# InverseFourier

InverseFourier[list]

finds the discrete inverse Fourier transform of a list of complex numbers.

InverseFourier[list,{p1,p2,}]

returns the specified positions of the discrete inverse Fourier transform.

# Details and Options • The inverse Fourier transform of a list of length is defined to be . »
• Note that the zero frequency term must appear at position 1 in the input list.
• Other definitions are used in some scientific and technical fields.
• Different choices of definitions can be specified using the option FourierParameters.
• With the setting FourierParameters->{a,b} the discrete Fourier transform computed by InverseFourier is .
• Some common choices for {a,b} are {0,1} (default), {-1,1} (data analysis), {1,-1} (signal processing).
• The setting b=-1 effectively corresponds to conjugating both input and output lists.
• To ensure a unique discrete Fourier transform, Abs[b] must be relatively prime to .
• The list of data need not have a length equal to a power of two.
• The list given in InverseFourier[list] can be nested to represent an array of data in any number of dimensions.
• The array of data must be rectangular.
• InverseFourier[list,{p1,p2,}] is typically equivalent to Extract[InverseFourier[list],{p1,p2,}]. Cases with just a few positions p are computed using an algorithm that takes less time and memory but is more subject to numerical error, particularly when the length of list is long.
• If the elements of list are exact numbers, InverseFourier begins by applying N to them.

# Examples

open allclose all

## Basic Examples(2)

Inverse Fourier transform of a real list:

Inverse Fourier transform of a complex list:

## Scope(3)

x is a list of real values:

Compute the inverse Fourier transform with machine arithmetic:

Compute using 24-digit precision arithmetic:

Compute a 2D inverse Fourier transform:

x is a rank-4 tensor with a single nonzero entry:

Compute the 4D inverse Fourier transform:

## Options(3)

### FourierParameters(3)

No normalization:

Normalization by :

Normalization by :

For real data, InverseFourier is the same as Fourier with parameter :

Data from a sinc function with noise:

Get the Fourier transform:

Reconstruct the signal from part of the spectrum: ## Applications(1)

Some Gaussian data:

The multiplication of each mode to get the first derivative:

Approximate the first derivative of the data:

Note the derivative approximation implicitly assumes periodicity:

## Properties & Relations(2) is given by :

InverseFourier is equivalent to matrix multiplication:

The conjugate transpose of the matrix is equivalent to Fourier:

## Possible Issues(1)

InverseFourier uses an efficient algorithm when only a small number of coefficients is needed:

The fast and efficient implementation may result in significant numerical error: