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ch6_ztrans

# ch6_ztrans - MIT OpenCourseWare http/ocw.mit.edu HST.582J...

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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 . Harvard-MIT 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- Z-TRANSFORMS 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 discrete-time Fourier transform to determine the fre- quency response of linear, time-invariant systems. The limitation of that approach is that the DTFT only exists if the signal/unit-sample response is absolutely summable or contains finite energy. The Z-transform is a generalization of the DTFT and applies to signals/unit-sample responses that do not meet either of these criteria. Therefore, the Z-transform 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 Z-transform can be used to characterize the response of linear, time-invariant filters to complex exponential signals. Specifically, consider the response of a filter with unit-sample 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, time-invariant system h [ n ]. When considered as a function of z , the eigenvalue H ( z ) is the Z-transform 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 unit-sample 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 Z-transform, 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 Z-transforms of filters defined by a difference equation Consider the class of digital filters described by linear, constant-coeﬃcient 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 Z-transform of digital filters defined by this equation follows directly from the...
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ch6_ztrans - MIT OpenCourseWare http/ocw.mit.edu HST.582J...

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