FourierMatrix

Updated in 14

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FourierMatrix[n]

returns an n×n Fourier matrix.

Details and Options

FourierMatrix of order n returns a list of the length-n discrete Fourier transform's basis sequences.
Each entry of the Fourier matrix is by default defined as , where .

Rows of the FourierMatrix are basis sequences of the discrete Fourier transform.
The result F of FourierMatrix[n] is complex symmetric and unitary, meaning that F^-1 is Conjugate[F].
The following options can be given:

FourierParameters	{0,1}	parameters to define the Fourier transform
TargetStructure	Automatic	the structure of the returned matrix
WorkingPrecision	Infinity	precision at which to create entries

Different choices of definitions for the Fourier matrix can be specified using the option FourierParameters. With the setting FourierParameters->{a,b}, the Fourier matrix has entries defined as , where .

Some common choices for {a,b} are {0,1} (physics), {-1,1} (data analysis), {1,-1} (signal processing).
Possible settings for TargetStructure include:

	Automatic	automatically choose the representation returned
	"Dense"	represent the matrix as a dense matrix
	"Structured"	represent the matrix as a structured array
	"Symmetric"	represent the matrix as a symmetric matrix
	"Unitary"	represent the matrix as a unitary matrix

With FourierMatrix[…,TargetStructureAutomatic], a dense matrix is returned if the number of matrix entries is less than a preset threshold, and a structured array is returned otherwise.
The result of FourierMatrix[n].list is equivalent to Fourier[list] when list has length n. However, the computation of Fourier[list] is much faster and has less numerical error, unless FourierMatrix is kept as a structured array. »
For a structured FourierMatrix sa, the following properties "prop" can be accessed as sa["prop"]:

	"FourierParameters"	parameters {a,b}
	"WorkingPrecision"	precision used internally
	"Properties"	list of supported properties
	"Structure"	type of structured array
	"StructuredData"	internal data stored by the structured array
	"StructuredAlgorithms"	list of functions with special methods for the structured array
	"Summary"	summary information, represented as a Dataset

Examples

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Basic Examples (2)Summary of the most common use cases

A Fourier matrix:

In[1]:=1

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https://wolfram.com/xid/0i1l28x3m-coc1q3

A large Fourier matrix:

In[1]:=1

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https://wolfram.com/xid/0i1l28x3m-uz99x

Out[1]=1

Scope (2)Survey of the scope of standard use cases

The real and imaginary parts of the Fourier's basis sequences of length 128:

In[1]:=1

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https://wolfram.com/xid/0i1l28x3m-gqda46

Out[2]=2

Construct a structured Fourier matrix using the option setting TargetStructure"Structured":

In[1]:=1

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https://wolfram.com/xid/0i1l28x3m-mnqv1t

Out[1]=1

The structured representation saves a significant amount of memory for larger matrices:

In[2]:=2

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https://wolfram.com/xid/0i1l28x3m-c83gou

Out[2]=2

Options (3)Common values & functionality for each option

FourierParameters (1)

The default definition of the Fourier matrix:

In[1]:=1

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https://wolfram.com/xid/0i1l28x3m-h91cff

Use the definition of the Fourier matrix used in signal processing:

In[2]:=2

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https://wolfram.com/xid/0i1l28x3m-fhinsc

Use the definition of the Fourier matrix used in data analysis:

In[3]:=3

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https://wolfram.com/xid/0i1l28x3m-oj27x

TargetStructure (1)

Return the Fourier matrix as a dense matrix:

In[1]:=1

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https://wolfram.com/xid/0i1l28x3m-e3t5fj

Out[1]=1

Return the Fourier matrix as a structured array:

In[2]:=2

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https://wolfram.com/xid/0i1l28x3m-numczz

Out[2]=2

Return the Fourier matrix as a symmetric matrix:

In[1]:=1

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https://wolfram.com/xid/0i1l28x3m-gmicjr

Out[1]=1

Return the Fourier matrix as a unitary matrix:

In[2]:=2

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https://wolfram.com/xid/0i1l28x3m-fs8nhv

Out[2]=2

WorkingPrecision (1)

Use machine precision:

In[1]:=1

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https://wolfram.com/xid/0i1l28x3m-ghan4a

Use arbitrary precision:

In[2]:=2

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https://wolfram.com/xid/0i1l28x3m-oncmum

Applications (3)Sample problems that can be solved with this function

The efficiency of the fast Fourier transform (FFT) relies on being able to form a larger Fourier matrix from two smaller ones. Generate two small Fourier matrices of sizes p and q:

In[1]:=1

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https://wolfram.com/xid/0i1l28x3m-cooljk

In[3]:=3

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https://wolfram.com/xid/0i1l28x3m-bjetwu

The Fourier matrix of size p q can be expressed as a product of four simpler matrices:

In[5]:=5

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https://wolfram.com/xid/0i1l28x3m-fibd5d

Show that the resulting matrix is equivalent to the result of FourierMatrix:

In[10]:=10

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https://wolfram.com/xid/0i1l28x3m-ly5xff

Out[10]=10

The discrete Fourier transform of a vector can be computed by successively multiplying the factors of the Fourier matrix to the vector:

In[11]:=11

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https://wolfram.com/xid/0i1l28x3m-h9ve36

Out[12]=12

The result is equivalent to applying Fourier to the vector:

In[13]:=13

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https://wolfram.com/xid/0i1l28x3m-c46typ

Out[13]=13

Define a function for constructing a circulant matrix from a vector:

In[1]:=1

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https://wolfram.com/xid/0i1l28x3m-e5fgw9

In[2]:=2

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https://wolfram.com/xid/0i1l28x3m-daijy3

Circulant matrices can be diagonalized by the Fourier matrix:

In[3]:=3

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https://wolfram.com/xid/0i1l28x3m-fjcxww

The diagonal elements of the resulting diagonal matrix are the same as the product of the Fourier matrix and the starting vector, up to a constant scaling factor:

In[4]:=4