ps1soln - 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 1 Solution: Convolution and Fourier Transforms Problem 1: Use the convolution definition ! " " # # = $ = % d t h f h f t y ) ( ) ( ) ( (a) ) 75 . 0 ( ) ( ) 5 . 1 ( ) ( ! + + + = t t t t f " ! " # $ $ % % = otherwise t t t h , 0 1 1 |, | 1 ) ( ( ) , ) ( ) 75 . 0 ( ) ( ) ( ) ( ) 5 . 1 ( ) ( ) 75 . 0 ( ) ( ) 5 . 1 ( ) ( ! ! ! ! " " # " " # " " # " " # # # + # + # + = # # + + + = $ d t h d t h d t h d t h t y and using the sifting property of the impulse function, ) 75 . 0 ( ) ( ) 5 . 1 ( ) ( ! + + + = t h t h t h t y -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 0 0.2 0.4 0.6 0.8 1 1.2 1.4 (b) h(t) is the as same used in part (a) ! " # $ $ % = otherwise t t f , 0 75 . 0 5 . 1 , 1 ) ( ,
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-4 -3 -2 -1 0 1 0 0.5 1 2 ) 75 . 1 ( 2 )) ( 1 ( 2 ) 5 . 2 ( 2 )) ( 1 ( 2 75 . 0 1
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ps1soln - MIT OpenCourseWare http/ocw.mit.edu 2.161 Signal...

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