naginterfaces.library.sum.fft_​hermitian_​1d_​rfmt

naginterfaces.library.sum.fft_hermitian_1d_rfmt(x)

fft_hermitian_1d_rfmt calculates the discrete Fourier transform of a Hermitian sequence of $n$ complex data values (using a work array for extra speed).

For full information please refer to the NAG Library document for c06fb

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/c06/c06fbf.html

Parameters

xfloat, array-like, shape $\left(n\right)$

The sequence to be transformed stored in Hermitian form. If the data values $z_j$ are written as $x_j+i{y}_j$, and if $x$ is declared with bounds $(0:n-1)$ in the function from which fft_hermitian_1d_rfmt is called, then for $0\le j\le n/2$, $x_j$ is contained in $x[j-1]$, and for $1\le j\le (n-1)/2$, $y_j$ is contained in $x[n-j]$. (See also the C06 Introduction.)

Returns

xfloat, ndarray, shape $\left(n\right)$

The components of the discrete Fourier transform ${\hat{x}}_k$. If $x$ is declared with bounds $(0:n-1)$ in the function from which fft_hermitian_1d_rfmt is called, ${\hat{x}}_{\text{k}}$ is stored in $x\left[\text{k}\right]$, for $\text{k}=0,1,\dots ,n-1$.

Raises

NagValueError

(errno $3$)

On entry, $n=⟨value⟩$.

Constraint: $n>1$.

Notes

No equivalent traditional C interface for this routine exists in the NAG Library.

Given a Hermitian sequence of $n$ complex data values $z_{\text{j}}$ (i.e., a sequence such that $z_0$ is real and $z_{n-\text{j}}$ is the complex conjugate of $z_{\text{j}}$, for $\text{j}=1,2,\dots ,n-1$), fft_hermitian_1d_rfmt calculates their discrete Fourier transform defined by

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

(Note the scale factor of $\frac{1}{\sqrt{n}}$ in this definition.) The transformed values ${\hat{x}}_k$ are purely real (see also the C06 Introduction).

To compute the inverse discrete Fourier transform defined by

$${\hat{y}}_k=\frac{1}{\sqrt{n}}\sum _{j=0}^{n-1}{z}_j\times exp\left(+i\frac{2\pi jk}{n}\right)\text{,}$$

this function should be preceded by forming the complex conjugates of the ${\hat{z}}_k$; that is, $x\left(\text{k}\right)=-x\left(\text{k}\right)$, for $\text{k}=n/2+2,\dots ,n$.

fft_hermitian_1d_rfmt uses the fast Fourier transform (FFT) algorithm (see Brigham (1974)).

References

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