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 h English (en)[[File:Segf18.jpg|center|thumb|600px|FIG. F-18. (<b>a</b>) <b>Fourier analysis</b> involves finding the amplitude of frequency components for a waveform. The frequency-domain representation or spectrum ''G''(''f'') of a discrete time function ''g''<sub>''t''</sub> (waveform, seismic record trace, etc.) can be decomposed into a series of sinusoids by any of the following equivalent equations: <center><math>\begin{align}
g_{t} & =a_{0}/2+\sum [a_{n}\cos (2\pi f_{n}t)+b_{n}\cos (2\pi ft)] \\
& =c_{0}/2+\sum c_{n}\cos (2\pi f_{n}t-\gamma_{n})=\sum \alpha_{n} exp[j2\pi f_{n}t]
\end{align}</math></center>
Where <center><math>\begin{align}
a_{n} &=(2/T)\sum g_{i}\cos (2\pi f_{i}t),\\  b_{n} &=(2/T)\sum g_{i}\sin (2\pi f_{i}t), \\
c_{n} &=(2/T)\sum g_{i}\cos (2\pi f_{i}t-\gamma _{i}),\\  \gamma _{n}&=0,\; \gamma _{n}=\tan ^{-1}(b_{n}/a_{n}), \\ & n>0,\; \alpha =(2/T)\sum g_{i} exp[j2\pi f_{i}t]
\end{align}</math></center>
If <math>g(t)</math> is a continuous waveform, the sum signs become integrals. 
(<b>b</b>) <b>Fourier synthesis</b> involves superimposing the components to reconstitute the waveform. For an antisymmetric sawtooth waveform, the first four components are: 
<center><math>\sin x; -(1/2)\sin 2x; (1/3)\sin 3x; -(1/4)\sin 4x</math></center>.
For a Fourier transform the limits are <math>0</math> and <math>\pm \infty,</math> and <math>G(f)</math> and <math>g(t)</math> constitute a Fourier-transform pair; see Figure [[Special:MyLanguage/Dictionary:Fig_F-19|F-19]].]]
 h Spanish (es)[[File:Segf18.jpg|center|thumb|600px|FIG. F-18. (<b>a</b>) <b>Análisis de Fourier</b> implica encontrar la amplitud de los componentes de frecuencia de una onda. La representación en el dominio de la frecuencia o espectro ''G''(''f'') de una función en tiempo discreta ''g''<sub>''t''</sub> (onda, traza sísmica, etc.) puede ser descompuesta en una serie de sinusoides por cualquiera de las siguientes ecuaciones equivalentes: <center><math>\begin{align}
g_{t} & =a_{0}/2+\sum [a_{n}\cos (2\pi f_{n}t)+b_{n}\cos (2\pi ft)] \\
& =c_{0}/2+\sum c_{n}\cos (2\pi f_{n}t-\gamma_{n})=\sum \alpha_{n} exp[j2\pi f_{n}t]
\end{align}</math></center>
Donde <center><math>\begin{align}
a_{n} &=(2/T)\sum g_{i}\cos (2\pi f_{i}t),\\  b_{n} &=(2/T)\sum g_{i}\sin (2\pi f_{i}t), \\
c_{n} &=(2/T)\sum g_{i}\cos (2\pi f_{i}t-\gamma _{i}),\\  \gamma _{n}&=0,\; \gamma _{n}=\tan ^{-1}(b_{n}/a_{n}), \\ & n>0,\; \alpha =(2/T)\sum g_{i} exp[j2\pi f_{i}t]
\end{align}</math></center>
Si <math>g(t)</math> es una onda continua, los signos sumatoria se convierten en integrales.
(<b>b</b>) <b>Síntesis de Fourier</b> implica la superposición de componentes para reconstruir la forma de la onda. Para una onda diente de sierra antisimétrica, los primeros cuatro componentes son: 
<center><math>\sin x; -(1/2)\sin 2x; (1/3)\sin 3x; -(1/4)\sin 4x</math></center>
Para una transformada de Fourier los limites son <math>0</math> y <math>\pm \infty,</math> y <math>G(f)</math> y constituye un par transformada de Fourier; ver Figura [[Special:MyLanguage/Dictionary:Fig_F-19|F-19]].]]