naginterfaces.library.sum.fft_​complex_​1d_​sep

naginterfaces.library.sum.fft_complex_1d_sep(x, y)

fft_complex_1d_sep calculates the discrete Fourier transform of a sequence of $n$ complex data values (using a work array for extra speed).

For full information please refer to the NAG Library document for c06fc

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/c06/c06fcf.html

Parameters

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

If $x$ is declared with bounds $(0:n-1)$ in the function from which fft_complex_1d_sep is called, $x[\text{j}-1]$ must contain $x_{\text{j}}$, the real part of $z_{\text{j}}$, for $\text{j}=0,1,\dots ,n-1$.

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

If $y$ is declared with bounds $(0:n-1)$ in the function from which fft_complex_1d_sep is called, $y[\text{j}-1]$ must contain $y_{\text{j}}$, the imaginary part of $z_{\text{j}}$, for $\text{j}=0,1,\dots ,n-1$.

Returns

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

The real parts $a_k$ of the components of the discrete Fourier transform. If $x$ is declared with bounds $(0:n-1)$ in the function from which fft_complex_1d_sep is called, for $0\le k\le n-1$, $a_k$ is contained in $x[k-1]$.

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

The imaginary parts $b_k$ of the components of the discrete Fourier transform. If $y$ is declared with bounds $(0:n-1)$ in the function from which fft_complex_1d_sep is called, then for $0\le k\le n-1$, $b_k$ is contained in $y[k-1]$.

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 sequence of $n$ complex data values $z_{\text{j}}$, for $\text{j}=0,1,\dots ,n-1$, fft_complex_1d_sep calculates their discrete Fourier transform defined by

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

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

To compute the inverse discrete Fourier transform defined by

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

this function should be preceded and followed by the complex conjugation of the data values and the transform (by negating the imaginary parts stored in $y$).

fft_complex_1d_sep uses the fast Fourier transform (FFT) algorithm (see Brigham (1974)).

References

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