Discrete Fourier transform – Wikipedia

Type of Fourier transform in discrete mathematics

Fig 1: Relationship between the (continuous) Fourier transform and the discrete Fourier transform. Left column: A continuous function (top) and its Fourier transform (bottom). Center-left column: Periodic summation of the original function (top). Fourier transform (bottom) is zero except at discrete points. The inverse transform is a sum of sinusoids called Fourier series. Center-right column: Original function is discretized (multiplied by a Dirac comb) (top). Its Fourier transform (bottom) is a periodic summation (DTFT) of the original transform. Right column: The DFT (bottom) computes discrete samples of the continuous DTFT. The inverse DFT (top) is a periodic summation of the original samples. The FFT algorithm computes one cycle of the DFT and its inverse is one cycle of the DFT inverse.

Fig 2: Depiction of a Fourier transform (upper left) and its periodic summation (DTFT) in the lower left corner. The spectral sequences at (a) upper right and (b) lower right are respectively computed from (a) one cycle of the periodic summation of s(t) and (b) one cycle of the periodic summation of the s(nT) sequence. The respective formulas are (a) the Fourier series integral and (b) the DFT summation. Its similarities to the original transform, S(f), and its relative computational ease are often the motivation for computing a DFT sequence.

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. 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 the most important discrete transform, used to perform Fourier analysis in many practical applications.^[1] 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^[2]). 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;^[3] 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”.

Definition[edit]

The discrete Fourier transform transforms a sequence of N complex numbers

into another sequence of complex numbers,

which is defined by

The transform is sometimes denoted by the symbol

, as in

or

or

.^[A]

The DFT has many applications, including purely mathematical ones with no physical interpretation. But physically it can be related to signal processing as a discrete version (i.e. samples) of the discrete-time Fourier transform (DTFT), which is a continuous and periodic function. The DFT computes N equally-spaced samples of one cycle of the DTFT. (see Fig.2 and § Sampling the DTFT)

Motivation[edit]

Eq.1 can also be evaluated outside the domain

, and that extended sequence is

-periodic. Accordingly, other sequences of

indices are sometimes used, such as

(if

is even) and

(if

is odd), which amounts to swapping the left and right halves of the result of the transform.^[4]

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

The normalization factor multiplying the DFT and IDFT (here 1 and

) and the signs of the exponents are merely conventions, and differ in some treatments. The only requirements of these conventions are that the DFT and IDFT have opposite-sign exponents and that the product of their normalization factors be

. A normalization of

for both the DFT and IDFT, for instance, makes the transforms unitary. A discrete impulse,

at n = 0 and 0 otherwise; might transform to

for all k (use normalization factors 1 for DFT and

for IDFT). A DC signal,

at k = 0 and 0 otherwise; might inversely transform to

for all

(use

for DFT and 1 for IDFT) which is consistent with viewing DC as the mean average of the signal.

Example[edit]

This example demonstrates how to apply the DFT to a sequence of length

and the input vector

Calculating the DFT of

using Eq.1

results in

Inverse transform[edit]

The discrete Fourier transform is an invertible, linear transformation

${displaystyle {mathcal {F}}colon mathbb {C} ^{N}to mathbb {C} ^{N}}$

with

denoting the set of complex numbers. Its inverse is known as Inverse Discrete Fourier Transform (IDFT). In other words, for any

-dimensional complex vectors.

The inverse transform is given by:

Properties[edit]

Linearity[edit]

The DFT is a linear transform, i.e. if

and

, then for any complex numbers

:

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

Time and frequency reversal[edit]

Reversing the time (i.e. replacing

by

)^[C] in

corresponds to reversing the frequency (i.e.

by

).^[5]: p.421 Mathematically, if

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[edit]

If

then

.^[5]: p.423

Real and imaginary part[edit]

This table shows some mathematical operations on

in the time domain and the corresponding effects on its DFT

in the frequency domain.

Orthogonality[edit]

The vectors

form an orthogonal basis over the set of N-dimensional complex vectors:

${displaystyle u_{k}^{mathsf {T}}u_{k’}^{*}=sum _{n=0}^{N-1}left(e^{{frac {i2pi }{N}}kn}right)left(e^{{frac {i2pi }{N}}(-k’)n}right)=sum _{n=0}^{N-1}e^{{frac {i2pi }{N}}(k-k’)n}=N~delta _{kk’}}$

where

is the Kronecker delta. (In the last step, the summation is trivial if

, 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[edit]

If

and

are the DFTs of

and

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[edit]

The periodicity can be shown directly from the definition:

${displaystyle X_{k+N} triangleq sum _{n=0}^{N-1}x_{n}e^{-{frac {i2pi }{N}}(k+N)n}=sum _{n=0}^{N-1}x_{n}e^{-{frac {i2pi }{N}}kn}underbrace {e^{-i2pi n}} _{1}=sum _{n=0}^{N-1}x_{n}e^{-{frac {i2pi }{N}}kn}=X_{k}.}$

Similarly, it can be shown that the IDFT formula leads to a periodic extension.

