Real FFT

The real FFT functions exploit the Hermitian symmetry of the spectrum of real-valued signals. Instead of returning the full n-point spectrum, rfft returns only the first n//2 + 1 unique complex coefficients, cutting storage roughly in half.

rfft

Compute the 1-D FFT of a real-valued array.

function rfft(
  a: ArrayLike,
  n?: number,
  axis?: number,
  norm?: 'backward' | 'ortho' | 'forward'
): NDArray

Name	Type	Default	Description
`a`	`ArrayLike`	—	Real-valued input array.
`n`	`number`	`a.shape[axis]`	Number of points in the transform.
`axis`	`number`	`-1`	Axis along which to compute the FFT.
`norm`	`'backward' \| 'ortho' \| 'forward'`	`'backward'`	Normalization mode.

Returns: NDArray — Complex128 array of length n//2 + 1 along the transformed axis.

import * as np from 'numpy-ts';

const signal = np.array([0, 1, 0, 0]);
const spectrum = np.fft.rfft(signal);
// array([1+0j, 0-1j, -1+0j])  length = 4//2 + 1 = 3

irfft

Compute the inverse of rfft. Reconstructs the full real-valued signal from its positive-frequency spectrum.

function irfft(
  a: ArrayLike,
  n?: number,
  axis?: number,
  norm?: 'backward' | 'ortho' | 'forward'
): NDArray

Name	Type	Default	Description
`a`	`ArrayLike`	—	Input array from `rfft` (complex, length `n//2 + 1`).
`n`	`number`	`(a.shape[axis] - 1) * 2`	Length of the output along the transformed axis.
`axis`	`number`	`-1`	Axis along which to compute the inverse FFT.
`norm`	`'backward' \| 'ortho' \| 'forward'`	`'backward'`	Normalization mode.

Returns: NDArray — Real-valued float64 array of length n.

import * as np from 'numpy-ts';

const signal = np.array([0, 1, 0, 0]);
const spectrum = np.fft.rfft(signal);
const recovered = np.fft.irfft(spectrum);
// array([0, 1, 0, 0])

rfft2

Compute the 2-D FFT of a real-valued array. Applies fft along the first axis and rfft along the last axis, so the last axis is truncated to n//2 + 1.

function rfft2(
  a: ArrayLike,
  s?: [number, number],
  axes?: [number, number],
  norm?: 'backward' | 'ortho' | 'forward'
): NDArray

Name	Type	Default	Description
`a`	`ArrayLike`	—	Real-valued input array (at least 2-D).
`s`	`[number, number]`	shape of `a` along `axes`	Shape of the output along the transform axes.
`axes`	`[number, number]`	`[-2, -1]`	Axes over which to compute the 2-D FFT.
`norm`	`'backward' \| 'ortho' \| 'forward'`	`'backward'`	Normalization mode.

Returns: NDArray — Complex128 array. The last transform axis has length s[1]//2 + 1.

import * as np from 'numpy-ts';

const a = np.array([[1, 2, 3, 4], [5, 6, 7, 8]]);
const F = np.fft.rfft2(a);
// Last axis: length 4//2 + 1 = 3

irfft2

Compute the inverse of rfft2.

function irfft2(
  a: ArrayLike,
  s?: [number, number],
  axes?: [number, number],
  norm?: 'backward' | 'ortho' | 'forward'
): NDArray

Name	Type	Default	Description
`a`	`ArrayLike`	—	Input array (complex, from `rfft2`).
`s`	`[number, number]`	inferred	Shape of the real-valued output along the transform axes.
`axes`	`[number, number]`	`[-2, -1]`	Axes over which to compute the inverse FFT.
`norm`	`'backward' \| 'ortho' \| 'forward'`	`'backward'`	Normalization mode.

Returns: NDArray — Real-valued float64 array.

rfftn

Compute the N-dimensional FFT of a real-valued array. Applies fft along all axes except the last, then rfft along the last axis.

function rfftn(
  a: ArrayLike,
  s?: number[],
  axes?: number[],
  norm?: 'backward' | 'ortho' | 'forward'
): NDArray

Name	Type	Default	Description
`a`	`ArrayLike`	—	Real-valued input array.
`s`	`number[]`	shape of `a` along `axes`	Shape of the output along the transform axes.
`axes`	`number[]`	all axes	Axes over which to compute the FFT.
`norm`	`'backward' \| 'ortho' \| 'forward'`	`'backward'`	Normalization mode.

Returns: NDArray — Complex128 array. The last transform axis has length s[-1]//2 + 1.

irfftn

Compute the inverse of rfftn.

function irfftn(
  a: ArrayLike,
  s?: number[],
  axes?: number[],
  norm?: 'backward' | 'ortho' | 'forward'
): NDArray

Name	Type	Default	Description
`a`	`ArrayLike`	—	Input array (complex, from `rfftn`).
`s`	`number[]`	inferred	Shape of the real-valued output along the transform axes.
`axes`	`number[]`	all axes	Axes over which to compute the inverse FFT.
`norm`	`'backward' \| 'ortho' \| 'forward'`	`'backward'`	Normalization mode.

Returns: NDArray — Real-valued float64 array.

hfft

Compute the FFT of a signal with Hermitian symmetry (i.e., the spectrum is real-valued). This is the inverse of ihfft.

function hfft(
  a: ArrayLike,
  n?: number,
  axis?: number,
  norm?: 'backward' | 'ortho' | 'forward'
): NDArray

Name	Type	Default	Description
`a`	`ArrayLike`	—	Input array with Hermitian symmetry.
`n`	`number`	`(a.shape[axis] - 1) * 2`	Length of the output.
`axis`	`number`	`-1`	Axis along which to compute.
`norm`	`'backward' \| 'ortho' \| 'forward'`	`'backward'`	Normalization mode.

Returns: NDArray — Real-valued float64 array of length n.

import * as np from 'numpy-ts';

// Signal with Hermitian symmetry
const a = np.array([1, 2, 3, 2]);
const spectrum = np.fft.hfft(a);

ihfft

Compute the inverse of hfft. Takes a real-valued signal and returns the Hermitian-symmetric spectrum.

function ihfft(
  a: ArrayLike,
  n?: number,
  axis?: number,
  norm?: 'backward' | 'ortho' | 'forward'
): NDArray

Name	Type	Default	Description
`a`	`ArrayLike`	—	Real-valued input array.
`n`	`number`	`a.shape[axis]`	Length of the input to use.
`axis`	`number`	`-1`	Axis along which to compute.
`norm`	`'backward' \| 'ortho' \| 'forward'`	`'backward'`	Normalization mode.

Returns: NDArray — Complex128 array of length n//2 + 1.

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rfft

irfft

rfft2

irfft2

rfftn

irfftn

hfft

ihfft

​rfft

​irfft

​rfft2

​irfft2

​rfftn

​irfftn

​hfft

​ihfft

rfft

irfft

rfft2

irfft2

rfftn

irfftn

hfft

ihfft