Discrete Fourier transform

Fourier transforms
Fourier transform Fourier series Discrete-time Fourier transform Discrete Fourier transform Discrete Fourier transform over a ring Fourier transform on finite groups Fourier analysis Related transforms

In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a complex-valued function of frequency. The interval at which the DTFT is sampled is the reciprocal of the duration of the input sequence.^[A][1]  An inverse DFT (IDFT) is a Fourier series, using the DTFT samples as coefficients of complex sinusoids at the corresponding DTFT frequencies. It has the same sample-values as the original input sequence. The DFT is therefore said to be a frequency domain representation of the original input sequence. If the original sequence spans all the non-zero values of a function, its DTFT is continuous (and periodic), and the DFT provides discrete samples of one cycle. If the original sequence is one cycle of a periodic function, the DFT provides all the non-zero values of one DTFT cycle.

The DFT is used in the Fourier analysis of many practical applications.^[2] In digital signal processing, the function is any quantity or signal that varies over time, such as the pressure of a sound wave, a radio signal, or daily temperature readings, sampled over a finite time interval (often defined by a window function^[3]). In image processing, the samples can be the values of pixels along a row or column of a raster image. The DFT is also used to efficiently solve partial differential equations, and to perform other operations such as convolutions or multiplying large integers.

Since it deals with a finite amount of data, it can be implemented in computers by numerical algorithms or even dedicated hardware. These implementations usually employ efficient fast Fourier transform (FFT) algorithms;^[4] so much so that the terms "FFT" and "DFT" are often used interchangeably. Prior to its current usage, the "FFT" initialism may have also been used for the ambiguous term "finite Fourier transform".

The discrete Fourier transform transforms a sequence of N complex numbers ${\displaystyle \left\{\mathbf {x} _{n}\right\}:=x_{0},x_{1},\ldots ,x_{N-1}}$ into another sequence of complex numbers, ${\displaystyle \left\{\mathbf {X} _{k}\right\}:=X_{0},X_{1},\ldots ,X_{N-1},}$ which is defined by:

The transform is sometimes denoted by the symbol ${\displaystyle {\mathcal {F}}}$, as in ${\displaystyle \mathbf {X} ={\mathcal {F}}\left\{\mathbf {x} \right\}}$ or ${\displaystyle {\mathcal {F}}\left(\mathbf {x} \right)}$ or ${\displaystyle {\mathcal {F}}\mathbf {x} }$.^[B]

Eq.1 can be interpreted or derived in various ways, for example:

Eq.1 can also be evaluated outside the domain ${\displaystyle k\in [0,N-1]}$, and that extended sequence is ${\displaystyle N}$-periodic. Accordingly, other sequences of ${\displaystyle N}$ indices are sometimes used, such as ${\textstyle \left[-{\frac {N}{2}},{\frac {N}{2}}-1\right]}$ (if ${\displaystyle N}$ is even) and ${\textstyle \left[-{\frac {N-1}{2}},{\frac {N-1}{2}}\right]}$ (if ${\displaystyle N}$ is odd), which amounts to swapping the left and right halves of the result of the transform.^[5]

The inverse transform is given by:

Eq.2. is also ${\displaystyle N}$-periodic (in index n). In Eq.2, each ${\displaystyle X_{k}}$ is a complex number whose polar coordinates are the amplitude and phase of a complex sinusoidal component ${\displaystyle \left(e^{i2\pi {\tfrac {k}{N}}n}\right)}$ of function ${\displaystyle x_{n}.}$ (see Discrete Fourier series) The sinusoid's frequency is ${\displaystyle k}$ cycles per ${\displaystyle N}$ samples.

The normalization factor multiplying the DFT and IDFT (here 1 and ${\displaystyle {\tfrac {1}{N}}}$) and the signs of the exponents are the most common conventions. The only actual requirements of these conventions are that the DFT and IDFT have opposite-sign exponents and that the product of their normalization factors be ${\displaystyle {\tfrac {1}{N}}.}$ An uncommon normalization of ${\displaystyle {\sqrt {\tfrac {1}{N}}}}$ for both the DFT and IDFT makes the transform-pair unitary.

