HELP MATLAB The Fourier series for a sawtooth wave with amplitude A and period T

Question

HELP MATLAB

The Fourier series for a sawtooth wave with amplitude A and period To is 2tnt x(t) = ? bn sin(2 ) where N = oo and bn = (-1)"+1 1. Write a MATLAB script to compute and plot this function. Instead of infinity, use a finite value of N which is input by the user. The plot time axis should contain two periods. You may make A and To constants in the script, rather than having the user input those values. 2. Your script probably uses one loop. Write a new version which does not use any loops. Hint: observe that the sum of bn times the sine (which is a function of n and t) can be rewritten as a vector times a matrix where one dimension of the matrix corresponds to n and the other to t. Procedure: (a) create vectors n and t, (b) create vector bn, (c) via appropriate vector multiplication of n and t create matrix where one dimension corresponds to n and the other to t and use that to compute sine, (d) perform matrix multiplication. Label the plot axes and title the plots. Test each version using N-5 and N-50.

Explanation / Answer

function sawtooth_wave_fourier_series_demo

clc; close all; clear all;

%parameter of input sawtooth wave
T0 = 10;    % period
A = 1;
dc = 0;     % dc level
ts = 0;     % time shift. positive: move right; negative: move left
M = 3;      % How many period are shown

Nf = M*T0;       %number of FS components, i.e., C(-Nf),...,C(-1),C(0),C(1),...C(Nf).
w0 = 2*pi/T0;   % frequency

%Figure 1, plot sawtooth wave

x = -M/2*T0:0.01:M/2*T0;

syms t n y a1 a2
y = (sym('Heaviside(t+a1)')-sym('Heaviside(t-a2)')) * (A/T0*t);
y = subs(y,a1,0);
y = subs(y,a2,T0);
y = simple(y);
f = subs(y,t,x);

%plot sawtooth wave
figure1 = figure(1);
axes1 = axes('FontSize',14,'Parent',figure1);
box(axes1,'on');
hold(axes1,'all');
ylim(axes1,[dc-2*A dc+2*A]);
grid;
title(['Sawtooth wave']);
xlabel('t (seconds)'); ylabel('Amplitude');

xx = (x>=0).*(x-ts-fix((x-ts)/T0).*T0) + (x<0).*(x-ts-fix((x-ts)/T0).*T0+T0);
yy = double(subs(y,t,xx))+dc;
yy(isnan(yy)) = dc;
plot(x,yy,'LineWidth',2,'color','b');

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% closed-form of Cn (KEY EQUATION) %%%%%%%%%%
%Compute the closed-form of Cn; i.e.,
%C0 = 1/T0 * integral(f(t)) over [-T0/2, T0/2]
%Cn = 1/T0 * integral(f(t)*exp(-j*n*w0*t)) over [-T0/2, T0/2]

C0=int(y,t,0,T0)/T0;
Cs=int(y*exp(-j*w0*n*t)/T0,t,0,T0);
C0=simple(C0);
Cs=simple(Cs);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% closed-form of Cn (KEY EQUATION) %%%%%%%%%%

%Figure 2, plot amplitude spectrum
figure2 = figure(2);
axes2 = axes('FontSize',14,'Parent',figure2);
box(axes2,'on');
hold(axes2,'all');
title(['Amplitude Spectrum']);
xlabel('n'); ylabel('Amplitude');

%Compute Cn and plot its amplitude
for k=-Nf:1:Nf
    if k==0
        c = abs(double(C0));        %C0
    else
        c = double(subs(Cs,n,k));   %Cn, substitue n=k into the above closed-form of Cn
    end

    stem(k, abs(c),'color','b','MarkerSize',5);                  %plot

end

%Figure 3, plot phase
figure3 = figure(3);
axes3 = axes('FontSize',14,'Parent',figure3);
box(axes3,'on');
hold(axes3,'all');
title(['Phase Spectrum']);
xlabel('n'); ylabel('Phase (degrees)');

%Compute Cn and plot its angle
for k=-Nf:1:Nf
    if k==0
        c = abs(double(C0));        %C0
    else
        c = double(subs(Cs,n,k));   %Cn, substitue n=k into the above closed-form of Cn
    end

    stem(k, angle(c)*180/pi,'color','b','MarkerSize',5);    %plot
   
end

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