InverseFourier

The numeric inverse discrete Fourier transform — Fourier with the FourierParameters sign flipped, $\{a, b\} \to \{-a, -b\}$ (confirmed against wolframscript), so it shares that head's kernel exactly.

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

InverseFourier(list, FourierParameters -> {a, b})equivalent to Fourier(list, FourierParameters -> {-a, -b}).

Typed signatures

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

Domain: Transforms

Details

• Everything Fourier's own details say about numeric-only evaluation, the separable 2D case, and $O(n^2)$ summation applies here too — this head is that same kernel with the parameters negated.

Examples

Basic #

The inverse DFT of $\{1, 2, 3, 4\}$ — the conjugate kernel of Fourier

live ↗

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

Round-trips exactly through Fourier

live ↗

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

Scope #

The inverse of Fourier's $2\times2$ matrix example

live ↗

In$\mathrm{InverseFourier}(\bigl\lbrack\bigl\lbrack5, -1\bigr\rbrack, \bigl\lbrack-2, 0\bigr\rbrack\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack1.0, 2.0\bigr\rbrack, \bigl\lbrack3.0, 4.0\bigr\rbrack\bigr\rbrack$

A custom FourierParameters -> {1, -1}, equivalent to Fourier(list, FourierParameters -> {-1, 1})

live ↗

In$\mathrm{InverseFourier}(\bigl\lbrack1, 2, 3, 4\bigr\rbrack, \operatorname{KeyValuePair}(\mathrm{FourierParameters}, \bigl\lbrack1, -1\bigr\rbrack))$Out$\bigl\lbrack2.5, -0.5-0.5i, -0.5, -0.5+0.5i\bigr\rbrack$

A numeric symbol is treated symbolically, so the result stays unevaluated

live ↗

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

Possible issues #

Declines a symbolic list, exactly like Fourier

live ↗

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

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: Fourier, InverseFourierTransform