This example demonstrates how to apply the DFT to a sequence of length ${\displaystyle N=4}$ and the input vector

$${\displaystyle \mathbf {x} ={\begin{pmatrix}x_{0}\\x_{1}\\x_{2}\\x_{3}\end{pmatrix}}={\begin{pmatrix}1\\2-i\\-i\\-1+2i\end{pmatrix}}.}$$

Calculating the DFT of ${\displaystyle \mathbf {x} }$ using Eq.1

$${\displaystyle {\begin{aligned}X_{0}&=e^{-i2\pi 0\cdot 0/4}\cdot 1+e^{-i2\pi 0\cdot 1/4}\cdot (2-i)+e^{-i2\pi 0\cdot 2/4}\cdot (-i)+e^{-i2\pi 0\cdot 3/4}\cdot (-1+2i)=2\\X_{1}&=e^{-i2\pi 1\cdot 0/4}\cdot 1+e^{-i2\pi 1\cdot 1/4}\cdot (2-i)+e^{-i2\pi 1\cdot 2/4}\cdot (-i)+e^{-i2\pi 1\cdot 3/4}\cdot (-1+2i)=-2-2i\\X_{2}&=e^{-i2\pi 2\cdot 0/4}\cdot 1+e^{-i2\pi 2\cdot 1/4}\cdot (2-i)+e^{-i2\pi 2\cdot 2/4}\cdot (-i)+e^{-i2\pi 2\cdot 3/4}\cdot (-1+2i)=-2i\\X_{3}&=e^{-i2\pi 3\cdot 0/4}\cdot 1+e^{-i2\pi 3\cdot 1/4}\cdot (2-i)+e^{-i2\pi 3\cdot 2/4}\cdot (-i)+e^{-i2\pi 3\cdot 3/4}\cdot (-1+2i)=4+4i\end{aligned}}}$$

results in $${\displaystyle \mathbf {X} ={\begin{pmatrix}X_{0}\\X_{1}\\X_{2}\\X_{3}\end{pmatrix}}={\begin{pmatrix}2\\-2-2i\\-2i\\4+4i\end{pmatrix}}.}$$

The DFT is a linear transform, i.e. if ${\displaystyle {\mathcal {F}}(\{x_{n}\})_{k}=X_{k}}$ and ${\displaystyle {\mathcal {F}}(\{y_{n}\})_{k}=Y_{k}}$, then for any complex numbers ${\displaystyle a,b}$:

${\displaystyle {\mathcal {F}}(\{ax_{n}+by_{n}\})_{k}=aX_{k}+bY_{k}}$

Reversing the time (i.e. replacing ${\displaystyle n}$ by ${\displaystyle N-n}$)^[D] in ${\displaystyle x_{n}}$ corresponds to reversing the frequency (i.e. ${\displaystyle k}$ by ${\displaystyle N-k}$).^[6]: p.421 Mathematically, if ${\displaystyle \{x_{n}\}}$ represents the vector x then

if ${\displaystyle {\mathcal {F}}(\{x_{n}\})_{k}=X_{k}}$

then ${\displaystyle {\mathcal {F}}(\{x_{N-n}\})_{k}=X_{N-k}}$

If ${\displaystyle {\mathcal {F}}(\{x_{n}\})_{k}=X_{k}}$ then ${\displaystyle {\mathcal {F}}(\{x_{n}^{*}\})_{k}=X_{N-k}^{*}}$.^[6]: p.423

This table shows some mathematical operations on ${\displaystyle x_{n}}$ in the time domain and the corresponding effects on its DFT ${\displaystyle X_{k}}$ in the frequency domain.

Property	Time domain ${\displaystyle x_{n}}$	Frequency domain ${\displaystyle X_{k}}$
Real part in time	${\displaystyle \operatorname {Re} {\left(x_{n}\right)}}$	${\displaystyle {\frac {1}{2}}\left(X_{k}+X_{N-k}^{*}\right)}$
Imaginary part in time	${\displaystyle \operatorname {Im} {\left(x_{n}\right)}}$	${\displaystyle {\frac {1}{2i}}\left(X_{k}-X_{N-k}^{*}\right)}$
Real part in frequency	${\displaystyle {\frac {1}{2}}\left(x_{n}+x_{N-n}^{*}\right)}$	${\displaystyle \operatorname {Re} {\left(X_{k}\right)}}$
Imaginary part in frequency	${\displaystyle {\frac {1}{2i}}\left(x_{n}-x_{N-n}^{*}\right)}$	${\displaystyle \operatorname {Im} {\left(X_{k}\right)}}$

The vectors ${\displaystyle u_{k}=\left[\left.e^{{\frac {i2\pi }{N}}kn}\;\right|\;n=0,1,\ldots ,N-1\right]^{\mathsf {T}}}$, for ${\displaystyle k=0,1,\ldots ,N-1}$, form an orthogonal basis over the set of N-dimensional complex vectors:

