an evd algorithm for para hermitian polynomial matrices an algorithm for computing the eigenvalue decomposition of a para hermitian polynomial matrix is described this amounts to diagonalizing the polynomial matrix by means of a paraunitary similarity transformation the algorithm makes use of elementary paraunitary transformations and constitutes a generalization of the classical jacobi algorithm for conventional hermitian matrix diagonalization a proof of convergence is presented the application to signal processing is highlighted in terms of strong decorrelation and multichannel data compaction some simulated results are presented to demonstrate the capability of the algorithm

multistability analysis of delayed quaternion valued neural networks with nonmonotonic piecewise nonlinear activation functions abstract this paper deals with the multistability problem of the quaternion valued neural networks qvnns with nonmonotonic piecewise nonlinear activation functions and unbounded time varying delays by virtue of the non commutativity of quaternion multiplication resulting from hamilton rules the qvnns can be separated into four real valued systems by using the fixed point theorem and other analytical tools some novel algebraic criteria are established to guarantee that the qvnns can have 54n equilibrium points 34n of which are locally stable some criteria that guarantee the multiple exponential stability multiple power stability multiple log stability multiple log log stability are also derived as special cases the obtained results reveal that the introduced qvnns in this paper can have larger storage capacity than the complex valued ones finally one numerical example is presented to clarify the validity of the theoretical results

vacuum expectation values of products of chiral currents in 3 1 dimensions an algebraic rule is presented for computing expectation values of products of local nonabelian charge operators for fermions coupled to an external vector potential in 3 1 space time dimensions the vacuum expectation value of a product of four operators is closely related to a cyclic cocycle in noncommutative geometry of alain connes the relevant representation of the current is constructed using kirillov s method of coadjoint orbits

properness and widely linear processing of quaternion random vectors in this paper the second order circularity of quaternion random vectors is analyzed unlike the case of complex vectors there exist three different kinds of quaternion properness which are based on the vanishing of three different complementary covariance matrices the different kinds of properness have direct implications on the cayley dickson representation of the quaternion vector and also on several well known multivariate statistical analysis methods in particular the quaternion extensions of the partial least squares pls multiple linear regression mlr and canonical correlation analysis cca techniques are analyzed showing that in general the optimal linear processing is full widely linear however in the case of jointly q proper or c proper vectors the optimal processing reduces respectively to the conventional or semi widely linear processing finally a measure for the degree of improperness of a quaternion random vector is proposed which is based on the kullback leibler divergence between two zero mean gaussian distributions one of them with the actual augmented covariance matrix and the other with its closest proper version this measure quantifies the entropy loss due to the improperness of the quaternion vector and it admits an intuitive geometrical interpretation based on kullback leibler projections onto sets of proper augmented covariance matrices

theory and design of signal adapted fir paraunitary filter banks we study the design of signal adapted fir paraunitary filter banks using energy compaction as the adaptation criterion we present some important properties that globally optimal solutions to this optimization problem satisfy in particular we show that the optimal filters in the first channel of the filter bank are spectral factors of the solution to a linear semi infinite programming sip problem the remaining filters are related to the first through a matrix eigenvector decomposition we discuss uniqueness and sensitivity issues the sip problem is solved using a discretization method and a standard simplex algorithm we also show how regularity constraints may be incorporated into the design problem to obtain globally optimal in the energy compaction sense filter banks with specified regularity we also consider a problem in which the polyphase matrix implementation of the filter bank is constrained to be dct based such constraints may also be incorporated into our optimization algorithm therefore we are able to obtain globally optimal filter banks subject to regularity and or computational complexity constraints numerous experiments are presented to illustrate the main features that distinguish adapted and nonadapted filters as well as the effects of the various constraints the conjecture that energy compaction and coding gain optimization are equivalent design criteria is shown not to hold for fir filter banks

on factorization of m channel paraunitary filterbanks we systematically investigate the factorization of causal finite impulse response fir paraunitary filterbanks with given filter length based on the singular value decomposition of the coefficient matrices of the polyphase representation a fundamental order one factorization form is first proposed for general paraunitary systems then we develop a new structure for the design and implementation of paraunitary system based on the decomposition of hermitian unitary matrices within this framework the linear phase filterbank and pairwise mirror image symmetry filterbank are revisited their structures are special cases of the proposed general structures compared with the existing structures more efficient ones that only use approximately half the number of free parameters are derived the proposed structures are complete and minimal although the factorization theory with or without constraints is discussed in the framework of m channel filterbanks the results can be applied to wavelets and multiwavelet systems and could serve as a general theory for paraunitary systems

on the gaussian measure of the intersection the gaussian correlation conjecture states that for any two symmetric convex sets in n dimensional space and for any centered gaussian measure on that space the measure of the intersection is greater than or equal to the product of the measures in this paper we obtain several results which substantiate this conjecture for example in the standard gaussian case we show there is a positive constant c such that the conjecture is true if the two sets are in the euclidean ball of radius c n further we show that if for every n the conjecture is true when the sets are in the euclidean ball of radius n then it is true in general our most concrete result is that the conjecture is true if the two sets are arbitrary centered ellipsoids

quaternion singular value decomposition based on bidiagonalization to a real or complex matrix using quaternion householder transformations we present a practical and efficient means to compute the singular value decomposition svd of a real or complex quaternion matrix a based on bidiagonalization of a to a real or complex bidiagonal matrix b using quaternionic householder transformations computation of the svd of b using an existing subroutine library such as lapack provides the singular values of a the singular vectors of a are obtained trivially from the product of the householder transformations and the real or complex singular vectors of b we show in the paper that left and right quaternionic householder transformations are different because of the non commutative multiplication of quaternions and we present formulae for computing the householder vector and matrix in each case

index formulas for generalized wiener hopf operators and boson fermion correspondence in 2n dimensions the kernels of operators associated with special chiral gauge trans formations kinks in the 2 dimensional dirac theory are explicitly deter mined the result is used to obtain index formulas for fredholm operators corresponding to continuous chiral gauge transformations moreover the fock space quadratic forms corresponding to the kinks are proved to converge to the dirac field as the kink size goes to zero it is also shown that for n i 2 mod4 the majorana field can be reached in a similar fashion

