Sunday, July 5, 2009

The poetic beauty of Fourier series

Mathematics is the essence of life, which has helped in all the human endeavors by establishing an order among the mystical set of parameters governing the universe. There are umpteen instances in nature which explicitly display the presence of mathematics making us believe that god created the universe based on the result of an infinite set of complicated equations. To give you a whiff of what I intend to say, I will give you an example of a beautiful equation.

Generalized Fourier series:

Y(t) = A1*X1(t) + A2*X2(t) + A3*X3(t) + ……………………… + An*Xn(t);
where the vectors Xr(t) are independent with each other.

The equation simply means that any signal can be represented by weighted sum of the N number of independent vectors. These vectors can uniquely represent any kind of signal. For a layman the equation is simple in its essence and possibly may not interest him. However looking deeper, it is a beautiful piece of poem expressed not in words but in terms of variables.
Essence of the poem:

* Every human being is a wonderful creation of nature with unique characteristics that define him and only himself.
* As in the case of the equation, the number of parameters should tend to infinity to completely describe the person.
* A list of these parameters would include the physical, emotional and intellectual aspects of a person.
* The low frequency contents, which contain most of the energy, are the ones given most importance in mathematics. In humanistic terms the low frequency content represents the obvious parameters such as the height, weight, accomplishments in life etc.
* These high frequency contents are analogous to the finer aspects of a human being and are the ones that truly define a person’s identity and his wonderful idiosyncrasies. They are generally overshadowed by the other cruder parameters (in cases such as compression in mathematical terms).

For those who in the same frequency band in which I am in would be able to enjoy the essence of this article and for others my advice is: Try to modulate or demodulate yourself to required frequency band to enjoy the spirit of mathematics.