naginterfaces.library.sum.fft_​real_​3d

naginterfaces.library.sum.fft_real_3d(n1, n2, n3, x)

fft_real_3d computes the three-dimensional discrete Fourier transform of a trivariate sequence of real data values.

For full information please refer to the NAG Library document for c06py

https://support.nag.com/numeric/nl/nagdoc_30.2/flhtml/c06/c06pyf.html

Parameters

n1int

$n_1$, the first dimension of the transform.

n2int

$n_2$, the second dimension of the transform.

n3int

$n_3$, the third dimension of the transform.

xfloat, array-like, shape $(n1\times n2\times n3)$

The real input dataset $x$, where $x_{{\text{j}}_1{\text{j}}_2{\text{j}}_3}$ is stored in $x[{\text{j}}_3\times n_1{n}_2+{\text{j}}_2\times n_1+{\text{j}}_1+1-1]$, for ${\text{j}}_3=0,1,\dots ,n_3-1$, for ${\text{j}}_2=0,1,\dots ,n_2-1$, for ${\text{j}}_1=0,1,\dots ,n_1-1$.

Returns

ycomplex, ndarray, shape $((n1/2+1)\times n2\times n3)$

The complex output dataset $\hat{z}$, where ${\hat{z}}_{{\text{k}}_1{\text{k}}_2{\text{k}}_3}$ is stored in $y[{\text{k}}_3\times (n_1/2+1)n_2+{\text{k}}_2\times (n_1/2+1)+{\text{k}}_1+1-1]$, for ${\text{k}}_3=0,1,\dots ,n_3-1$, for ${\text{k}}_2=0,1,\dots ,n_2-1$, for ${\text{k}}_1=0,1,\dots ,n_1/2$. Note the first dimension is cut roughly by half to remove the redundant information due to conjugate symmetry.

Raises

NagValueError

(errno $1$)

On entry, $n1=⟨value⟩$.

Constraint: $n1\ge 1$.

(errno $2$)

On entry, $n2=⟨value⟩$.

Constraint: $n2\ge 1$.

(errno $3$)

On entry, $n3=⟨value⟩$.

Constraint: $n3\ge 1$.

(errno $4$)

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_3d computes the three-dimensional discrete Fourier transform of a trivariate sequence of real data values $x_{j_1{j}_2{j}_3}$, for ${\text{j}}_3=0,1,\dots ,n_3-1$, for ${\text{j}}_2=0,1,\dots ,n_2-1$, for ${\text{j}}_1=0,1,\dots ,n_1-1$.

The discrete Fourier transform is here defined by

$${\hat{z}}_{k_1{k}_2{k}_3}=\frac{1}{\sqrt{n_1{n}_2{n}_3}}\sum _0^{n_1-1}\sum _0^{n_2-1}\sum _0^{n_3-1}{x}_{j_1{j}_2{j}_3}\times exp(-2\pi i(\frac{j_1{k}_1}{n_1}+\frac{j_2{k}_2}{n_2}+\frac{j_3{k}_3}{n_3}))\text{,}$$

where $k_1=0,1,\dots ,n_1-1$, $k_2=0,1,\dots ,n_2-1$ and $k_3=0,1,\dots ,n_3-1$. (Note the scale factor of $\frac{1}{\sqrt{n_1{n}_2{n}_3}}$ in this definition.)

The transformed values ${\hat{z}}_{k_1{k}_2{k}_3}$ are complex. Because of conjugate symmetry (i.e., ${\hat{z}}_{k_1{k}_2{k}_3}$ is the complex conjugate of ${\hat{z}}_{(n_1-k_1)(n_2-k_2)(n_3-k_3)}$), only slightly more than half of the Fourier coefficients need to be stored in the output.

A call of fft_real_3d followed by a call of fft_hermitian_3d() 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