# Dictionary:Fourier transform

Formulas that convert a time function *g*(*t*) (waveform, seismic record trace, etc.) into its frequency-domain representation *G*(*f*) and vice versa. *G*(*f*) and *g*(*t*) constitute a Fourier-transform pair; see Figure F-19 below. An example is

The inverse transform is

Finding *G*(*f*) from *g*(*t*) is called Fourier analysis and finding *g*(*t*) from *G*(*f*) is called Fourier synthesis. *G*(*f*) is the complex spectrum, the real part being the cosine transform and the imaginary part the sine transform whenever *g*(*t*) is real. Another expression for *G*(*f*) is

where the functions *A*(*f*) and γ(*f*) are real. They are, respectively, the amplitude spectrum and the phase spectrum of *g*(*t*):

γ(*f*) is in the first or second quadrant if the imaginary part is positive, in the first or fourth quadrant if the real part is positive. A record trace *h*(*t*) that extends only from 0 to *T* may be assumed to be repeated indefinitely and so expanded in a Fourier series of period *T*:

where

and

The same rules for quadrants apply to γ_{n} as expressed for γ(*f*); *a*_{0} is the zero-frequency component (or dc shift). The frequency spectrum is discrete if the function is periodic. If *h*_{t} is a sampled time series sampled at intervals of time *t*_{2}, then we can stop summing when *n*>2*T*/*t*_{2} (see sampling theorem). In this case *a*_{n} and *b*_{n} can be expressed as sums:

and

Also see phase response and fast Fourier transform.

Operations in one domain have equivalent operations in the transform domain (see Figure F-20). Computations can sometimes be carried out more economically in one domain than the other and Fourier transforms provide a means of accomplishing this. The Fourier-transform relations can be generalized for more than one dimension (see Figure F-21). For example,

and

The 1/4π factor is sometimes distributed between the two integrals; where calculations involve an arbitrary scaling factor, the 1/4π factor may be dropped entirely.

Fourier transforms are discussed in Sheriff and Geldart (1995, 277, 532–533)^{[1]}. Theorems relating to Fourier transforms are shown in Figure F-22.

## Contents

## Notations and sign conventions

The notations and sign conventions used above are common in the electrical engineering world. Many exploration geophysicists may be familiar, however, with the conventions used by physicists and mathematicians. These differ subtly from those used above. Many geophysicists whose education may in those fields, or who draw from the scientific results from those fields prefer conventions different from those of electrical engineers. It is, therefore, important to include a short discussion of these conventions.

It is more common in the world of mathematics and theoretical physics for the convention of rather than .
Electrical engineers prefer to use the letter because the letter is reserved for current. Mathematicians and mathematical
physicists prefer using the angular frequency where the units are radians per time, rather than cycles per time. This means
that there may be a factor of discrepancy between computations using the differing conventions. Finally, there may be different sign
conventions on the exponent of the exponentials in the Fourier transform definitions. Because signals recorded in the space-time domain are *causal,* meaning
that there are no arrivals before time the forward temporal transform integration will start at rather than .

## 1D transforms

Putting all of these together, we obtain a common notational convention for the *forward* Fourier transform in time as

and the *inverse* Fourier transform

The 1D spatial forward Fourier transform differs from the temporal transform in that the integration is on infinite limits and the sign of the exponent in the exponential is negative

and the *inverse* spatial Fourier transform is similarly different

## Transforms in spatial dimensions

In dimensions, the spatial transforms become

and the *inverse* spatial Fourier transform is similarly different

Here we have used the conventions that and

## Transforms in 1 temporal and 3 spatial dimensions

In 3 dimensions of space and 1 dimension of time, as is encountered in problems dealing with the wave equation, we have the forward Fourier transform from

and the corresponding inverse Fourier transform

Here, and

## References

- ↑ Sheriff, R. E. and Geldart, L. P., 1995, Exploration Seismology, 2nd Ed., Cambridge Univ. Press.