Why did Fourier choose sinusoids as the basis functions in Fourier series?

[klen]

Fourier said that any periodic signal can be represented as sum of harmonics i.e., containing frequencies which are integral multiples of fundamental frequncies. Why did he chose the basis functions i.e., the functions which are added to make the original signal to be sinusoidal? I know sinusoids are orthogonal functions but could we use some other basis functions for this representation?

 

[milesyoung]

Fourier said that any periodic signal can be represented as sum of harmonics i.e., containing frequencies which are integral multiples of fundamental frequncies. Why did he chose the basis functions i.e., the functions which are added to make the original signal to be sinusoidal? I know sinusoids are orthogonal functions but could we use some other basis functions for this representation?

Well, I think Fourier saw the value in being able to express any periodic function as a sum of very simple terms. This is, for instance, extremely practical if you're solving linear differential equations (like Fourier did with the heat equation). Due to the property of superposition, you can solve your differential equation "one simple term at a time" and them recombine all these to form a solution to your original problem. This is a very powerful tool to have at your disposal, especially since you don't have to include all the terms of the Fourier series, i.e. you could find an approximate solution using a partial sum instead.

There are other sets of functions that you can use to form an analog to the Fourier series, but the Fourier series is, arguably, "special" in its mathematical simplicity. Sines and cosines have a tendency to pop up naturally in a lot of places and they're very easy functions to work with.

 

[MagicianT]

There are Walsh series, essential part of encrypted digital communications
http://en.wikipedia.org/wiki/Walsh_function

 

[analogdesign]

In practice decaying exponentials are used instead of raw sinusoids (although this is just a book-keeping convention as either formulation can be recast as the other).

It isn't enough for the functions to be orthogonal. They also have to span the space. Any good book for a second course on signals and systems will talk about this.

 

[jim hardy]

I remember reading a Hewlett Packard instruction manual for a function generator that described in great detail how it synthesized sine waves from square waves.
Ever since i retired I've sought a reference to that instrument.

But:
As i get older, the number things i can remember grows, and the number of things i can imagine grows also,
but the distinction between them grows less clear .

So I too await an answer to your query .

 

[analogdesign]

I remember reading a Hewlett Packard instruction manual for a function generator that described in great detail how it synthesized sine waves from square waves.
Ever since i retired I've sought a reference to that instrument.

That one's easy. A narrowband filter to pick out on of the harmonics (or dominant) tone in the square wave will do the trick. ;)

I suspect you meant synthesize square waves from sine waves. Since to get a square wave you add as many odd harmonics of the final wave as needed (keeping in mind Gibb's phenomenon will limit the purity of the square wave) the speed you need determines the method. At slow speeds, instruments keep the square wave in ROM (or RAM) and play it back. At faster speeds they can use sinewave ROMS and then offset them with frequency doublers and add them back in. Fascinating stuff.

 

[jim hardy]

No, that's why the gizmo was so fascinating - it went the other way.

Found a patent that describes a technique
but they have to low - pass to remove last 3% of harmonics

https://www.google.com/patents/US3512092

The theoretical basis of the synthesis of sine waves from square waves will now be described.

A square wave of angular repetition frequency to has the following components:

Sin wt+ /a sin 3 wt+ /5 sin 5 wt sin 7 wt+% sin wt-I- where the amplitude of the fundamental component, sin wt is one. Similarly the components of a square wave of amplitude one third of the fundamental component of the first square wave and three times its frequency are:

/3 sin 3 wt+ ,4 sin 9 wt+ sin 15 wt and those of a square wave of a fifth of the amplitude of the fundamental component of the first square wave and five times its frequency are:

/5 sin 5 wt+ sin 15 wt If the second square wave, i.e. that of angular frequency 3w, is subtracted from first or primary square wave no components of angular frequencies 3w, 9w and 15w will be present in the resultant signal. Similarly the components of angular frequency SW and 7w can be eliminated by taking the third square Wave, i.e. that of angular frequency SW, and one of angular frequency 7w from the first square wave. The process cannot be carried out indefinitely since some high frequency components do not continue to cancel. For example subtraction of the third square 'wave restores a component at angular frequency 15w which was eliminated by subtracting the second wave. However by subtracting square waves of angular frequency 3w, SW and 7w a sine wave with harmonic below 3% can be produced. 'Further harmonics may be removed by filters. Since the remaining harmonies are well removed in frequency from the fundamental and are already of low level in comparison with the fundamental, the filters required present little difliculty of design and require only a comparatively low loss at the frequency to be rejected. Moreover a single filter will suflice even where the fundamental frequency varies over a range of 2 to 1 or more.

block diagram here but it's too wide to post
https://patentimages.storage.googleapis.com/pages/US3512092-0.png

might work well with a switched capacitor filter?

Here's a clever one that works as you suggested, remove harmonics from a square wave with a sharp filter:
http://electronicdesign.com/analog/standalone-circuit-converts-square-waves-sine-waves

looks like a pure digital synthesis is not possible, just an approximation that can be refined by old fashioned filtering.

This is more than i was able to find a few years ago.

Thanks, guys !

old jim

 

[f95toli]

Fourier said that any periodic signal can be represented as sum of harmonics i.e., containing frequencies which are integral multiples of fundamental frequncies. Why did he chose the basis functions i.e., the functions which are added to make the original signal to be sinusoidal? I know sinusoids are orthogonal functions but could we use some other basis functions for this representation?

Sine is a nice easy, function to use; but there are many others can that can -and ARE- used.
Any book on the more "general" theory of Fourier analysis will introduce you to a whole families of functions are can be used to e.g. solve differential equations and/or in signal analysis.
You also have things like Wavelet's that can used.