supersymmetric quantum mechanics with levy disorder in one dimension we consider the schrodinger equation with a random potential of the form open image in new window where w is a levy noise we focus on the problem of computing the so called complex lyapunov exponent open image in new window where n is the integrated density of states of the system and is the lyapunov exponent in the case where the levy process is non decreasing we show that the calculation of reduces to a stieltjes moment problem we ascertain the low energy behaviour of the density of states in some generality and relate it to the distributional properties of the levy process we review the known solvable cases where can be expressed in terms of special functions and discover a new one

on the distribution of a random variable occurring in 1d disordered systems the authors consider the random variable z 1 x1 x1x2 x1x2x3 where the xi are independent identically distributed variables they derive some asymptotic properties of the distribution of z which are related e g to the low temperature behaviour of the random field ising chain for a special class of distributions of the xi exact solutions are presented they also study the cases where the distribution function of z exhibits a power law fall off modulated by a periodic critical amplitude

exactly soluble random field ising models in one dimension the authors solve exactly the one dimensional random field ising model for two classes of magnetic field distributions symmetric exponential model i and non symmetric exponential model ii for both models expressions for the free energy at all finite temperatures are presented the low temperature region is examined in more detail they obtain the zero temperature energy and entropy in closed form it is shown that the free energy of both models has an expansion in integer powers of temperature model i has a non vanishing zero point entropy for all values of the parameters as soon as randomness is diluted in model ii the zero point entropy is zero except for a discrete sequence of values of one parameter in some cases the zero temperature magnetisation is positive whereas the average magnetic field is negative the magnetisation may also change sign as a function of temperature

products of random matrices and generalised quantum point scatterers to every product of 2 2 matrices there corresponds a one dimensional schrodinger equation whose potential consists of generalised point scatterers products of random matrices are obtained by making these interactions and their positions random we exhibit a simple one dimensional quantum model corresponding to the most general product of matrices in sl 2 we use this correspondence to find new examples of products of random matrices for which the invariant measure can be expressed in simple analytical terms

fluctuations of the product of random matrices and generalized lyapunov exponent i present a general framework allowing to carry out explicit calculation of the moment generating function of random matrix products varpi n m nm n 1 ldots m 1 where m i s are i i d following tutubalin theor probab appl 10 1 15 27 1965 the calculation of the generating function is reduced to finding the largest eigenvalue of a certain transfer operator associated with a family of representations of the group the formalism is illustrated by considering products of random matrices from the group mathrm sl 2 mathbb r where explicit calculations are possible for concreteness i study in detail transfer matrix products for the one dimensional schrodinger equation where the random potential is a levy noise derivative of a levy process in this case i obtain a general formula for the variance of ln varpi n and for the variance of ln psi x where psi x is the wavefunction in terms of a single integral involving the fourier transform of the invariant density of the matrix product finally i discuss the continuum limit of random matrix products matrices close to the identity in particular i investigate a simple case where the spectral problem providing the generalized lyapunov exponent can be solved exactly

shift variance measures for multirate lpsv filter banks with random input signals the linear periodically shift variant lpsv properties within critically sampled multirate fir filter banks are generally analysed using deterministic signals periodic shift variance is however closely related to cyclostationarities introduced by the lpsv system into originally wide sense stationary wss random signals passing through the system we provide first a unified framework to measure both shift variance of the lpsv system and the amount of cyclostationarity it generates in this respect the key concept is the covariance operator associated to a random variable cyclostationarity of the variable translates to lpsv properties of the operators and vice versa we study several related concepts for the quantification of shift variance in operators and their interpretation in the stochastic setting we then introduce a new concept called expected shift variance fourier analytic expressions for the various measures are given and subsequently used to derive explicit formulae for the case of critically sampled two channel filter banks as well as sharp upper bounds for unitary two channel filter banks numerical evaluations of the measures show that they provide largely consistent rankings between various critically sampled two channel filter banks

statistical tests for presence of cyclostationarity the presence of kth order cyclostationarity is defined in terms of nonvanishing cyclic cumulants or polyspectra relying upon the asymptotic normality and consistency of kth order cyclic statistics asymptotically optimal spl chi sup 2 tests are developed to detect the presence of cycles in the kth order cyclic cumulants or polyspectra without assuming any specific distribution on the data constant false alarm rate tests are derived in both time and frequency domain and yield consistent estimates of possible cycles present in the kth order cyclic statistics explicit algorithms for k spl les 4 are discussed existing approaches are rather empirical and deal only with k spl les 2 case simulation results are presented to confirm the performance of the given tests

