Under the topic of
“complex variables and transforms”, you approach the Fourier series using a method
where you consider a function f(x) defined on a specific interval, let’s say
[-p, p] and it can be expanded on an orthogonal series consisting of
trigonometric function:
And in form of
summation, the above series can be represented as:
The above expression is
known as the Fourier series representation. But what is the Fourier series
actually and why do we need it?
Well, in order to
understand that, consider the following graph of a function f(t)
The above function is
apparently periodic having a period of 2
because the wave is completing one cycle every
2 seconds and hence the frequency f = 1/ 2. Now, what we are
about to explore is if we can take a periodic signal/wave like this and
represent it as a sum of sines and cosines like:
Now, why sine and cosines?
Because the original function f(t) has a period of 2, it means it must be involving some functions having a period of 2(sin(t) and cos(t) have a period of 2).
In the above equation, a1 and b1 are the weights that will tell how much cosine or sine is involved to make up the original function f(t). For example, if a1 is a lot larger than b1, it means that cos (t) is composed of more parts of f(t) compared to sin(t).
But, only this cannot
fully describe the function f(t) because if it was only composed of cos(t) and
sin(t), it would look like a clean sinusoid but that is apparently not the case
so, it must be involving other cosines with different frequencies like cos(2t),
cos(3t), sin(2t), sin(3t) and so on.
The expression is known as the “Fourier series”, which is the representation of any signal/wave as a sum of sines and cosines.
Why do we use the Fourier
series?
We use the Fourier series because most of the differential equations are very difficult to solve with just a signal like the above. But if the signal can be represented in the form of sines and cosines, we can easily generalize a solution for our given differential system of equations.
Definition:
For any periodic
function f(t) with fundamental period T, f(t) can be represented in form of
Fourier series as:
