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[note 3] Still further generalization is possible to functions on groups, which, besides the original Fourier transform on R or R n, notably includes the discrete-time Fourier transform (DTFT, group = Z), the discrete Fourier transform (DFT, group = Z mod N) and the Fourier series or circular Fourier transform (group = S 1, the unit circle ≈ ...
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 ...
The term discrete-time refers to the fact that the transform operates on discrete data, often samples whose interval has units of time. From uniformly spaced samples it produces a function of frequency that is a periodic summation of the continuous Fourier transform of the original continuous function.
A fast Fourier transform (FFT) is an algorithm that computes the discrete Fourier transform (DFT) of a sequence, or its inverse (IDFT). A Fourier transform converts a signal from its original domain (often time or space) to a representation in the frequency domain and vice versa.
This is, of course, shorthand for the assertion that the Fourier transform of the tempered distribution = is ^ = which again follows by imposing self-adjointness of the Fourier transform. By analytic continuation of the Fourier transform, the Laplace transform of the delta function is found to be [ 66 ] ∫ 0 ∞ δ ( t − a ) e − s t d t ...
In the latter case, the exponential form of Fourier series synthesizes a discrete-time Fourier transform where variable represents frequency instead of time. In general, the coefficients are determined by analysis of a given function s ( x ) {\displaystyle s(x)} whose domain of definition is an interval of length P {\displaystyle P} .
Simply, in the continuous-time case, the function to be transformed is multiplied by a window function which is nonzero for only a short period of time. The Fourier transform (a one-dimensional function) of the resulting signal is taken, then the window is slid along the time axis until the end resulting in a two-dimensional representation of the signal.
These can be generalizations of the Fourier transform, such as the short-time Fourier transform, the Gabor transform or fractional Fourier transform (FRFT), or can use different functions to represent signals, as in wavelet transforms and chirplet transforms, with the wavelet analog of the (continuous) Fourier transform being the continuous ...