Saturday, 21 December 2013

The legacy of Jean Baptiste Joseph Fourier

Vladimir Nabokov, a Russian novelist once said:
"The breaking of a wave cannot explain the whole sea"

I tend to disagree and today I'm gonna try and convince you that Vladimir couldn't be any further from the truth. To prove my point I would like to call the Ghost of Jean Baptiste Joseph Fourier into the witness box.

ME: Tell us Joseph, do you agree with Vladimir?

JOSEPH: Well...I think our friend's remark might explain a lot of philosophical questions as it is indeed true that in order to get the complete picture it is simply not enough to understand a few bits and pieces of it. To be called the master of one's creation, a person has to understand all the pieces of the puzzle and how they fit together.

ME: Did Vladimir pay you for saying all this?

JOSEPH: Oh no, no! I was just getting to the point that while his words have great value in the world of Philosophy, it stands no chance in the ruthless world of mathematics. Through more than 200 years of study, initiated by my work on 'Fourier Series', it has been rigorously proved that periodic signals can be expressed as a sum of very simple oscillating functions, namely sines and cosines (or complex exponential) each oscillating at an integer multiple of the fundamental frequency.

ME: Did you copy that line from Wikipedia or something? Anyways, continue.

JOSEPH: If you have a continuous time periodic function \(f(x)\) (satisfying some criteria for convergence, of course) with period 2L, then it can be written as a linear combination or a weighted sum of complex exponential functions as follows
\begin{equation}
f(x)=\sum_{n=-\infty}^{\infty}A_n e^{\frac{\iota 2 \pi n x}{2 L}} \\
where \: A_n = \frac{1}{2L} \int_{-L}^{L}f(x)e^{\frac{\iota 2 \pi n x}{2 L}} dx
\end{equation}
ME: Dude, you're talking French.

JOSEPH: Let me explain with the help of an example. Consider a square wave with unit amplitude and a period of 2 seconds. A quick calculation shows that the Fourier Series expansion of this particular signal is given by
\begin{equation}
f(x)=\sum_{n=1}^{\infty} \frac{2}{\pi n}(1-(-1)^n) sin({\frac{ \pi n x}{ L}})
\end{equation}
ME: Yeah...right, that would've been a very quick calculation :p

JOSEPH: You can take a look at both my code and calculations if you are inquisitive enough. Click here, c'mon now don't be shy!

ME: Alright I got the calculation but what's the meaning of all this.

JOSEPH: I'm glad you asked that. What the above equation says, is that if you know the Fourier coefficients, not one or two, but all of them then you know the signal completely and it is possible to reconstruct the signal from them.

ME: But doesn't that justify Vladimir's statement then....the need to know all the pieces of the puzzle....all the waves in the sea :-(

JOSEPH: Not quite :-). Look at the image below where I've taken the sum of the first few terms in the Fourier expansion. I want you to notice that as the number of terms are increased from 10 to 1000 the quality of the reconstructed signal increases greatly and I can bet on the Eiffel Tower that with just a 1000 coefficients you won't be able to distinguish between a true square wave and a reconstructed one with just your naked eye.


ME: Ok...that kinda makes sense. But why is there an overshoot at the point of discontinuity.

JOSEPH: Ahem....I was.....umm....I was trying to avoid this discussion but now that you have asked, it is because of something called the Gibbs phenomenon. It happens whenever you try to take the partial sum of the Fourier series of a piecewise continuous signal. Don't give me that look....it is not my mistake. I bet if the computers you've made had infinite precision and were capable of adding the infinitely many terms in the series then the results would have been perfect.

ME: Alright Joseph, but you know there is a fundamental problem. Many signals in nature are continuous time but we must use digital computers to process them, so what use is all that you've told us.

JOSEPH: I think I must leave now for the gates of heavens are closing. Some bloke called Einstein is going to deliver a lecture on some hypothetical theory of Relativity. But two of my proponents Prof. Alan. V. Oppenhiem (MIT) and Prof. Ronald. W. Schafer (Hewlett Packard) have written a book on Digital Signal Processing which addresses this issue in great details.

ME: Your Honor, my witness has given enough evidence to suggest that, breaking a signal into its component harmonics often gives valuable insight about the signal itself. Even today, in applications like speech recognition and image compression Fourier Coefficients are used to encode the signals. With this note I rest my case.

In my next blog, I will skip a few steps i.e. the subtle relationship between Fourier Series, Discrete Time Fourier Transform and Discrete Fourier Transform and really get into how breaking a signal into component frequencies finds use in noise filtering. I know I promised to do it in this blog itself but some foundation had to be established. Anyways its my blog and I can change my mind whenever I want :p.

Despite all my efforts to be accurate and technically correct, some mistakes might have crept in. Please feel free to point them out in the comments bellow.

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