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| Multi-dimensional Fourier transforms in the helical coordinate
system | |
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Next: Speed comparison
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Previous: Linking 1-D and 2-D
With the understanding that the 1-D FFT of a multi-dimensional signal
in helical coordinates is equivalent to the 2-D FFT, a natural
question to ask is: how does the helical wavenumber, , relate to
spatial wavenumbers, and ?
The helical delay operator, , is related to
through the equation,
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(8) |
In the discrete frequency domain this becomes
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(9) |
where is the integer frequency index that lies in the range,
.
The uncertainty relationship,
, allows this to be
simplified still further, leaving
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(10) |
If we find a form of in terms of Fourier indices,
and , that can be plugged into equation (10)
in order to satisfy equations (4)
and (5), this will provide the link between and
spatial wavenumbers, and .
The idea that -axis wavenumbers will have a higher frequency than
-axis wavenumbers, leads us to try a of the form,
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(11) |
Substituting this into equation (10) leads to
Since is bounded by , for large the second term in
braces
, and this
reduces to
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(14) |
which satisfies equation (4).
Substituting equation (11) into
equation (10), and raising it to the power of leads
to:
Since is an integer,
, and this reduces to
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(17) |
which satisfies equation (5).
Equation (11), therefore, provides the link we are
looking for between , , and . It is interesting to
note that not only is there a one-to-one mapping between 1-D and 2-D
Fourier components, but equation (11) describes helical
boundaries in Fourier space: however, rather than wrapping around the
-axis as it does in physical space, the helix wraps around the
-axis in Fourier space (Figure 2). This provides
the link that is missing in Figure 1, but shown in
Figure 3.
transp
Figure 2. Fourier dual of helical boundary
conditions is also helical boundary conditions with axis of helix
transposed.
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ill2
Figure 3. Relationship between 1-D and 2-D
convolution, FFT's and the helix, illustrating the Fourier dual of
helical boundary conditions.
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As with helical coordinates in physical space,
equation (11) can easily be inverted to yield
where denotes the integer part of .
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| Multi-dimensional Fourier transforms in the helical coordinate
system | |
|
Next: Speed comparison
Up: Theory
Previous: Linking 1-D and 2-D
2013-03-03