Discrete Fourier transform
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.
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.
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.
Prior to its current usage, the "FFT" initialism may have also been used for the ambiguous term "finite Fourier transform".
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
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.
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.
First, we can compute the inverse DFT by reversing all but one of the inputs (Duhamel et al., 1988): (As usual, the subscripts are interpreted modulo N; thus, for
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.
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).
[15] A straightforward approach to obtain DFT eigenvectors is to discretize an eigenfunction of the continuous Fourier transform, of which the most famous is the Gaussian function.
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.
In the discrete case, the Shannon entropies are defined as and and the entropic uncertainty principle becomes[17] The equality is obtained for
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.
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).
This decomposition is of great importance for everything from digital image processing (two-dimensional) to solving partial differential equations.
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).
into a discrete-time Fourier transform (DTFT), which generally entails a type of distortion called aliasing.
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.
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.
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.
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 field of digital signal processing relies heavily on operations in the frequency domain (i.e. on the Fourier transform).
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.)
The ordinary product expression for the coefficients of c involves a linear (acyclic) convolution, where indices do not "wrap around."
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).
Left: A continuous function (top) and its Fourier transform (bottom).
Center-left: 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: Original function is discretized (multiplied by a Dirac comb ) (top). Its Fourier transform (bottom) is a periodic summation ( DTFT ) of the original transform.
Right: 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.