optimal design of multichannel equalizers for the structural similarity index the optimization of multichannel equalizers is studied for the structural similarity ssim criteria the closed form formula is provided for the optimal equalizer when the mean of the source is zero the formula shows that the equalizer with maximal ssim index is equal to the one with minimal mean square error mse multiplied by a positive real number which is shown to be equal to the inverse of the achieved ssim index the relation of the maximal ssim index to the minimal mse is also established for given blurring filters and fixed length equalizers an algorithm is also presented to compute the suboptimal equalizer for the general sources various numerical examples are given to demonstrate the effectiveness of the results

lyapunov exponents one dimensional anderson localization and products of random matrices the concept of lyapunov exponent has long occupied a central place in the theory of anderson localisation its interest in this particular context is that it provides a reasonable measure of the localisation length the lyapunov exponent also features prominently in the theory of products of random matrices pioneered by furstenberg after a brief historical survey we describe some recent work that exploits the close connections between these topics we review the known solvable cases of disordered quantum mechanics involving random point scatterers and discuss a new solvable case finally we point out some limitations of the lyapunov exponent as a means of studying localisation properties

bifrequency and bispectrum maps a new look at multirate systems with stochastic inputs in multirate digital signal processing we often encounter decimators interpolators and complicated interconnections of these with lti filters we also encounter cyclo wide sense stationary cwss processes and linear periodically time varying ptv systems it is often necessary to understand the effects of multirate systems on the statistical properties of their input signals some of these issues have been addressed earlier for example it has been shown that a necessary and sufficient addition for the output of an l fold interpolation filter to be wide sense stationary wss for all wss inputs is that the filter was an alias free l support however several questions of this nature remain unanswered for example what is the necessary and sufficient condition on a pair or more generally a bank of interpolation filters so that their outputs are jointly wss jwss for all jointly wss inputs what is the condition if only the sum of their outputs is required to be wss when is the output of an lptv system for example a uniform filter bank wss for all wss inputs some of these questions may appear to be simple generalizations of the above mentioned result for a single interpolation filter however the frequency domain approaches that proved this result are quite difficult to generalize to answer these questions the purpose of this paper is to provide these answers using analysis based on bifrequency maps and bispectra these tools are two dimensional 2 d fourier transforms that characterize linear time varying ltv systems and nonstationary random processes respectively we show that the questions raised above are addressed elegantly and in a geometrically insightful way using these tools we also derive a bifrequency characterization lossless ltv systems this may potentially lead to an increased understanding of these systems

the lyapunov exponent of products of random 2 2 matrices close to the identity we study products of arbitrary random real 2 2 matrices that are close to the identity matrix using the iwasawa decomposition of sl 2 we identify a continuum regime where the mean values and the covariances of the three iwasawa parameters are simultaneously small in this regime the lyapunov exponent of the product is shown to assume a scaling form in the general case the corresponding scaling function is expressed in terms of gauss hypergeometric function a number of particular cases are also considered where the scaling function of the lyapunov exponent involves other special functions airy bessel whittaker elliptic the general solution thus obtained allows us among other things to recover in a unified framework many results known previously from exactly solvable models of one dimensional disordered systems

singularities in spectra of disordered systems abstract this paper starts with a review of work on singularities in the spectral density of random harmonic chains detailed descriptions of singular behavior near the maximal frequency near special frequencies and near island frequencies are discussed next the site disordered anderson model is considered here the discussion is limited to the singularity in the density of states near the low energy band edge for systems with arbitrary disorder in dimensions d 1 2 3 in a specific example it is shown that renormalization improves the prediction for the band tail quantitatively

the distribution of the reflection phase of disordered conductors the reflection coefficients of one dimensional disordered conductors is asymptotically a pure phase when the sample length is much larger than the localisation length the authors study the stationary distribution of this reflection phase for tight binding chains with diagonal disorder i e random site potentials they derive the dyson integral equation obeyed by this distribution and relate it to the lyapunov exponent and the integrated density of states this general approach allows them to consider then more specifically two types of disorder in the case of a binary alloy where the potentials take two values the distribution of the reflection phase is shown to have generically an infinity of power law singularities related to the occurrence of periodic patterns of potentials this phenomenon produces divergences in the distribution when the strength of disorder exceeds some energy dependent threshold they also obtain an exact expression for the distribution of the reflection phase for a symmetric exponential distribution of potentials this non trivial solvable model allows them to examine analytically various properties of the phase distribution such as the anomalies which occur at weak disorder

