Question

1. We have a signal, x (t), with period of T - 2 second with the signal in one period given below: x(t)- ^x,(t+nT) wherex,) 1-2 (t - ) Kt<1 n=-oo 0 1/2 (a) Find the Fourier series coefficients for this signal. That is, find the values of ak so that x (t) Hint: te-jwt teJwt dt (b) Write some MATLAB code which will plot the signal resulting from a truncated Fourier Series using the coefficients you calculated in part (a) above for the range -1<ts 1. That is, plot the values of XK (t) defined below: 27T Plot this function for K -1, 3, and 5.

Answer 1

Solution: Given χpは defined as- Here Tasee 2.

K17 -0.5 not | 0.5 е ће ㄦ

,'"-.nw.rdt 0.5 プnwit

-Int If we replace variable 'n' by 'k', we get lcnce, Complex exponcntial Fouricer series ofxp(t) is given by-

MATLAB CODE FOR SOLUTION clc clear all close all To 2; % Time period of the given signal wo= ( 2 *pi)/To; fundamental frequency in rad/sec t linspace(-4,4, 10000); % creating the time vector % creates a zero valued function co-0.75; D.C term of given wave N=input ('Enter the Number of truncated terms you want to approximate \n: '); % number of harmonics for k--N:1:N % skip the zeroth term continue; end; Fourier coefficient of exponential form C-k 2*(cos ( k *pǐ/2)-cos ( k *pǐ))/(k*k*pǐ*pi); % the k-th the computes = % kth term of the series f=f+fk; % adds the k-th term to xp(t) end Xpt-f+co; % Approximated representation of fourier series plot (t, Xpt, 'LineWidth', 2) grid on xlabel 't') ylabel ('Amplitude) title (strcat ('Truncated Fourier series of the Xp (t) with -', int2str (N), ' harmonics.'))

Truncated Fourier series of the Xptt) with K-1 harmonics. 1.3 1.2 0.9 0.8 0.7 0.6 0.5 .3 .2 Approximated Fourier series for K-1 Harmonic using Matlab

1.4 Truncated Fourier series of the Xpit) with K-3 harmonics 0.8 0.6 0.4 -2 Truncated Fourier series of xp(t) for k-3 harmonics using Matlab

Truncated Fourier series of the Xp(t) with K-5 harmonics 0.6 0.2 -2 -3 Truncated Fourier series for k 5 harmonics using Matlab