quiz2 - MIT OpenCourseWare http://ocw.mit.edu 2.161 Signal...

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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 . MASSACHUSETTS INSTITUTE OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING 2.161 Signal Processing- Continuous and Discrete Fall Term, 2008 Quiz #2 (Take Home) Due in class (1pm) on December 9, 2008 Notes: This is a take-home quiz. Collaboration (with anybody) is expressly forbidden. Do not spend excessive time on this quiz! There are 4 problems, answer them all. Partial credit will be given. You may ask the teaching staff for help- but that help will be generally limited to ensuring that you have access to the required data files. 1 M Problem 1: (20 points) Here is a simple way to design a digital high-pass filter from a low-pass design (this is not a method we talked about in class). Consider a prototype causal digital low-pass filter H lp ( e j ). The figure below shows how a digital low-pass filter may be transformed to a high-pass filter by simply translating (shifting) its frequency response H ( e j ) by , that is we create the high-pass filter from the prototype: H hp ( e j ) = H lp ( e j ( ) ) . | H ( e ) | l p j M M F | H ( e ) | h p j M F p r o t o t y p e l o w- p a s s f i l t e r h i g h- p a s s f i l t e r (a) Assume that the original low-pass filter has a recursive difference equation N M y ( n ) = a k y ( n k ) + b k x ( n k ) k =1 k =0 corresponding to the discrete-time transfer function M b k z k k =0 H lp ( z ) = N 1 + k =1 a k z k Show that the difference equation of the new high-pass filter is N M y ( n ) = ( 1) k a k y ( n k ) + ( 1) k b k x ( n k ) k =1 k =0 In other words, a high-pass filter may be constructed by simply alternating the signs of the coecients in the low-pass difference equation! (b) Show that the two discrete-time impulse responses are related by h hp ( n ) = ( 1) n h lp ( n ) (c) How is the phase response H hp ( e j ) of the high-pass filter related to the phase response H lp ( e j ) of the prototype low-pass filter?...
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quiz2 - MIT OpenCourseWare http://ocw.mit.edu 2.161 Signal...

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