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Topic: The Hartley Transform

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	Frequency transform
		A frequency transform is the mapping of functions of a function space[?] on the coefficients of basis functions, where the basis functions must have a locality in the frequency spectrum.
		The result of the transform are the coefficients of the components (basis functions), i.e.
		Frequency transforms are often used as part of the process of transform coding, but have many other uses, including scientific and engineering analysis.
	www.ebroadcast.com.au /lookup/encyclopedia/fr/Frequency_transform.html (105 words)

	Hartley (Site not responding. Last check: 2007-11-07)
		Hartley oscillator are inductively coupled, variable frequency oscillators where the oscillator may be series or shunt fed. Hartley oscillators have the advantage of having one centre tapped inductor and one tuning capacitor.
		Hartley was aware of a relationship between the amount of energy in an information system and the amount of information that could be transmitted.
		One drawback of using the Hartley transform is that it is difficult to extend to higher dimensions because it is not a separable transform.
	www.geocities.com /neveyaakov/electro_science/hartley.html (878 words)

	Fast Fourier Transform
		A Fast Fourier Transform (FFT) is an efficient algorithm to compute the discrete Fourier transform (DFT) and its inverse.
		It is of great importance to a wide variety of applications, from digital signal processing to solving partial differential equations to algorithms for quickly multiplying large integers.
		It was once believed that real-input DFTs could be more efficiently computed by means of the Discrete Hartley transform (DHT), but this was subsequently disproved: 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.
	www.ebroadcast.com.au /lookup/encyclopedia/ff/FFT.html (2315 words)

	Automatic Generation of Transform Algorithms
		The transform is given in the form of a defining matrix.
		The methods have been implemented in the library AREP and used to generate fast algorithms for a class of transforms including the discrete Fourier, cosine, sine, and Hartley transform, automatically.
		We have successfully demonstrated the method by computing automatically efficient transforms in several important cases: for the DFT, we obtain the Cooley/Tukey FFT; for a class of transforms including the DCT, type 2, the number of arithmetic operations for our fast transforms is the same as for the best known algorithms.
	www.ece.cmu.edu /~smart/papers/autgen.html (615 words)

	Discrete Hartley transform biography .ms (Site not responding. Last check: 2007-11-07)
		A discrete Hartley transform (DHT) is a Fourier-related transform of discrete, periodic data similar to the discrete Fourier transform (DFT), with analogous applications in signal processing etcetera.
		Its main distinction from the DFT is that it transforms real inputs to real outputs, with no intrinsic involvement of complex numbers.
		Just as the DFT is the discrete analogue of the continuous Fourier transform, the DHT is the discrete analogue of the continuous Hartley transform, introduced by R.
	www.biography.ms /Discrete_Hartley_transform.html (994 words)

	Ralph Hartley (Site not responding. Last check: 2007-11-07)
		He invented the Hartley oscillator, the Hartley transform, and contributed to the foundations of information theory.
		Hartley was born in Spruce, Nevada and attended the University of Utah, receiving an A.B. degree in 1909.
		Hartley, Ralph Hartley, Ralph Hartley, Ralph Hartley, Ralph
	ralph-hartley.kiwiki.homeip.net (407 words)

	[Abstract] Phase Estimation of Minimum Phase Systems using the Hartley Phase Cepstrum
		In section 2 of this paper, the Hartley phase spectrum and the Hartley phase cepstrum are defined and their application to phase analysis is explained.
		In this paper the Hartley Transform is presented as a novel alternative to the conventional Fourier Transform, for the estimation of the phase response of linear discrete minimum phase systems.
		The representation of phase via the Hartley transform avoids the need for a phase unwrapping algorithm and the Hartley phase spectrum has the added advantage of being a function bounded by Â±â2.
	www.actapress.com /Abstract.aspx?paperId=23478 (363 words)

	[No title] (Site not responding. Last check: 2007-11-07)
		The output of the r2c transform is a two-dimensional complex array of size `nx' by `ny/2+1', where the `y' dimension has been cut nearly in half because of redundancies in the output.
		Half of these column transforms, however, are of imaginary parts, and should therefore be multiplied by i and combined with the r2hc transforms of the real columns to produce the 2d DFT amplitudes; FFTW's r2r transform does _not_ perform this combination for you.
		Note that the boundary conditions of the transform output array are given by the input boundary conditions of the inverse transform.
	www.math.temple.edu /doc/packages/fftw3/fftw3.info-1 (5887 words)

