lecture_19 - MIT OpenCourseWare http://ocw.mit.edu 2.161...

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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 z-Transform (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 z-transform 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 s-plane poles and zeros must lie in the primary strip in the s-plane (that is | ( s ) | < /T ). Poles/zeros on the s-plane imaginary axis will map to the unit circle, and left-half s-plane 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 z-plane using z = e sT . 3. Form the z-plane transfer function with the transformed poles/zeros. 4. Determine the gain constant K by matching gains at some frequency (for a low-pass 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 z-transform method to design a filter based on the prototype first-order low-pass 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.

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lecture_19 - MIT OpenCourseWare http://ocw.mit.edu 2.161...

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