Linearity

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 ${\displaystyle a,b}$:

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

Time and frequency reversal

Reversing the time (i.e. replacing ${\displaystyle n}$ by ${\displaystyle N-n}$)^{[upper-alpha 3]} in ${\displaystyle x_{n}}$ corresponds to reversing the frequency (i.e. ${\displaystyle k}$ by ${\displaystyle N-k}$).^[5]: p.421 Mathematically, if ${\displaystyle \{x_{n}\}}$ represents the vector x then

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

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

Conjugation in time

If ${\displaystyle {\mathcal {F}}(\{x_{n}\})_{k}=X_{k}}$ then ${\displaystyle {\mathcal {F}}(\{x_{n}^{*}\})_{k}=X_{N-k}^{*}}$.^[5]: p.423

Real and imaginary part

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	${\displaystyle {\frac {1}{2i}}\left(x_{n}-x_{N-n}^{*}\right)}$	${\displaystyle \operatorname {Im} {\left(X_{k}\right)}}$

Orthogonality

The vectors ${\displaystyle u_{k}=\left[\left.e^{{\frac {i2\pi }{N}}kn}\;\right|\;n=0,1,\ldots ,N-1\right]^{\mathsf {T}}}$ 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.

The Plancherel theorem and Parseval's theorem

If ${\displaystyle X_{k}}$ and ${\displaystyle Y_{k}}$ are the DFTs of ${\displaystyle x_{n}}$ and ${\displaystyle y_{n}}$ respectively then the 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. Plancherel theorem is a special case of the 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.

Periodicity

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.

Shift theorem

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

Circular convolution theorem and cross-correlation theorem

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 ${\displaystyle y.}$^[6][7] 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:

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

Uniqueness of the Discrete Fourier Transform

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 ^[8][9] 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 exists exactly n! linear and invertible maps with the same fundamental property as the DFT with respect to convolution.

Convolution theorem duality

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 ${\displaystyle \mathbf {Y} }$.

Trigonometric interpolation polynomial

The trigonometric interpolation polynomial

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

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.

The unitary DFT

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.

Expressing the inverse DFT in terms of the DFT

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.

Eigenvalues and eigenvectors

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.

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:

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

Therefore, the eigenvalues of ${\displaystyle \mathbf {U} }$ are the fourth roots of unity: ${\displaystyle \lambda }$ is +1, −1, +i, or −i.

Since there are only four distinct eigenvalues for this ${\displaystyle N\times N}$ matrix, they have some multiplicity. The multiplicity gives the number of linearly independent eigenvectors corresponding to each eigenvalue. (There are N independent eigenvectors; a unitary matrix is never defective.)

The problem of their multiplicity was solved by McClellan and Parks (1972), although it was later shown to have been equivalent to a problem solved by Gauss (Dickinson and Steiglitz, 1982). The multiplicity depends on the value of N modulo 4, and is given by the following table:

Otherwise stated, the characteristic polynomial of ${\displaystyle \mathbf {U} }$ is:

${\displaystyle \det(\lambda I-\mathbf {U} )=(\lambda -1)^{\left\lfloor {\tfrac {N+4}{4}}\right\rfloor }(\lambda +1)^{\left\lfloor {\tfrac {N+2}{4}}\right\rfloor }(\lambda +i)^{\left\lfloor {\tfrac {N+1}{4}}\right\rfloor }(\lambda -i)^{\left\lfloor {\tfrac {N-1}{4}}\right\rfloor }.}$

No simple analytical formula for general eigenvectors is known. Moreover, the eigenvectors are not unique because any linear combination of eigenvectors for the same eigenvalue is also an eigenvector for that eigenvalue. Various researchers have proposed different choices of eigenvectors, selected to satisfy useful properties like orthogonality and to have "simple" forms (e.g., McClellan and Parks, 1972; Dickinson and Steiglitz, 1982; Grünbaum, 1982; Atakishiyev and Wolf, 1997; Candan et al., 2000; Hanna et al., 2004; Gurevich and Hadani, 2008).

A straightforward approach is to discretize an eigenfunction of the continuous Fourier transform, of which the most famous is the Gaussian function. Since periodic summation of the function means discretizing its frequency spectrum and discretization means periodic summation of the spectrum, the discretized and periodically summed Gaussian function yields an eigenvector of the discrete transform:

• ${\displaystyle F(m)=\sum _{k\in \mathbb {Z} }\exp \left(-{\frac {\pi \cdot (m+N\cdot k)^{2}}{N}}\right).}$

The closed form expression for the series can be expressed by Jacobi theta functions as

• ${\displaystyle F(m)={\frac {1}{\sqrt {N}}}\vartheta _{3}\left({\frac {\pi m}{N}},\exp \left(-{\frac {\pi }{N}}\right)\right).}$

