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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:FIG.F-18.png|thumb|center|450px|FIG. F-18. (a) El análisis de Fourier involucra encontrar la amplitud de las componentes de frecuencia de una forma de onda. La representación del dominio de frecuencia o el espectro G(f) de una función discreta en el tiempo <math>g_t</math> (forma de onda, traza sísmica, etc.) puede ser descompuesta en una serie de sinusoidales con cualquiera de las siguientes ecuaciones equivalentes:


<center><math>g_t = a_{0}/2 + \sum \left [ a_{n} cos(2\pi f_{n}t) + b_{n}cos(2\pi ft)\right ]</math>


<center>= <math>c_{0}/2 + \sum {c_{n}cos(2\pi {f_{n}}t - \gamma _{n})} = \sum {\alpha _{n}}exp\left [ 2\pi f_{n}t \right ]</math>


donde  <math>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),</math>

<math>c_{n}=(2/T)\sum {g_{i}}  cos(2\pi f_{i}t - {\gamma _{i}}), \gamma _{n}= 0, </math>

<math>\gamma _{n}= tan^{-1}({b_{n}}/a_{n}),  n>0,  \alpha _{n}= (2/T)\sum g_{i} exp[j2\pi f_{i}t]</math>





.Si g(t) es una forma de onda continua, el signo de suma se convierte en una integral. (b) La síntesis de Fourier involucre superponer las componentes para restituir la forma de onda. Para una forma de onda antisimétrica con forma de cierra, las primeras cuatro componentes son:


<center><math>sin x ; -(1/2)sin 2x; (1/3)sin 3x; -(1/4)sin 4x .</math>



Para un transformada de Fourier los limites son 0 y 
<math>\pm \infty </math>  y G(f) y g(t) constituyen un par de una transformada de Fourier (ver figura F-19)]]