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Topic: Discrete Fourier transform

Related Topics

Fourier transform

Continuous Fourier transform

Fast Fourier transform

Discrete sine transform

Fractional Fourier transform

Discrete cosine transform

FFT multiplication

Fourier series

Convolution theorem

Frequency spectrum

Linear transformation

System of linear equations

Digital signal processing

Autocorrelation

Fourier analysis

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	ipedia.com: Fourier transform Article (Site not responding. Last check: 2007-11-07)
		Fourier transforms have many scientific applications — in physics, number theory, combinatorics, signal processing, probability theory, statistics, cryptography, acoustics, oceanography, optics, geometry, and other areas.
		The transforms are linear operators and, with proper normalization, are unitary as well (a property known as Parseval's theorem or, more generally, as the Plancherel theorem, and most generally via Pontryagin duality).
		These Fourier variants can also be generalized to Fourier transforms on arbitrary locally compact abelian topological groups, which are studied in harmonic analysis; there, one transforms from a group to its dual group.
	www.ipedia.com /fourier_transform.html (779 words)

	PlanetMath: discrete Fourier transform (Site not responding. Last check: 2007-11-07)
		If you take the limit of the discrete Fourier transform as the number of time divisions increases without bound, you get the integral form of the continuous Fourier transform.
		The justification for this comes from the fact that the set of matrix elements of representations spans the space of functions on the group and the orthogonality relation for matrix elements.
		This is version 7 of discrete Fourier transform, born on 2002-05-23, modified 2005-05-03.
	planetmath.org /encyclopedia/DiscreteFourierTransform.html (443 words)

	3.5 Discrete Fourier Transform (Site not responding. Last check: 2007-11-07)
		The Fourier transform of the sampled version of s(t) in Figure 3.11c is from equation 3.19 continuous since S(f) is continuous, but an unexpected result is that S(f) is periodic as is illustrated in the sketch in Figure 3.11c.
		Its inverse Fourier transform is, of course, another Dirac delta function in the time domain with teeth separated by T as shown on the left side of Figure 3.11f.
		The extent to which the single period of the DFT approximates the true, continuous Fourier transform depends on the degree of aliasing and leakage (as shown in the frequency domain columns in Figure 3.11c and e, respectively).
	www-rohan.sdsu.edu /~jiracek/digital/spectralanalysis/dft.html (1548 words)

	Discrete Fourier Transform (Site not responding. Last check: 2007-11-07)
		The transform of an infinite train of delta functions spaced by T is an infinite train of delta functions spaced by 1/T. The transform of a cos function is a positive delta at the appropriate positive and negative frequency.
		The transform of a sin function is a negative complex delta function at the appropriate positive frequency and a negative complex delta at the appropriate negative frequency.
		For example the transform of a truncated sin function are two delta functions convolved with a sinc function, a truncated sin function is a sin function multiplied by a square pulse.
	astronomy.swin.edu.au /~pbourke/analysis/dft (1075 words)

	Discrete Fourier Transform and the FFT (Site not responding. Last check: 2007-11-07)
		The Fourier Transform provides the means of transforming a signal defined in the time domain into one defined in the frequency domain (see Tutorial 2 on time and frequency representation).
		The DFT is usually used to approximate the Fourier transform of a continuous time process, and it is necessary to understand some of the limitations inherent in this approach.
		This effect is produced by the inability of the DFT to observe the spectrum as a continuous function, since computation of the spectrum is limited to integer multiples of the fundamental frequency F (reciprocal of the sample length).
	www.cage.curtin.edu.au /mechanical/info/vibrations/tut4.htm (935 words)

	Discrete Fourier Transform
		If you examine the Fourier transform at multiples of 125Hz, you will find a correspondance between what is seen in the DFT frequency domain and the Fourier transform frequency domain.
		Thus the DFT may be considered to be a sampled version of a Fourier transform that has been made periodic with a period of 8kHz in this example.
		Compare this with the Fourier transform of a 333Hz cosine wave and you should see that it has been sampled not at the zero amplitude values of the frequency domain as the 500Hz sine wave was, but at non-zero amplitudes for all the samples spaced by 125Hz.
	www.see.ed.ac.uk /~mjj/dspDemos/EE4/tutDFT.html?http://oldeee.see.ed.ac.uk/~mjj/dspDemos/EE4/tutDFT.html (1333 words)

	What is the Discrete Fourier Transform? (Site not responding. Last check: 2007-11-07)
		Fourier Transforms are a very powerful tool used in physics to determine for example frequency components of a time signal, momentum distributions of particles and many other applications.
		The discrete Fourier transform does not act on signals that exist at all time and continue to time infinity, the DFT applies to signals that exist at a finite number of time points and products a finite number of frequency points.
		The DFT approximates the DTFT (the discrete-Time Fourier Transform).
	www.physlink.com /Education/AskExperts/ae704.cfm (440 words)