multiple emitter location and signal parameter estimation processing the signals received on an array of sensors for the location of the emitter is of great enough interest to have been treated under many special case assumptions the general problem considers sensors with arbitrary locations and arbitrary directional characteristics gain phase polarization in a noise interference environment of arbitrary covariance matrix this report is concerned first with the multiple emitter aspect of this problem and second with the generality of solution a description is given of the multiple signal classification music algorithm which provides asymptotically unbiased estimates of 1 number of incident wavefronts present 2 directions of arrival doa or emitter locations 3 strengths and cross correlations among the incident waveforms 4 noise interference strength examples and comparisons with methods based on maximum likelihood ml and maximum entropy me as well as conventional beamforming are included an example of its use as a multiple frequency estimator operating on time series is included

the construction of filtrations on abstract wiener space abstract the abstract wiener space unlike the classical wiener space does not possess a natural notion of time the introduction of such a notion through a resolution of the identity on the cameron martin space associated with the abstract wiener space is considered in this paper the notion of vector fields adapted to this filtration and causal linear operators are introduced and geometric characterizations of these notions are derived the notion of equivalence between two abstract wiener spaces endowed with filtrations is defined it is shown that under some general conditions every abstract wiener space endowed with a filtration is equivalent to a classical wiener space of a finite or countably infinite collection of independent one dimensional wiener processes let u be an adapted vector field and u the divergence of u some sufficient conditions for the exponential integrability of u are derived

a system of generalized sylvester quaternion matrix equations and its applications let h m n be the set of all m n matrices over the real quaternion algebra we call that a h n n is hermitian if a a i j k where i j k are the quaternion units denote a a in this paper we derive some necessary and sufficient conditions for the solvability to the system of generalized sylvester real quaternion matrix equations a i x i y i b i c i z d i e i i 1 2 and give an expression of the general solution to the above mentioned system as applications we give some solvability conditions and general solution for the generalized sylvester real quaternion matrix equation a 1 x y b 1 c 1 z d 1 e 1 where z is required to be hermitian we also present some solvability conditions and general solution for the system of real quaternion matrix equations involving hermicity a i x i a i x i b i y b i c i i 1 2 where y is required to be hermitian our results include some well known results as special cases

design of fir paraunitary filter banks for subband coding using a polynomial eigenvalue decomposition the problem of paraunitary pu filter bank design for subband coding has received considerable attention in recent years not least because of the energy preserving property of this class of filter banks in this paper we consider the design of signal adapted finite impulse response fir pu filter banks using polynomial matrix evd pevd techniques modifications are proposed to an iterative time domain pevd method known as the sequential best rotation sbr2 algorithm which enables its effective application to the problem of fir orthonormal filter bank design for efficient subband coding by choosing an optimization scheme that maximizes the coding gain at each stage of the algorithm it is shown that the resulting filter bank behaves more and more like the infinite order principle component filter bank pcfb the proposed method is compared to state of the art techniques namely the iterative greedy algorithm iga the approximate evd aevd standard sbr2 and a fast algorithm for fir compaction filter design called the window method wm we demonstrate that for the calculation of the subband coder the wm approach offers a low cost alternative at lower coding gains while at moderate to high complexity the proposed approach outperforms the benchmarkers in terms of run time complexity aevd performs well at low orders while the proposed algorithm offers a better coding gain than the benchmarkers at moderate to high filter order for a number of simulation scenarios

algorithms for interpolation based qr decomposition in mimo ofdm systems detection algorithms for multiple input multiple output mimo wireless systems based on orthogonal frequency division multiplexing ofdm typically require the computation of a qr decomposition for each of the data carrying ofdm tones the resulting computational complexity will in general be significant motivated by the fact that the channel matrices arising in mimo ofdm systems result from oversampling of a polynomial matrix we formulate interpolation based qr decomposition algorithms an in depth complexity analysis based on a metric relevant for very large scale integration vlsi implementations shows that the proposed algorithms for a sufficiently large number of data carrying tones and sufficiently small channel order provably exhibit significantly smaller complexity than brute force per tone qr decomposition

solution of the monge ampere equation on wiener space for general log concave measures abstract in this work we prove that the unique 1 convex solution of the monge kantorovitch measure transportation problem between the wiener measure and a target measure which has an h log concave density in the sense of feyel and ustunel j funct anal 176 2000 400 428 w r t the wiener measure is also the strong solution of the monge ampere equation in the frame of infinite dimensional frechet spaces we further enhance the polar factorization results of the mappings which transform a spread measure to another one in terms of the measure transportation of monge kantorovitch and clarify the relation between this concept and the ito solutions of the monge ampere equation

optimization in quaternion dynamic systems gradient hessian and learning algorithms the optimization of real scalar functions of quaternion variables such as the mean square error or array output power underpins many practical applications solutions typically require the calculation of the gradient and hessian however real functions of quaternion variables are essentially nonanalytic which are prohibitive to the development of quaternion valued learning systems to address this issue we propose new definitions of quaternion gradient and hessian based on the novel generalized hamilton real ghr calculus thus making a possible efficient derivation of general optimization algorithms directly in the quaternion field rather than using the isomorphism with the real domain as is current practice in addition unlike the existing quaternion gradients the ghr calculus allows for the product and chain rule and for a one to one correspondence of the novel quaternion gradient and hessian with their real counterparts properties of the quaternion gradient and hessian relevant to numerical applications are also introduced opening a new avenue of research in quaternion optimization and greatly simplified the derivations of learning algorithms the proposed ghr calculus is shown to yield the same generic algorithm forms as the corresponding real and complex valued algorithms advantages of the proposed framework are illuminated over illustrative simulations in quaternion signal processing and neural networks

corrections to on the existence and uniqueness of the eigenvalue decomposition of a parahermitian matrix presents corrections to the paper on the existence and uniqueness of the eigenvalue decomposition of a paraher mitian matrix weiss s et al ieee trans signal process vol 66 no 10 pp 2659 2672 may 2018

