naginterfaces.library.sum.fft_​complex_​2d

naginterfaces.library.sum.fft_complex_2d(direct, m, n, x)

fft_complex_2d computes the two-dimensional discrete Fourier transform of a bivariate sequence of complex data values (using complex data type).

For full information please refer to the NAG Library document for c06pu

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/c06/c06puf.html

Parameters

directstr, length 1

If the forward transform as defined in Notes is to be computed, $direct$ must be set equal to ‘F’.

If the backward transform is to be computed, $direct$ must be set equal to ‘B’.

mint

$m$, the first dimension of the transform.

nint

$n$, the second dimension of the transform.

xcomplex, array-like, shape $(m\times n)$

The complex data values. $x[m\times {\text{j}}_2+{\text{j}}_1]$ must contain $z_{{\text{j}}_1{\text{j}}_2}$, for ${\text{j}}_2=0,\dots ,n-1$, for ${\text{j}}_1=0,\dots ,m-1$.

Returns

xcomplex, ndarray, shape $(m\times n)$

The corresponding elements of the computed transform.

Raises

NagValueError

(errno $1$)

On entry, $m=⟨value⟩$.

Constraint: $m\ge 1$.

(errno $2$)

On entry, $n=⟨value⟩$.

Constraint: $n\ge 1$.

(errno $3$)

On entry, $direct=⟨value⟩$.

Constraint: $direct=\text{'F'}$ or $\text{'B'}$.

(errno $6$)

An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.

Notes

fft_complex_2d computes the two-dimensional discrete Fourier transform of a bivariate sequence of complex data values $z_{{\text{j}}_1{\text{j}}_2}$, for ${\text{j}}_2=0,1,\dots ,n-1$, for ${\text{j}}_1=0,1,\dots ,m-1$.

The discrete Fourier transform is here defined by

$${\hat{z}}_{k_1{k}_2}=\frac{1}{\sqrt{mn}}\sum _{j_1=0}^{m-1}\sum _{j_2=0}^{n-1}{z}_{j_1{j}_2}\times exp\left(\pm 2\pi i(\frac{j_1{k}_1}{m}+\frac{j_2{k}_2}{n})\right)\text{,}$$

where $k_1=0,1,\dots ,m-1$ and $k_2=0,1,\dots ,n-1$.

(Note the scale factor of $\frac{1}{\sqrt{mn}}$ in this definition.) The minus sign is taken in the argument of the exponential within the summation when the forward transform is required, and the plus sign is taken when the backward transform is required.

A call of fft_complex_2d with $direct=\text{'F'}$ followed by a call with $direct=\text{'B'}$ will restore the original data.

This function calls fft_complex_1d_multi_row() to perform multiple one-dimensional discrete Fourier transforms by the fast Fourier transform (FFT) algorithm in Brigham (1974).

References

Brigham, E O, 1974, The Fast Fourier Transform, Prentice–Hall

Temperton, C, 1983, Self-sorting mixed-radix fast Fourier transforms, J. Comput. Phys. (52), 1–23