# Sawtooth Wave Fourier Series - General derivation and Matlab issue

I am working on one of my first Matlab projects. I am trying to graph a sawtooth wave with 10-V Pk-Pk, 0-V average value. 50% duty cycle, 2.5 khz.

I first attempted to find a general equation for such a sawtooth wave.

$$\frac{-V_{pp}}{2} + \frac{jV_{pp}}{2\pi} \sum_{n=-\infty}^{\infty} \frac{1}{n}e^{jnw_0t}$$

From this, for the particular function I am trying to plot I have the equation

-5 + j*5/pi Sum (from n=-inf to inf) (1/n)e(j*5000*pint) $$\ -5 + \frac{j5}{\pi} \sum_{n=-\infty}^{\infty} \frac{1}{n}e^{j5000\pi nt}$$

And my code:

%% SAWTOOTH WAVE

t = linspace(-2,10,1e4); % time vector [s]

%% constructing approximation x_N(t), N = 100

n = -100:100;                            % indices included in the summation, n is a 101 element vector
X = (5*1i)./(pi*n);                          % this constructs the coefficients of e^(*) terms
X(n==0) = 5;                             % C_0

% X is a horizontal vector with the coefficients of the exponential term
% corresponding each n-value from -100 to 100.

XM = repmat(X.',[1, length(t)]);         % This is a matrix that replicates the transpose of the
% vector X as many times as t is long
% now, XM is a matrix with identical values in each row.
% Each column has a coefficient of an exponential term corresponding to an
% n-value, repeated throughout
basis = exp(1i*pi*5000*n.'*t);              % basis is a matrix containing the exponential part,
% each column corresponding with each n-value
% and each row corresponding time

x100 = -5 + sum(XM.*basis);                   % x100 is matrix resulting from summing down the rows of element-wise
% multiplacation of XM and basis

if max(imag(x100))<1e-15  % this checks to see if the imaginary part of x
x100 = real(x100);      % is below Matlab's noise floor (approx 10^-15).
end                     % If so, we get rid of the imaginary part.

%% plots

% we will use these variables to keep the limits on our plots uniform
tmin = min(t); tmax = max(t); xmin = -6; xmax = 6;

figure(1);
%subplot(3,1,1);     % this command allows you to plot several graphs
%                    % on one page
plot(t,x100);
axis([tmin tmax xmin xmax]);    % sets the plot's axis limits
title('N=100 approximation of Triangle Wave');
xlabel('time [s]'); ylabel('amplitude [a.u.]');


However the period here is 2 seconds, which seems totally wrong to me. I'm having trouble figuring out where I went wrong.

Also - I apologize, I am also having trouble getting MathJax to preview my math on my browser, I don't know what's wrong there. I am sorry for the messy looking representation of my math.

• fixed up your first equation, at least into legal mathjax anyway, check it still expresses what you mean mathematically, I may have garbled it. You do the same for the second equation. Apr 17 '18 at 6:20

The linspace is probably not what you want it to be. A 2.5kHz signal has a period of 0.4ms

tmin = 1/2500; t = linspace(0,tmin*5,2000); % time vector [s]

or

tmin = 1/2500; t = linspace(-tmin,tmin*4,2000); % time vector [s]

I don't know why the amplitude is off, but at least the period is right. • Ah OK, great, thank you! Simpler than I thought. I guess with the larger linspace I just expected that it would look almost like a fuzzy line. This gives me something to think about, thank you. Apr 17 '18 at 0:37
• If you like the answer upvote Apr 17 '18 at 0:50
• I did - upvotes are not public unless you have a certain reputation. Apr 17 '18 at 19:26
• Why do you say the amplitude is off? The overshoot is expected for the truncated fourier series. en.wikipedia.org/wiki/Gibbs_phenomenon Apr 17 '18 at 19:55
• @RussellBorogove the OP wanted a 10V-Pk Pk signal, I didn't have time to look at the amplitude. Apr 17 '18 at 20:16