Two other simple closed-form analytical eigenvectors for special DFT period N were found (Kong, 2008):

For DFT period N = 2L + 1 = 4K + 1, where K is an integer, the following is an eigenvector of DFT:

• ${\displaystyle F(m)=\prod _{s=K+1}^{L}\left[\cos \left({\frac {2\pi }{N}}m\right)-\cos \left({\frac {2\pi }{N}}s\right)\right]}$

For DFT period N = 2L = 4K, where K is an integer, the following is an eigenvector of DFT:

• ${\displaystyle F(m)=\sin \left({\frac {2\pi }{N}}m\right)\prod _{s=K+1}^{L-1}\left[\cos \left({\frac {2\pi }{N}}m\right)-\cos \left({\frac {2\pi }{N}}s\right)\right]}$

The choice of eigenvectors of the DFT matrix has become important in recent years in order to define a discrete analogue of the fractional Fourier transform—the DFT matrix can be taken to fractional powers by exponentiating the eigenvalues (e.g., Rubio and Santhanam, 2005). For the continuous Fourier transform, the natural orthogonal eigenfunctions are the Hermite functions, so various discrete analogues of these have been employed as the eigenvectors of the DFT, such as the Kravchuk polynomials (Atakishiyev and Wolf, 1997). The "best" choice of eigenvectors to define a fractional discrete Fourier transform remains an open question, however.

Uncertainty principles

DFT of real and purely imaginary signals

• If ${\displaystyle x_{0},\ldots ,x_{N-1}}$ are real numbers, as they often are in practical applications, then the DFT ${\displaystyle X_{0},\ldots ,X_{N-1}}$ is even symmetric:

${\displaystyle x_{n}\in \mathbb {R} \quad \forall n\in \{0,\ldots ,N-1\}\implies X_{k}=X_{-k\mod N}^{*}\quad \forall k\in \{0,\ldots ,N-1\}}$, where ${\displaystyle X^{*}\,}$ denotes complex conjugation.

It follows that for even ${\displaystyle N}$ ${\displaystyle X_{0}}$ and ${\displaystyle X_{N/2}}$ are real-valued, and the remainder of the DFT is completely specified by just ${\displaystyle N/2-1}$ complex numbers.

• If ${\displaystyle x_{0},\ldots ,x_{N-1}}$ are purely imaginary numbers, then the DFT ${\displaystyle X_{0},\ldots ,X_{N-1}}$ is odd symmetric:

${\displaystyle x_{n}\in i\mathbb {R} \quad \forall n\in \{0,\ldots ,N-1\}\implies X_{k}=-X_{-k\mod N}^{*}\quad \forall k\in \{0,\ldots ,N-1\}}$, where ${\displaystyle X^{*}\,}$ denotes complex conjugation.

The real-input multidimensional DFT

For input data ${\displaystyle x_{n_{1},n_{2},\dots ,n_{d}}}$ consisting of real numbers, the DFT outputs have a conjugate symmetry similar to the one-dimensional case above:

${\displaystyle X_{k_{1},k_{2},\dots ,k_{d}}=X_{N_{1}-k_{1},N_{2}-k_{2},\dots ,N_{d}-k_{d}}^{*},}$

where the star again denotes complex conjugation and the ${\displaystyle \ell }$-th subscript is again interpreted modulo ${\displaystyle N_{\ell }}$ (for ${\displaystyle \ell =1,2,\ldots ,d}$).

Spectral analysis

Discrete transforms embedded in time & space.

When the DFT is used for signal spectral analysis, the ${\displaystyle \{x_{n}\}}$ sequence usually represents a finite set of uniformly spaced time-samples of some signal ${\displaystyle x(t)\,}$, where ${\displaystyle t}$ represents time. The conversion from continuous time to samples (discrete-time) changes the underlying Fourier transform of ${\displaystyle x(t)}$ into a discrete-time Fourier transform (DTFT), which generally entails a type of distortion called aliasing. Choice of an appropriate sample-rate (see Nyquist rate) is the key to minimizing that distortion. Similarly, the conversion from a very long (or infinite) sequence to a manageable size entails a type of distortion called leakage, which is manifested as a loss of detail (a.k.a. resolution) in the DTFT. Choice of an appropriate sub-sequence length is the primary key to minimizing that effect. When the available data (and time to process it) is more than the amount needed to attain the desired frequency resolution, a standard technique is to perform multiple DFTs, for example to create a spectrogram. If the desired result is a power spectrum and noise or randomness is present in the data, averaging the magnitude components of the multiple DFTs is a useful procedure to reduce the variance of the spectrum (also called a periodogram in this context); two examples of such techniques are the Welch method and the Bartlett method; the general subject of estimating the power spectrum of a noisy signal is called spectral estimation.