new optimal z complementary code sets based on generalized paraunitary matrices the concept of paraunitary pu matrices arose in the early 1990 s in the study of multi rate filter banks these matrices have found wide applications in cryptography digital signal processing and wireless communications existing pu matrices are subject to certain constraints on their existence and hence their availability is not guaranteed in practice motivated by this for the first time we introduce a novel concept called z paraunitary zpu matrix whose orthogonality is defined over a matrix of polynomials with identical degree not necessarily taking the maximum value we show that there exists an equivalence between a zpu matrix and a z complementary code set when the latter is expressed as a matrix with polynomial entries furthermore we investigate some important properties of zpu matrices which are useful for the extension of matrix sizes and sequence lengths we propose a new construction framework for optimal zpu matrices which includes existing pu matrices as special cases finally we give the enumeration formula for the constructed z complementary sequences from the optimal zpu matrices

source localization using vector sensor array in a multipath environment coherent signals from distinct directions is a natural characterization of the multipath propagation effect this paper addresses the problem of coherent fully correlated source localization using vector sensor arrays the maximum likelihood ml and minimum variance distortionless response mvdr estimators for source direction of arrival doa and signal polarization parameters are derived these estimators require no search over the polarization parameters in addition a novel method for decorrelating the incident signals is presented this method is based on the polarization smoothing algorithm psa and enables the use of eigenstructure based techniques which assume uncorrelated or partially correlated signals the method is implemented as a preprocessing stage before applying eigenstructure based techniques such as music unlike other existing preprocessing techniques such as spatial smoothing and forward backward fb averaging this method is not limited to any specific array geometry the performance of the proposed psa preprocessing combined with music is evaluated and compared to the crame spl acute r rao bound crb and the ml and mvdr estimators simulation results show that the mvdr and psa music asymptotically achieve the crb for a scenario with two coherent sources with and without an uncorrelated interference source a sensitivity study of psa music to source polarization was also conducted via simulations

jacobi method for quaternion matrix singular value decomposition the study of quaternion matrices has gained interest in many areas in recent years and the problem of diagonalizing such matrices has also attracted attention in this article we present an algorithm for computing the svd of a matrix with quaternion coefficients directly in quaternion arithmetic using a generalization of classical complex jacobi methods the extension of the jacobi transformation to the quaternion case is introduced for the diagonalization of a hermitian quaternion valued matrix based on this an implicit jacobi algorithm is proposed for computing the svd of a quaternion matrix the performance of the proposed algorithm is presented and compared with an already known algorithm using a complex equivalent of the quaternion matrix and shown to be superior in execution time and accuracy

quaternion valued echo state networks quaternion valued echo state networks qesns are introduced to cater for 3 d and 4 d processes such as those observed in the context of renewable energy 3 d wind modeling and human centered computing 3 d inertial body sensors the introduction of qesns is made possible by the recent emergence of quaternion nonlinear activation functions with local analytic properties required by nonlinear gradient descent training algorithms to make qenss second order optimal for the generality of quaternion signals both circular and noncircular we employ augmented quaternion statistics to introduce widely linear qesns to that end the standard widely linear model is modified so as to suit the properties of dynamical reservoir typically realized by recurrent neural networks this allows for a full exploitation of second order information in the data contained both in the covariance and pseudocovariances and a rigorous account of second order noncircularity improperness and the corresponding power mismatch and coupling between the data components simulations in the prediction setting on both benchmark circular and noncircular signals and on noncircular real world 3 d body motion data support the analysis

representation of the distributions on wiener space and stochastic calculus of variations abstract in this work we construct projections and dual projections with respect to the past of the wiener process for the cameron martin space valued distributions on the wiener space with their help we then prove on the one hand an extension of clark s formula to the representation of the scalar distributions with the extended ito integral and on the other hand we give a decomposition of the vector valued distributions by exactly calculating the kernel of the divergence operator

cramer rule for the unique solution of restricted matrix equations over the quaternion skew field in this paper we establish the determinantal representations of the generalized inverses a r t 1 s 1 2 a l t 2 s 2 2 and a t 1 t 2 s 1 s 2 2 over the quaternion skew field by the theory of the column and row determinants in addition we derive some generalized cramer rules for the unique solution of some restricted quaternion matrix equations the findings of this paper extend some known results in the literature

a novel block truncation coding of color images using a quaternion moment preserving principle block truncation coding btc is an efficient tool for image compression to compress color pixel blocks a novel color btc algorithm called quaternion moment block truncation coding qmbtc is presented analytical formulas for qmbtc whose computation time is on the order of pixel block size are derived by using quaternion arithmetic and the moment preserving principle the proposed color btc algorithm can adaptively truncate a pixel block into one or two output classes according to the distribution of the color values inside the blocks the experimental results show that the compression ratio is increased as compared with existing color btc algorithms and the picture quality of the reconstructed images is satisfactory in addition a post btc data compression scheme is proposed to further compress the subimage constructed by reproduction colors of truncated pixel blocks using a lookup table to display decoded data this postprocessing scheme can output images acceptable to human eyes

a new real structure preserving quaternion qr algorithm abstract new real structure preserving decompositions are introduced to develop fast and robust algorithms for the right eigenproblem of general quaternion matrices under the orthogonally j r s symplectic transformations the francis j r s qr step and the j r s qr algorithm are firstly proposed for j r s symmetric matrices and then applied to calculate the schur form of quaternion matrices a novel quaternion givens matrix is defined and utilized to compute the qr factorization of quaternion hessenberg matrices an implicit double shift quaternion qr algorithm is presented with a technique for automatically choosing shifts and within real operations numerical experiments are provided to demonstrate the efficiency and accuracy of newly proposed algorithms

some covariance inequalities in wiener space in this work we give an account of some covariance inequalities in abstract wiener space an fkg inequality is obtained with positivity and monotonicity being defined in terms of a given cone in the underlying cameron martin space the last part is dedicated to convex and log concave functionals including a proof of the gaussian conjecture for a particular class of log concave wiener functionals

