Showing papers on "Hadamard transform published in 1996"

Micro-local approach to the Hadamard condition in quantum field theory on curved space-time

Abstract: For the two-point distribution of a quasi-free Klein-Gordon neutral scalar quantum field on an arbitrary four dimensional globally hyperbolic curved space-time we prove the equivalence of (1) the global Hadamard condition, (2) the property that the Feynman propagator is a distinguished parametrix in the sense of Duistermaat and Hormander, and (3) a new property referred to as the wave front set spectral condition (WFSSC), because it is reminiscent of the spectral condition in axiomatic quantum field theory on Minkowski space. Results in micro-local analysis such as the propagation of singularities theorem and the uniqueness up toC∞ of distinguished parametrices are employed in the proof. We include a review of Kay and Wald's rigorous definition of the global Hadamard condition and the theory of distinguished parametrices, specializing to the case of the Klein-Gordon operator on a globally hyperbolic space-time. As an alternative to a recent computation of the wave front set of a globally Hadamard two-point distribution on a globally hyperbolic curved space-time, given elsewhere by Kohler (to correct an incomplete computation in [32]), we present a version of this computation that does not use a deformation argument such as that used in Fulling, Narcowich and Wald and is independent of the Cauchy evolution argument of Fulling, Sweeny and Wald (both of which are relied upon in Kohler's proof). This leads to a simple micro-local proof of the preservation of Hadamard form under Cauchy evolution (first shown by Fulling, Sweeny and Wald) relying only on the propagation of singularities theorem. In another paper [33], the equivalence theorem is used to prove a conjecture by Kay that a locally Hadamard quasi-free Klein-Gordon state on any globally hyperbolic curved space-time must be globally Hadamard.

Well-posedness of characteristic symmetric hyperbolic systems

Abstract: We consider the initial-boundary-value problem for quasi-linear symmetric hyperbolic systems with characteristic boundary of constant multiplicity. We show the well-posedness in Hadamard's sense (i.e., existence, uniqueness and continuous dependence of solutions on the data) of regular solutions in suitable functions spaces which take into account the loss of regularity in the normal direction to the characteristic boundary.

The Hadamard transform-a tool for index assignment

Abstract: We show that the channel distortion for maximum-entropy encoders, due to noise on a binary-symmetric channel, is minimized if the vector quantizer can be expressed as a linear transform of a hypercube. The index assignment problem is regarded as a problem of linearizing the vector quantizer. We define classes of index assignments with related properties, within which the best index assignment is found by sorting, not searching. Two powerful algorithms for assigning indices to the codevectors of nonredundant coding systems are presented. One algorithm finds the optimal solution in terms of linearity, whereas the other finds a very good, but suboptimal, solution in a very short time.

Method and apparatus for joint transmission of multiple data signals in spread spectrum communication systems

Abstract: A more efficient and less complex structure and method for combining multiple user channels onto a single carrier frequency in spread spectrum communication systems. A single transformation element, such as a fast Hadamard Transformation processor, is used to orthogonally encode and combine a series of subscriber digital data signals in parallel. Portions of the orthogonally encoded and combined data signals are then selected over preselected time intervals to generate a single encoded data signal which is subsequently PN spread, subjected to analog signal processing, and transmitted to system subscribers. This is typically effected by summing digital values in an array of data signal combiners. An exemplary data transformation device uses at least one of a preselected set of orthogonal functions as a controlling pattern for interconnection of the combining elements. Each of the combining elements receives at least two input signals, either data or previously combined data, and produces a combined encoded signal output. The order of combination maps the input data signals into an orthogonal combined signal. Portions of the combined data signals are then selected and transferred in order as a serialized data stream for subsequent spreading and transmission as a communication signal.

Multi-Terminal Binary Decision Diagrams and Hybrid Decision Diagrams

Abstract: Functions that map vectors with binary values into the integers are important for the design and verification of arithmetic circuits We demonstrate how multi-terminal binary decision diagrams (MTBDDs) can be used to represent such functions concisely The Walsh transform and Reed-Muller transform have numerous applications in computer-aided design, but the usefulness of these techniques in practice has been limited by the size of the binary valued functions that can be transformed We show how to compute the MTBDD representations of the Walsh transform and Reed-Muller transform for functions with several hundred variables Bryant and Chen have proposed binary moment diagrams (BMDs) for representing the class of functions that we have considered We discuss the relationship between these methods and describe a generalization called hybrid decision diagrams which is often much more concise

Fast hadamard transform within a code division, multiple access communication system

Abstract: In a receiver (10) of a code division multiple access system, a method of processing a received signal including reading a first and second data item from a first memory (50), at least one of the first and second data items associated with Walsh data; performing a first arithmetic operation on the first and second data items to produce a first result; performing a second arithmetic operation on the first and second data items to produce a second result; and storing the first and second result into a second memory (52) while reading a third and fourth data item from the first memory (50).

