Discrete Fourier Transform oor Grieks

Discrete Fourier Transform

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discrete Fourier transform

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(mathematics) The process of converting a discrete time-based function into its frequency-based representation.

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DFT: Discrete Fourier Transform
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For more information, see number-theoretic transform and discrete Fourier transform (general).
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For details, see comparison of the discrete wavelet transform with the discrete Fourier transform.
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The "best" choice of eigenvectors to define a fractional discrete Fourier transform remains an open question, however.
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This was followed in 1754 by some lunar tables, which he computed using a form of the discrete Fourier transform.
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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.
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In particular for very large numbers, methods based on the discrete Fourier transform can reduce the number of simple multiplications to the order of n log(n) log log(n).
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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).
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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.
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The discrete-time Fourier transform is a periodic function, often defined in terms of a Fourier series.
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The conversion from continuous time to samples (discrete-time) changes the underlying Fourier transform of x(t) into a discrete-time Fourier transform (DTFT), which generally entails a type of distortion called aliasing.
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For this reason, the discrete Fourier transform can be defined by using roots of unity in fields other than the complex numbers, and such generalizations are commonly called number-theoretic transforms (NTTs) in the case of finite fields.
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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.
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There are many concepts in continuous mathematics which have discrete versions, such as discrete calculus, discrete probability distributions, discrete Fourier transforms, discrete geometry, discrete logarithms, discrete differential geometry, discrete exterior calculus, discrete Morse theory, difference equations, discrete dynamical systems, and discrete vector measures.
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Mathematicians define a quasicrystal as a set of discrete points whose Fourier transform is also a set of discrete points.
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The Riemann hypothesis implies that the zeros of the zeta function form a quasicrystal, meaning a distribution with discrete support whose Fourier transform also has discrete support.
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The discrete version of the Fourier transform (see below) can be evaluated quickly on computers using Fast Fourier Transform (FFT) algorithms.
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For example, JPEG compression uses a variant of the Fourier transformation (discrete cosine transform) of small square pieces of a digital image.
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The DFT is the most important discrete transform, used to perform Fourier analysis in many practical applications.
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A straightforward approach is to discretize an eigenfunction of the continuous Fourier transform, of which the most famous is the Gaussian function.
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Convolution, filters, two-dimensional discrete Fourier transform, and discrete wavelet transform.
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‘Real-time bandwidth’ is the widest frequency range for which the analyser can continuously transform time-domain data entirely into frequency-domain results, using a Fourier or other discrete time transform that processes every incoming time point, without a reduction of measured amplitude of more than 3 dB below the actual signal amplitude caused by gaps or windowing effects, while outputting or displaying the transformed data.
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'Real-time bandwidth' is the widest frequency range for which the analyser can continuously transform time-domain data entirely into frequency-domain results, using a Fourier or other discrete time transform that processes every incoming time point, without a reduction of measured amplitude of more than 3 dB below the actual signal amplitude caused by gaps or windowing effects, while outputting or displaying the transformed data.
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A sequence of samples from a measuring device produces a temporal or spatial domain representation, whereas a discrete Fourier transform produces the frequency domain representation.
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A sequence of samples from a measuring device produces a time or spatial domain representation, whereas a discrete Fourier transform produces the frequency domain information.
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A sequence of samples from a measuring device produces a temporal or spatial domain representation, whereas a discrete Fourier transform produces the frequency domain information, that is, the frequency spectrum.
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