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Wavenumber in helical coordinates

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, kh, relate to spatial wavenumbers, kx and ky?

The helical delay operator, Zh, is related to kh through the equation,
Z_h = e^{i k_h \Delta x}.\end{displaymath} (8)
In the discrete frequency domain this becomes
Z_h = e^{i q_h \Delta k_h \Delta x},\end{displaymath} (9)
where qh is the integer frequency index that lies in the range, $0 \leq q_h < N_x N_y$. The uncertainty relationship, $\Delta k_h \Delta x = \frac{2 \pi}{N_x N_y}$, allows this to be simplified still further, leaving  
Z_h = e^{2 \pi i \; \frac{q_h}{N_x N_y}}.\end{displaymath} (10)
If we find a form of qh in terms of Fourier indices, qx and qy, that can be plugged into equation (10) in order to satisfy equations (4) and (5), this will provide the link between kh and spatial wavenumbers, kx and ky.

The idea that x-axis wavenumbers will have a higher frequency than y-axis wavenumbers, leads us to try a qh of the form,

qh = Ny qx + qy.


Substituting this into equation (10) leads to
Z_h & = & e^{2 \pi i \; \frac{(N_y q_x + q_y)}{N_x N_y}} \ & =...
 ...2 \pi i \; \left( \frac{q_x}{N_x} + \frac{q_y}{N_x N_y}
\right)}. \end{eqnarray} (12)
Since qy is bounded by Ny, for large Nx the second term in braces $\frac{q_y}{N_x N_y} \approx 0$, and this reduces to
Z_h \approx e^{2 \pi i \frac{q_x}{N_x}} = Z_x,\end{displaymath} (14)
which satisfies equation (4).

Substituting equation (11) into equation (10), and raising it to the power of Nx leads to:
Z_h^{N_x} & = & e^{2 \pi i \frac{(N_y q_x + q_y)}{N_y}} \ & = & e^{2 \pi i \; \left( q_x + \frac{q_y}{N_y} \right)}.\end{eqnarray} (15)
Since qx is an integer, $e^{2 \pi i q_x} = 1$, and this reduces to
Z_h^{N_x} = e^{2 \pi i \frac{q_y}{N_y}} = Z_y,\end{displaymath} (17)
which satisfies equation (5).

Equation (11), therefore, provides the link we are looking for between qx, qy, and qh. 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 x-axis as it does in physical space, the helix wraps around the ky-axis in Fourier space (Figure 2). This provides the link that is missing in Figure 1, but shown in Figure 3.

Figure 2
Fourier dual of helical boundary conditions is also helical boundary conditions with axis of helix transposed.

Figure 3
Relationship between 1-D and 2-D 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
k_x & = & \Delta k_x \, q_x = \frac{2 \pi}{N_x \Delta x} 
 ...y \Delta y}
q_h - N_y \left[ \frac{q_h}{N_y}\right]
\right)\end{eqnarray} (18)
where [x] denotes the integer part of x.

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