an algorithm for calculating the qr and singular value decompositions of polynomial matrices in this paper a new algorithm for calculating the qr decomposition qrd of a polynomial matrix is introduced this algorithm amounts to transforming a polynomial matrix to upper triangular form by application of a series of paraunitary matrices such as elementary delay and rotation matrices it is shown that this algorithm can also be used to formulate the singular value decomposition svd of a polynomial matrix which essentially amounts to diagonalizing a polynomial matrix again by application of a series of paraunitary matrices example matrices are used to demonstrate both types of decomposition mathematical proofs of convergence of both decompositions are also outlined finally a possible application of such decompositions in multichannel signal processing is discussed

covariance identities and inequalities for functionals on wiener and poisson spaces we present covariance identities and inequalities for functionals of the wiener and the poisson processes using malliavin calculus techniques an expansion with a remainder term is obtained for the covariance of such functionals our results extend known identities and inequalities for functions of multivariate random vectors

a system of real quaternion matrix equations with applications abstract let h be the real quaternion algebra and h n m denote the set of all n m matrices over h let p h n n and q h m m be involutions i e p 2 i q 2 i a matrix a h n m is said to be p q symmetric if a paq this paper studies the system of linear real quaternion matrix equations a 1 x 1 c 1 x 1 b 1 c 2 a 2 x 2 c 3 x 2 b 2 c 4 a 3 x 1 b 3 a 4 x 2 b 4 c c we present some necessary and sufficient conditions for the existence of a solution to this system and give an expression of the general solution to the system when the solvability conditions are satisfied as applications we discuss the necessary and sufficient conditions for the system a a x c a xb b c b a c xb c c c to have a p q symmetric solution we also show an expression of the p q symmetric solution to the system when the solvability conditions are met moreover we provide an algorithm and a numerical example to illustrate our results the findings of this paper extend some known results in the literature

commutator anomalies and the fock bundle we show that the anomalous finite gauge transformations can be realized as linear operators acting on sections of the bundle of fermionic fock spaces parametrized by vector potentials and more generally by splittings of the fermionic one particle space into a pair of complementary subspaces on the lie algebra level we show that the construction leads to the standard formula for the relevant commutator anomalies

bezout space time precoders and equalizers for mimo channels the transmitter receiver diversities in p input q output multiple input multiple output mimo channels will play a key role in future high rate wireless communication the major challenge in mimo signal recovery is the mitigation of the inevitable intersymbol interference isi and interchannel interference ici in multipath mimo channels the bezout system theory provides a simple solution for isi ici cancellation and thus can serve as a powerful mathematical foundation for mimo systems if p q and the mimo system h d is left coprime the system is perfectly recoverable via a bezout precoder assuming the channel information is available at the transmitter for a robust channel design a quantitative analysis of signal to noise ratio snr of the optimal bezout equalizer precoder can be derived from the bezout null space compared with the alamouti see ieee j select areas commun vol 16 p 1451 58 oct 1998 space time coding the bezout precoder is shown to be more appealing in slowly time varying channels where the feedback of the channel information induces minor overhead the bezout system can work well jointly with the space time block coding stbc the stbc can be adopted to artificially construct a larger q spl times p q p virtual transfer function without the channel information a necessary and sufficient condition is provided for perfect recoverability pr for the stbc induced virtual mimo systems based on the virtual transfer function both channel independent and channel dependent precoding strategies are feasible via a singular value decomposition svd analysis the bezout precoder can be shown to outperform the orthogonal frequency division multiplexing ofdm precoder in bit error rate ber transmission rate and receiver implementation the interplay of the two design parameters n and spl rho is analyzed to provide a simple guideline for an optimal configuration the combined stbc and bezout equalization techniques offer a broader spectrum of transceiver system configuration to achieve optimal design tradeoff among ber transmission rate and implementation complexity

measures on infinite dimensional grassmann manifolds abstract a 1 parameter family of quasi invariant measures is presented these measures are cylinder measures in graph coordinates their characteristic functions are represented as integrals relative to an infinite product measure this is applied to the problem of determining the support properties of the measures one of the measures can be used to define the unitary structure for the basic representation of the affine extension of the restricted unitary group

quaternion reproducing kernel hilbert spaces existence and uniqueness conditions the existence and uniqueness conditions of quaternion reproducing kernel hilbert spaces qrkhs are established in order to provide a mathematical foundation for the development of quaternion valued kernel learning algorithms this is achieved through a rigorous account of left quaternion hilbert spaces which makes it possible to generalise standard rkhs to quaternion rkhs quaternion versions of the riesz representation and moore aronszajn theorems are next introduced thus underpinning kernel estimation algorithms operating on quaternion valued feature spaces the difference between the proposed quaternion kernel concept and the existing real and vector approaches is also established in terms of both theoretical advantages and computational complexity the enhanced estimation ability of the so introduced quaternion valued kernels over their real and vector valued counterparts is validated through kernel ridge regression applications simulations on real world 3d inertial body sensor data and nonlinear channel equalisation using novel quaternion cubic and gaussian kernels support the approach

computing the zeros of quaternion polynomials abstract a method is developed to compute the zeros of a quaternion polynomial with all terms of the form q k x k this method is based essentially in niven s algorithm 1 which consists of dividing the polynomial by a characteristic polynomial associated to a zero the information about the trace and the norm of the zero is obtained by an original idea which requires the companion matrix associated to the polynomial the companion matrix is represented by a matrix with complex entries three numerical examples using mathematica 2 2 version are given

a class of quaternion kalman filters the existing kalman filters for quaternion valued signals do not operate fully in the quaternion domain and are combined with the real kalman filter to enable the tracking in 3 d spaces using the recently introduced hr calculus we develop the fully quaternion valued kalman filter qkf and quaternion extended kalman filter qekf allowing for the tracking of 3 d and 4 d signals directly in the quaternion domain to consider the second order noncircularity of signals we employ the recently developed augmented quaternion statistics to derive the widely linear qkf wl qkf and widely linear qekf wl qekf to reduce computational requirements of the widely linear algorithms their efficient implementation are proposed and it is shown that the quaternion widely linear model can be simplified when processing 3 d data further reducing the computational requirements simulations using both synthetic and real world circular and noncircular signals illustrate the advantages offered by widely linear over strictly linear quaternion kalman filters