	Mathematical Operations (Site not responding. Last check: 2007-11-07)
		Mathematical transformations of images may be as simple as image arithmetic or as complex as an iterating Fourier transform.
		where O is the Fourier transform of the convolved image, H is the Fourier transform of the convolution filter, Hc is the complex conjugate of H, and fn/fs is the ratio of the noise to signal power spectra.
		The wavelet transform is used primarily for smoothing, noise reduction and lossy compression.
	www.wavemetrics.com /products/igorpro/imageprocessing/imagetransforms/arithematic.htm (722 words)

	The HARTLEY Surname Hall of Fame 2 M-R
		Marcellus HARTLEY DODGE was a descendant of Marcellus HARTLEY.
		Marie Hartley and Joan Ingilby have researched and chronicled life in the Dales, and particularly Wensleydale, from prehistoric times to the present day, it is a unique collection.
		Hartley's first solo exhibition at 291 in 1909, led to his long-standing affiliation with the Stieglitz circle of artists, writers, and cultural critics.
	www.hartleyfamily.org.uk /Fame2MR.htm (1087 words)

	The Hartley transform applied to particle image velocimetry
		The Hartley transform is an integral transform similar to the Fourier transform and has most of the characteristics of the Fourier transform.
		The cross-correlation property of the Hartley transform based on separable kernels is presented in detail and the application to PIV analysis is introduced.
		The advantage of the Hartley transform can be shown from the comparison of numbers of operations in theory and computation time in practice.
	stacks.iop.org /0957-0233/13/1996 (309 words)

	CAP 6416 Fall 1994 -- Midterm Examination (Site not responding. Last check: 2007-11-07)
		Described how the separability property of the Fourier transform is used to generate an algorithm for computing the FFT of an N by N image, and show how many additions and multiplications are required to compute the 2-D FFT of such an image.
		The Fourier Transform displays what is referred to as conjugate symmetry because it is formed from the sum of real cos terms and imaginary sin terms, where cos and sin are (respectively) even and odd functions.
		The Hartley transform avoids the use of imaginary sin terms, instead just adding the cos and sin terms as real quantities.
	www.cise.ufl.edu /~jnw/VisionCourse/Examinations/midterm.html (396 words)

	Calculating the FHT in Hardware
		The fast Hartley transform has attracted considerable research interest as an alternative to the FFT.
		The processor can be programmed to transform sequence lengths of any power of two from 4 to 1024 points, thereby overcoming an inflexibility found in some dedicated hardware processors.
		Larger transforms are a simple extension of the processor’s address generation unit and an increase in the depth of the memories.
	www.faginfamily.net /barry/Papers/ieeetsp.htm (3295 words)

	Fast Fourier transform (Site not responding. Last check: 2007-11-07)
		This article describes the algorithms, of which there are many; see discrete Fourier transform for properties and applications of the transform.
		This method (and the general idea of an FFT) was popularized by a publication of J. Cooley and J. Tukey in 1965, but it was later discovered that those two authors had independently re-invented an algorithm known to Carl Friedrich Gauss around 1805 (and subsequently rediscovered several times in limited forms).
		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.
	fast-fourier-transform.iqnaut.net (1604 words)

	FFTW 3.0.1 (Site not responding. Last check: 2007-11-07)
		The discrete Hartley transform (DHT) is an invertible linear transform closely related to the DFT.
		In FFTW, the DHT is actually computed by post-processing an r2hc transform, so there is ordinarily no reason to prefer it from a performance perspective.
		However, we have heard rumors that the DHT might be the most appropriate transform in its own right for certain applications, and we would be very interested to hear from anyone who finds it useful.
	www.uic.edu /depts/accc/hardware/argo/fftw/The-Discrete-Hartley-Transform.html (474 words)

