Fourier Transform and Fourier Calculator in Matlab

Table Of Contents

Ø Introduction to Fourier Transform and Series

Ø Concept of function Vs vectors

Ø Dot Product of Vectors and Orthogonality

Ø Dot Product of Analogue functions and Orthogonality

Ø Fourier Series

Ø Fourier Transform

Ø Fourier Series Vs Fourier Transform

Ø Introduction to MATLAB

Ø Fourier Transform in MATLAB

Ø Using MATLAB for Fourier Calculator

Ø GUI of Fourier Calculator

Introduction to Fourier Transform and Series

The concept of Fourier transform is very simple. It is used to find the frequency component of the any electrical (analogue) signal. Whenever I read Fourier transform I always ask questions from myself that how Joseph Fourier came up with the Fourier series. These types of stupid questions arise in my mind (not only in case of Fourier transform but also in every type invention and discovery). And the answer that I found is that they think, think and think and then try, try and try to implement those thoughts and came up with something innovative. And with the passage of time revisions of their equations by other mathematicians adds different flavors to their work. So Joseph Fourier simply shows that any function can be written as the series of sinusoidal components because they are orthogonal.

In my point of view the every function in time can be written as summation of infinite series of any function with variable coefficients and variable phase for every term in the series i.e. if we have two functions f(t) and g(t)

Then we can write f(t) in term of g(t)

f(t)=c₀ g(t – x₀) + c₂ g(t – x₁) + c₂ g(t – x₂) + . . . . . . . . .

c is the coefficient

x is the phase difference or time delay

Concept of function Vs vectors

The concept of Fourier transform is very simple. It is used to find the frequency component of the any electrical (analogue) signal. It all starts with the discovery that every function can considered as vector. As three dimensional vector is written as A=(2,3,1). So function f(t) can be represented as n-dimensional vector e.g.f(t)=x² can be written as f(t)=(1, 4, 9, 16, 25 . . . . ) , for better approximations we can decrease the step size from 1 to 0.001 or more less, here domain of f(t) is [1 to infinity] with step size 1.

As i said earlier function can be represented as vectors, and also functions behavior is very much same as that of vectors, then we can say that every vector can be represented with the help of basis or orthogonal components e.g A=(1,2,3) can be written as A=1i+2j+3k where i,j,k are unit vectors.

So orthogonal components found in analog domain are sine and cosine functions. The orthogonality of two vectors can be tested as if there do product is zero i.e. A.B=0 where A and B are vectors and they have discrete values. if A is n-dimensional vector then A has components a1,a2,a3. . . . and B has b1,b2,b3 . . . . then There dot product can also be expressed with the help of following expression.

As i explained earlier that analogue functions are also treated as vectors so the orthogonality test can also be applied on these functions. If we A and B are function of time in the above expression the summation sign will be replaced by the integral sign.

F=3.75 * 1000000;
T=1/F;
v=[-0.000001:.00000001:0.000001];
n=1;
ans=3.3*0.4;

an=(3.3/(n*pi) * sin (n* 0.4 *2*pi));
bn=(3.3/(n*pi) * (1-cos(n* 0.4 *2*pi)) );
ans=ans+an*cos(n* (2*pi/T)*v)+bn*sin(n* (2*pi/T)*v);
n=n+1;
end
plot (v,ans)

this is a result of fourier series in matlab with 40% duty cycle
replece 0.4 with desired duty cycle of square wave

UNDER CONSTRUCTION