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With the understanding that the 1D FFT of a multidimensional signal
in helical coordinates is equivalent to the 2D FFT, a natural
question to ask is: how does the helical wavenumber, k_{h}, relate to
spatial wavenumbers, k_{x} and k_{y}?
The helical delay operator, Z_{h}, is related to k_{h}
through the equation,
 
(8) 
In the discrete frequency domain this becomes
 
(9) 
where q_{h} is the integer frequency index that lies in the range,
.
The uncertainty relationship,
, allows this to be
simplified still further, leaving
 
(10) 
If we find a form of q_{h} in terms of Fourier indices,
q_{x} and q_{y}, that can be plugged into equation (10)
in order to satisfy equations (4)
and (5), this will provide the link between k_{h} and
spatial wavenumbers, k_{x} and k_{y}.
The idea that xaxis wavenumbers will have a higher frequency than
yaxis wavenumbers, leads us to try a q_{h} of the form,

q_{h} = N_{y} q_{x} + q_{y}.

(11) 
Substituting this into equation (10) leads to
 
(12) 
 (13) 
Since q_{y} is bounded by N_{y}, for large N_{x} the second term in
braces , and this
reduces to
 
(14) 
which satisfies equation (4).
Substituting equation (11) into
equation (10), and raising it to the power of N_{x} leads
to:
 
(15) 
 (16) 
Since q_{x} is an integer, , and this reduces to
 
(17) 
which satisfies equation (5).
Equation (11), therefore, provides the link we are
looking for between q_{x}, q_{y}, and q_{h}. It is interesting to
note that not only is there a onetoone mapping between 1D and 2D
Fourier components, but equation (11) describes helical
boundaries in Fourier space: however, rather than wrapping around the
xaxis as it does in physical space, the helix wraps around the
k_{y}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.

 
ill2
Figure 3 Relationship between 1D and 2D
convolution, FFT's and the helix, illustrating the Fourier dual of
helical boundary conditions.

 
As with helical coordinates in physical space,
equation (11) can easily be inverted to yield
 
(18) 
 (19) 
where [x] denotes the integer part of x.
Next: Speed comparison
Up: Theory
Previous: Linking 1D and 2D
Stanford Exploration Project
9/5/2000