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

naginterfaces.library.sum.fft_realherm_1d_multi_col(direct, n, m, x)

fft_realherm_1d_multi_col computes the discrete Fourier transforms of $m$ sequences, each containing $n$ real data values or a Hermitian complex sequence stored column-wise in a complex storage format.

For full information please refer to the NAG Library document for c06pq

https://support.nag.com/numeric/nl/nagdoc_30.2/flhtml/c06/c06pqf.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 real or complex values in each sequence.

mint

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

xfloat, array-like, shape $((n+2)\times m)$

The $m$ real or Hermitian data sequences to be transformed.

if $direct=\text{'F'}$, the $m$ real data sequences, $x^{\text{p}}=(x_1^{\text{p}},x_2^{\text{p}},\dots ,x_n^{\text{p}})$, for $\text{p}=1,2,\dots ,m$, should be stored sequentially in $x$, with a stride of $n+2$ between sequences.

if $direct=\text{'B'}$, the $m$ Hermitian data sequences, ${\hat{z}}^{\text{p}}=({\hat{z}}_1^{\text{p}},{\hat{z}}_2^{\text{p}},\dots ,{\hat{z}}_{n/2+1}^{\text{p}})=(Re\left({\hat{z}}_1^{\text{p}}\right),Im\left({\hat{z}}_1^{\text{p}}\right),Re\left({\hat{z}}_2^{\text{p}}\right),Im\left({\hat{z}}_2^{\text{p}}\right),\dots ,Re\left({\hat{z}}_{n/2+1}^{\text{p}}\right),Im\left({\hat{z}}_{n/2+1}^{\text{p}}\right))$, for $\text{p}=1,2,\dots ,m$, should be stored sequentially in $x$, with a stride of $n+2$ between sequences.

In other words:

if $direct=\text{'F'}$, $x[(\text{p}-1)\times (n+2)+\text{j}-1]$ must contain $x_{\text{j}}^{\text{p}}$, for $\text{p}=1,2,\dots ,m$, for $\text{j}=1,2,\dots ,n$;

if $direct=\text{'B'}$, $x[(\text{p}-1)\times (n+2)+(2\times \text{k}-1)-1]$ and $x[(\text{p}-1)\times (n+2)+(2\times \text{k})-1]$ must contain the real and imaginary parts respectively of ${\hat{z}}_{\text{k}}^{\text{p}}$, for $\text{p}=1,2,\dots ,m$, for $\text{k}=1,2,\dots ,n/2+1$. (Note that for the sequence ${\hat{z}}_k^p$ to be Hermitian, the imaginary part of ${\hat{z}}_1^p$, and of ${\hat{z}}_{n/2+1}^p$ for $n$ even, must be zero.)

Returns

xfloat, ndarray, shape $((n+2)\times m)$

If $direct=\text{'F'}$ then the $m$ sequences, ${\hat{z}}^{\text{p}}$, for $\text{p}=1,2,\dots ,m$ stored as described on entry for $direct=\text{'B'}$

if $direct=\text{'B'}$ then the $m$ sequences, $x^{\text{p}}$, for $\text{p}=1,2,\dots ,m$ stored as described on entry for $direct=\text{'F'}$

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 $4$)

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

Given $m$ sequences of $n$ real data values $x_{\text{j}}^{\text{p}}$, for $\text{p}=1,2,\dots ,m$, for $\text{j}=0,1,\dots ,n-1$, fft_realherm_1d_multi_col simultaneously calculates the Fourier transforms of all the sequences defined by

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

The transformed values ${\hat{z}}_k^p$ are complex, but for each value of $p$ the ${\hat{z}}_k^p$ form a Hermitian sequence (i.e., ${\hat{z}}_{n-k}^p$ is the complex conjugate of ${\hat{z}}_k^p$), so they are completely determined by $mn$ real numbers (since ${\hat{z}}_0^p$ is real, as is ${\hat{z}}_{n/2}^p$ for $n$ even).

Alternatively, given $m$ Hermitian sequences of $n$ complex data values $z_j^p$, this function simultaneously calculates their inverse (backward) discrete Fourier transforms defined by

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

The transformed values ${\hat{x}}_k^p$ are real.

(Note the scale factor $\frac{1}{\sqrt{n}}$ in the above definition.)

A call of fft_realherm_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 coding is provided for the factors $2$, $3$, $4$ and $5$.

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