${\displaystyle u_{k}^{\mathsf {T}}u_{k'}^{*}=\sum _{n=0}^{N-1}\left(e^{{\frac {i2\pi }{N}}kn}\right)\left(e^{{\frac {i2\pi }{N}}(-k')n}\right)=\sum _{n=0}^{N-1}e^{{\frac {i2\pi }{N}}(k-k')n}=N~\delta _{kk'}}$

where ${\displaystyle \delta _{kk'}}$ is the Kronecker delta. (In the last step, the summation is trivial if ${\displaystyle k=k'}$, where it is 1 + 1 + ⋯ = N, and otherwise is a geometric series that can be explicitly summed to obtain zero.) This orthogonality condition can be used to derive the formula for the IDFT from the definition of the DFT, and is equivalent to the unitarity property below.

If ${\displaystyle X_{k}}$ and ${\displaystyle Y_{k}}$ are the DFTs of ${\displaystyle x_{n}}$ and ${\displaystyle y_{n}}$ respectively then Parseval's theorem states:

${\displaystyle \sum _{n=0}^{N-1}x_{n}y_{n}^{*}={\frac {1}{N}}\sum _{k=0}^{N-1}X_{k}Y_{k}^{*}}$

where the star denotes complex conjugation. The Plancherel theorem is a special case of Parseval's theorem and states:

${\displaystyle \sum _{n=0}^{N-1}|x_{n}|^{2}={\frac {1}{N}}\sum _{k=0}^{N-1}|X_{k}|^{2}.}$

These theorems are also equivalent to the unitary condition below.

The periodicity can be shown directly from the definition:

${\displaystyle X_{k+N}\ \triangleq \ \sum _{n=0}^{N-1}x_{n}e^{-{\frac {i2\pi }{N}}(k+N)n}=\sum _{n=0}^{N-1}x_{n}e^{-{\frac {i2\pi }{N}}kn}\underbrace {e^{-i2\pi n}} _{1}=\sum _{n=0}^{N-1}x_{n}e^{-{\frac {i2\pi }{N}}kn}=X_{k}.}$

Similarly, it can be shown that the IDFT formula leads to a periodic extension of ${\displaystyle x_{n}}$.

Multiplying ${\displaystyle x_{n}}$ by a linear phase ${\displaystyle e^{{\frac {i2\pi }{N}}nm}}$ for some integer m corresponds to a circular shift of the output ${\displaystyle X_{k}}$: ${\displaystyle X_{k}}$ is replaced by ${\displaystyle X_{k-m}}$, where the subscript is interpreted modulo N (i.e., periodically). Similarly, a circular shift of the input ${\displaystyle x_{n}}$ corresponds to multiplying the output ${\displaystyle X_{k}}$ by a linear phase. Mathematically, if ${\displaystyle \{x_{n}\}}$ represents the vector x then

if ${\displaystyle {\mathcal {F}}(\{x_{n}\})_{k}=X_{k}}$

then ${\displaystyle {\mathcal {F}}\left(\left\{x_{n}\cdot e^{{\frac {i2\pi }{N}}nm}\right\}\right)_{k}=X_{k-m}}$

and ${\displaystyle {\mathcal {F}}\left(\left\{x_{n-m}\right\}\right)_{k}=X_{k}\cdot e^{-{\frac {i2\pi }{N}}km}}$

The convolution theorem for the discrete-time Fourier transform (DTFT) indicates that a convolution of two sequences can be obtained as the inverse transform of the product of the individual transforms. An important simplification occurs when one of sequences is N-periodic, denoted here by ${\displaystyle y_{_{N}},}$ because ${\displaystyle \scriptstyle {\text{DTFT}}\displaystyle \{y_{_{N}}\}}$ is non-zero at only discrete frequencies (see DTFT § Periodic data), and therefore so is its product with the continuous function ${\displaystyle \scriptstyle {\text{DTFT}}\displaystyle \{x\}.}$  That leads to a considerable simplification of the inverse transform.

