ps6soln - MIT OpenCourseWare http:/ocw.mit.edu 2.161 Signal...

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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 Problem Set 6 Solution Assigned: October 23, 2008 Due: October 30, 2008 Problem 1: Given the difference equation, y n = 0 . 5 y n 1 + 0 . 5 u n + u n 1 (a) The transfer function is given by, Y ( z ) 0 . 5 + 1 z 1 0 . 5 z + 1 H ( z ) = = = U ( z ) 1 + 0 . 5 z 1 z + 0 . 5 (b) Pole zero map Pole−Zero Map Imaginary Axis −1 −0.5 0 0.5 1 unit circle z−plane x o −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 Real Axis (c) The system is causal, therefore the ROC includes all z > 0 . 5 – which includes the unit circle. | The system is therefore stable. (d) Let Ω = ω Δ T . The system frequency response magnitude is given by 0 . 5 e j Ω + 1 H ( e j Ω ) = | H ( z ) | z = e j Ω | = e j Ω + 0 . 5 1 . 25 + cos(Ω) = 1 . 25 + cos(Ω) = 1 . and the system is an all-pass ±lter. When ω = 0 (or Ω = 0), H ( ) = 0. When ω = π/T (or Ω = π ), H ( ) = π . 1
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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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ps6soln - MIT OpenCourseWare http:/ocw.mit.edu 2.161 Signal...

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