DFT methods can be very accurate for little computational cost.

A key enabling factor for these applications is the fact that the DFT can be computed efficiently in practice using a fast Fourier transform ( FFT ) algorithm.

The transform is sometimes denoted by the symbol, as in or or < ref group =" note "> As a linear transformation on a finite-dimensional vector space, the DFT expression can also be written in terms of a DFT matrix ; when scaled appropriately it becomes a unitary matrix and the X < sub > k </ sub > can thus be viewed as coefficients of x in an orthonormal basis .</ ref >

) 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.

An important simplification occurs when the sequences are of finite length, N. In terms of the DFT and inverse DFT, it can be written as follows:

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

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.

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 ".

First, we can compute the inverse DFT by reversing the inputs:

The conjugation trick can also be used to define a new transform, closely related to the DFT, that is involutary — that is, which is its own inverse.

An FFT is a way to compute the same result more quickly: computing a DFT of N points in the naive way, using the definition, takes O ( N < sup > 2 </ sup >) arithmetical operations, while an FFT can compute the same result in only O ( N log N ) operations.

Since the inverse DFT is the same as the DFT, but with the opposite sign in the exponent and a 1 / N factor, any FFT algorithm can easily be adapted for it.

Also, because the Cooley – Tukey algorithm breaks the DFT into smaller DFTs, it can be combined arbitrarily with any other algorithm for the DFT, such as those described below.

For with coprime and, one can use the Prime-Factor ( Good-Thomas ) algorithm ( PFA ), based on the Chinese Remainder Theorem, to factorize the DFT similarly to Cooley – Tukey but without the twiddle factors.

Indeed, Winograd showed that the DFT can be computed with only irrational multiplications, leading to a proven achievable lower bound on the number of multiplications for power-of-two sizes ; unfortunately, this comes at the cost of many more additions, a tradeoff no longer favorable on modern processors with hardware multipliers.

Density functional theory ( DFT ) methods are often considered to be ab initio methods for determining the molecular electronic structure, even though many of the most common functionals use parameters derived from empirical data, or from more complex calculations.

The inverse DFT cannot reproduce the entire time domain, unless the input happens to be periodic.

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 1 / N.

Formally, there is a clear distinction: " DFT " refers to a mathematical transformation or function, regardless of how it is computed, whereas " FFT " refers to a specific family of algorithms for computing DFTs.

Rader's algorithm, exploiting the existence of a generator for the multiplicative group modulo prime, expresses a DFT of prime size as a cyclic convolution of ( composite ) size, which can then be computed by a pair of ordinary FFTs via the convolution theorem ( although Winograd uses other convolution methods ).

It was once believed that real-input DFTs could be more efficiently computed by means of the discrete Hartley transform ( DHT ), but it was subsequently argued that a specialized real-input DFT algorithm ( FFT ) can typically be found that requires fewer operations than the corresponding DHT algorithm ( FHT ) for the same number of inputs.

The coefficients in the upper and lower figures are respectively computed by the Fourier series integral and the discrete Fourier transform | DFT summation.

In this way, they argued that a DHT of power-of-two length can be computed with, at best, 2 more additions than the corresponding number of arithmetic operations for the real-input DFT.

In particular, a DCT is a Fourier-related transform similar to the discrete Fourier transform ( DFT ), but using only real numbers.

In mathematics, the discrete sine transform ( DST ) is a Fourier-related transform similar to the discrete Fourier transform ( DFT ), but using a purely real matrix.

Similarly k = g < sup >– p </ sup > ( mod N ) for any non-zero index k and for a unique p in 0 ,..., N – 2, where the negative exponent denotes the multiplicative inverse of g < sup > p </ sup > modulo N. That means that we can rewrite the DFT using these new indices p and q as:

If only a small number of ω are desired, or if the STFT is desired to be evaluated for every shift m of the window, then the STFT may be more efficiently evaluated using a sliding DFT algorithm.

You simply compute the DFT of an initial filter design that you have using the FFT algorithm ( if you don't have an initial estimate you can start with h = delta ).

* Discrete cosine transform, a Fourier-related transform similar to the discrete Fourier transform ( DFT ), but using only real numbers.

One prominent model is the so-called half-metallic ferromagnetic model, which is based on spin-polarized ( SP ) band structure calculations using the local spin-density approximation ( LSDA ) of the density functional theory ( DFT ) where separate calculations are carried out for spin-up and spin-down electrons.

Recently the DNA fluorescent probe, Thiazole Orange ( TO ) has been effectively modeled, using the Density Functional Theory ( DFT ) with the M06-2X functional coupled with the Polarisable Continuum model ( PCM ).

Crystal structures determined via electron crystallography can be checked for their quality by using first-principles calculations within density functional theory ( DFT ).

The common understanding of DFT in the context of Electronic Design Automation ( EDA ) for modern microelectronics is shaped to a large extent by the capabilities of commercial DFT software tools as well as by the expertise and experience of a professional community of DFT engineers researching, developing, and using such tools.

The terminology is further blurred by the ( now rare ) synonym finite Fourier transform for the DFT, which apparently predates the term " fast Fourier transform " ( Cooley et al., 1969 ) but has the same initialism.

A fast Fourier transform ( FFT ) is an efficient algorithm to compute the discrete Fourier transform ( DFT ) and its inverse.

Rader's algorithm ( 1968 ) is a fast Fourier transform ( FFT ) algorithm that computes the discrete Fourier transform ( DFT ) of prime sizes by re-expressing the DFT as a cyclic convolution.

Bluestein's FFT algorithm ( 1968 ), commonly called the chirp z-transform algorithm ( 1969 ), is a fast Fourier transform ( FFT ) algorithm that computes the discrete Fourier transform ( DFT ) of arbitrary sizes ( including prime sizes ) by re-expressing the DFT as a convolution.

The prime-factor algorithm ( PFA ), also called the Good – Thomas algorithm ( 1958 / 1963 ), is a fast Fourier transform ( FFT ) algorithm that re-expresses the discrete Fourier transform ( DFT ) of a size N = N < sub > 1 </ sub > N < sub > 2 </ sub > as a two-dimensional N < sub > 1 </ sub >× N < sub > 2 </ sub > DFT, but only for the case where N < sub > 1 </ sub > and N < sub > 2 </ sub > are relatively prime.

This category is for fast Fourier transform ( FFT ) algorithms, i. e. algorithms to compute the discrete Fourier transform ( DFT ) in O ( N log N ) time ( or better, for approximate algorithms ), where is the number of discrete points.

In order to take advantage of a fast Fourier transform algorithm for computing the DFT, the summation is usually performed over all N terms, even though N-L of them are zeros.

In the context of fast Fourier transform algorithms, a butterfly is a portion of the computation that combines the results of smaller discrete Fourier transforms ( DFTs ) into a larger DFT, or vice versa ( breaking a larger DFT up into subtransforms ).