Spectral Transform Decision Diagrams

Abstract: This chapter proposes spectral decision diagrams (STDDs), that are graphical representations of spectral transforms of switching functions and integer-valued functions. Binary decision diagrams (BDDs) and functional decision diagrams (FDDs) are graphical representations for switching functions and their Reed-Muller transforms, respectively. Multi-terminal decision diagrams (MTBDDs), arithmetic transform decision diagrams (ACDDs), and Walsh transform decision diagrams (WDDs) are graphical representations for integer-valued functions, their arithmetic transforms, and their Walsh transforms, respectively. This chapter shows that an STDD represents a function and its spectral transform at the same time. As for n-bit adders, ACDDs and WDDs require O(n) nodes while MTBDDs require O(2 n ) nodes. As for n-bit multipliers, ACDDs and WDDs require O(n 2) nodes while MTBDDs require O(4 n ) nodes.

Linear Complexity of Periodic Sequences: A General Theory

Abstract: The linear complexity of an N-periodic sequence with components in a field of characteristic p, where N = npϕ and gcd(n, p) = 1, is characterized in terms of the nth roots of unity and their multiplicities as zeroes of the polynomial whose cofficients are the first N digits of the sequence. Hasse derivatives are then introduced to quantify these multiplicities and to define a new generalized discrete Fourier transform that can be applied to sequences of arbitrary length N with components in a field of characteristic p, regardless of whether or not gcd(N, p) = 1. This generalized discrete Fourier transform is used to give a simple proof of the validity of the well-known Games-Chan algorithm for finding the linear complexity of an N-periodic binary sequence with N = 2ϕ and to generalize this algorithm to apply to N-periodic sequences with components in a finite field of characteristic p when N = pϕ. It is also shown how to use this new transform to study the linear complexity of Hadamard (i.e., component-wise) products of sequences.

Hadamard states, adiabatic vacua and the construction of physical states for scalar quantum fields on curved spacetime

Abstract: Quasifree states of a linear Klein-Gordon quantum field on globally hyperbolic spacetime manifolds are considered. After a short mathematical review techniques from the theory of pseudodifferential operators and wavefront sets on manifolds are used to develop a criterion for a state to be an Hadamard state. It is proven that ground- and KMS-states on certain static spacetimes and adiabatic vacuum states on Robertson-Walker spaces are Hadamard states. A counterexample is given which shows that the idea of instantaneous positive energy states w.r.t. a Cauchy surface does in general not yield physical states. Finally, the problem of constructing Hadamard states on arbitrary curved spacetimes is solved in principle.

RMS bandwidth constrained signature waveforms that maximize the total capacity of PAM-synchronous CDMA channels

Abstract: Optimum time-limited signal sets of equal and unequal energies are obtained under root mean square (RMS) bandwidth constraints. The total capacity and the total asymptotic efficiency of the PAM synchronous Gaussian CDMA (PSG-CDMA) channel are considered as the optimality criteria. The latter measure is monotonic with the determinant of the correlation matrix, R, and the former is monotonic with det(I+/spl sigma//sup -2/R), where /spl sigma//sup 2/ represents the noise level. Average as well as maximum RMS bandwidth constraints are considered in the equal-energy case, and the energy-weighted RMS bandwidth constraint is considered for unequal energy signals. For the equal-energy problem, signal sets are found that simultaneously optimize the total asymptotic efficiency under both average and maximum RMS bandwidth constraints. For the total capacity measure, such simultaneously optimal signal sets are also obtained, albeit under the restriction that the number of signals n be a Hadamard matrix dimension. When the Hadamard dimension is in particular a power of two, we obtain optimum signal sets that are shown to yield equal optimum multiuser detector asymptotic efficiencies for all users of an uncoded PSG-CDMA channel. Unequal energy signal sets are also found under an energy-weighted RMS bandwidth constraint for both optimality criteria.

A survey of Hadamard difference sets

Abstract: difference set, Hadamard A (u,k,A) difference set is a k-element subset D of a group G of order u for which the multiset {dldl2: dl,d2 E D, dl =t= d2} contains each nonidentity element of G exactly A times. A Hadamard difference set (HDS) has parameters of the form (u,k, A) = (4N2,2N2_N, N2·N). The Hadamard parameters provide the richest source of known examples of difference sets. The central question is: for each integer N, which groups of order 4N2 support a HDS? This question remains open, for abelian and nonabelian groups, despite an extensive literature. We survey the current state of knowledge of the subject.