	Transforms VIs (Not in Base Package) - LabVIEW 8 Help
		Use the Transforms VIs to implement common transforms used in signal processing.
		Computes the inverse of the wavelet transform based on the Daubechies4 function of the input sequence X.
		Computes the wavelet transform based on the Daubechies4 function of the input sequence X.
	zone.ni.com /reference/en-XX/help/371361A-01/lvanls/transforms_vis (403 words)

	Hartley Transform (Site not responding. Last check: 2007-11-07)
		(They are near kin of the Fourier transform.) A good test is the double-transform of a Kronecker delta function in the specified position; the double transform should recover the original function, and it does so very well.
		The "Hartley Transform" button makes the code print out the entire transform, which might be more than you want to see for size 8192 so in general use the "Double Hartley Transform" button when you play with this applet, since it will only print out the one value that isn't down in the mud.
		Especially if your computer is an old, slow one like mine, you'll really see that the slow, direct calculation of the Hartley transform takes more time than the fast one.
	www.mindspring.com /~hamill4/hamillnumerics/java/funapplets/hartley/HartleyApplet1.html (227 words)

	Matrix Algorithms of Accelerated Computation of Fast Fourier Transform and Fast Hartley Transform - Begell House Inc. ... (Site not responding. Last check: 2007-11-07)
		Besides, operation of "running" spectral analysis can be implemented not only via Fourier transform, but on the basis of recently proposed Hartley transform too.
		Usually, in practice, fast transform algorithms are employed: FFT or FHT.
		A new matrix algorithm of accelerated computation of "running" ("sliding") Fourier or Hartley spectrum is described in the article.
	www.begellhouse.com.cob-web.org:8888 /journals/0632a9d54950b268,36892d6f2b71a131,4a8a63fb55ba1479.html (221 words)

	Radix-4 Fast Cosine Transform
		as far as I know, FFT is an algorithm to compute transforms (DFT, discrete sine transform, discrete cosine transform, hartley transform, etc) faster than if we use the original formula of the aforementioned transform.
		I am looking for Radix-4 algorithms, because for N being a power of four, Radix-4 algorithms are faster than radix-2 algorithms.
		In the radix-4 algorithm, the transform is split into a number of these trivial four-point transforms, and non-trivial multiplications only have to be performed between stages of these fourpoint transforms.
	www.edaboard.com /ftopic159869.html (798 words)

	2005, 18 - Electronic Journal "Technical Acoustics"
		The discrete Hartley transform (DHT) is a real-valued transform and is closely related to the familiar Fourier transform (FT).
		Because the Hartley transform is a real transform, it is more computationally efficient than the Fourier and Laplace transforms.
		Application of Hartley Transform for the Analysis of the Propagation of the Non-Sinusoidal Waveforms in Power Systems, IEEE Transaction on Power Delivery, vol.6, N°4, 1862–1868, 1991.
	ejta.org /en/soliman1 (479 words)

	Migration by Hartley Transform , by Rick Ottolini (Site not responding. Last check: 2007-11-07)
		A Hartley transform is a variation of the Fourier transform with an integration kernal cos(omega)t + sin(omega)t.
		Computationally useful properties include that the forward and inverse transforms are identical and the transform of real numbers remain real numbers.
		Solutions to the wave equation can be derived in Hartley transform coordinates and used to image seismic data.
	sepwww.stanford.edu /oldreports/oldreports/sep38/38_16_abs.html (78 words)

	Multidimensional Fast Hartley Transform Into Simd Hypercubes (ResearchIndex) (Site not responding. Last check: 2007-11-07)
		Abstract: We present a parallel algorithm for performing multidimensional fast Hartley transforms (FHTs) on hypercube SIMD computers with unshared local memory.
		The flexibility of the algorithm derives from the partition of the dimensions of the hypercube in subsets associated with each of the dimensions of the transform, the pure binary processor indexing, and the consecutive distribution of data in the processors' local memory, which facilitates the parallel performance of unidimensional FHTs.
		8 The discrete Hartley transform (context) - Bracewell - 1983
	citeseer.ist.psu.edu /34508.html (491 words)