Shift theorem[edit]

Multiplying

by a linear phase

for some integer m corresponds to a circular shift of the output

:

is replaced by

, where the subscript is interpreted modulo N (i.e., periodically). Similarly, a circular shift of the input

corresponds to multiplying the output

by a linear phase. Mathematically, if

represents the vector x then

if

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

then

${displaystyle {mathcal {F}}left(left{x_{n}cdot e^{{frac {i2pi }{N}}nm}right}right)_{k}=X_{k-m}}$

and

${displaystyle {mathcal {F}}left(left{x_{n-m}right}right)_{k}=X_{k}cdot e^{-{frac {i2pi }{N}}km}}$

Circular convolution theorem and cross-correlation theorem[edit]

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

because

is non-zero at only discrete frequencies (see DTFT § Periodic data), and therefore so is its product with the continuous function

  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

is a periodic summation of the

sequence:

Customarily, the DFT and inverse DFT summations are taken over the domain

. Defining those DFTs as

and

, the result is:

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

In practice, the

sequence is usually length N or less, and

is a periodic extension of an N-length

-sequence, which can also be expressed as a circular function:

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

Then the convolution can be written as:

which gives rise to the interpretation as a circular convolution of

and

^[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

and

is given by:

${displaystyle (xstar y_{_{N}})_{n}triangleq sum _{ell =-infty }^{infty }x_{ell }^{*}cdot (y_{_{N}})_{n+ell }={mathcal {F}}^{-1}left{X^{*}cdot Yright}_{n}.}$

It has been shown ^[8] that any linear transform that turns convolution into pointwise product is the DFT (up to a permutation of coefficients).

Convolution theorem duality[edit]

It can also be shown that:

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

${displaystyle ={frac {1}{N}}(mathbf {X*Y_{N}} )_{k},}$

which is the circular convolution of

${displaystyle mathbf {X} }$

and

${displaystyle mathbf {Y} }$

.

Trigonometric interpolation polynomial[edit]

The trigonometric interpolation polynomial

${displaystyle p(t)={begin{cases}{frac {1}{N}}left[X_{0}+X_{1}e^{i2pi t}+cdots +X_{N/2-1}e^{i2pi (N/2-1)t}+X_{N/2}cos(Npi t)+X_{N/2+1}e^{-i2pi (N/2-1)t}+cdots +X_{N-1}e^{-i2pi t}right]&N{text{ even}}\{frac {1}{N}}left[X_{0}+X_{1}e^{i2pi t}+cdots +X_{(N-1)/2}e^{i2pi (N-1)t}+X_{(N+1)/2}e^{-i2pi (N-1)t}+cdots +X_{N-1}e^{-i2pi t}right]&N{text{ odd}}end{cases}}}$

where the coefficients X_k are given by the DFT of x_n above, satisfies the interpolation property

for

.

For even N, notice that the Nyquist component

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

to

) without changing the interpolation property, but giving different values in between the

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

are real numbers, then

is real as well.

In contrast, the most obvious trigonometric interpolation polynomial is the one in which the frequencies range from 0 to

(instead of roughly

to

as above), similar to the inverse DFT formula. This interpolation does not minimize the slope, and is not generally real-valued for real

; its use is a common mistake.

The unitary DFT[edit]

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,

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

where

is a primitive Nth root of unity.

For example, in the case when

,

, and

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

(which is a Hadamard matrix) or when

as in the Discrete Fourier transform § Example above,

, 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

, the DFT becomes a unitary transformation, defined by a unitary matrix:

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

where

is the determinant function. The determinant is the product of the eigenvalues, which are always

or

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

, 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[edit]

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

, we have

.)

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

as

with its real and imaginary parts swapped—that is, if

then

is

. Equivalently,

equals

. 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,

is clearly its own inverse:

. A closely related involutory transformation (by a factor of

) is

, since the

factors in

cancel the 2. For real inputs

, the real part of

is none other than the discrete Hartley transform, which is also involutory.

