Joseph Fourier submitted a paper in 1807 to the Academy of Sciences of Paris. The paper was a mathematical description of problems involving heat conduction, and was at first rejected for lack of mathematical rigour. However, it contained ideas which have developed into an important area of mathematics named in his honour, Fourier analysis.
According to the theory developed by Fourier, any periodic function F(t), with period T, may be represented by an infinite series of the form
where the coefficients a0, an, and bn for a given periodic function F(t) are calculated by the formulas
It should be noted that the first coefficent a0 is twice the average of the function F(t) over one period. This series is called the Fourier series and the coefficents are called the Fourier coefficients.
The Fourier series expansion of a continuous and periodic waveform provides a means of expanding a function into its major sine / cosine or complex exponential terms. These individual terms represent various frequency components which make up the original waveform.
A line graph of the amplitudes of the Fourier series components can be drawn as a function of frequency. Such a graph is called a spectrum or frequency spectrum.
The periodic time T defines completely which spectral components occur in the spectrum as f0 = 1 / T. The component f0 is called the fundamental frequency or first harmonic. All the higher components are multiples of f0 and are called the higher harmonics. The smallest spacing that can occur between frequency components in the spectrum is f0 = 1 / T.
Near a point where f has a jump discontinuity, the partial sums of the Fourier series exhibit very interesting and unusual behaviour known as the Gibbs phenomenon. Taking a partial sum of the Fourier series of a function with a jump discontinuity will result in relatively high peaks near the discontinuity. Intuition might suggest that these peaks would decrease as the number of terms included in the partial sum was increased, but this is not the case.
These figures were generated using a MATLAB M-file written to illustrate this phenomenon. The first figure is the approximation of a square wave using 2 fourier series terms, and the second figure is the approximation when 16 terms are included. You should download the M-file and experiment with the number of terms (see the M-file list for instructions on how to download and use the M-file).
The amplitude of these peaks approach a limit which is approximately 9 percent more than the magnitude of the jump at the discontinuity.
By using Euler's formula to derive complex expressions for sin(t) and cos(t), and substituting these into the Fourier series it can be shown that the complex form of the Fourier series is
where the coefficents cn are the complex Fourier coefficents.
