Fourier

The numeric discrete Fourier transform of a list, or of a rectangular matrix (a separable 2D transform), under Wolfram's default $\mathrm{FourierParameters} \to \{0, 1\}$ — $f_s = \frac{1}{\sqrt n}\sum_r \mathrm{list}_r\, e^{2\pi i (r-1)(s-1)/n}$.

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Fourier(list)the discrete Fourier transform of a numeric list, or a rectangular numeric matrix.

Fourier(list, FourierParameters -> {a, b})the general two-parameter transform $f_s = \sqrt{|b|/n^{1-a}}\sum_r \mathrm{list}_r\, e^{2\pi i b (r-1)(s-1)/n}$.

Typed signatures

Type	Package	Applies
(list<list<number>> | list<number>, expression?) -> list<list<number>> | list<number>	analytic	always

Domain: Transforms

Details

• Always numeric, matching Wolfram exactly — even an all-exact input like Fourier({0, 0, 0}) comes back as machine floats there, never a symbolic or exact result, so this declines (stays unevaluated) rather than fabricate an exact answer for anything that isn't already a concrete numeric list or matrix.
• A rectangular matrix gets the 2D transform, separable into the 1D transform along each dimension in turn; both dimensions' FourierParameters prefactors are folded into a single multiplication at the end rather than applied one dimension at a time, which avoids the extra rounding two separate Sqrts would otherwise leave in every entry.
• Direct $O(n^2)$ summation, not an FFT — correct for any $n$ and fast enough at the sizes these heads see in practice; a genuinely large transform would want a radix-2 or Bluestein implementation instead.
• Declined: a symbolic entry, an empty list, a ragged nested list, a list more than two levels deep, or an FourierParameters option that isn't a 2-element real list.

Examples

Basic #

The DFT of $\{1, 2, 3, 4\}$

live ↗

In$\mathrm{Fourier}(\bigl\lbrack1, 2, 3, 4\bigr\rbrack)$Out$\bigl\lbrack5.0, -1.0-i, -1.0, -1.0+i\bigr\rbrack$

Round-trips exactly through InverseFourier

live ↗

In$\mathrm{InverseFourier}(\mathrm{Fourier}(\bigl\lbrack1, 2, 3, 4\bigr\rbrack))$Out$\bigl\lbrack1.0, 2.0, 3.0, 4.0\bigr\rbrack$

Scope #

A $2\times2$ matrix — the separable 2D transform

live ↗

In$\mathrm{Fourier}(\bigl\lbrack\bigl\lbrack1, 2\bigr\rbrack, \bigl\lbrack3, 4\bigr\rbrack\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack5.0, -1.0\bigr\rbrack, \bigl\lbrack-2.0, 0.0\bigr\rbrack\bigr\rbrack$

A custom FourierParameters -> {1, -1}

live ↗

In$\mathrm{Fourier}(\bigl\lbrack1, 2, 3, 4\bigr\rbrack, \operatorname{KeyValuePair}(\mathrm{FourierParameters}, \bigl\lbrack1, -1\bigr\rbrack))$Out$\bigl\lbrack10.0, -2.0+2.0i, -2.0, -2.0-2.0i\bigr\rbrack$

A numeric symbol is treated symbolically and the transform stays unevaluated

live ↗

In$\mathrm{Fourier}(N(x, 24))$Out$\mathrm{Fourier}({{x}})$

Possible issues #

Declines a symbolic list, as Wolfram's Fourier::fftl does

live ↗

In$\mathrm{Fourier}(\bigl\lbrack a, b\bigr\rbrack)$Out$\mathrm{Fourier}(\bigl\lbrack a, b\bigr\rbrack)$

About

Examples	6
Checked against	Wolfram Language 6 agree
Crosswalk	Wolfram Language

Implementation

primitive · kernela kernel — an algorithm over mutable state that would not be readable as a tree

nativeenginetypescriptpackages/symbols/analysis/analytic/src/fourier-transform.ts

See also: InverseFourier, FourierSeries, FourierTransform