extreme ranks of a linear quaternion matrix expression subject to triple quaternion matrix equations with applications abstract in this paper we establish the formulas of the maximal and minimal ranks of the quaternion matrix expression c 4 a 4 xb 4 where x is a variant quaternion matrix subject to quaternion matrix equations a 1 x c 1 xb 2 c 2 a 3 xb 3 c 3 as applications we give a new necessary and sufficient condition for the existence of solutions to the system of matrix equations a 1 x c 1 xb 2 c 2 a 3 xb 3 c 3 a 4 xb 4 c 4 which was investigated by wang q w wang a system of four matrix equations over von neumann regular rings and its applications acta math sin 21 2 2005 323 334 by rank equalities in addition extremal ranks of the generalized schur complement d ca b with respect to an inner inverse a of a which is a common solution to quaternion matrix equations a 1 x c 1 xb 2 c 2 are also considered some previous known results can be viewed as special cases of the results of this paper

simultaneous real diagonalization of rectangular quaternionic matrix pairs and its algorithm the simultaneous real diagonalization srd of a pair of rectangular quaternionic matrices is introduced and some necessary and sufficient conditions for the existence of the srd are derived based on one of the necessary and sufficient condition an algorithm for gaining the srd of a given matrix pair is presented

the p q skew symmetric extremal rank solutions to a system of quaternion matrix equations abstract let h m n denote the set of all m n matrices over the quaternion algebra h and p h m m q h n n be involutions we say that a h m n is p q symmetric or p q skewsymmetric if a paq or a paq we in this paper mainly investigate the p q skew symmetric maximal and minimal rank solutions to the system of quaternion matrix equations ax b xc d we present necessary and sufficient conditions for the existence of the maximal and minimal rank solutions with p q symmetry and p q skewsymmetry to the system the expressions of such solutions to this system are also given when the solvability conditions are satisfied a numerical example is presented to illustrate our results the findings of this paper extend some known results in this literature

gridless line spectrum estimation and low rank toeplitz matrix compression using structured samplers a regularization free approach this paper considers the problem of compressively sampling wide sense stationary random vectors with a low rank toeplitz covariance matrix certain families of structured deterministic samplers are shown to efficiently compress a high dimensional toeplitz matrix of size n times n producing a compressed sketch of size o sqrt r times o sqrt r the reconstruction problem can be cast as that of line spectrum estimation whereby in absence of noise toeplitz matrices of any size n can be exactly recovered from compressive sketches of size o sqrt r times o sqrt r no matter how large n is in the presence of noise and finite data the line spectrum estimation algorithm is combined with a novel denoising technique that only exploits a positive semidefinite psd toeplitz constraint to denoise the compressed sketch using a simple least squares minimization framework a major advantage of the algorithm is that it does not require any regularization parameter it also enjoys lower computational complexity owing to its ability to predict the unobserved entries of the low rank toeplitz matrix explicit bounds on the reconstruction error are established and it is shown that the psd constraint on the denoiser is sufficient to ensure stable reconstruction from a sketch of size o sqrt r times o sqrt r extensive simulations demonstrate that the proposed algorithm provides better performance over random samplers and algorithms that use nuclear norm based regularizers

proper quaternion gaussian graphical models in this paper we extend gaussian graphical models to proper quaternion gaussian distributions the properness assumption reduces the number of unknowns by a factor of four while graphical models reduce the number of degrees of freedom via sparsity each of the methods allows accurate estimation using a small number of samples to enjoy both gains we show that the proper quaternion gaussian inverse covariance estimation problem is convex and has a closed form solution we proceed to demonstrate that the additional sparsity constraints on the inverse covariance matrix also lead to a convex problem and the optimizations can be efficiently solved by standard numerical methods in the special but practical case of a chordal graph we provide a closed form solution we demonstrate the improved performance of our suggested estimators on both synthetic and real data

current algebras in d 1 dimensions and determinant bundles over infinite dimensional grassmannians we extend the methods of pressley and segal for constructing cocycle representations of the restricted general linear group in infinite dimensions to the case of a larger linear group modeled by schatten classes of rank 1 p an essential ingredient is the generalization of the determinant line bundle over an infinite dimensional grassmannian to the case of an arbitrary schatten rank p 1 the results are used to obtain highest weight representations of current algebras with the operator schwinger terms ind 1 dimensions when the space dimensiond is any odd number

a fast structure preserving method for computing the singular value decomposition of quaternion matrices abstract in this paper we propose a fast structure preserving algorithm for computing the singular value decomposition of quaternion matrices the algorithm is based on the structure preserving bidiagonalization of the real counterpart for quaternion matrices by applying orthogonal jrs symplectic matrices the algorithm is efficient and numerically stable

rank equalities related to the generalized inverse at s 2 with applications abstract a variety of rank formulas of some matrix expressions and certain partitioned matrices with respect to the generalized inverse a t s 2 are established some necessary and sufficient conditions are given by using the rank formulas presented in this paper for two three and four ordered matrices to be independent in the generalized inverse a t s 2 as special cases necessary and sufficient conditions are derived for two three and four ordered matrices to be independent in the weighted moore penrose inverse and the drazin inverse some known results can be regarded as the special cases of the results in this paper

quaternionic eigenvalue problem we discuss the right eigenvalue equation for h c and r linear quaternionic operators the possibility to introduce an isomorphism between these operators and real complex matrices allows us to translate the quaternionic problem into an equivalent real or complex counterpart interesting applications are found in solving differential equations within quaternionic formulations of quantum mechanics

quaternions and matrices of quaternions abstract we give a brief survey on quaternions and matrices of quaternions present new proofs for certain known results and discuss the quaternionic analogues of complex matrices the methods of converting a quaternion matrix to a pair of complex matrices and homotopy theory are emphasized

