naginterfaces.library.sum.fft_​complex_​1d_​multi_​col

naginterfaces.library.sum.fft_complex_1d_multi_col(direct, n, m, x)

fft_complex_1d_multi_col computes the discrete Fourier transforms of $m$ sequences, stored as columns of an array, each containing $n$ complex data values.

For full information please refer to the NAG Library document for c06ps

https://support.nag.com/numeric/nl/nagdoc_30.2/flhtml/c06/c06psf.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’.

nint

$n$, the number of complex values in each sequence.

mint

$m$, the number of sequences to be transformed.

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

The complex data values $z_{\text{j}}^p$ stored in $x[(\text{p}-1)\times n+\text{j}]$, for $\text{p}=1,2,\dots ,m$, for $\text{j}=0,1,\dots ,n-1$.

Returns

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

Is overwritten by the complex transforms.

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 $5$)

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

Given $m$ sequences of $n$ complex data values $z_{\text{j}}^{\text{p}}$, for $\text{p}=1,2,\dots ,m$, for $\text{j}=0,1,\dots ,n-1$, fft_complex_1d_multi_col simultaneously calculates the (forward or backward) discrete Fourier transforms of all the sequences defined by

$${\hat{z}}_k^p=\frac{1}{\sqrt{n}}\sum _{j=0}^{n-1}{z}_j^p\times exp\left(\pm i\frac{2\pi jk}{n}\right)\text{,}k=0,1,\dots ,n-1\text{and}p=1,2,\dots ,m\text{.}$$

(Note the scale factor $\frac{1}{\sqrt{n}}$ 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_1d_multi_col with $direct=\text{'F'}$ followed by a call with $direct=\text{'B'}$ will restore the original data.

The function uses a variant of the fast Fourier transform (FFT) algorithm (see Brigham (1974)) known as the Stockham self-sorting algorithm, which is described in Temperton (1983). Special code is provided for the factors $2$, $3$ and $5$.

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