Eigenvalues and eigenvectors[edit]

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

defined above for the DFT of length N, where

${displaystyle mathbf {U} _{m,n}={frac {1}{sqrt {N}}}omega _{N}^{(m-1)(n-1)}={frac {1}{sqrt {N}}}e^{-{frac {i2pi }{N}}(m-1)(n-1)}.}$

This matrix satisfies the matrix polynomial equation:

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

This can be seen from the inverse properties above: operating

twice gives the original data in reverse order, so operating

four times gives back the original data and is thus the identity matrix. This means that the eigenvalues

satisfy the equation:

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

Therefore, the eigenvalues of

are the fourth roots of unity:

is +1, −1, +i, or −i.

Since there are only four distinct eigenvalues for this

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

is:

${displaystyle det(lambda I-mathbf {U} )=(lambda -1)^{leftlfloor {tfrac {N+4}{4}}rightrfloor }(lambda +1)^{leftlfloor {tfrac {N+2}{4}}rightrfloor }(lambda +i)^{leftlfloor {tfrac {N+1}{4}}rightrfloor }(lambda -i)^{leftlfloor {tfrac {N-1}{4}}rightrfloor }.}$

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:

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

  

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:

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

  

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

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

  

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[edit]

Probabilistic uncertainty principle[edit]

If the random variable X_k is constrained by

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

then

${displaystyle P_{n}=|X_{n}|^{2}}$

may be considered to represent a discrete probability mass function of n, with an associated probability mass function constructed from the transformed variable,

${displaystyle Q_{m}=N|x_{m}|^{2}.}$

For the case of continuous functions

and

, the Heisenberg uncertainty principle states that

${displaystyle D_{0}(X)D_{0}(x)geq {frac {1}{16pi ^{2}}}}$

where

and

are the variances of

and

respectively, with the equality attained in the case of a suitably normalized Gaussian distribution. Although the variances may be analogously defined for the DFT, an analogous uncertainty principle is not useful, because the uncertainty will not be shift-invariant. Still, a meaningful uncertainty principle has been introduced by Massar and Spindel.^[9]

However, the Hirschman entropic uncertainty will have a useful analog for the case of the DFT.^[10] The Hirschman uncertainty principle is expressed in terms of the Shannon entropy of the two probability functions.

In the discrete case, the Shannon entropies are defined as

${displaystyle H(X)=-sum _{n=0}^{N-1}P_{n}ln P_{n}}$

and

${displaystyle H(x)=-sum _{m=0}^{N-1}Q_{m}ln Q_{m},}$

and the entropic uncertainty principle becomes^[10]

${displaystyle H(X)+H(x)geq ln(N).}$

The equality is obtained for

equal to translations and modulations of a suitably normalized Kronecker comb of period

where

is any exact integer divisor of

. The probability mass function

will then be proportional to a suitably translated Kronecker comb of period

.^[10]

Deterministic uncertainty principle[edit]

There is also a well-known deterministic uncertainty principle that uses signal sparsity (or the number of non-zero coefficients).^[11] Let

and

be the number of non-zero elements of the time and frequency sequences

and

, respectively. Then,

${displaystyle Nleq left|xright|_{0}cdot left|Xright|_{0}.}$

As an immediate consequence of the inequality of arithmetic and geometric means, one also has

. Both uncertainty principles were shown to be tight for specifically-chosen “picket-fence” sequences (discrete impulse trains), and find practical use for signal recovery applications.^[11]

DFT of real and purely imaginary signals[edit]

• 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:

${displaystyle x_{n}in mathbb {R} quad forall nin {0,ldots ,N-1}implies X_{k}=X_{-kmod N}^{*}quad forall kin {0,ldots ,N-1}}$

, where

${displaystyle X^{*},}$

denotes complex conjugation.

It follows that for even

and

are real-valued, and the remainder of the DFT is completely specified by just

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:

${displaystyle x_{n}in imathbb {R} quad forall nin {0,ldots ,N-1}implies X_{k}=-X_{-kmod N}^{*}quad forall kin {0,ldots ,N-1}}$

, where

${displaystyle X^{*},}$

denotes complex conjugation.

Generalized DFT (shifted and non-linear phase)[edit]

It is possible to shift the transform sampling in time and/or frequency domain by some real shifts a and b, respectively. This is sometimes known as a generalized DFT (or GDFT), also called the shifted DFT or offset DFT, and has analogous properties to the ordinary DFT:

${displaystyle X_{k}=sum _{n=0}^{N-1}x_{n}e^{-{frac {i2pi }{N}}(k+b)(n+a)}quad quad k=0,dots ,N-1.}$

Most often, shifts of

(half a sample) are used.
While the ordinary DFT corresponds to a periodic signal in both time and frequency domains,

produces a signal that is anti-periodic in frequency domain (

) and vice versa for

.
Thus, the specific case of

is known as an odd-time odd-frequency discrete Fourier transform (or O² DFT).
Such shifted transforms are most often used for symmetric data, to represent different boundary symmetries, and for real-symmetric data they correspond to different forms of the discrete cosine and sine transforms.