Operational Use of the Hadamard Variance in GPS

Abstract: : With upcoming GPS Block IIR hunches scheduled, rubidium clock estimation will require more attention than ever before during the next decade of GPS operations GPS Master Control Station (MCS) estimation architecture relies on a three-state polynomial clock model, which does not include a time-variant decay parameter for frequency drift Since current GPS rubidium frequency standard exhibit signifiant time-dependent frequency drift changes, the MCS is compelled to make precise utilization of the random run process noise parameter, known as q sub 3 The work of various scientists over the past three decades has shown the Hadamard variance to converge for random run FM At PTTI '95, the 2d Space Operations Squadron (2 SOPS) introduced an algorithm that presented a simple, convergent polynomial relationship between the Hadamard variance and the MCS's Kalman filter process noise parameters Until recently, however, neither the Hadamard variance nor the Hadamard-Q equation had actually been put to use in GPS The Naval Research Laboratory (NRL) has now created analysis software designed to employ the Hadamard variance in their GPS clock analyses, to supplement their already existing software, which makes use of the Allan variance This paper presents results of the NRL analysis using both the Allan and Hadamard variances for several operational GPS rubidium frequency standard, as well as results from the recent operational use of the Hadamard-Q equation, by 2 SOPS personnel, based on the NRL analysis data

Inverse aeroacoustic problem for a streamlined body. I - Basic formulation

Abstract: The inverse aeroacoustic problem associated with a flat-plate airfoil in unsteady compressible flow is formulated in terms of a Fredholm integral equation of the first kind. The feasibility of determining the unsteady pressure along the airfoil surface from the radiated acoustic signal is demonstrated. It is shown that the Hadamard conditions of existence and uniqueness of the inverse solution can be satisfied. The third Hadamard condition for the continuous dependence of the solution on the input acoustic data shows the problem to be ill-posed. The singular value decomposition method with regularization is used to treat the associated ill-conditioned algebraic problem. Discretization and collocation techniques are used to represent the unsteady pressure on the airfoil. Both methods give very accurate reconstructions when "perfect" input data are used. The collocation method requires only a small amount of input data either from the far field or the near field, thus making it more suitable for applications.

Corrected Parsimony, Minimum Evolution, and Hadamard Conjugations

Abstract: Phylogenetic spectral analysis based on Hadamard transforms is an interesting new development for the inference of evo? lutionary trees and for understanding the interrelationships of existing methods (Hendy and Penny, 1989, 1993; Steel et al., 1993a; Hendy et al., 1994; Lento et al., 1994). We use it to interconvert between an underlying evolutionary model and the ex? pected frequencies of patterns observed in sequences. Because the Hadamard is a dis? crete Fourier transform on a finite group (Diaconis, 1988), we call this approach spectral analysis. It gives a mathematical description of an invertible relationship be? tween data and model: the expected fre? quencies of patterns in the data can be cal? culated from the model, and the model can be recovered consistently from the fre? quencies of patterns in the data. Recently, Takezaki and Nei (1994) que? ried four aspects of the utility of spectral analysis: the use of the parsimony criterion after correcting for multiple changes (cor? rected parsimony), the current limitation of spectral analysis on nucleotide sequenc? es to the Kimura 3-ST mechanism of evo? lution, the relationship between a Hada? mard transform and the minimum

Complex Hadamard matrices related to Bent sequences

Abstract: Hadamard matrices are often used for some applications, such as error-correcting codes and spread sequences. This article gives the construction of complex Hadamard matrices of order p/sup n/, where p is prime and n is even. The complex Hadamard matrices include bi-phase Hadamard matrices whose elements take {-1, +1}, and four-phase Hadamard matrices whose elements take {/spl plusmn/1, /spl plusmn/j} with j=/spl radic/(-1).

Some properties of analytic functions on the unit ball with hadamard gaps

Abstract: In this paper, we describe some criteria for an analytic function f on the unit ball B, with Hadamard gaps, to belong to the Besov space X p and the space B p, respectively. The special case of the space B p is BMOA. We use these results, together with Ryll-Wojtaszczyk polynomials, to show that the following containments and are proper.

Hadamard's Finite Part

Abstract: A systematic way of ascribing values to integrals which diverge at infinity is described For integrals with algebraic or logarithmic behaviour it is akin to Hadamard's finite part for integrands which have singularities at finite points The power of the method is illustrated by examples of particular integrals and the convolution of distributions which are beyond the range of standard theory Both one-dimensional and multi-dimensional integrals are included The theory also enables the definition of products of distributions which are considered normally to be too singular for multiplication

Generation of Hadamard matrices for phase-code-multiplexed holographic memories.

Abstract: To accommodate the available spatial light modulators and to achieve the maximum possible storage capacity of a phase-code-multiplexed holographic memory, it is often necessary to generate an orthogonal phase code whose length is not a power of 2. We describe an algorithm for the generation of basic Hadamard matrices of order M = 4p (with p an odd number) and show that Hadamard matrices of order N = 4t (with t a positive integer) can be constructed by combination of the algorithms presented with the tensor-product extension method.

Complex spectral decision diagrams

Abstract: Different decision diagrams for representations of binary and multiple-valued functions in the form of Complex Hadamard Transforms and Spectra are introduced in this paper. Since Complex Hadamard Transform matrix can be recursively expanded through Kronecker products, complex hybrid decision diagrams can be easily derived. Other types of decision diagrams introduced are: complex multi-terminal decision diagrams, complex algebraic decision diagrams, real and imaginary decision diagrams and complex edge-valued decision diagrams. The latter decision diagrams are derived from partial Complex Hadamard Transform. Introduction of different efficient representations of Complex Hadamard Transforms and spectra in the form of decision diagrams with their useful properties as the discrete transformations should open the possibility of new applications of spectral techniques based on such transformations in the design of binary and multiple-valued logic systems.

Complex decision diagrams

Abstract: Complex decision diagrams to represent integer-valued functions in the form of complex Hadamard transforms and spectra are introduced. With the distinctive and unique properties of the transform, the novel complex decision diagrams could be further simplified by reduction rules and a half-spectra theorem that will lead to a more compact representation.

Curvature bound and trajectories for magnetic fields on a Hadamard surfacs

Abstract: On a complete oriented Riemannian manifold M, a closed 2-form B is called a magnetic field. Let Q denote the skew symmetric operator Q,: TM ―>TM defined by (u,Q(v)) = B(u, v) for every u, veTM. We call a smooth curve7 a trajectory for J? if it satisfies the equation V^7 = Q(y). Since O is skew symmetric, we find that every trajectory has constant speed and is defined for ―00< t< 00. We shall call a trajectory normal if it is parametrized by its arc length. When y is a trajectory for B, the curve G defined by R(t) = y(Xt) with some constant A is a trajectory for AB. We call the norm \\\\BX\\\\ of the operator Bx:TxMxTxM ―>R the strength of the magnetic field at the point x. For the trivialmagnetic field B = 0, the case without the force of a magnetic field, trajectories are nothing but geodesies. In term of physics it is a trajectory of a charged particleunder the action of the magnetic field.For a classical treatment and physical meaning of magnetic fieldssee [8]. On a Riemann surface M we can write down B = f -Vo\\M with a smooth function/and the volum form VolM on M. When/is a constant function, the case

Composite complex Hadamard spectra of Boolean functions

Abstract: A new class of orthogonal transform, Complex Hadamard Transform is proposed as a spectral technique for analysis, synthesis and testing of Boolean functions. The method is presented to evaluate Complex Hadamard Spectra of AND, OR and XOR of Boolean functions directly front the spectra of the separate Boolean functions. The results are given using a general coding scheme, and different possible coding of Boolean functions are discussed. Moreover, new definition of the convolution operation called complex convolution is derived. Different properties of such a convolution are presented. Theorem giving final formulae for composite complex Hadamard spectra of Boolean functions is stated in terms of complex convolution. By using presented methods, many of the Boolean operations in original domain can be represented much simpler in terms of their composite complex spectra.

On Hadamard stability for compressible viscoelastic constitutive equations

Abstract: In this paper, the necessary and sufficient condition of the global Hadamard stabiltiy is formulated for isothermal and isotropic compressible viscoelastic constitutive equations. Two broad classes of quasilinear differential models (especially the Leonov class) and time-strain separable single integral models of either hyper- or nonhyper-viscoelastic type are considered. In order to derive an algebraic form of the stability condition, we modigy and then combine two mathematical procedures employed in formulating the Hadamard stability criteria for incompressible viscoelastic liquids and the strong ellipticity conditions for compressible elastic solids. We also briefly show that the concept of Hadamard stability is equivalent to the strong ellipticity of compressible elastic field equations. After restating inequalities for stability in terms of such separated variables as equivolumetric shear and volumetric components, we conclude that the stability criteria impose much more restrictive constraints on constitutive relations for compressible materials than for incompressible ones.

Hadamard transform coding/decoding device for image signals

Abstract: In a Hadamard transform coding device, input image signals are first subjected to Hadamard transform. After the Hadamard transform, the DC coefficients are subjected to variable-length coding, and the AC coefficients are subjected to variable-length coding after the output of an AC coefficient predictor is subtracted therefrom. The AC coefficient predictor predicts the AC coefficients of a center block on the basis of the DC coefficients of the adjacent blocks. In a Hadamard transform decoding device, the DC coefficients are first subjected to variable-length decoding. An AC coefficient predictor predicts the AC coefficients of the center block on the basis of the DC efficient of the adjacent blocks. Thereafter, the AC coefficient prediction error is subjected to variable-length decoding, and added with the output of the AC coefficient predictor to obtain the AC coefficients. The transform coefficients thus obtained are subjected to Hadamard inverse-transform to obtain image signals.