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Hw1_sol - ECE 220 Spring'07 Signals and Information By Prof...

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1 ECE 220 Spring’07 Signals and Information By Prof. Rick Johnson, Prof. Adam Bojanczyk Solution of Homework # 1 This solution was written by Tae Eung Sung Problem 1 The general representation for a cosine function is as follows: ) 2 cos( ) ( 0 φ π + = t f A t x where A , 0 f , φ and 0 ω are the amplitude, (cyclic) frequency, phase shift and radian frequency, respectively. 1) 8 . 2 = A , 8765 0 = f , 256 π φ = , π ω 17530 0 = 2) Since ϑ π ϑ cos ) 2 sin( = + , then ) 30 2 cos( 25 . 5 ) ( t t x π = . Hence, 25 . 5 = A , 30 0 = f , π ω 60 0 = , 0 = φ . 3) Since 2 ) 1 )( 1 ( = + j j and ϑ ϑ 2 cos 1 cos 2 2 + = , then ) 16384 2 cos( 2 ) 16384 2 cos( 2 ))) 8192 2 2 cos( 1 ( 1 ( 2 ) ( π π π π + = = + = t t t t x . Hence, 2 = A , 16384 0 = f , π ω 32768 0 = , π φ = . Problem 2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 -1 -0.5 0 0.5 1 1.5 2 time t x(t),y(t),z(t) Plots for Problem 2 x(t) y(t) z(t)
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2 Problem 3 By Euler’s formula ϑ ϑ ϑ sin cos j e j + = , 1) 0 2 2 2 ) ( ) 2 sin 2 (cos j j j j j e e e e j j π π π π π = = = + = , where 2 π = e A and we will consider over π θ π . 2) 3 2 2 4 ) 3 sin 3 (cos 4 )) 6 sin 6 (cos 2 ( ) 3 ( π π π π π j e j j j = + = + = + NOTE: For integer, ϑ ϑ ϑ ϑ ϑ ϑ n j n e e j jn
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