BUILT-IN MATHEMATICA SYMBOL

InverseFourier

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

MORE INFORMATION

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 the discrete Fourier transform computed by Fourier is .

Some common choices for are (default), (data analysis), (signal processing).

The setting 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.

If the elements of list are exact numbers, InverseFourier begins by applying N to them.

EXAMPLES

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Basic Examples (2)

Inverse Fourier transform of a real list:

Inverse Fourier transform of a complex list:

Inverse Fourier transform of a real list:

In[1]:=

Out[1]=

Inverse Fourier transform of a complex list:

In[1]:=

Out[1]=

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)

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: