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Topic: Discrete sine transform

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	Discrete cosine transform
		A related transform is the discrete sine transform (DST), which is equivalent to a DFT of real and odd functions.
		Formally, the discrete cosine transform is a linear, invertible function F : R
		A related transform, the modified discrete cosine transform (MDCT), is used in AAC, Vorbis, and MP3 audio compression.
	www.ebroadcast.com.au /lookup/encyclopedia/dc/DCT.html (792 words)

	Computer-based video compression system - United States Patent 4,897,717
		The transform coefficients are reconstructed by the inverse quantizer 36.
		4, the one dimensional discrete sine transforms 25a, 25b, 25c, the quantizers 26a, 26b and the adaptive vector coders 27a, 27b are discrete sine transform 25, quantizer 26 and adaptive vector coder 27, respectively, of FIG.
		The variable Q is given by Q1 for the transform coefficients of the right edge vector and the lower edge vector and by Q2 for transform coefficients generated by the two dimensional discrete sine transform of the interior block.
	xrint.com /patents/us/4897717 (16889 words)

	Discrete cosine transform - Biocrawler (Site not responding. Last check: 2007-10-08)
		The most common variant of discrete cosine transform is the type-II DCT, which is often called simply "the DCT"; its inverse, the type-III DCT, is correspondingly often called simply "the inverse DCT" or "the IDCT".
		Two related transforms are the discrete sine transform (DST), which is equivalent to a DFT of real and odd functions, and the modified discrete cosine transform (MDCT), which is based on a DCT of overlapping data.
		The result is an 8x8 transform coefficient array in which the (0,0) element is the DC (zero-frequency) component and entries with increasing vertical and horizontal index values represent higher vertical and horizontal spatial frequencies.
	www.biocrawler.com /encyclopedia/DCT (1024 words)

	The Finite Element Method (Partial Differential Equation Toolbox)
		The key to the solution of the problem Kv = F is that the problem Tw = f is possible to solve using the discrete sine transform.
		Solving Tw = f using the discrete sine transform would not be an advantage in itself, since the system is tridiagonal and should be solved as such.
		Reverse the reordering, and perform N2 - 1 discrete sine transforms on the blocks of length N1 - 1.
	www-rohan.sdsu.edu /doc/matlab/toolbox/pde/4fem11.html (512 words)

	Define DCT - Discrete Cosine Transform
		In the decoder, an inverse discrete transform is used to reverse the process.
		Discrete Cosine Transform is a Fourier-related transform similar to the discrete Fourier transform (DFT), but using only real numbers.
		The result is an 8 × 8 transform coefficient array in which the (0,0) element is the DC (zero-frequency) component and entries with increasing vertical and horizontal index values represent higher vertical and horizontal spatial frequencies.
	www.birds-eye.net /definition/d/dct-discrete_cosine_transform.shtml (923 words)

	Modified discrete cosine transform
		The modified discrete cosine transform (MDCT) is a frequency transform based on the type-IV discrete cosine transform (DCT-IV), with the additional property of being lapped: it is designed to be performed on consecutive blocks of a larger dataset, where subsequent blocks are 50% overlapped.
		As a lapped transform, the MDCT is a bit unusual compared to other frequency transforms in that it has half as many outputs as inputs (instead of the same number).
		In principle, x and y could have different window functions, and the window function could also change from one block to the next (especially for the case where data blocks of different sizes are combined), but for simplicity we consider the common case of identical window functions for equal-sized blocks.
	www.ebroadcast.com.au /lookup/encyclopedia/md/MDCT.html (1295 words)

	Discrete sine transform
		A related transform is the discrete cosine transform (DCT), which is equivalent to a DFT of real and even functions.
		The inverse of DST-I is DST-I multiplied by 2/(n+1).
		DSTs are widely employed in solving partial differential equations by spectral methods, where the different variants of the DST correspond to slightly different odd/even boundary conditions at the two ends of the array.
	www.mik.fastload.org /di/Discrete_sine_transform.html (575 words)

	Discrete Fourier Transform
		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.
	local.wasp.uwa.edu.au /~pbourke/other/dft (1075 words)

	Discrete Fourier Transform
		Transformation is an effort to represent an arbitrary signal with a set of signals (called basis functions) with known characteristics like amplitude, frequency and phase.
		Thus, each X(k) representing the magnitude of sine and cosine components is obtained by the spot correlation of the N samples of the signal segment and N samples of the sine and cosine components.
		The representation through samples of the Fourier transform is in effect a representation of the finite-duration sequence by a periodic sequence, one period of which is the finite-duration sequence we wish to represent.
	rfdesign.com /mag/radio_understanding_discrete_fourier (2043 words)

	Contour approximation apparatus for representing a contour of an object - Patent 6259818
		It is, therefore, a primary object of the invention to provide a novel contour approximation apparatus by employing a polygonal approximation and discrete sine transform(DST), thereby providing a representation of an contour image with an increased accuracy and a reduced overall computational complexity.
		The errors calculated by the error detector 300 are supplied to the DST and quantization block 400 for generating quantized DST coefficients.
		At the contour coder 500, each set of the quantized DST coefficients is encoded, e.g., by using the binary arithmetic code of JPEG(Joint Photographic Experts Group), while the vertex information from the polygonal approximation section 100 is encoded by using, e.g., a fixed length code without compression since the vertices are sparsely correlated.
	www.freepatentsonline.com /6259818.html (1689 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.
		Note that the discrete cosine transform computation can be based on the Fourier transform - all N coefficients of the discrete cosine transform may be computed using a 2N -point fast Fourier transform.
	www.icaen.uiowa.edu /~dip/LECTURE/LinTransforms.html (1478 words)

	Optimal Unified Architectures (Site not responding. Last check: 2007-10-08)
		Discrete sinusoidal transforms play significant roles in various digital signal processing applications, such as spectrum analysis, image and speech signal processing, computer tomography, data compression, and signal reconstruction.
		Among different discrete sinusoidal transforms, the discrete consine transform (DCT), the discrete sine transform (DST), discrete Hartley transform (DHT), and the discrete Fourier transform (DFT), are widely used because of their efficient performance.
		The transfer functions of various discrete transforms may be generalized and used to implement a universal filter module.
	www.isr.umd.edu /ISR/accomplishments/043_UnifiedArchitectures (1188 words)

	DCT - DiCTionary, Discrete Cosine Transform, Display Compression Technology
		Discrete Co-sine Transform: used to transform discrete data from the domain of time or space to the frequency domain, without analyzing the phase of the signal
		DCT is a process that converts images from three-dimensions (3D) to two-dimensions (2D) by using the Discrete Cosine (DC) coefficient to examine the luminance of each block of pixels used to form an image.
		A mathematical transform that can provide aliasing cancellation and good frequency resolution, used in some codecs to convert the audio or video signal from the time domain to the frequency domain.
	www.auditmypc.com /acronym/DCT.asp (392 words)

	Apparatus for encoding a contour of an object (US5691769)
		A set of first approximation errors is calculated at a predetermined number of sample points on each first line segment between two vertex points, and a first set of discrete sine transform coefficients is obtained by discrete sine transforming the set of first approximation errors for each first line segment.
		A set of second approximation errors is calculated at the predetermined number of sample points on each second line segment between two predicted vertex points, and a second set of discrete sine transform coefficients is obtained by discrete sine transforming the set of second approximation errors for each second line segment.
		After determining a set of differences by subtracting the second set of discrete sine transform coefficients from the corresponding first set of discrete sine transform coefficients, the set of differences are encoded for transmission to thereby reduce the volume of transmission data.
	www.delphion.com /details?pn10=US05691769 (568 words)