Another interesting choice is

, which is called the centered DFT (or CDFT). The centered DFT has the useful property that, when N is a multiple of four, all four of its eigenvalues (see above) have equal multiplicities (Rubio and Santhanam, 2005)^[12]

The term GDFT is also used for the non-linear phase extensions of DFT. Hence, GDFT method provides a generalization for constant amplitude orthogonal block transforms including linear and non-linear phase types. GDFT is a framework
to improve time and frequency domain properties of the traditional DFT, e.g. auto/cross-correlations, by the addition of the properly designed phase shaping function (non-linear, in general) to the original linear phase functions (Akansu and Agirman-Tosun, 2010).^[13]

The discrete Fourier transform can be viewed as a special case of the z-transform, evaluated on the unit circle in the complex plane; more general z-transforms correspond to complex shifts a and b above.

Multidimensional DFT[edit]

The ordinary DFT transforms a one-dimensional sequence or array

that is a function of exactly one discrete variable n. The multidimensional DFT of a multidimensional array

that is a function of d discrete variables

for

in

is defined by:

${displaystyle X_{k_{1},k_{2},dots ,k_{d}}=sum _{n_{1}=0}^{N_{1}-1}left(omega _{N_{1}}^{~k_{1}n_{1}}sum _{n_{2}=0}^{N_{2}-1}left(omega _{N_{2}}^{~k_{2}n_{2}}cdots sum _{n_{d}=0}^{N_{d}-1}omega _{N_{d}}^{~k_{d}n_{d}}cdot x_{n_{1},n_{2},dots ,n_{d}}right)right),}$

where

as above and the d output indices run from

. This is more compactly expressed in vector notation, where we define

and

as d-dimensional vectors of indices from 0 to

, which we define as

:

${displaystyle X_{mathbf {k} }=sum _{mathbf {n} =mathbf {0} }^{mathbf {N} -1}e^{-i2pi mathbf {k} cdot (mathbf {n} /mathbf {N} )}x_{mathbf {n} },,}$

where the division

is defined as

to be performed element-wise, and the sum denotes the set of nested summations above.

The inverse of the multi-dimensional DFT is, analogous to the one-dimensional case, given by:

${displaystyle x_{mathbf {n} }={frac {1}{prod _{ell =1}^{d}N_{ell }}}sum _{mathbf {k} =mathbf {0} }^{mathbf {N} -1}e^{i2pi mathbf {n} cdot (mathbf {k} /mathbf {N} )}X_{mathbf {k} },.}$

As the one-dimensional DFT expresses the input

as a superposition of sinusoids, the multidimensional DFT expresses the input as a superposition of plane waves, or multidimensional sinusoids. The direction of oscillation in space is

. The amplitudes are

. This decomposition is of great importance for everything from digital image processing (two-dimensional) to solving partial differential equations. The solution is broken up into plane waves.

The multidimensional DFT can be computed by the composition of a sequence of one-dimensional DFTs along each dimension. In the two-dimensional case

the

independent DFTs of the rows (i.e., along

) are computed first to form a new array

. Then the

independent DFTs of y along the columns (along

) are computed to form the final result

. Alternatively the columns can be computed first and then the rows. The order is immaterial because the nested summations above commute.

An algorithm to compute a one-dimensional DFT is thus sufficient to efficiently compute a multidimensional DFT. This approach is known as the row-column algorithm. There are also intrinsically multidimensional FFT algorithms.

The real-input multidimensional DFT[edit]

For input data

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

-th subscript is again interpreted modulo

(for

).

Applications[edit]

The DFT has seen wide usage across a large number of fields; we only sketch a few examples below (see also the references at the end). All applications of the DFT depend crucially on the availability of a fast algorithm to compute discrete Fourier transforms and their inverses, a fast Fourier transform.

Spectral analysis[edit]

Discrete transforms embedded in time & space.

When the DFT is used for signal spectral analysis, the

sequence usually represents a finite set of uniformly spaced time-samples of some signal

, where

represents time. The conversion from continuous time to samples (discrete-time) changes the underlying Fourier transform of

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[edit]

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[edit]

See § FFT filter banks and § Sampling the DTFT.

Data compression[edit]

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[edit]

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

, which are eigenfunctions of differentiation:

. Thus, in the Fourier representation, differentiation is simple—we just multiply by

. (However, the choice of

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[edit]

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

is defined so

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