a3_solution

# A3_solution - Instructor Solutions for Assignment 3 Problem 5.2(a By definition X 1 3e jt dt 3(e j jt 0 0 3 j e j 1 3 j 2 e j e j 2 e j 2 3 j 2 e j

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Instructor Solutions for Assignment 3 Problem 5.2 (a) By definition,             ). 2 / ( sinc 3 6 6 ) 2 / sin( 2 1 3 3 ) ( 2 / 2 / ) 2 / sin( / 2 1 2 / ) 2 / sin( 2 / 2 / 3 0 2 / 2 / 2 / 3 3 0 ) ( 1  j j j j j j j j j j j j e t j e e e j e e e e e dt e X t j (b) By definition,     0.5 1.5 0.5 1.5 2 ( ) ( ) 0.5 0.5 0.5 0.5 0.5 0.5 1.5 0.5 0.5 1 0.5 1 sin 0.5 0.5 ( ) 0.5 0.5 2 sin(0.5 ) 2 sin(0.5 ) j t j t TT j t j t ee jj j T j T j T j T jT T T X e dt e dt e e e e j T e j T             (0.5 ) sin(0.5 ) 0.5 0.5 0.5 T 0.5 T 2 0.5 sinc sinc . T T e T Te    (c) By definition,             . 1 0 1 1 ) ( 2 2 2 2 2 2 1 1 1 1 1 ) ( 1 1 ) ( 1 ) ( 1 0 ) ( 1 ) ( 0 3 T j T j T j T j T j T j j e T T j e T j e T t T t j T t e e dt e X T j t j t j For = 0,     2 2 0 2 2 0 3 0 1 1 ) ( T T T T t T T T t dt X . (d) By definition,

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Assignment 3 CSE 3451: Signals and Systems 2                       . ) ( sinc ) cos( 1 0 0 1 1 1 1 ) ( 5 . 0 2 ) 5 . 0 ( ) 5 . 0 ( sin ) 5 . 0 /( 1 4 ) 5 . 0 ( sin 2 2 2 ) ( 1 1 ) ( 1 ) ( 1 ) ( 1 ) ( 1 1 ) ( 1 0 ) ( 1 ) ( 0 ) ( 1 ) ( 0 0 4 2 2 2 2 2 2 2 2 2 2 2 2 T T T T T T T j T j j e T j e T j T j T j e T j e T t T j e T j e T t T t j T t T t j T t T T dt e dt e X T j T j t j t j t j t j (e) By definition,     5 0 0 0 ( ) 1 0.5sin 0.5 sin T T T j t j t j t tt TT AB X e dt e dt e dt     We consider different cases for the above integral. Case I: ( = 0)       5 00 0 2 / 0.5 1 (0) ( ) 1 0.5sin 0.5 sin cos( ) cos( ) cos(0) (1 ) T t T T X x t dt dt dt dt T T T T   Case II: ( ≠ 0, / T ):   0 0 1 1 1 1 1 0 T T j t j t j T j T j j j A e dt e e e            2 2 2 2 2 2 2 2 2 2 2 2 2 2 0 0.5 0 0, 0 0.5 0.5 0.5 sin cos for 0, sin cos 1 jt T T e t t T T T T T T t t T T T T at t T j T j T Bj ej ee
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## This note was uploaded on 12/11/2010 for the course CSE CSE 3451 taught by Professor A during the Fall '10 term at York University.

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A3_solution - Instructor Solutions for Assignment 3 Problem 5.2(a By definition X 1 3e jt dt 3(e j jt 0 0 3 j e j 1 3 j 2 e j e j 2 e j 2 3 j 2 e j

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