	Image Transforms - Fourier Transform
		The Fourier Transform is an important image processing tool which is used to decompose an image into its sine and cosine components.
		The output of the transformation represents the image in the Fourier or frequency domain, while the input image is the spatial domain equivalent.
		The DFT is the sampled Fourier Transform and therefore does not contain all frequencies forming an image, but only a set of samples which is large enough to fully describe the spatial domain image.
	homepages.inf.ed.ac.uk /rbf/HIPR2/fourier.htm (2172 words)

	55:148 Dig. Image Proc. Chapter 11
		The discrete Fourier transform is analogous to the continuous one and may be efficiently computed using the fast Fourier transform algorithm.
		The properties of linearity, shift of position, modulation, convolution, multiplication, and correlation are analogous to the continuous case, with the difference of the discrete periodic nature of the image and its transform.
		Fourier functions are localized in frequency but not in space, in the sense that they isolate frequencies, but not isolate occurrences of those frequencies.
	www.icaen.uiowa.edu /~dip/LECTURE/LinTransforms.html (1478 words)

	The Discrete Fourier Transform (Site not responding. Last check: 2007-11-07)
		The Fourier transform is among the most widely used tools for transforming data sequences and functions, from the time domain to their representation in the frequency domain.
		The DFT maps a discrete sequence in the time domain (observations) to a discrete sequence in the frequency domain (frequency coefficients).
		mini tutorial on the Fourier transform provides a quick introduction to the continuous and the discrete transforms, sampling, the FFT (Fast Fourier Transform), and some of the applications in which the DFT is used.
	www.cs.brown.edu /research/ai/dynamics/tutorial/Documents/DiscreteFourierTransform.html (199 words)

	An application of Discrete Fast Fourier Transform algorithm
		The Fourier transform, in essence, decomposes or separates a waveform or function into sinusoids of different frequency which sum to the original waveform.
		In addition, the Fourier transform of the complex conjugate of a function f(x) is F(-s), the reflection of the conjugate of the transform*.
		Since the Fourier transform F(s) is a frequency domain representation of a function f(x), the s characterizes the frequency of the decomposed cosinusoids and sinusoids and is equal to the number of cycles per unit of x.
	www.bridgeport.edu /sed/projects/cs597/Summer_2002/kunhlee (2639 words)

	The Discrete Fourier Transform (DFT)
		Instead of having a signal magnitude 1.0 value at an x-axis location this case corresponds to the number of sin(x) periods, there is a magnitude that is slightly less than 1.0 and several sub-magnitudes.
		The windowing functions reduce DFT leakage to close to zero in all three cases, except in the immediate vicinity of the sin(x) frequency.
		The DFT leakage magnitudes at the far ends of the graph are removed.
	www.bearcave.com /misl/misl_tech/signal/dftwin (564 words)

	PROPERTIES OF DISCRETE FOURIER TRANSFORM
		For instance, the fact that real data input produces a transform with real part even and imaginary part odd means that the transform is symmetric and only half the output produces new information.
		Thus the result of entering only real values (only half of the input the transform expects since all the imaginary components are zero) is that only half the number of harmonics are useful.
		The frequency shifting property is widely used to produce a display of the Fourier transform which has the (0,0) frequency located in the center of the image, rather than in the lower left corner.
	www.marquette.edu /courses/phys/matthysd/L1980317.htm (310 words)

	FFTW Home Page
		FFTW is a C subroutine library for computing the discrete Fourier transform (DFT) in one or more dimensions, of arbitrary input size, and of both real and complex data (as well as of even/odd data, i.e.
		The paper "A Fast Fourier Transform Compiler," by Matteo Frigo, appears in the Proceedings of the 1999 ACM SIGPLAN Conference on Programming Language Design and Implementation (PLDI '99), Atlanta, Georgia, May 1999.
		The slides from the 7/28/98 talk "The Fastest Fourier Transform in the West," by M. Frigo, are also available, along with the slides from a shorter 1/14/98 talk on the same subject by S. Johnson.
	www.fftw.org (908 words)

	Discrete Fourier Transform
		The Fourier Transform depends on a theory that says that a signal can be represented by a infinite summation of sine waves (also called sinusoids).
		The Fourier Transform (FT) takes a continuous signal (the time domain) and tells you which sinusoids are required to represent it (the frequency domain).
		The Inverse Fourier Transform (IFT) goes from the frequency domain to the time domain, that is it takes a set of sinusoids and tells you what signal they represent.
	www.arrizza.com /articles/dft.html (1176 words)

	The discrete Fourier transform
		The DFT provides information over a discrete number of frequencies, so we need to determine precisely which frequencies these are.
		Alternatively, if one is seeking to describe a function by a set of discrete values, we must sample the function at 2 times the highest frequency in the function.
		Note that the only differences between the forward and inverse transforms are (i) changing the sign in the exponential, and (ii) dividing the answer by N.
	homepages.inf.ed.ac.uk /rbf/CVonline/LOCAL_COPIES/OWENS/LECT4/node3.html (473 words)

