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ECE220_HWsol_1_08

# ECE220_HWsol_1_08 - ECE220 Signals and Information Spring...

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1 ECE220 Signals and Information Spring 2008 Homework 1 Solutions by Tae Eung Sung Problem 1 (15 pts) The general representation for a cosine function is as follows: )) ( cos( ) 2 cos( ) ( 0 0 s t t A t f A t x = + = ω φ π where 0 f (Hz), 0 ω (rad/s), A , φ and s t (sec) are the cyclic frequency, radian frequency, amplitude, phase shift and time shift, respectively. 1) 438 0 = f , π ω 876 0 = , 8 . 4 = A , 128 π φ = , 112128 1 = s t 2) Since ) 2 cos( ) sin( π φ ω φ ω + = + t t , then )) 120 1 ( 20 cos( 25 . 0 ) 2 3 10 2 cos( 25 . 0 ) ( = + = t t t x π π π π . Hence, 10 0 = f , π ω 20 0 = , 25 . 0 = A , 6 π φ = and 120 1 = s t 3) Since 2 ) 1 )( 1 ( = + j j and ϑ ϑ 2 cos 1 cos 2 2 + = , then )) 8192 1 ( 8192 cos( 2 ) 4096 2 cos( 2 ) 8192 cos( 2 ))) 4096 2 cos( 1 ( 1 ( 2 ) ( + = + = = + = t t t t t x π π π π π . Hence, 4096 0 = f , π ω 8192 0 = , 2 = A , π φ = and 8192 1 = s t Problem 2 (10 pts) -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 (sec) x(t),y(t),z(t) Plot for Problem 2 x(t) y(t) z(t)

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2 Problem 3 (16 pts) By Euler’s formula ϑ ϑ ϑ sin cos j e j + = , 1) 5 3 5 3 5 5 5 3 5 4 5 2 6180 . 1 ) 5 cos( 2 ) ( π π π π π π π π j j j j j j j e e e e e e e = = + = + .
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