	Search SPIE Papers - Publications - SPIE Web
		The main difficulty in practical processing of the complex Fourier-like transforms (Laplace, Mellin transforms, etc.) despite of their convenience in analytical calculations and algebra becomes the insufficient speed of data processing for reception of the dynamic distribution image of an investigated physical characteristic.
		In the present work we examined the possibilities of application in various areas of a reconstructive tomography of the real-domain integral transform offered by Ralph Vinton Lyon Hartley in 1942 for study of a spectra of electrosignals, lastly named in his honor by Hartley transform.
		From middle of the 1960-the years there were offered various fast algorithms of calculation of discrete Fourier transform (FFT) which characterized by some advantage in speed of data processing in comparison with discrete FT, but however owing to its complexity and asymmetry FFT concedes in speed of processing to fast algorithms based on Hartley transform.
	www.spie.org /scripts/abstract.pl?bibcode=1999SPIE.3732..353K&page=1&qs=spie (302 words)

	1d Discrete Hartley Transforms (DHTs) - FFTW 3.1.2 (Site not responding. Last check: 2007-11-07)
		The discrete Hartley transform (DHT) of a 1d real array X of size n computes a real array Y of the same size, where:
		FFTW computes an unnormalized transform, in that there is no coefficient in front of the summation in the DHT.
		In other words, applying the transform twice (the DHT is its own inverse) will multiply the input by n.
	www.fftw.org /doc/1d-Discrete-Hartley-Transforms-_0028DHTs_0029.html (83 words)

	[No title] (Site not responding. Last check: 2007-11-07)
		Below is a Pascal program of a fast Hartley transform...
		If my memory serves me it was originally derived from a BYTE magazine article on the Hartley transform about...
		This routine re-orders the data before the butterly transform routine is called...
	www.dsv.su.se /~fk/DSP/Hartley.fft.txt (299 words)

	E71 Final 2000
		The Inverse Discrete Hartely Transform is the same except for a division by N (eq 3.167),
		You showed, as part of the first exam, that the DHT (and IDHT) can be more efficient for certain operations than the DFT (and IDFT) because all multiplications are with real numbers instead of complex numbers (recall that a complex multiplication generally uses four real multiplications).
		Since the DHT is so closely linked to the DFT it seems reasonable to to try to develop a Fast Hartley Transform, or FHT.
	www.swarthmore.edu /NatSci/echeeve1/Class/e71/FinalExam/Final.html (892 words)

	Theußl Thomas
		This paper first gives an overview of the theoretical principles, whereas it also concentrates on the Hartley transform as a means of changing between spatial and frequency domain.
		The Hartley transform is more suitable in this context as the more familiar Fourier transform, since it generates real output for real data.
		Either the data is inherently three-dimensional, e.g., flow simulation data or medical data, or 3D is chosen as the biggest reasonable projection space for high-dimensional data, e.g., dynamical systems or databases.
	www.cg.tuwien.ac.at /~theussl (725 words)

	Systems Thinkers
		All of these areas, and their interrelationships, underlie complex systems, as they are studied today.
		[ Ralph V. Hartley ] Ralph V.L. Hartley was IRE Fellow, American Association for the Advancement of Science Fellow and 1946 IRE Medal of Honor.
		535-563) Possibly his greatest contribution was the creation of a new transform: The Hartley Transform which has many practical applications.
	mathcs.wilkes.edu /~rpryor/systhinkers.html (791 words)

	Fast Fourier Transforms
		FFTW - Fastest Fourier Transform in the West - A collection of C subroutines.
		GPFA - Routines for the generalized prime factor fast Fourier transform (written by C. Temperton).
		The routines use table look-up of sines and cosines and incorporate a bit reversal table for the in-place bit reversal.
	faculty.prairiestate.edu /skifowit/fft (708 words)

	ij.process.FHT (Java2HTML)
		The Fast Hartley Transform was restricted by U.S. Patent No. 4,646,256, but was placed in the public domain by Stanford University in 1995 and is now freely available.
		The image contained in this FHT must be square and its width must be a power of 2.
		/ Returns an inverse transform** of this image, which is assumed to be in the frequency domain.
	rsb.info.nih.gov /ij/docs/source/ij/process/FHT.java.html (1042 words)

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