${\displaystyle x*y_{_{N}}\ =\ \scriptstyle {\rm {DTFT}}^{-1}\displaystyle \left[\scriptstyle {\rm {DTFT}}\displaystyle \{x\}\cdot \scriptstyle {\rm {DTFT}}\displaystyle \{y_{_{N}}\}\right]\ =\ \scriptstyle {\rm {DFT}}^{-1}\displaystyle \left[\scriptstyle {\rm {DFT}}\displaystyle \{x_{_{N}}\}\cdot \scriptstyle {\rm {DFT}}\displaystyle \{y_{_{N}}\}\right],}$

where ${\displaystyle x_{_{N}}}$ is a periodic summation of the ${\displaystyle x}$ sequence: ${\displaystyle (x_{_{N}})_{n}\ \triangleq \sum _{m=-\infty }^{\infty }x_{(n-mN)}.}$

Customarily, the DFT and inverse DFT summations are taken over the domain ${\displaystyle [0,N-1]}$. Defining those DFTs as ${\displaystyle X}$ and ${\displaystyle Y}$, the result is:

${\displaystyle (x*y_{_{N}})_{n}\triangleq \sum _{\ell =-\infty }^{\infty }x_{\ell }\cdot (y_{_{N}})_{n-\ell }=\underbrace {{\mathcal {F}}^{-1}} _{\rm {DFT^{-1}}}\left\{X\cdot Y\right\}_{n}.}$

In practice, the ${\displaystyle x}$ sequence is usually length N or less, and ${\displaystyle y_{_{N}}}$ is a periodic extension of an N-length ${\displaystyle y}$-sequence, which can also be expressed as a circular function:

${\displaystyle (y_{_{N}})_{n}=\sum _{p=-\infty }^{\infty }y_{(n-pN)}=y_{(n\operatorname {mod} N)},\quad n\in \mathbb {Z} .}$

Then the convolution can be written as:

which gives rise to the interpretation as a circular convolution of ${\displaystyle x}$ and ${\displaystyle y.}$^[7][8] It is often used to efficiently compute their linear convolution. (see Circular convolution, Fast convolution algorithms, and Overlap-save)

Similarly, the cross-correlation of ${\displaystyle x}$ and ${\displaystyle y_{_{N}}}$ is given by:

${\displaystyle (x\star y_{_{N}})_{n}\triangleq \sum _{\ell =-\infty }^{\infty }x_{\ell }^{*}\cdot (y_{_{N}})_{n+\ell }={\mathcal {F}}^{-1}\left\{X^{*}\cdot Y\right\}_{n}.}$

As seen above, the discrete Fourier transform has the fundamental property of carrying convolution into componentwise product. A natural question is whether it is the only one with this ability. It has been shown ^[9][10] that any linear transform that turns convolution into pointwise product is the DFT up to a permutation of coefficients. Since the number of permutations of n elements equals n!, there exist exactly n! linear and invertible maps with the same fundamental property as the DFT with respect to convolution.

It can also be shown that:

${\displaystyle {\mathcal {F}}\left\{\mathbf {x\cdot y} \right\}_{k}\ \triangleq \sum _{n=0}^{N-1}x_{n}\cdot y_{n}\cdot e^{-i{\frac {2\pi }{N}}kn}}$

${\displaystyle ={\frac {1}{N}}(\mathbf {X*Y_{N}} )_{k},}$ which is the circular convolution of ${\displaystyle \mathbf {X} }$ and ${\displaystyle \mathbf {Y} }$.

The trigonometric interpolation polynomial

${\displaystyle p(t)={\begin{cases}\displaystyle {\frac {1}{N}}\left[{\begin{alignedat}{3}X_{0}+X_{1}e^{i2\pi t}+\cdots &+X_{{\frac {N}{2}}-1}e^{i2\pi {\big (}\!{\frac {N}{2}}-1\!{\big )}t}&\\&+X_{\frac {N}{2}}\cos(N\pi t)&\\&+X_{{\frac {N}{2}}+1}e^{-i2\pi {\big (}\!{\frac {N}{2}}-1\!{\big )}t}&+\cdots +X_{N-1}e^{-i2\pi t}\end{alignedat}}\right]&N{\text{ even}}\\\displaystyle {\frac {1}{N}}\left[{\begin{alignedat}{3}X_{0}+X_{1}e^{i2\pi t}+\cdots &+X_{\frac {N-1}{2}}e^{i2\pi {\frac {N-1}{2}}t}&\\&+X_{\frac {N+1}{2}}e^{-i2\pi {\frac {N-1}{2}}t}&+\cdots +X_{N-1}e^{-i2\pi t}\end{alignedat}}\right]&N{\text{ odd}}\end{cases}}}$

where the coefficients X_k are given by the DFT of x_n above, satisfies the interpolation property ${\displaystyle p(n/N)=x_{n}}$ for ${\displaystyle n=0,\ldots ,N-1}$.

