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

naginterfaces.library.sum.fft_hermitian_2d(m, n, y)

fft_hermitian_2d computes the two-dimensional inverse discrete Fourier transform of a bivariate Hermitian sequence of complex data values.

For full information please refer to the NAG Library document for c06pw

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/c06/c06pwf.html

Parameters

mint

$m$, the first dimension of the transform.

nint

$n$, the second dimension of the transform.

ycomplex, array-like, shape $((m/2+1)\times n)$

The Hermitian sequence of complex input dataset $z$, where $z_{{\text{j}}_1{\text{j}}_2}$ is stored in $y[{\text{j}}_2\times (m/2+1)+j_1]$, for ${\text{j}}_2=0,1,\dots ,n-1$, for ${\text{j}}_1=0,1,\dots ,m/2$.

Returns

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

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

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_hermitian_2d computes the two-dimensional inverse discrete Fourier transform of a bivariate Hermitian 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

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

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.)

Because the input data satisfies conjugate symmetry (i.e., $z_{j_1{j}_2}$ is the complex conjugate of $z_{(m-j_1)(n-j_2)}$, the transformed values ${\hat{x}}_{k_1{k}_2}$ are real.

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