# Fourier synthesis

Fourier synthesis is a method of electronically constructing a signal with a specific, desired periodic waveform . It works by combining a sine wave signal and sine-wave or cosine-wave harmonics (signals at multiples of the lowest, or fundamental, frequency) in certain proportions. The scheme gets its name from a French mathematician and physicist named Jean Baptiste Joseph, Baron de Fourier, who lived during the 18th and 19th centuries.

Many waveforms represent signal energy at a fundamental frequency and also at harmonic frequencies (whole-number multiples of the fundamental). The relative proportions of energy concentrated at the fundamental and harmonic frequencies determine the shape of the wave. The wave function (usually amplitude , frequency , or phase versus time ) can be expressed as of a sum of sine and cosine functions called a Fourier series , uniquely defined by constants known as Fourier coefficient s. If these coefficients are represented by *a* _{} , *a* _{1} , *a* _{2} , *a* _{3} , ..., *a _{n}* , ... and

*b*

_{1},

*b*

_{2},

*b*

_{3}, ...,

*b*, ..., then the Fourier series

_{n}*F*(

*x*), where

*x*is an independent variable (usually time), has the following form:

*F* ( *x* ) = *a* _{} /2 + *a* _{1} cos *x* + *b* _{1} sin *x* + *a* _{2} cos 2 *x* + *b* _{2} sin 2 *x* + ...

+ *a _{n}* cos

*nx*+

*b*sin

_{n}*nx*+ ...

In Fourier synthesis, it is necessary to know, or to determine, the coefficients *a* _{} , *a* _{1} , *a* _{2} , *a* _{3} , ..., *a _{n}* , ... and

*b*

_{1},

*b*

_{2},

*b*

_{3}, ...,

*b*, ... that will produce the waveform desired when "plugged into" the generalized formula for the Fourier series, as defined above. Then, sine and cosine waves with the proper amplitudes (as defined by the coefficients) must be electronically generated and combined, up to the highest possible value of

_{n}*n*. The larger the value of

*n*for which sine-wave and cosine-wave signals are generated, the more nearly the synthesized waveform matches the desired waveform.

Fourier synthesis is used in electronic music applications to generate waveforms that mimic the sounds of familiar musical instruments. It is also employed in laboratory instruments known as waveform generators or function generators. These devices are used to test communication systems.

Compare Fourier analysis .