User:IMB2007/FourierSeries

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Consider the function f given by f(x) = x for x from −π to π. Next extend this function to the entire real line i.e. (-\infty,\infty), by repeating it in either direction. The new function, which we will also call f, is called the "sawtooth wave" or "sawtooth function". Clearly, it is periodic of period 2π.

Plot of a periodic function defined by f(x) = x in [ − π,π] - a sawtooth wave.
Plot of a periodic function defined by f(x) = x in [ − π,π] - a sawtooth wave.

Note that this (infinitely extending) periodic function f is not continuous (it breaks every 2π units; however, it nonetheless has a Fourier expansion. This is given by:

f(x)=a_0 + \left[\sum_{n=1}^{\infty}a_n\sin\left(nx\right)\right]
=2\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n} \sin(nx), \quad \ \mathrm{for \ all\ }x\in [-\pi,\pi].

The first five terms of this expansion will look like:

f(x) = ( − 1)22sin(x) + ( − 1)32 / 2sin(2x) + ( − 1)42 / 3sin(3x) + ( − 1)52 / 4sin(4x) + ( − 1)62 / 5sin(5x)
= 2\left[\sin(x) - 1/2\sin(2x) + 1/3\sin(3x) - 1/4\sin(4x) + 1/5\sin(5x)\right]

As one begins to add, these ("pure tone") sine terms, the sum on the right hand side above begins to look more and more like the saw-tooth function, although, clearly, it is still an approximation. See the diagram below for an animation.

Animated plot of the first five successive partial Fourier series
Animated plot of the first five successive partial Fourier series

The next diagram shows the sum of the first 25 terms. Notice that even though the approximation is continuous, in contrast to the original sawtooth wave which has breaks, it looks like a pretty good approximation.

Animation of the additive synthesis of a sawtooth wave with an increasing number of terms (up to 25) of the Fourier expansion.
Animation of the additive synthesis of a sawtooth wave with an increasing number of terms (up to 25) of the Fourier expansion.

[edit] Square Wave

Sine, square, triangle, and sawtooth waveforms
Sine, square, triangle, and sawtooth waveforms
Animation of the additive synthesis of a square wave with an increasing number of sin(nt) terms for n odd, up to 25
Animation of the additive synthesis of a square wave with an increasing number of sin(nt) terms for n odd, up to 25
For other senses of the word, see square.

A square wave is a basic kind of non-sinusoidal waveform encountered in electronics and signal processing. An ideal square wave alternates regularly and instantaneously between two levels. Square waves are universally encountered in digital switching circuits and are naturally generated by binary (two-level) logic devices.

In contrast to the sawtooth wave, which contains sin(nx) terms for all n, the square wave contains only terms for odd values of n.

Using Fourier series we can write an ideal square wave as an infinite series of the form

x_{\mathrm{square}}(t) = \frac{4}{\pi} \sum_{k=1}^\infty {\sin{\left ((2k-1)t \right )}\over(2k-1)} = \frac{4}{\pi}\left[\sin(t)+{1\over3}\sin(3t)+{1\over5}\sin(5t) + ...\right]


[edit] Some Killer Whale Songs for Possible Fourier Analysis

See also: List of whale songs