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

naginterfaces.library.sum.fft_real_1d_rfmt(x)

fft_real_1d_rfmt calculates the discrete Fourier transform of a sequence of $n$ real data values (using a work array for extra speed).

For full information please refer to the NAG Library document for c06fa

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/c06/c06faf.html

Parameters

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

If $x$ is declared with bounds $(0:n-1)$ in the function from which fft_real_1d_rfmt is called, $x\left[\text{j}\right]$ must contain $x_{\text{j}}$, for $\text{j}=0,1,\dots ,n-1$.

Returns

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

The discrete Fourier transform stored in Hermitian form. If the components of the transform ${\hat{z}}_k$ are written as $a_k+i{b}_k$, and if $x$ is declared with bounds $(0:n-1)$ in the function from which fft_real_1d_rfmt is called, then for $0\le k\le n/2$, $a_k$ is contained in $x[k-1]$, and for $1\le k\le (n-1)/2$, $b_k$ is contained in $x[n-k]$. (See also the C06 Introduction.)

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 sequence of $n$ real data values $x_{\text{j}}$, for $\text{j}=0,1,\dots ,n-1$, fft_real_1d_rfmt calculates their discrete Fourier transform defined by

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

(Note the scale factor of $\frac{1}{\sqrt{n}}$ in this definition.) The transformed values ${\hat{z}}_k$ are complex, but they form a Hermitian sequence (i.e., ${\hat{z}}_{n-k}$ is the complex conjugate of ${\hat{z}}_k$), so they are completely determined by $n$ real numbers (see also the C06 Introduction).

To compute the inverse discrete Fourier transform defined by

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

this function should be followed 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_real_1d_rfmt uses the fast Fourier transform (FFT) algorithm (see Brigham (1974)).

References

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