Question

2.10-3 Using direct integration, numerically derive and plot the exponential Fourier series coefficients of the following periodic signals: (a) The signal waveform of Figure P2.1-5 (b) The signal waveform of Figure P2.1-10(a) (c) The signal waveform of Figure P2.1-10(f) 2.10-4 Using the FFT method, repeat Problem 2.10-3. e P2.1-5 g (t) -6 4 -8

Answer 1

MATLAB code is given below in bold letters.

clc;
close all;
clear all;

% define the signal parameters
Vm = 13;
T = 4;
w0 = 2*pi/T;

% define the symbolic variables
syms n t;

% define the signal over one period
v1 = t^3;

% evaluate the fourier series integral
c = 1/T*int(v1*exp(-1j*n*w0*t),-T/2,T/2);
c0 = 1/T*int(v1,-T/2,T/2);

% obtain C by substituting n in c[n]
nmax = 100;
n = -nmax:nmax;
c = subs(c);
c0 = subs(c0);
c(nmax+1) = c0;

% plot the fourier coefficients
figure;
stem(n,abs(c));grid on;xlabel('n');ylabel('Amplitude');
title('fourier series coefficients of g(t): G[n]');

% fft method
% define the signal first over one period
t = -2:0.01:2;
g = t.^3;

% find fft of g(t)
G = fftshift(fft(g))/length(g);
% plot fft
figure;plot(abs(G));grid on; xlabel('frequency bins');title('fft of g(t)');

Plots are given below:

fourier series coefficients of g( G[n] 1.2 0.6 0.4 0.2 00 80 0 40-20 0 20 460 80 100

ft of g(t) 1.4 1.2 0.8 0.6 0.4 0.2 0 50 100 150 200 250300 350 400 450 frequency bins