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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 takehome 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 highpass filter from a lowpass design (this is not a method we talked about in class). Consider a prototype causal digital lowpass filter H lp ( e j ). The figure below shows how a digital lowpass filter may be transformed to a highpass filter by simply translating (shifting) its frequency response H ( e j ) by , that is we create the highpass 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 lowpass 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 discretetime 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 highpass 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 highpass filter may be constructed by simply alternating the signs of the coecients in the lowpass difference equation! (b) Show that the two discretetime 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 highpass filter related to the phase response H lp ( e j ) of the prototype lowpass filter?...
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 Fall '08
 DerekRowell

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