a structure preserving algorithm for the quaternion cholesky decomposition in this paper we discuss the cholesky decomposition of the hermitian positive definite quaternion matrix for the first time the structure preserving gauss transformation is defined and then a novel structure preserving algorithm which is applied to its real representation matrix is proposed our algorithm needs only real number operations does not depend on the quaternion toolbox for matlab qtfm and has more portability although the flops of our algorithm are theoretically about the same as those based on quaternion arithmetic operations or qtfm numerical experiments show that our algorithm runs faster

the common solution to six quaternion matrix equations with applications abstract we in this paper establish necessary and sufficient conditions for the existence of and an explicit expression for a common solution to six classical linear quaternion matrix equations a 1 x c 1 xb 1 c 2 a 2 x c 3 xb 2 c 4 a 3 xb 3 c 5 and a 4 xb 4 c 6 as applications symmetric persymmetric and centrosymmetric solutions to certain systems of quaternion matrix equations are presented furthermore we give an example to illustrate the result of this paper some known results can be regarded as the special cases of this paper

theory of optimal orthonormal subband coders the theory of the orthogonal transform coder and methods for its optimal design have been known for a long time we derive a set of necessary and sufficient conditions for the coding gain optimality of an orthonormal subband coder for given input statistics we also show how these conditions can be satisfied by the construction of a sequence of optimal compaction filters one at a time several theoretical properties of optimal compaction filters and optimal subband coders are then derived especially pertaining to behavior as the number of subbands increases significant theoretical differences between optimum subband coders transform coders and predictive coders are summarized finally conditions are presented under which optimal orthonormal subband coders yield as much coding gain as biorthogonal ones for a fixed number of subbands

music algorithm for vector sensors array using biquaternions in this paper we use a biquaternion formalism to model vector sensor signals carrying polarization information this allows a concise and elegant way of handling signals with eight dimensional 8 d vector valued samples using this model we derive a biquaternionic version of the well known array processing music algorithm and we show its superiority to classically used long vector approach new results on biquaternion valued matrix spectral analysis are presented of particular interest for the biquaternion music bq music algorithm is the decomposition of the spectral matrix of the data into orthogonal subspaces we propose an effective algorithm to compute such an orthogonal decomposition of the observation space via the eigenvalue decomposition evd of a hermitian biquaternionic matrix by means of a newly defined quantity the quaternion adjoint matrix the bq music estimator is derived and simulation results illustrate its performances compared with two other approaches in polarized antenna processing lv music and psa music the proposed algorithm is shown to be superior in several aspects to the existing approaches compared with lv music the bq music algorithm is more robust to modelization errors and coherent noise while it can detect less sources in comparaison with psa music our approach exhibits more accurate estimation of direction of arrival doa for a small number of sources while keeping the polarization information accessible

on the existence and uniqueness of the eigenvalue decomposition of a parahermitian matrix this paper addresses the extension of the factorization of a hermitian matrix by an eigenvalue decomposition evd to the case of a parahermitian matrix that is analytic at least on an annulus containing the unit circle such parahermitian matrices contain polynomials or rational functions in the complex variable z and arise e g as cross spectral density matrices in broadband array problems specifically conditions for the existence and uniqueness of eigenvalues and eigenvectors of a parahermitian matrix evd are given such that these can be represented by a power or laurent series that is absolutely convergent at least on the unit circle permitting a direct realization in the time domain based on an analysis of the unit circle we prove that eigenvalues exist as unique and convergent but likely infinite length laurent series the eigenvectors can have an arbitrary phase response and are shown to exist as convergent laurent series if eigenvalues are selected as analytic functions on the unit circle and if the phase response is selected such that the eigenvectors are holder continuous with alpha frac 1 2 on the unit circle in the case of a discontinuous phase response or if spectral majorisation is enforced for intersecting eigenvalues an absolutely convergent laurent series solution for the eigenvectors of a parahermitian evd does not exist we provide some examples comment on the approximation of a parahermitian matrix evd by laurent polynomial factors and compare our findings to the solutions provided by polynomial matrix evd algorithms

classical mechanics and geometric quantization on an infinite dimensional disc and grassmannian we discuss the classical mechanics on the grassmannian and the disc modeled on the ideal l 2 we apply methods of geometric quantization to these systems their relation to a flat symplectic space is also discussed

observability of linear discrete time systems of algebraic and difference equations abstractthe notion of observability for higher order discrete time systems of algebraic and difference equations is studied such systems are also known as polynomial matrix descriptions attention is first given to a special form of descriptor systems with a state lead in the output this system is transformed into its causal and noncausal subsystems and observability criteria are given in terms of the subsystem s matrices and the fundamental matrix sequence of the matrix pencil e a afterwards the higher order system is studied by transforming it into a first order descriptor system of the above form an observability criterion is provided for the higher order system in terms of the laurent expansion at infinity of the system s polynomial matrix in addition observability is connected with the coprimeness of the polynomial matrices of the higher order system and the coprimeness of the matrix pencils of the descriptor system

the theory of quaternion matrix derivatives a systematic framework for the calculation of the derivatives of quaternion matrix functions with respect to quaternion matrix variables is introduced the proposed approach is equipped with the matrix product and chain rules and applies to both analytic and nonanalytic functions of quaternion variables this rectifies a mathematical shortcut in the existing methods which incorrectly use the traditional product rule we also show that within the proposed framework the derivatives of quaternion matrix functions can be calculated directly without using quaternion differentials or resorting to the isomorphism with real vectors illustrative examples show how the proposed quaternion matrix derivatives can be used as an important tool for solving optimization problems in signal processing applications