For even N, notice that the Nyquist component ${\textstyle {\frac {X_{N/2}}{N}}\cos(N\pi t)}$ is handled specially.

This interpolation is not unique: aliasing implies that one could add N to any of the complex-sinusoid frequencies (e.g. changing ${\displaystyle e^{-it}}$ to ${\displaystyle e^{i(N-1)t}}$) without changing the interpolation property, but giving different values in between the ${\displaystyle x_{n}}$ points. The choice above, however, is typical because it has two useful properties. First, it consists of sinusoids whose frequencies have the smallest possible magnitudes: the interpolation is bandlimited. Second, if the ${\displaystyle x_{n}}$ are real numbers, then ${\displaystyle p(t)}$ is real as well.

In contrast, the most obvious trigonometric interpolation polynomial is the one in which the frequencies range from 0 to ${\displaystyle N-1}$ (instead of roughly ${\displaystyle -N/2}$ to ${\displaystyle +N/2}$ as above), similar to the inverse DFT formula. This interpolation does not minimize the slope, and is not generally real-valued for real ${\displaystyle x_{n}}$; its use is a common mistake.

Another way of looking at the DFT is to note that in the above discussion, the DFT can be expressed as the DFT matrix, a Vandermonde matrix, introduced by Sylvester in 1867,

${\displaystyle \mathbf {F} ={\begin{bmatrix}\omega _{N}^{0\cdot 0}&\omega _{N}^{0\cdot 1}&\cdots &\omega _{N}^{0\cdot (N-1)}\\\omega _{N}^{1\cdot 0}&\omega _{N}^{1\cdot 1}&\cdots &\omega _{N}^{1\cdot (N-1)}\\\vdots &\vdots &\ddots &\vdots \\\omega _{N}^{(N-1)\cdot 0}&\omega _{N}^{(N-1)\cdot 1}&\cdots &\omega _{N}^{(N-1)\cdot (N-1)}\\\end{bmatrix}}}$

where ${\displaystyle \omega _{N}=e^{-i2\pi /N}}$ is a primitive Nth root of unity.

For example, in the case when ${\displaystyle N=2}$, ${\displaystyle \omega _{N}=e^{-i\pi }=-1}$, and

${\displaystyle \mathbf {F} ={\begin{bmatrix}1&1\\1&-1\\\end{bmatrix}},}$

(which is a Hadamard matrix) or when ${\displaystyle N=4}$ as in the Discrete Fourier transform § Example above, ${\displaystyle \omega _{N}=e^{-i\pi /2}=-i}$, and

${\displaystyle \mathbf {F} ={\begin{bmatrix}1&1&1&1\\1&-i&-1&i\\1&-1&1&-1\\1&i&-1&-i\\\end{bmatrix}}.}$

The inverse transform is then given by the inverse of the above matrix,

${\displaystyle \mathbf {F} ^{-1}={\frac {1}{N}}\mathbf {F} ^{*}}$

With unitary normalization constants ${\textstyle 1/{\sqrt {N}}}$, the DFT becomes a unitary transformation, defined by a unitary matrix:

${\displaystyle {\begin{aligned}\mathbf {U} &={\frac {1}{\sqrt {N}}}\mathbf {F} \\\mathbf {U} ^{-1}&=\mathbf {U} ^{*}\\\left|\det(\mathbf {U} )\right|&=1\end{aligned}}}$

where ${\displaystyle \det()}$ is the determinant function. The determinant is the product of the eigenvalues, which are always ${\displaystyle \pm 1}$ or ${\displaystyle \pm i}$ as described below. In a real vector space, a unitary transformation can be thought of as simply a rigid rotation of the coordinate system, and all of the properties of a rigid rotation can be found in the unitary DFT.

The orthogonality of the DFT is now expressed as an orthonormality condition (which arises in many areas of mathematics as described in root of unity):

${\displaystyle \sum _{m=0}^{N-1}U_{km}U_{mn}^{*}=\delta _{kn}}$

If X is defined as the unitary DFT of the vector x, then

${\displaystyle X_{k}=\sum _{n=0}^{N-1}U_{kn}x_{n}}$

and the Parseval's theorem is expressed as

${\displaystyle \sum _{n=0}^{N-1}x_{n}y_{n}^{*}=\sum _{k=0}^{N-1}X_{k}Y_{k}^{*}}$

If we view the DFT as just a coordinate transformation which simply specifies the components of a vector in a new coordinate system, then the above is just the statement that the dot product of two vectors is preserved under a unitary DFT transformation. For the special case ${\displaystyle \mathbf {x} =\mathbf {y} }$, this implies that the length of a vector is preserved as well — this is just Plancherel theorem,

${\displaystyle \sum _{n=0}^{N-1}|x_{n}|^{2}=\sum _{k=0}^{N-1}|X_{k}|^{2}}$

A consequence of the circular convolution theorem is that the DFT matrix F diagonalizes any circulant matrix.