A final source of distortion (or perhaps illusion) is the DFT itself, because it is just a discrete sampling of the DTFT, which is a function of a continuous frequency domain. That can be mitigated by increasing the resolution of the DFT. That procedure is illustrated at § Sampling the DTFT.

• The procedure is sometimes referred to as zero-padding, which is a particular implementation used in conjunction with the fast Fourier transform (FFT) algorithm. The inefficiency of performing multiplications and additions with zero-valued "samples" is more than offset by the inherent efficiency of the FFT.
• As already stated, leakage imposes a limit on the inherent resolution of the DTFT, so there is a practical limit to the benefit that can be obtained from a fine-grained DFT.

Optics, diffraction, and tomography

The discrete Fourier transform is widely used with spatial frequencies in modeling the way that light, electrons, and other probes travel through optical systems and scatter from objects in two and three dimensions. The dual (direct/reciprocal) vector space of three dimensional objects further makes available a three dimensional reciprocal lattice, whose construction from translucent object shadows (via the Fourier slice theorem) allows tomographic reconstruction of three dimensional objects with a wide range of applications e.g. in modern medicine.

Filter bank

See § FFT filter banks and § Sampling the DTFT.

Data compression

The field of digital signal processing relies heavily on operations in the frequency domain (i.e. on the Fourier transform). For example, several lossy image and sound compression methods employ the discrete Fourier transform: the signal is cut into short segments, each is transformed, and then the Fourier coefficients of high frequencies, which are assumed to be unnoticeable, are discarded. The decompressor computes the inverse transform based on this reduced number of Fourier coefficients. (Compression applications often use a specialized form of the DFT, the discrete cosine transform or sometimes the modified discrete cosine transform.) Some relatively recent compression algorithms, however, use wavelet transforms, which give a more uniform compromise between time and frequency domain than obtained by chopping data into segments and transforming each segment. In the case of JPEG2000, this avoids the spurious image features that appear when images are highly compressed with the original JPEG.

Partial differential equations

Discrete Fourier transforms are often used to solve partial differential equations, where again the DFT is used as an approximation for the Fourier series (which is recovered in the limit of infinite N). The advantage of this approach is that it expands the signal in complex exponentials ${\displaystyle e^{inx}}$, which are eigenfunctions of differentiation: ${\displaystyle {{\text{d}}{\big (}e^{inx}{\big )}}/{\text{d}}x=ine^{inx}}$. Thus, in the Fourier representation, differentiation is simple—we just multiply by ${\displaystyle in}$. (However, the choice of ${\displaystyle n}$ is not unique due to aliasing; for the method to be convergent, a choice similar to that in the trigonometric interpolation section above should be used.) A linear differential equation with constant coefficients is transformed into an easily solvable algebraic equation. One then uses the inverse DFT to transform the result back into the ordinary spatial representation. Such an approach is called a spectral method.

Polynomial multiplication

Suppose we wish to compute the polynomial product c(x) = a(x) · b(x). The ordinary product expression for the coefficients of c involves a linear (acyclic) convolution, where indices do not "wrap around." This can be rewritten as a cyclic convolution by taking the coefficient vectors for a(x) and b(x) with constant term first, then appending zeros so that the resultant coefficient vectors a and b have dimension d > deg(a(x)) + deg(b(x)). Then,

${\displaystyle \mathbf {c} =\mathbf {a} *\mathbf {b} }$

Where c is the vector of coefficients for c(x), and the convolution operator ${\displaystyle *\,}$ is defined so

${\displaystyle c_{n}=\sum _{m=0}^{d-1}a_{m}b_{n-m\ \mathrm {mod} \ d}\qquad \qquad \qquad n=0,1\dots ,d-1}$

But convolution becomes multiplication under the DFT:

${\displaystyle {\mathcal {F}}(\mathbf {c} )={\mathcal {F}}(\mathbf {a} ){\mathcal {F}}(\mathbf {b} )}$

Here the vector product is taken elementwise. Thus the coefficients of the product polynomial c(x) are just the terms 0, ..., deg(a(x)) + deg(b(x)) of the coefficient vector

${\displaystyle \mathbf {c} ={\mathcal {F}}^{-1}({\mathcal {F}}(\mathbf {a} ){\mathcal {F}}(\mathbf {b} )).}$

With a fast Fourier transform, the resulting algorithm takes O(N log N) arithmetic operations. Due to its simplicity and speed, the Cooley–Tukey FFT algorithm, which is limited to composite sizes, is often chosen for the transform operation. In this case, d should be chosen as the smallest integer greater than the sum of the input polynomial degrees that is factorizable into small prime factors (e.g. 2, 3, and 5, depending upon the FFT implementation).