filterbank optimization with convex objectives and the optimality of principal component forms this paper proposes a general framework for the optimization of orthonormal filterbanks fbs for given input statistics this includes as special cases many previous results on fb optimization for compression it also solves problems that have not been considered thus far fb optimization for coding gain maximization for compression applications has been well studied before the optimum fb has been known to satisfy the principal component property i e it minimizes the mean square error caused by reconstruction after dropping the p weakest lowest variance subbands for any p we point out a much stronger connection between this property and the optimality of the fb the main result is that a principal component fb pcfb is optimum whenever the minimization objective is a concave function of the subband variances produced by the fb this result has its grounding in majorization and convex function theory and in particular explains the optimality of pcfbs for compression we use the result to show various other optimality properties of pcfbs especially for noise suppression applications suppose the fb input is a signal corrupted by additive white noise the desired output is the pure signal and the subbands of the fb are processed to minimize the output noise if each subband processor is a zeroth order wiener filter for its input we can show that the expected mean square value of the output noise is a concave function of the subband signal variances hence a pcfb is optimum in the sense of minimizing this mean square error the above mentioned concavity of the error and hence pcfb optimality continues to hold even with certain other subband processors such as subband hard thresholds and constant multipliers although these are not of serious practical interest we prove that certain extensions of this pcfb optimality result to cases where the input noise is colored and the fb optimization is over a larger class that includes biorthogonal fbs we also show that pcfbs do not exist for the classes of dft and cosine modulated fbs

vector sensor array processing for electromagnetic source localization the authors present a new approach for localizing electromagnetic sources using sensors where the output of each is a vector consisting of the complete six electric and magnetic field components two types of source transmissions are considered 1 single signal transmission sst and 2 dual signal transmission dst the model is given in terms of several parameters including the wave direction of arrival doa and state of polarization a compact expression is derived for the cramer rao bound crb on the estimation errors of these parameters for the multi source multi vector sensor model quality measures including mean square angular error msae and covariance of vector angular error cvae are introduced and their lower bounds are derived the advantage of using vector sensors is highlighted by explicit evaluation of the msae and cvae bounds for source localization with a single vector sensor a simple algorithm for estimating the source doa with this sensor is presented along with its statistical performance analysis

estimating the doa and the polarization of a polynomial phase signal using a single polarized vector sensor this paper introduces a novel algorithm to estimate the direction of arrival doa and the polarization of a completely polarized polynomial phase signal of an arbitrary degree the algorithm utilizes a polarized vector sensor comprising a spatially collocated six component electromagnetic vector sensor a dipole triad or a loop triad this esprit based algorithm is based on a time invariant matrix pencil pair derived from the time delayed data sets collected by a single polarized vector sensor the high order difference function of the signal s phase constructs the invariant factor used in the esprit algorithm the steering vector is estimated from the signal subspace eigenvector of the data correlation matrix following which the closed form doa and polarization can be obtained given the degree of the polynomial phase signal this approach resolves the two dimensional azimuth elevation angle and the polarization of the source and requires neither a priori knowledge of the polynomial phase signal s coefficients nor a priori knowledge of the polynomial phase signal s frequency spectrum the efficacy of the proposed algorithm is verified by monte carlo simulations estimation accuracies of the doa and the polarization parameters are evaluated by the closed form cramer rao bounds which are independent of the polynomial coefficients the degree of the polynomial phase signal and the azimuth angle of the source

hypercomplex signals a novel extension of the analytic signal to the multidimensional case the construction of gabor s 1946 complex signal which is also known as the analytic signal provides direct access to a real one dimensional 1 d signal s local amplitude and phase the complex signal is built from a real signal by adding its hilbert transform which is a phase shifted version of the signal as an imaginary part to the signal since its introduction the complex signal has become an important tool in signal processing with applications for example in narrowband communication different approaches to an n d analytic or complex signal have been proposed in the past we review these approaches and propose the hypercomplex signal as a novel extension of the complex signal to n d this extension leads to a new definition of local phase which reveals information on the intrinsic dimensionality of the signal the different approaches are unified by expressing all of them as combinations of the signal and its partial and total hilbert transforms examples that clarify how the approaches differ in their definitions of local phase and amplitude are shown an example is provided for the two dimensional 2 d hypercomplex signal which shows how the novel phase concept can be used in texture segmentation

real structure preserving algorithms of householder based transformations for quaternion matrices in this paper we survey three different forms of householder based transformations for quaternion matrices in the literature and propose a new form of quaternion householder based transformation we propose real structure preserving algorithms of these householder based transformations which make the procedure computationally more flexible and efficient we compare the computation counts and assignment numbers of these algorithms we also compare the effectiveness of these real structure preserving algorithms applying to the quaternion qrd and the quaternion svd all these four real structure preserving algorithms are more efficient comparing to the algorithms which apply quaternion toolbox using quaternion arithmetics or algorithms which directly performs real householder transformations on the real representation of a quaternion matrix among these four real structure preserving algorithms the most efficient ones are based on quaternion householder reflection and new proposed householder based transformation

monge kantorovitch measure transportation and monge ampere equation on wiener space let w h be an abstract wiener space assume two i i 1 2 probabilities on w w we give some conditions for the wasserstein distance between 1 and 2 with respect to the cameron martin space to be finite where the infimum is taken on the set of probability measures on w w whose first and second marginals are 1 and 2 in this case we prove the existence of a unique cyclically monotone map t iw with w h such that t maps 1 to 2 moreover if 2 then t is stochastically invertible i e there exists s w w such that s t iw 1 a s and t s iw 2 a s if in addition 1 then there exists a 1 convex function in the gaussian sobolev space such that these results imply that the quasi invariant transformations of the wiener space with finite wasserstein distance from can be written as the composition of a transport map t and a rotation i e a measure preserving map we give also 1 convex sub solutions and ito type solutions of the monge ampere equation on w

on left eigenvalues of a quaternionic matrix abstract in this paper we introduce the concept of left and right eigenvalues for a quaternionic matrix and investigate their properties quantities and relationship

geometric algebra of euclidean 3 space for electromagnetic vector sensor array processing part i modeling a new mathematical tool the geometric algebra of euclidean 3 space g3 is introduced for electromagnetic vector sensor array processing herein this paper focuses on modeling the six component outputs of a vector sensor holistically by an entry called as a multivector in g3 a compact polarized model for the array termed as a geometric algebra model g model is then presented using the g model a novel data covariance matrix model is defined by the geometric products in g3 and then analyzed the analytical results show that the six component measurement noise of a vector sensor can naturally be whitened if the noise cross correlations between the different axial electric and magnetic components are equal to one another compared with the known best quad quaternion model the new covariance matrix model results in a reduction of half memory requirements while the amount of divisions is reduced to 1 2 multiplications and additions reduced to almost 1 7

