDEs) of the form K M y [ n ] = a k y [ n − k ] + b m x [ n − m ] (6.2) k =1 m =0 The Ztransform of digital filters defined by this equation follows directly from the...
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 Spring '07
 JulieGreenberg
 Image processing, Digital Signal Processing, Complex number, Infinite impulse response, Bertrand Delgutte

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