This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.

# InverseFourier

 InverseFourier[list]finds the discrete inverse Fourier transform of a list of complex numbers.
• 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.
• 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.
Inverse Fourier transform of a real list:
Inverse Fourier transform of a complex list:
Inverse Fourier transform of a real list:
 Out[1]=

Inverse Fourier transform of a complex list:
 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:
is given by :
InverseFourier is equivalent to matrix multiplication:
The conjugate transpose of the matrix is equivalent to Fourier: