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, 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,

$$Z_h = e^{i k_h \Delta x}.$$ (8)

In the discrete frequency domain this becomes

$$Z_h = e^{i q_h \Delta k_h \Delta x},$$ (9)

where q_h 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  

$$Z_h = e^{2 \pi i \; \frac{q_h}{N_x N_y}}.$$ (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 x-axis wavenumbers will have a higher frequency than y-axis wavenumbers, leads us to try a q_h of the form,

 

q_h = N_y q_x + q_y.

Substituting this into equation (10) leads to

$$\begin{eqnarray} Z_h & = & e^{2 \pi i \; \frac{(N_y q_x + q_y)}{N_x N_y}} \ & =... ...2 \pi i \; \left( \frac{q_x}{N_x} + \frac{q_y}{N_x N_y} \right)}. \end{eqnarray}$$ (12) (13)

Since q_y is bounded by N_y, for large N_x the second term in braces $\frac{q_y}{N_x N_y} \approx 0$, and this reduces to

$$Z_h \approx e^{2 \pi i \frac{q_x}{N_x}} = Z_x,$$ (14)

which satisfies equation (4).

Substituting equation (11) into equation (10), and raising it to the power of N_x leads to:

$$\begin{eqnarray} Z_h^{N_x} & = & e^{2 \pi i \frac{(N_y q_x + q_y)}{N_y}} \ & = & e^{2 \pi i \; \left( q_x + \frac{q_y}{N_y} \right)}.\end{eqnarray}$$ (15) (16)

Since q_x is an integer, $e^{2 \pi i q_x} = 1$, and this reduces to

$$Z_h^{N_x} = e^{2 \pi i \frac{q_y}{N_y}} = Z_y,$$ (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 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 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 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

$$\begin{eqnarray} k_x & = & \Delta k_x \, q_x = \frac{2 \pi}{N_x \Delta x} \left... ...y \Delta y} \left( q_h - N_y \left[ \frac{q_h}{N_y}\right] \right)\end{eqnarray}$$ (18) (19)

where [x] denotes the integer part of x.