Fourier Synthesis of Periodic Waveforms 

Theory 

According to the important theorem formulated by the French mathematician Jean Baptiste Joseph Baron Fourier, any periodic function, no matter how trivial or complex, can be expressed in terms of converging series of combinations of sines and/or cosines, known as Fourier series. Therefore, any periodic signal is a sum of discrete sinusoidal components.

   Jean Baptiste Joseph
Baron Fourier (1768-1830) 

The Fourier theorem is fairly general and also applies to periodic functions that have discontinuities and cannot be represented by a single analytic expression. For a periodic function f(x), provided that the following conditions are satisfied (Dirichlet conditions): 

(a)    f(x) is defined and single-valued except (perhaps) at a finite number in (-T, T),
(b)    f(x) is periodic outside (-T, T) with period 2T,
(c)    f(x) and f΄(x) are piecewise continuous in (-T, T), then f(x) can be expressed by the following series: 

where

Although the calculation of a₀, a₁, b₁, a₂, b₂, … is a mathematically straightforward process, it may become rather tedious depending on the complexity and the discontinuities of f(x). The Fourier theorem is particularly useful in scientific instrumentation, where f(x) functions may represent actual periodic signals f(t). Some simple examples of Fourier series are those of square, triangular and sawtooth waveforms: 

Square waveform

Triangular waveform

Sawtooth waveform

Applet 

This applet demonstrates the gradual formation of a periodic function by successive additions of sinus and/or cosinus terms (i.e. in the aforementioned definition of f(x), n becomes: 1, 2, …). The user can select up to 8 different periodic waveforms, and starting from the first term he/she can add the subsequent terms one after the other. In the upper plot area the current sum of terms is shown. In the same plot (in faint red color} the previous sum is also shown for comparison. In the lower plot area, the next sinusoidal term to be added (point by point) is displayed. It is of interest to note that waveforms containing sharp transitions (e.g. the square or the sawtooth waveforms) require many more terms in order to be synthesized with some fidelity. This happens because these signals are "rich" in high frequencies and the a and b coefficients do not quickly converge to zero. The absence (or clipping) of these high order terms (i.e. the high frequency components in real signals) is testified by the "ringing" effect close to these transitions. On the other hand, waveforms with relatively smooth transitions (e.g. the triangular or the fully rectified sinusoidal waveforms) require very few terms to be accurately synthesized.