lecture_15

# Lecture_15 - MIT OpenCourseWare http/ocw.mit.edu 2.161 Signal Processing Continuous and Discrete Fall 2008 For information about citing these

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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 .

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Massachusetts Institute of Technology Department of Mechanical Engineering 2.161 Signal Processing - Continuous and Discrete Fall Term 2008 Lecture 15 1 Reading: Proakis & Manolakis, Ch. 7 Oppenheim, Schafer & Buck. Ch. 10 Cartinhour, Chs. 6 & 9 1 Frequency Response and Poles and Zeros As we did for the continuous case, factor the discrete-time transfer functions into as set of poles and zeros; b 0 z 0 + b 1 z 2 + + b M z M M i =1 ( z z i ) H ( z ) = ··· = K a 0 z 0 + a 1 z 2 + + a N z N N i =1 ( z p i ) where Z i are the system zeros , the p i are the system poles , and K = b 0 /a 0 is the overall gain . We note, as in the continuous case that the polse and zeros must be either real, or appear in complex conjugate pairs. As in the continuous case, we can draw a set of vectors from the poles and zeros to a test point in the z -plane, and evaluate H ( z ) in terms of the lengths and angles of these vectors. In particular, we choose to evaluate H (e j ω ) on the unit circle, M ± e ² H j ω ) = K i =1 j ω z i N i =1 j ω p i ) and M i =1 ³ ³ e z i ³ ³ M i =1 q i ³ ³ H j ω ) ³ ³ = K M N i =1 | e j j ω ω p i | = K N i =1 r i M N N j ω j ω H j ω ) = ´ ± e z i ² ´ ± e p i ² = ´ θ i ´ φ i i =1 i =1 i =1 i =1 where the q i and θ i are the lengths and angles of the vectors from the zeros to the point z = e j ω , the r i and φ i are the lengths and angles of the vectors from the poles to the point z = e j ω , as shown below: 1 copyright c D.Rowell 2008 15–1
X X X O O 1 - 1 - j 1 j 1 Â { z } Á { z } f f f q q p p p z z 1 1 1 1 2 2 2 2 3 3 r r r q q 1 1 2 2 3 z = e j w A t f r e q u e n c y w : | H ( e ) | = K q q r r j w 1 1 2 2 3 H ) = ( + q ) - ( f + f ) 1 2 j w 1 2 3 We can interpret the eﬀect of pole and zero locations on the frequency response as follows: (a) A pole (or conjugate pole pair) on the unit circle will cause H ( e j ω ) to become inﬁnite at frequency ω .

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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_15 - MIT OpenCourseWare http/ocw.mit.edu 2.161 Signal Processing Continuous and Discrete Fall 2008 For information about citing these

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