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![]() | Waves and Fourier sums | ![]() |
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Let us review some basic facts about
two-dimensional Fourier transform.
A two-dimensional function is represented in a computer as numerical
values in a matrix, whereas
a one-dimensional Fourier transform in a computer
is an operation on a vector.
A 2-D Fourier transform can be computed by a sequence
of 1-D Fourier transforms.
We can first transform each column vector of the matrix and then
each row vector of the matrix.
Alternately, we can first do the rows and later do the columns.
This is diagrammed as follows:
The diagram has the
notational problem that we cannot maintain
the usual convention of using a lower-case letter for the domain
of physical space and an upper-case letter for the Fourier domain,
because that convention cannot include
the mixed objects and
.
Rather than invent some new notation, it seems best to let the reader
rely on the context:
the arguments of the function must help name the function.
An example of two-dimensional Fourier transforms on typical deep-ocean data is shown in Figure 4.
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plane4
Figure 4. A deep-marine dataset ![]() ![]() |
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Altogether, the
two-dimensional Fourier transform
of a collection of seismograms
involves only twice as much computation as the one-dimensional
Fourier transform of each seismogram.
This is lucky.
Let us write some equations to establish that the asserted procedure
does indeed do a 2-D Fourier transform.
Say first that any function of and
may
be expressed as a superposition of sinusoidal functions:
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![]() | Waves and Fourier sums | ![]() |
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