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hw5-sol

# hw5-sol - Homework 5 Problem 1(Chapter 8 P 22 T0 = 2 f 0 =...

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Homework 5 Problem 1 (Chapter 8, P. 22) 0 2 T = , 0 1 2 f = , 0 ω π = , sin(2 ), 1/ 2 ( ) 0, 1/ 2 1 t t x t t π < = < < , as shown in figures (dotted line). Find the harmonic function [ ] X k by: 0 0 0 0 1 [ ] ( ) T jk t X k x t e dt T ω = 1 1 1 ( ) 2 jk t x t e dt π = 1/2 1/2 1 sin(2 ) 2 jk t t e dt π π = 2 2 1/2 1/2 1 2 2 j t j t jk t e e e dt j π π π = 1/2 (2 ) (2 ) 1/2 1 ( ) 4 j k t j k t e e dt j π π + = 1/2 1/2 (2 ) (2 ) 1/2 1/2 1 4 (2 ) (2 ) j k t j k t e e j j k j k π π π π + = + + sin( (2 )) sin( (2 )) 1 2 2 2 (2 ) (2 ) k k j k k π π π π + = + 2 2 sin c( )-sinc( ) 4 2 2 j k k + = − So, the complex CTFS description is: ( ) [ ] k jk t k x t X k e π =∞ =−∞ = Approximation to the signal: ( ) [ ] k N jk t N k N x t X k e π = =− = , as shown is figures (solid line) Code: clear all T0 = 2; icount = 0; for t = -3:0.01:3; % To get x(t) icount = icount+1; if abs(t-round(t/T0)*T0) <= 1/2; x_t(icount) = sin(2*pi*t);

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