Fundamentals of Statistics contains material of various lectures and courses of H. Lohninger on statistics, data analysis and here for more.

Fourier Series

Any periodic signal y(t) may be constructed from an infinite sum of sine and cosine terms:

with An and Bn being the Fourier coefficients, and T the cycle time of the function y(t). The coefficient A0 represents the aperiodic fraction of the signal (i.e. A0 is equal to the average amplitude of the signal). Alternatively, the signal y(t) may be described by the magnitudes Dn and the phase angles φn:

where the magnitude and the phase angle can be calculated from the Fourier coefficients as follows:

φn = tan-1 (Bn/An).

The following  interactive example  shows you how to combine sines and cosines to form a signal y(t). By setting the coefficients Ai and Bi one can immediately see their effect on the resulting function. Click the image to start the simulation.
The set of cosine and sine functions is complete and orthogonal, which guarantees that any periodic function f(t) can be represented, and that the Fourier coefficients are independent of each other.

The index n in the equations above can be assigned to a specific frequency fn:

fn = nf0

where the fundamental frequency f0 is related to the cycle time T of the function y(t). Thus, the signal y(t) can be represented by two (in general infinitely broad) spectra simply by plotting the coefficients An and Bn against n. The resulting spectra are called the real (An) and the imaginary (Bn) part of the Fourier spectrum.

In practical settings one usually takes the magnitude and the phase angle to get the magnitude spectrum and the phase spectrum. If we plot the square of the magnitude against the frequency we get the power spectrum of the signal.

Examples of Fourier series

The following table shows a few examples of simple periodic functions. The displayed magnitude spectra are normalized to a maximum value of 1.0 and ignore the actual signal amplitudes.

Function Fourier series Magnitude spectrum (normalized)
square signal
sawtooth function
half wave sine
(full wave rectifier)