Fourier Analysis
Fourier Analysis
Hi !
Today i begin to share a new area , a genius theory from Joseph Fourier. Fourier says that any engineering signals can be represented as a series sum of sine , even square and triangle waves .
Sine waves have many interesting properties , many natural operations deal with a set of differing frequency sine waves as if they were processed individually .
We will learn the technique of expressing waves in terms of sine,this will allow us to use them as we want them to be.
Suppose your signal has noise at a particular frequency (may be 3040 HZ) , we can pick apart the data to its constituent parts,remove the noise frequency , then stictch the rest back together to get a signal without the noise . This kind of notch filtering is very useful in audio signals .
In image processing,it is possible to look at an image and distil it down to small blocks , then look at each of the line;
now we can treat each line as a 1D signal,it’s then easy to see how you would apply the same techniques to the data that you would with any other signal . Let’s begin our theoric expansions.
sine wave
Trigonometric Fourier Series
What it means to be periodic ? When we use periodic for a signal , we say that you get the same samples in the same intervals ;
f (t) = f (t+nT) means that for every period T .
Any periodic function of frequency wo can be expressed as an infinite sum of sine and cosine functions that are integral multiples of wo.
f(t) = a0 + a1cos(w0) + b1sin (w0)+…..
in this equation,we can take (a0) as DC part of our signal,which is linear and not wavy .(a0) is also called the average value of f(t) (remember the average value of sinusoids are zero)
Also ;
w0 = 2*pi/T; which is called “fundamental frequency in radians per second”
Another important term is harmonic; The sin(nwot) or cos(nwot) is called nth harmonic of f(t).
(an) and (bn) are called “Fourier Coeficcients .” Our work is to find these coefficients to transform our time domain signal through frequency domain.
so ,let’s ask; Why we need Fourier Series of a function,why function itself is not enough for us ?
The fourier series of a periodic function f(t) is a representation that analyzes f(t) into a DC component and AC component comprising an infinite series of harmonic sinusoids.
Your fourier series must contain these conditions ;
1).f(t) is singlevalued everywhere in the domain
2).f(t) has a finite number of finite discontinuities in any one period .
3).f(t) has a finite number of maxima and minima in any one period .
As i say earlier , the process of determining the coefficients is called “Fourier Analysis” .
Let’s translate what we say into math !
I will share some trigonometric properties in the chart below , It tells you how to calculate ao,an and bn values for analysis .You will see that average values of sinusoids are zero in an interval.
Here is a useful Fourier Analysis video , that tells what it means to transform functions , A must see 3 video :
Fourier transform1
Fourier tranform2
Fourier Transform3
See al videos after reading my paper , See you next lesson
Take Care!
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