# 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

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## Basic Examples(2)

Inverse Fourier transform of a real list:

 In[1]:=
 Out[1]=

Inverse Fourier transform of a complex list:

 In[1]:=
 Out[1]=