Friday, 11 July 2014

FOURIER TRANSFORMS INTUITIVELY EXPLAINED




|||| Now each color of light has a distinct frequency, isnt it? 

|||| Now, lets say we want to arrange these colors in increasing order of their frequencies.

|||| It will look like this.,......below.

|||| Now lets see what we did around here,

|||| Now, lets take just one of them,...hmmmm.. RED for example.

|||| x(t) is the waveform which is a function of time.

|||| When this waveform i.e x(t) is rotated in a horizontal plane......

YES....
e^j---------> Rotation in the anticlockwise direction in a horizontal plane.
e^-j---------> Rotation in the clockwise direction in a horizontal plane. 

(By the way....e^i --------> Rotation in the vertical plane.)
                    

|||| Now e^ j(2.pie.freq.time) = cos(2.pie.freq.time) + j.sin(2.pie.freq.time)

        Where....cos(2.pie.freq.time) = Horizontal component of the waveform.
             and... sin(2.pie.freq.time) = Vertical component of the waveform.

|||| So now,.....if you are removing the horizontal as well as the vertical component component from a wave what remains?

|||| Well....lets see. Every wave can be expressed in the form A.e^(j0) . Now if you remove e^(i0) by dividing it by itself what remains? Only A right and A is the amplitude of the wave.

|||| So basically, when you divide a waveform by all its horizontal and vertical components,...you are always and always left only with its amplitude.

|||| This amplitude is what you plot in the frequency spectrum. You plot the amplitude at that point on the X-axis which co-incides with the frequency of the wave.

|||| Now lets say you want to plot the entire vibgyor on the frequency spectrum. So now for that you will have to keep on adding all the amplitudes of all the different colors onto the frequency spectrum graph.

|||| Integration is another term for adding.

|||| So,... Integration of (e^(2.pie.freq.time)) = adding all the amplitudes of all of colors (from Red to violet) one after the other(This is synonymus to plotting them on a graph one after the other).

|||| So basically if there was an algorithm for fourier transforms....it would be as follows..... 

Another way of putting it .......


ALTERNATE USEFUL EXPLANATION THROUGH WIKIPEDIA IMAGE
NOW WHAT IS THE APPLICATION OF FOURIER TRANSFORMS???????

|||| Truth being told, I myself dont know it fully.

|||| The below image from wikipedia is the best explanation I could get.




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ABOUT
I hardly understood Math in school. In fact was on the verge of dropping the subject I loved the most because as much as I loved the theory of it, I could not understand the math involved in it. However I loved the subject too much to be able to live without. Hopelessly, I was continuing my love-affair with it. 


Then one day....a miracle happened,....while applying a certain formula again and again.....I came to know its significance. Slowly and steadily....other equations also started clicking. I got to see a strong relationship between Maths and the Physics it was pointing towards. They both were the same. Maths was just an easy language to express a physical phenomenon. And a bit more to that in the sense that it could even predict the behaviour of a certain physical phenomenon. Equations now as if came to life.

 Every equation now had as if something to say. A burning urge to share these things with the world aflamed within me. So that no one has to give up the subject that he or she loves the most. 
The book on visualizing maths thus got written as a sprout of inspiration. The blog followed. Both these are dedicated to you and all such similar minds searching for answers.


Visualizing Math & Physics is a blog dedicated to you and all such similar minds searching for answers.


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