	Using Sun Performance Library Fast Fourier Transform Routines
		Because the cosine even-wave and sine odd-wave routines perform either the transform or inverse transform, depending upon whether the input array contains the Fourier coefficients or the periodic sequence, only the notation for the transform is shown in this table.
		19, the transform of the vector corresponding to 4
		The Fourier transform of the vector [1 2 3 4] is:
	docs.sun.com /source/806-6147/FFT.html (3873 words)

	SSRN-A Discrete Sine Transform Approach for Realized Volatility Measurement by Giuseppe Curci, Fulvio Corsi
		The motivation for this approach rests on the ability of the DST to diagonalize MA type of process which arises naturally in discrete time models of tick-by-tick returns with market microstructure noise.
		Hence, the DST provides a natural orthonormal basis decomposition of observed returns which permits to optimally disentangle the volatility signal of the underlying price process from the market microstructure noise.
		Robustness of the DST estimators with respect to more general dependent structure of the microstructure noise is also analytically shown.
	papers.ssrn.com /Sol3/papers.cfm?abstract_id=650504 (424 words)

	An application of Discrete Fast Fourier Transform algorithm
		Thus the cochlea serves to transform the air pressure signal experienced by the ear drum into frequency information which can be interpreted by the brain as tonality and texture.
		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)

	What's New in Mathematica 4.1: Descriptions
		The discrete sine and cosine transforms are real variants of the discrete Fourier transform and are widely applicable from signal processing to the numerical solution of partial differential equations.
		Here is the discrete sine transform of the data.
		Mathematica has normalized the discrete sine and cosine transforms so that they are each their own inverses.
	wolfram.com /products/mathematica/newin41/descriptions/fouriertrig.html (91 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.
		Transforms of real even/odd data: the discrete cosine transform (DCT) and the discrete sine transform (DST), types I-IV.
		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)

	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)

	Discrete Hankel Transform Definition - GNU Scientific Library -- Reference Manual
		It is this discrete expression which defines the discrete Hankel transform.
		The implementation allows a scaling of the fundamental interval, for convenience, so that one can assume the function is defined on the interval [0,X], rather than the unit interval.
		Therefore, this transform corresponds to an orthogonal expansion in eigenfunctions of the Dirichlet problem for the Bessel differential equation.
	www.gnu.org /software/gsl/manual/html_node/Discrete-Hankel-Transform-Definition.html (225 words)

	The Discrete Hartley Transform - FFTW 3.1.2
		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.fftw.org /doc/The-Discrete-Hartley-Transform.html (371 words)

	FFTPACK - Fast Fourier Transform Package (Site not responding. Last check: 2007-10-08)
		Sometimes, a routine has been so heavily optimized that it is impossible to determine if it is really a correct implementation of the relatively simple Fourier formulas.
		In some cases, a "slow" version of a transform routine has been supplied, simply to provide a simple check that the formulas are correct.
		DSINT computes the discrete Fourier sine transform of an odd sequence.
	www.csit.fsu.edu /~burkardt/f_src/fftpack/fftpack.html (974 words)

	5.3 Discrete Sine Transform
		Replaces the columns of a dense real matrix with their discrete sine transforms.
		The second argument, an integer between 1 and 4, denotes the type of transform (DST-I, DST-II, DST-III, DST-IV).
		These transforms are defined as follows (for a matrix with n rows).
	www.ee.ucla.edu /~vandenbe/cvxopt/doc/node31.html (69 words)

	1.2.11 Discrete Fourier Transform
		Problem: The discrete Fourier transform H of h, H_m = \sum_{k=0}^{n-1} h_k e^{2 \pi i k m / n}, 0 \leq m \leq n-1.
		For example, the sharp spike in the figure above describes the period of a single sine function that closely models the input data.
		By eliminating the coefficients of sine functions that contribute relatively little to the image, we can further reduce the size of the image, at little cost in image fidelity.
	www.cs.sunysb.edu /~algorith/files/fourier-transform.shtml (454 words)

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