A useful property of the DFT is that the inverse DFT can be easily expressed in terms of the (forward) DFT, via several well-known "tricks". (For example, in computations, it is often convenient to only implement a fast Fourier transform corresponding to one transform direction and then to get the other transform direction from the first.)

First, we can compute the inverse DFT by reversing all but one of the inputs (Duhamel et al., 1988):

${\displaystyle {\mathcal {F}}^{-1}(\{x_{n}\})={\frac {1}{N}}{\mathcal {F}}(\{x_{N-n}\})}$

(As usual, the subscripts are interpreted modulo N; thus, for ${\displaystyle n=0}$, we have ${\displaystyle x_{N-0}=x_{0}}$.)

Second, one can also conjugate the inputs and outputs:

${\displaystyle {\mathcal {F}}^{-1}(\mathbf {x} )={\frac {1}{N}}{\mathcal {F}}\left(\mathbf {x} ^{*}\right)^{*}}$

Third, a variant of this conjugation trick, which is sometimes preferable because it requires no modification of the data values, involves swapping real and imaginary parts (which can be done on a computer simply by modifying pointers). Define ${\textstyle \operatorname {swap} (x_{n})}$ as ${\displaystyle x_{n}}$ with its real and imaginary parts swapped—that is, if ${\displaystyle x_{n}=a+bi}$ then ${\textstyle \operatorname {swap} (x_{n})}$ is ${\displaystyle b+ai}$. Equivalently, ${\textstyle \operatorname {swap} (x_{n})}$ equals ${\displaystyle ix_{n}^{*}}$. Then

${\displaystyle {\mathcal {F}}^{-1}(\mathbf {x} )={\frac {1}{N}}\operatorname {swap} ({\mathcal {F}}(\operatorname {swap} (\mathbf {x} )))}$

That is, the inverse transform is the same as the forward transform with the real and imaginary parts swapped for both input and output, up to a normalization (Duhamel et al., 1988).

The conjugation trick can also be used to define a new transform, closely related to the DFT, that is involutory—that is, which is its own inverse. In particular, ${\displaystyle T(\mathbf {x} )={\mathcal {F}}\left(\mathbf {x} ^{*}\right)/{\sqrt {N}}}$ is clearly its own inverse: ${\displaystyle T(T(\mathbf {x} ))=\mathbf {x} }$. A closely related involutory transformation (by a factor of ${\textstyle {\frac {1+i}{\sqrt {2}}}}$) is ${\displaystyle H(\mathbf {x} )={\mathcal {F}}\left((1+i)\mathbf {x} ^{*}\right)/{\sqrt {2N}}}$, since the ${\displaystyle (1+i)}$ factors in ${\displaystyle H(H(\mathbf {x} ))}$ cancel the 2. For real inputs ${\displaystyle \mathbf {x} }$, the real part of ${\displaystyle H(\mathbf {x} )}$ is none other than the discrete Hartley transform, which is also involutory.

The eigenvalues of the DFT matrix are simple and well-known, whereas the eigenvectors are complicated, not unique, and are the subject of ongoing research. Explicit formulas are given with a significant amount of number theory.^[11]

Consider the unitary form ${\displaystyle \mathbf {U} }$ defined above for the DFT of length N, where

${\displaystyle \mathbf {U} _{m,n}={\frac {1}{\sqrt {N}}}\omega _{N}^{(m-1)(n-1)}={\frac {1}{\sqrt {N}}}e^{-{\frac {i2\pi }{N}}(m-1)(n-1)}.}$

This matrix satisfies the matrix polynomial equation:

${\displaystyle \mathbf {U} ^{4}=\mathbf {I} .}$

This can be seen from the inverse properties above: operating ${\displaystyle \mathbf {U} }$ twice gives the original data in reverse order, so operating ${\displaystyle \mathbf {U} }$ four times gives back the original data and is thus the identity matrix. This means that the eigenvalues ${\displaystyle \lambda }$ satisfy the equation:

${\displaystyle \lambda ^{4}=1.}$