Fourier Series

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

with A_n and B_n 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 D_n and the phase angles φ_n:

φ_n = tan^-1 (B_n/A_n).

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 f_n:

f_n = nf₀

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)