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Unformatted text preview: MIT OpenCourseWare http://ocw.mit.edu 2.161 Signal Processing: Continuous and Discrete Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms . j p / T j p / T o o x x x x o o o o x x s  p l a n e z  p l a n e s j W { z } { z } p r i m a r y s t r i p Massachusetts Institute of Technology Department of Mechanical Engineering 2.161 Signal Processing Continuous and Discrete Fall Term 2008 Lecture 19 1 Reading: Proakis and Manolakis: Sec. 10.3.3 Oppenheim, Schafer, and Buck: Sec. 7.1 1 The Design of IIR Filters (continued) 1.1 Design by the Matched zTransform (Root Matching) Given a prototype continuous filter H p ( s ), M H p ( s ) = K k =1 ( s z k ) N ( s p k ) k =1 with zeros z k , poles p k , and gain K , the matched ztransform method approximates the ideal mapping sT H p ( s ) H ( z )  z =e by mapping the poles and zeros M z k T ) H ( z ) = K k =1 ( z e N ( z e p k T ) k =1 where K must be determined from some empirical response comparison between the pro totype and digital filters. Note that an implicit assumption is that all splane poles and zeros must lie in the primary strip in the splane (that is  ( s )  < /T ). Poles/zeros on the splane imaginary axis will map to the unit circle, and lefthalf splane poles and zeros will map to the interior of the unit circle, preserving stability. 1 copyright c D.Rowell 2008 191 The steps in the design procedure are: 1. Determine the poles and zeros of the prototype filter H p ( s ). 2. Map the poles and zeros to the zplane using z = e sT . 3. Form the zplane transfer function with the transformed poles/zeros. 4. Determine the gain constant K by matching gains at some frequency (for a lowpass filter this is normally the low frequency response). 5. Add poles or zeros at z = to adjust the delay of the filter (while maintaining causal ity). Example 1 Use the matched ztransform method to design a filter based on the prototype firstorder lowpass filter a H p ( s ) = ....
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This note was uploaded on 02/27/2012 for the course MECHANICAL 2.161 taught by Professor Derekrowell during the Fall '08 term at MIT.
 Fall '08
 DerekRowell

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