Problem Set 1 solution

# Problem Set 1 solution - ECE 101 - Linear Systems, Winter...

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Unformatted text preview: ECE 101 - Linear Systems, Winter 2009 Problem Set # 1 Solutions (send comments/questions to kquest@ucsd.edu) Problem 1 Plots:-5-4-3-2-1 1-3-2-1 1 2 3 t x(-2-t)-3-2-1 1 2-1 1 2 3 4 t [x(t)+x(-t)]u(-t) 1-2-1 1 2 3-1 1 2 n x[1-2n]-4-3-2-1 1 2 3 4 5-2-1.5-1-0.5 0.5 1 1.5 2 n x[-n]u[3+n] Problem 2 (1.24(c) OW2) n x [ n ] x [- n ] even odd 1 1 1 1 2 1 3/2 1/2 2 1 2 3/2 -1/2 3-1 2 1/2 -3/2 4-1-1/2 1/2 2-5 5-2-1.5-1-0.5 0.5 1 1.5 2 n Even x[n]-5 5-2-1.5-1-0.5 0.5 1 1.5 2 n Odd x[n] Problem 3 (1.25(b,f), 1.26(d), and 1.26(c), modified, OW2) 1.25(b) x ( t ) = exp[ j ( t- 1)] =- exp[ jt ] 3 x ( t ) is periodic with fundamental period T = 2. 1.25(f) We will argue that x ( t ) is periodic with fundamental period T = 1 / 2. First, note that if 0 &lt; t &lt; 1 / 2, then x ( t ) = X n =- exp(- 2 t + n ) u (2 t- n ) = exp(- 2 t ) X n =0 (exp(- 1)) n = exp(- 2 t ) 1- exp(- 1) Next, any other time t can be modeled as t = t + m/ 2, where m is an integer. Thus, x ( t ) = X n =- exp(- 2 t- m...
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## Problem Set 1 solution - ECE 101 - Linear Systems, Winter...

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