Dictionary:Fourier transform

F-19. Fourier transform pairs. The time functions on the left are Fourier transforms of the frequency functions on the right and vice-versa. Many more transform pairs could be shown. The above are all even functions and hence have zero phase. Transforms for real odd functions are imaginary, i.e., they have a phase shift of +π/2. Transforms of functions that are neither odd nor even involve variations of phase with frequency. Note f=1/t.

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

$g(t)\leftrightarrow G(f)=\int \limits _{-\infty }^{\infty }g(t)e^{-j2\pi ft}dt=\int \limits _{-\infty }^{\infty }g(t)\cos(j2\pi ft)dt-j\int \limits _{-\infty }^{\infty }g(t)\sin(j2\pi ft)dt$

The inverse transform is

$g(t)=\int \limits _{-\infty }^{\infty }G(f)e^{j2\pi ft}dt=\int \limits _{-\infty }^{\infty }G(f)\cos(j2\pi ft)dt+j\int \limits _{-\infty }^{\infty }G(f)\sin(j2\pi ft)dt$

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

$G(f)=\left\vert A(f)\right\vert e^{j\gamma (f)}$

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

$A(f)={\sqrt {[{\text{real part of }}G(f)]^{2}+[{\text{imaginary part of }}G(f)]^{2}}}$ $\gamma (f)=\arctan {\bigg (}{\frac {{\text{imaginary part of }}G(f)}{{\text{real part of }}G(f)}}{\bigg )}$

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:

$h(t)=\sum _{n=0}^{\infty }a_{n}\cos(2\pi n({\frac {t}{T}}))+\sum _{n=1}^{\infty }b_{n}\sin(2\pi n({\frac {t}{T}}))$

where

$a_{n}={\frac {2}{T}}\int \limits _{0}^{T}h(t)\cos(2\pi n({\frac {t}{T}}))dt,$ $b_{n}={\frac {2}{T}}\int \limits _{0}^{T}h(t)\sin(2\pi n({\frac {t}{T}}))dt,$

and

$h(t)\leftrightarrow H_{n}=\left\vert A_{n}\right\vert e^{j\gamma _{n}}$ $A_{n}={\sqrt {a_{n}^{2}+b_{n}^{2}}},\gamma _{n}=\arctan {\Bigg (}{\frac {b_{n}}{a_{n}}}{\Bigg )}$

The same rules for quadrants apply to γ_n as expressed for γ(f); a₀ 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 t₂, then we can stop summing when n>2T/t₂ (see sampling theorem). In this case a_n and b_n can be expressed as sums:

$a_{n}={\frac {2}{T}}\sum _{n=0}^{2{\frac {T}{t_{2}}}}h_{t}\cos(2\pi n({\frac {t}{T}}))$

and

$b_{n}={\frac {2}{T}}\sum _{n=1}^{2{\frac {T}{t_{2}}}}h_{t}\sin(2\pi n({\frac {t}{T}}))$

Also see phase response and fast Fourier transform.

F-20. Equivalence of Fourier transform operations. Doing the time operation is equivalent to doing the frequency operation on the transform of the data. Note: g(t)↔G(f) and h(t)↔H(f).

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,

$G(k,\omega )=\int \limits _{-\infty }^{\infty }\int \limits _{-\infty }^{\infty }g(x,t)e^{-j(kx+\omega t)}dxdt$

and

$g(x,t)={\frac {1}{\pi }}\int \limits _{-\infty }^{\infty }\int \limits _{-\infty }^{\infty }G(k,\omega )e^{j(kx+\omega t)}d\omega dk$

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). Theorems relating to Fourier transforms are shown in Figure F-22.

F-22. Fourier transform theorems.

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 $i={\sqrt {-1}}$ rather than $j$. Electrical engineers prefer to use the letter $j$ because the letter $i$ is reserved for current. Mathematicians and mathematical physicists prefer using the angular frequency $\omega =2\pi f$ where the units are radians per time, rather than cycles per time. This means that there may be a factor of $2\pi$ 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 $t=0,$ the forward temporal transform integration will start at $0$ rather than $-\infty$.

1D transforms

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

${\tilde {f}}(\omega )=\int _{-\infty }^{\infty }f(t)e^{i\omega t}\;dt$

and the inverse Fourier transform

$f(t)={\frac {1}{2\pi }}\int _{-\infty }^{\infty }{\tilde {f}}(\omega )e^{-i\omega t}\;d\omega .$

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

${\tilde {f}}(k)=\int _{-\infty }^{\infty }f(x)e^{-ikx}\;dx$

and the inverse spatial Fourier transform is similarly different

$f(t)={\frac {1}{2\pi }}\int _{-\infty }^{\infty }{\tilde {f}}(k)e^{ikx}\;dk.$

Transforms in $n$ spatial dimensions

In $n$ dimensions, the spatial transforms become

${\tilde {f}}(\mathbf {k} )=\int _{-\infty }^{\infty }...{\mbox{n total integrations}}...\int _{-\infty }^{\infty }f(x)e^{-i\mathbf {k} \cdot \mathbf {x} }\;d\mathbf {x}$

and the inverse spatial Fourier transform is similarly different

$f(\mathbf {x} )={\frac {1}{(2\pi )^{n}}}\int _{-\infty }^{\infty }...{\mbox{n total integrations}}...\int _{-\infty }^{\infty }{\tilde {f}}(\mathbf {k} )e^{i\mathbf {k} \cdot \mathbf {x} }\;d\mathbf {k} .$

Here we have used the conventions that $\mathbf {x} =(x_{1},x_{2},...,x_{n}),$ $d\mathbf {x} =dx_{1}dx_{2}...dx_{n},$ $\mathbf {k} =(k_{1},k_{2},...,k_{n}),$ $d\mathbf {k} =dk_{1}dk_{2}...dk_{n},$ and $\mathbf {k} \cdot \mathbf {x} =k_{1}x_{1}+k_{2}x_{2}+...+k_{n}x_{n}.$

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 $(\mathbf {x} ,t)\rightarrow (\mathbf {k} ,\omega )$

${\tilde {F}}(\mathbf {k} ,\omega )=\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }\int _{0}^{\infty }f(\mathbf {x} ,t)e^{-i(\mathbf {k} \cdot \mathbf {x} -\omega t)}\;dt\,d^{3}x$

and the corresponding inverse Fourier transform

$f(\mathbf {x} ,t)={\frac {1}{(2\pi )^{n+1}}}\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }\int _{\infty }^{\infty }{\tilde {F}}(\mathbf {k} ,\omega )e^{i(\mathbf {k} \cdot \mathbf {x} -\omega t)}\;d\omega \,d^{3}k.$

Here, $d^{3}x=dx_{1}dx_{2}dx_{3}$ and $d^{3}k=dk_{1}dk_{2}dk_{3}.$