Calculating Fourier Series Coefficients

I am trying to compute the trigonometric fourier series coefficients of a periodic square wave time signal that has a value of 2 from time 0 to 3 and a value of -12 from time 3 to 6. It then repeats itself. I am trying to calculate in MATLAB the fourier series coefficients of this time signal and am having trouble on where to begin.

The equation is x(t) = a0 + sum(bk*cos(2*pi*f*k*t)+ck*sin(2*pi*f*k*t))

The sum is obviously from k=1 to k=infinity.

a0, bk, and ck are the coefficients I am trying to find. Thanks for the help.

[omar seraj]

How to plot in Fourier series given a time interval -1.5 to 2.5I need a step by step answer to this problem please

Answer 1

[Rick Rosson]

>> doc fft
>> doc real
>> doc imag

http://en.wikipedia.org/wiki/Fourier_series

Answer 2

[Youssef Khmou]

hi Jay , computing a0 bk and ck is bout theory i think, anyway try :

 You have first to construct the original signal "Square(t)" so as to compare it with Fourier approximation :

 clear , close all;
 Fs=60;
 t=0:1/Fs:20-1/Fs;
 y=square(t,50);
 y(y>0)=2;
 y(y<0)=-12;
 figure, plot(t,y);
 axis ([0 20 -20 10])

 % Fourier Series 
 a0=0;
 Fy=zeros(size(t));
 N=10;
 for n=1:2:N

      Fy=Fy+(4/n*pi)*sin(2*pi*n*t/(2*pi));
 end
 hold on,
 plot(t,Fy,'r')
 legend(' Square ','Fourier Approx');

Try now to to compute an, and bn and increase the number of iterations N and conclude

You have also to adjust the amplitudes

Answer 3

[Kamal Kaushal]

clear , close all;

Fs=60;

t=0:1/Fs:20-1/Fs;

y=square(t,50);

y(y>0)=2;

y(y<0)=-12;

figure, plot(t,y);

axis ([0 20 -20 10])

% Fourier Series

a0=0;

Fy=zeros(size(t));

N=10;

for n=1:2:N

Fy=Fy+(4/n*pi)*sin(2*pi*n*t/(2*pi));

end

hold on,

plot(t,Fy,'r')

legend(' Square ','Fourier Approx');

Answer 4

[Hemang Mehta]

clear , close all;

Fs=60;

t=0:1/Fs:20-1/Fs;

y=square(t,50);

y(y>0)=2;

y(y<0)=-12;

figure, plot(t,y);

axis ([0 20 -20 10])

% Fourier Series

a0=0;

Fy=zeros(size(t));

N=10;

for n=1:2:N

Fy=Fy+(4/n*pi)*sin(2*pi*n*t/(2*pi));

end

hold on,

plot(t,Fy,'r')

legend(' Square ','Fourier Approx');

[Aaron Vargas]

hello , is there any chance that you can explain to me the variables and how it works ? im stuck in a homework for college

[Rik]

Comment posted as flag by Hariharan Hariharan:

need the flowof code for case study

Answer 5

[Sikha ranjith kumar]

x(t)=cos(50t)

Answer 6

[Gabriele Bunkheila]

From a computational perspective, the Fourier Series is closely related to the Discrete Fourier Transform when talking about discrete signals (say sampled at a sampling rate fs, i.e. using a sampling period Ts = 1/fs). The Discrete Fourier Transform, in turn, is most often computed through the Fast Fourier Transform (or FFT, implemented by the fft function). That said, the Fourier Series only applies to periodic signals (say with period T).

Some of the key ideas to keep in mind when using the FFT to compute the Fourier Series include:

• The time-domain signal segment provided as input to the fft function should include exactly an integer number of periods or the signal. The simplest case is when that's exactly 1 period made of N samples, or T = N*Ts
• The coefficients returned by the FFT will be complex numbers. To obtain back the cosine and sine coefficients of the bk and ck coefficients, use cos and sin on the k-th complex coefficient returned by the FFT
• The FFT will only provide a "finite" sequence of separate complex coefficients, from k=0 to k equal just below T/(2*Ts). That is N/2-1 for N even and (N-1)/2 for N odd. The actual number of complex values returned by the FFT will be up to double that number, but the second half of those can be ignored if the input time-domain sequence is real

If you need a review of the basics on this topic, I recommend taking a look at the following resources:

• The fantastic Fourier Analysis learning module on MATLAB Central File Exchange (including the Sine and Cosine series App captured below)
• The super-popular video by Brian Douglas on Understanding the Discrete Fourier Transform and the FFT
• The section Fourier Analysis and Filtering from the MATLAB documentation

Answer 7

[NIHAD]

we know that fourier series coefficients are given by and To find them just write these formulas in matlab.

your give example is 2 for t=0-3 and -12 for t=3-6. So the time period is T=6.

syms t n

assume(k>0)

assume (k,‘integer’) %assume k is a positive integer

a_0=(1/T)*(int(2,t,0,3)+int(-12,t,3,6))

a_k=(2/T)*(int(2*cos(2*k*pi*t/T),t,0,3)+int(-12*cos(2*k*pi*t/T),t,3,6))

b_k=(2/T)*(int(2*sin(2*k*pi*t/T),t,0,3)+int(-12*sin(2*k*pi*t/T),t,3,6))