Fourier Synthesis Demo App with Coefficient Visual Display

Fourier Synthesis Demo App with Coefficient Display

Because of limitations regarding redistribution of code/functionality contained in the Symbolic Math Toolbox this app doesn't do coefficient calculation in real time. Working around the limitations leads to pre-calculated Fourier Coefficients being used. So we need to pre-calculate the coefficients (see steps below).
And yes, the functions/pulse trains are not running from t=-inf to t=+inf.

Deploying this as a stand-alone app (.exe) isn't possible because of a bug in "matlabFunction" (it will use rectangularPulse in the resulting .m-file, the compiler will accept it but the app will display an error); I've been in contact with support and the issue is noted. But that's about it...

Since this is more of a teaching tool these limitations are not a deal breaker. But you should be aware of it.
We're doing signals (and systems), that's why there is no units (at least on the y-axis). Imagine this being some voltage having been normalized to a maximum amplitude of "1".

Choosing a Waveform

You may choose a waveform by clicking the drop down menu in the upper left corner.

Changing the Number of Coefficients used for Approximation

There's a slider setting the number of coefficients used for calculating the approximation (marked in red). The color of the coefficients that are ununsed will change to grey. In the example above there's only the DC part (coefficient #0).

How Fourier Coefficients / Functions are prepared

This (unintentionally) turned into a nice run-down of using Matlab for Complex Fourier Series Computation. See the other App Repo (FourierSynthesis) for Real Coefficients (a-b-fashion with Cosine and Sine).

• Set up Matlab for symbolic computation of Fourier Coefficients:

syms x k L n
evalin(symengine,'assume(k,Type::Integer)');

• Define function that is to be approxiated:

% rectangular pulse train without dc offset
% amplitude between -.5 and +.5
f = rectangularPulse(-1,1,x)  + rectangularPulse(-1,1,x-3) + rectangularPulse(-1,1,x+3) -.5;
% Save function to Matlab file for later use in the app:
matlabFunction(f,'File','rectangular_1')

• Define function for symbolic coefficient calculation:

% complex coefficients
c = @(f,x,k,L) 1/(2*L) * int( f * exp(-j*2*pi*k*x/(2*L))  , x,-L,L);

• Define symbolic sum (n-th order) approximating the original function (pulse train):

% fc is computation via complex coefficients
% this is not als slick as it could be 
% symsum will fail at k=0 when running k from -n..+n
% so we split the computation into three parts (negative, zero, positive)
fc = @(f,x,n,L) symsum( c(f,x,k,L)* exp(j*k*2*pi*x/(2*L)) ,k,1,n)+ symsum( c(f,x,k,L)* exp(j*k*2*pi*x/(2*L)) ,k,-n,-1) + c(f,x,0,L)*exp(j*0*2*pi*x/(2*L));

• Pre-compute Fourier Coefficients up to order 20:

% compute complex fourier coefficients for pulse trains
% this one makes for 41 coefficients in total (-20..0..20)
for R = -20:20
 rect_c(R+21) = c(f,x,R,1.5); 
end

% Save to Matlab function file ('.m')
matlabFunction(rect_c,'File','koeff_rectangular_1_c')

• Do a quick check of the computations:

% plot complex coefficients 
figure;stem([-20:20],rect_c)

% symbolic calculation of approximating function fc
figure;fplot(fc(f,x,2,1.5)), hold on, fplot(f), hold off
legend("2nd Order Approximation","Function")

% quick manual computation of numeric first order approximation
% create time vector
tv = [-5:0.05:5];
% create numeric (matlab) function of pulse train
rect_1 = matlabFunction(f)
% manual summing, 
% rect_c(21) is coefficient #0,
% rect_c(20) is coefficient #-1
% rect_c(22) is coefficient #1
% and we know that our rectangular pulse train has a period of T=3
sn = rect_c(21) *exp(1i*2*pi*0*tv/3) + rect_c(20) *exp(1i*2*pi*-1*tv/3) +rect_c(22) *exp(1i*2*pi*1*tv/3)
% plot original function and manually summed first order approximation
figure;plot(tv,rect_1(tv)), hold on, plot(tv, sn), hold off
title("Numeric Computation of First Order Approximation")
legend("Function", "1st Order Approximation")

% do a full computation taking all (41) coefficients
% as close to 42 as it gets ;)
% we'll notice artifacts from loss of precision
% one symptom thereof will be plot complaining
% about not being able to display imaginary arguments...
% these imaginary parts result from numeric imprecision when
% doing coefficient conversion from symbolic to double
% sn means sum, numeric
sn = 0;
for R = 1:41
 sn = sn + double(rect_c(R))* exp(1i*2*pi*(R-21)*tv/3);
end
figure;plot(tv,sn), hold on, plot(tv, rect_1(tv)), hold off;
legend("20th Order Numeric Approximation", "Function")
title("Numeric Approximation, 20th Order")

% looks better, huh?
figure;fplot(f),hold on, fplot(fc(f,x,20,1.5)),hold off
title("Symbolic Approximation, 20th Order")
legend("Function", "20th Order Numeric Approximation")

• Save Latex string for later display in app:

% one coefficient per line
% complex coefficients
fid = fopen('rectangular_1_c_lstr.txt','wt');
for R=1:41
  fprintf(fid, '%s \n', latex(rectangular_1_c(R)));
end
fclose(fid)

• Now, there's Fourier Coefficients (complex) up to order 20 and the original function in a file for later use in the app.

Implemented Pulse Trains

Rectangular 1

Rectangular Pulse Train without/with DC Offset.
This one is an interesting case for teaching/discussion. Does it have an offset? If no, why not? If yes, why so?

f = rectangularPulse(-1,1,x)  + rectangularPulse(-1,1,x-3) + rectangularPulse(-1,1,x+3) -.5;

Period is 3.

Rectangular 2

Rectangular Pulse Train with DC Offset.

f = rectangularPulse(-1,1,x)  + rectangularPulse(-1,1,x-3) + rectangularPulse(-1,1,x+3);

Period is 3.