a quaternion qr algorithm this paper extends the francis qr algorithm to quaternion and antiquaternion matrices it calculates a quaternion version of the schur decomposition using quaternion unitary similarity transformations following a finite step reduction to a hessenberg like condensed form a sequence of implicit qr steps reduces the matrix to triangular form eigenvalues may be read off the diagonal eigenvectors may be obtained from simple back substitutions for serial computation the algorithm uses only half the work and storage of the unstructured francis qr iteration by preserving quaternion structure the algorithm calculates the eigenvalues of a nearby quaternion matrix despite rounding errors

on the mickelsson faddeev extension and unitary representations the mickelsson faddeev extension is a 3 space analogue of a kac moody group where the central charge is replaced by a space of functions of the gauge potential this extension is a pullback of a universal extension where the gauge potentials are replaced by operators in a schatten ideal as in non commutative differential geometry our main result is that the universal extension cannot be faithfully represented by unitary operators on a separable hilbert space we also examine potential consequences of the existence of unitary representations for the mickelsson faddeev extension

solutions to matrix equations x axb cy r and x axb cy r abstract the present work proposed an alternative approach to find the closed form solutions of the nonhomogeneous yakubovich matrix equation x a x b c y r based on the derived closed form solution to the nonhomogeneous yakubovich matrix equation the solutions to the nonhomogeneous yakubovich quaternion j conjugate matrix equation x a x b c y r are obtained by the use of the real representation of a quaternion matrix the existing complex representation method requires the coefficient matrix a to be a block diagonal matrix over complex field in contrast in this publication we allow a quaternion matrix of any dimension as an application eigenstructure assignment problem for descriptor linear systems is considered

sequential matrix diagonalization algorithms for polynomial evd of parahermitian matrices for parahermitian polynomial matrices which can be used for example to characterize space time covariance in broadband array processing the conventional eigenvalue decomposition evd can be generalized to a polynomial matrix evd pevd in this paper a new iterative pevd algorithm based on sequential matrix diagonalization smd is introduced at every step the smd algorithm shifts the dominant column or row of the polynomial matrix to the zero lag position and eliminates the resulting instantaneous correlation a proof of convergence is provided and it is demonstrated that smd establishes diagonalization faster and with lower order operations than existing pevd algorithms

generalized likelihood ratios for testing the properness of quaternion gaussian vectors in a recent paper the second order statistical analysis of quaternion random vectors has shown that there exist two different kinds of quaternion widely linear processing which are associated with the two main types of quaternion properness in this paper we consider the problem of determining from a finite number of independent vector observations whether a quaternion gaussian vector is proper or not specifically we derive three generalized likelihood ratio tests glrts for testing the two main kinds of quaternion properness and show that the glrts reduce to the estimation of three previously proposed quaternion improperness measures interestingly the three glrt statistics improperness measures can be interpreted as an estimate of the entropy loss due to the quaternion improperness additionally we analyze the case in which the orthogonal basis for the representation of the quaternion vector is unknown which results in the problem of estimating the principal bbc properness direction i e the pure unit quaternion minimizing the bbc improperness measure although this estimation problem is not convex we propose a technique based on successive convex approximations which can be solved in closed form finally some simulation examples illustrate the performance and practical application of the proposed tests

the qrd and svd of matrices over a real algebra abstract recent work in the field of signal processing has shown that the singular value decomposition of a matrix with entries in certain real algebras can be a powerful tool in this article we show how to generalise the qr decomposition and svd to a wide class of real algebras including all finite dimensional semi simple algebras twisted group algebras and clifford algebras two approaches are described for computing the qrd svd one jacobi method with a generalised givens rotation and one based on the artin wedderburn theorem

quaternion music for vector sensor array processing this paper considers the problem of direction of arrival doa and polarization parameters estimation in the case of multiple polarized sources impinging on a vector sensor array the quaternion model is used and a data covariance model is proposed using quaternion formalism a comparison between long vector orthogonality and quaternion vector orthogonality is also performed and its implications for signal subspace estimation are discussed consequently a music like algorithm is presented allowing estimation of wave s doas and polarization parameters the algorithm is tested in numerical simulations and performance analysis is conducted when compared with other music like algorithms for vector sensor array the newly proposed algorithm results in a reduction by half of memory requirements for representation of data covariance model and reduces the computational effort for equivalent performance this paper also illustrates a compact and elegant way of dealing with multicomponent complex valued data

rational subspace estimation using adaptive lossless filters rational models have been studied as a tractable attempt to account for frequency dependencies using a finite parameter description in this context the following problem is addressed given time domain measurements estimate rational orthonormal spanning vectors for the signal and noise subspaces it is shown that the problem can be rephrased as adapting a lossless transfer matrix so as to maximize the power split between two sets of output bins an efficient and numerically robust adaptive filtering algorithm is derived for lossless multivariable lattice filters and the system can be programmed in real time using cordic processors the adaptive filter equations are consistent with the proper subspace identification if the subspace filter satisfies a sufficient order condition in under modeled scenarios the stable stationary points are characterized by a minimized rayleigh quotient which leads to good subspace fits the proposed adaptive filter makes real time rational subspace estimation an accessible alternative to computationally expensive offline techniques

the notion of convexity and concavity on wiener space abstract we define in the frame of an abstract wiener space the notions of convexity and of concavity for the equivalence classes of random variables as application we show that some important inequalities of the finite dimensional case have their natural counterparts in this setting

spatial analysis using new properties of the cross spectral matrix a novel concept called the propagator based upon a partition of the steering vectors matrix and correlatively of the received signals cross spectral matrix csm is introduced for the characterization of source and noise subspaces without reference to the csm eigenstructure a localization technique can thereby be applied for uncorrelated or partially correlated sources with an arbitrary propagation model which is directly comparable to the music algorithm given the signals produced by one particular source on some of the sensors the propagator permits the extrapolation of the signals received by the remaining sensors various applications of this property can be found an example is given concerning the estimation of the sensors complex gains