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

naginterfaces.library.sum.fft_real_2d(m, n, x)

fft_real_2d computes the two-dimensional discrete Fourier transform of a bivariate sequence of real data values.

For full information please refer to the NAG Library document for c06pv

https://www.nag.com/numeric/nl/nagdoc_29.3/flhtml/c06/c06pvf.html

Parameters

mint

$m$, the first dimension of the transform.

nint

$n$, the second dimension of the transform.

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

The real input dataset $x$, where $x_{{\text{j}}_1{\text{j}}_2}$ is stored in $x[{\text{j}}_2\times m+{\text{j}}_1]$, for ${\text{j}}_2=0,1,\dots ,n-1$, for ${\text{j}}_1=0,1,\dots ,m-1$.

Returns

ycomplex, ndarray, shape $((m/2+1)\times n)$

The complex output dataset $\hat{z}$, where ${\hat{z}}_{{\text{k}}_1{\text{k}}_2}$ is stored in $y[{\text{k}}_2\times (m/2+1)+k_1]$, for ${\text{k}}_2=0,1,\dots ,n-1$, for ${\text{k}}_1=0,1,\dots ,m/2$. Note the first dimension is cut roughly by half to remove the redundant information due to conjugate symmetry.

Raises

NagValueError

(errno $1$)

On entry, $m=⟨value⟩$.

Constraint: $m\ge 1$.

(errno $2$)

On entry, $n=⟨value⟩$.

Constraint: $n\ge 1$.

(errno $3$)

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_real_2d computes the two-dimensional discrete Fourier transform of a bivariate sequence of real data values $x_{{\text{j}}_1{\text{j}}_2}$, 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

$${\hat{z}}_{k_1{k}_2}=\frac{1}{\sqrt{mn}}\sum _0^{m-1}\sum _0^{n-1}{x}_{j_1{j}_2}\times exp\left(-2\pi i(\frac{j_1{k}_1}{m}+\frac{j_2{k}_2}{n})\right)\text{,}$$

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 transformed values ${\hat{z}}_{k_1{k}_2}$ are complex. Because of conjugate symmetry (i.e., ${\hat{z}}_{k_1{k}_2}$ is the complex conjugate of ${\hat{z}}_{(m-k_1)(n-k_2)}$), only slightly more than half of the Fourier coefficients need to be stored in the output.

A call of fft_real_2d followed by a call of fft_hermitian_2d() will restore the original data.

This function calls fft_realherm_1d_multi_col() and fft_complex_1d_multi_row() to perform multiple one-dimensional discrete Fourier transforms by the fast Fourier transform (FFT) algorithm in Brigham (1974) and Temperton (1983).

References

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

Temperton, C, 1983, Fast mixed-radix real Fourier transforms, J. Comput. Phys. (52), 340–350