	FFTLog
		In the discrete case this remains true for odd N, but it is not generally true for even N (the usual choice) except in the important special case discussed in §8.
		From the definition (2) of the continuous Hankel transform, it can be seen that periodically replicating a function a(r) in logarithmic space lnr and then taking its continuous Hankel transform is equivalent to Hankel transforming the function a(r) and then periodically replicating the transform ã(k) in lnk.
		The ringing that results from taking the discrete transform of a finite segment of a function can be reduced by arranging that the function folds smoothly from large to small scales.
	casa.colorado.edu /~ajsh/FFTLog (3470 words)

	TechOnLine - The Discrete Fourier Transform (Site not responding. Last check: 2007-11-07)
		Fourier analysis is a family of mathematical techniques, all based on decomposing signals into sinusoids.
		The discrete Fourier transform (DFT) is the family member used with digitized signals.
		This is the first of four chapters on the real DFT, a version of the discrete Fourier transform that uses real numbers to represent the input and output signals.
	www.techonline.com /community/ed_resource/tech_paper/37728 (121 words)

	Discrete Fourier Transform (Site not responding. Last check: 2007-11-07)
		This can be back transformed both by DFT and by DFTINT to find d(r) at points other than those to which it was fitted.
		Thus the interval Fourier transform is a least squares fit to the data in the interval only if the series of f values is that given in equation
		The forward transform is first made between -10 <= r <= 10 and then converted to the range -256/20 <= f <= 256/20 by moving the upper half of the transformed values down.
	www.phys.ufl.edu /~coldwell/class2K/Fourier/DFT.htm (1327 words)

	Discrete Fourier Transform
		TV Well, normalizing I guess would be reasonable as normal part of the transform, or the backtransform or subsequent analysis may well be of.
		That's why often averaging is used of various repeated fourier transforms, and why 'frequency analysis' of many programs based on the (real or not) FFT is not the same as actual frequency analysis, and based on the observation of the limitations of the Niquist rate even possibly ambiguous.
		There the aforementioned (in the case of jpeg 2 dimenstional) is applied in a special way, and when you would include the phase in the transform (either by using an orthogonal basis of sine and cosine like in regular fourier analyis, or by transforming to norm/angle complex numbers), the results would become comparable.
	wiki.tcl.tk /11146 (666 words)

	Sampling and Discrete Time Fourier Transform (Site not responding. Last check: 2007-11-07)
		This discreteness gives rise to a periodic spectrum in the frequency domain with a period equal to the sampling frequency.
		Notice that the situation here is the "opposite" of the Fourier Series where the signal is periodic in the time domain and the spectrum is discrete in the frequency domain.
		This transform is intermediate between the Fourier Transform and the DFT and FFT.
	dspcan.homestead.com /files/Sdtft/dtftintr.htm (205 words)

	Notes for the Discrete Fourier Transform (DFT)
		We will next introduce the Discrete Fourier Transform and learn that it is nothing more than a particular representation of a vector in terms of a special basis (Fourier basis) for C[n].
		We denote the Fourier transform of v by v^(k) and of v[d] by v[d]^(k).
		The latter formula is referred to as the inverse discrete fourier transform.
	www.cs.colorado.edu /~mcbryan/3656.04/mail/93.htm (3587 words)

	Localised discrete Fourier transform
		We use a mixed space Fourier transform approach that is applicable to any Bravais lattice symmetry.
		Fourier transformation is a natural method to adopt for this task since in a total-energy calculation one computes other terms, such as the electron density and the Hartree energy, using reciprocal space techniques.
		This implicitly defines the basis set that we use to be plane-waves and for consistency we should calculate the kinetic energy using the same basis set, i.e.
	www.tcm.phy.cam.ac.uk /~pdh1001/papers/paper9/node4.html (987 words)

	THE DISCRETE FOURIER TRANSFORM
		It presents the latest and practically efficient DFT algorithms, as well as the computation of discrete cosine and Walsh—Hadamard transforms.
		Discrete Fourier analysis is covered first, followed by the continuous case, as the discrete case is easier to grasp and is very important in practice.
		This book will be useful as a text for regular or professional courses on Fourier analysis, and also as a supplementary text for courses on discrete signal processing, image processing, communications engineering and vibration analysis.
	www.worldscibooks.com /engineering/4610.html (184 words)

	The Discrete Fourier Transform (Site not responding. Last check: 2007-11-07)
		The discrete Fourier transform (DFT) provides a means for analyzing the frequency components of a discrete-time signal.
		The fast Fourier transform (FFT) reduces the number of operations required to calculate the DFT of a signal by factoring the dense kernel matrix into a number of sparse matrices.
		In effect, this approach reduces the complexity of calculating the DFT of a signal from O(N-squared) to O(n lg n) by reducing the number of operations needed to calculate the N roots of unity in the complex plane.
	www.cs.unc.edu /~parente/igv/hw3/parente_hw3.html (1208 words)

	FFT Links
		NFFT is a free library for non-equispaced discrete Fourier transforms, based on FFTW.
		Picture Book of Fourier Transforms by Kevin Cowtan gives an interesting graphical tutorial on the interpretation of 2D FFT output, with a special emphasis on crystallography.
		DFT Introduction by Paul Bourke, describing the discrete Fourier transform in terms of the continuous transform, with examples of the transforms of various functions.
	www.fftw.org /links.html (986 words)

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