Linking 1-D and 2-D FFT's

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Linking 1-D and 2-D FFT's

Taking the one-dimensional

transform of ${\bf b}$ in the helical coordinate system gives

$\begin{displaymath} B(Z_h) = \sum_{p_h=0}^{N_x N_y -1} b_{p_h} Z_h^{p_h}. \end{displaymath}$

(1)

Here,

represents the unit delay operator in the sampled (helical) coordinate system. The summation in equation (1) can be split into two components,

$\displaystyle B(Z_h)$	$\textstyle =$	$\displaystyle \sum_{p_y=0}^{N_y-1} \; \sum_{p_x=0}^{N_x-1} b_{p_x,p_y} Z_h^{p_x+ p_y N_x}$	(2)
	$\textstyle =$	$\displaystyle \sum_{p_y=0}^{N_y-1} \; \sum_{p_x=0}^{N_x-1} b_{p_x,p_y} \; Z_h^{p_x} \; Z_h^{N_x p_y}.$	(3)

Ignoring boundary effects, a single unit delay in the helical coordinate system is equivalent to a single unit delay on the -axis; similarly, but irrespective of boundary conditions, unit delays in the helical coordinate system are equivalent to a single delay on the -axis. This leads to the following definitions of and $Z_h^{N_x}$ in terms of delay operators, and , or wavenumbers, and :

$\displaystyle Z_h$	$\textstyle \approx$	$\displaystyle Z_x \; = \; e^{i k_x \Delta x},$	(4)
$\displaystyle Z_h^{N_x}$	$\textstyle =$	$\displaystyle Z_y \; = \; e^{i k_y \Delta y},$	(5)

where $\Delta x$ and $\Delta y$ define the grid-spacings along the

and

-axis respectively.

Substituting equations (4) and (5) into equation (3) leaves

$\displaystyle B(k_x,k_y) \; = \; B(Z_h)$	$\textstyle =$	$\displaystyle \sum_{p_y=0}^{N_y-1} \; \sum_{p_x=0}^{N_x-1} b_{p_x,p_y} \; Z_x^{p_x} \; Z_y^{p_y}$	(6)
	$\textstyle =$	$\displaystyle \sum_{p_y=0}^{N_y-1} \; \sum_{p_x=0}^{N_x-1} b_{p_x,p_y} \; e^{i k_x \Delta x p_x} \; e^{i k_y \Delta y p_y}.$	(7)

Equation (7) implies that, if we ignore boundary effects, the one-dimensional FFT of ${\bf b}(x,y)$ in helical coordinates is equivalent to its two-dimensional Fourier transform.

Next: Wavenumber in helical coordinates Up: Theory Previous: Theory

2013-03-03