a generalized thouless formula as a criterion for anderson localization in two and three dimensional systems abstract a generalized thouless formula is proposed as an approximate criterion for anderson localization in two and three dimensional systems the criterion is exact for ordered systems and in the mean field limit of infinite dimensionality for random systems it is expected that possible corrections are small ample numerical evidence is provided through an exact diagonalization of several large size hamiltonians the predictions of the thouless formula whose evaluation only needs the eigenvalues are critically checked against the inverse participation ratio ipr of the corresponding eigenvectors included are the lloyd model for sufficiently low disorder mobility edges cannot be excluded and surprising new results on the binary alloy problem the criterion predicts four as the upper and two as the lower critical dimension

on spectral theory of cyclostationary signals in multirate systems this paper studies two problems in the spectral theory of discrete time cyclostationary signals the cyclospectrum representation and the cyclospectrum transformation by linear multirate systems four types of cyclospectra are presented and their interrelationships are explored in the literature the problem of cyclospectrum transformation by linear systems was investigated only for some specific configurations and was usually developed with inordinate complexities due to lack of a systematic approach a general multirate system that encompasses most common systems linear time invariant systems and linear periodically time varying systems is proposed as the unifying framework more importantly it also includes many configurations that have not been investigated before e g fractional sample rate changers with cyclostationary inputs the blocking technique provides a systematic solution as it associates a multirate system with an equivalent linear time invariant system and cyclostationary signals with stationary signals thus the original problem is elegantly converted into a relatively simple one which is solved in the form of matrix multiplication

toeplitz and hankel kernels for estimating time varying spectra of discrete time random processes for a nonstationary random process the dual time correlation function and the dual frequency loeve spectrum are complete theoretical descriptions of second order behavior that is each may be used to synthesize the random process itself according to the cramer loeve spectral representation when suitably transformed on one of its two variables each of these descriptions produces a time varying spectrum this spectrum is in fact the expected value of the rihaczek distribution in this paper we derive two large families of estimators for this spectrum one based on a diagonal toeplitz diagonal dtd factorization of smoothing kernels and the other based on a diagonal hankel diagonal dhd factorization the dtd factorization produces noncoherent averages of the time varying spectrogram and the dhd factorization produces coherent averages some of the dtd estimators may be called time varying power spectrum estimators and some of the dhd estimators may be called time varying wigner ville wv estimators the former may always be implemented as multiwindow spectrum estimators and in some casts they are true time variations on the blackman tukey rosenblatt grenander btgr spectrogram the latter are variations on the stankovic class of wv estimators

special energies and special frequencies special frequencies have been asserted to be zeros of the density of frequencies corresponding to a random chain of coupled oscillators our investigation includes both this model and the random one dimensional schrodinger operator describing an alloy or its discrete analogue using the phase method we exactly determine a bilateral lifsic asymptotic of the integrated density of statesk e at special energiese s which is not only of the classical type exp c e e s 1 2 but also exp c e e s is a typical behaviour in addition other asymptotics occur e g e e s c which show thatk e need not bek

the laplacian on a random one dimensional lattice the authors study the spectrum of minus the laplacian on a random one dimensional lattice it extends monotonically throughout the positive real axis with a continuum limit behaviour for small values the authors find as expected localisation effects throughout the whole range they use a variety of techniques which are shown to be consistent and in agreement with a monte carlo simulation they include small and large disorder expansions direct integral equations for probability distributions as well as the replica method the latter is also used to investigate green functions

optimization of filter banks using cyclostationary spectral analysis this paper proposes a design method of optimal biorthogonal fir filter banks that minimize the time averaged mean squared error tamse when the high frequency subband signal is dropped to study filter banks from a statistical point of view cyclostationary spectral analysis is used since the output of the filter bank for a wide sense stationary input is cyclostationary first the cyclic spectral density of the output signal is derived and an expression for the tamse is presented then optimal filter banks are given by minimizing the tamse with respect to the coefficients of the filters under the biorthogonality condition by imposing the additional constraints on the coefficients the optimal biorthogonal linear phase filter bank can be obtained

lifshitz singularities in the total and the wavenumber dependent spectral density of random harmonic chains abstract lifshitz has shown that the density of states of random systems becomes exponentially small near the band edge we perform a detailed computation in a one dimensional system and calculate the periodic prefactor of this essential singularity the analysis is extended to the wavenumber dependent spectral density the periodic amplitude is derived for the exactly soluble cases where the distribution of masses is exponential for binary distributions the leading behavior in the concentration of light masses is also derived

identification of systems with cyclostationary input and correlated input output measurement noise an alternative approach to wiener s optimum filtering theory for stationary random time series that is applicable when the system input exhibits cyclostationarity is presented this alternative approach can in principle eliminate performance degradation due to input output corruption this method of system identification is called the speccorr spectral correlation ratio method since the cross spectra in the ratio are actually spectral correlation functions an application to interference tolerant time difference of arrival estimation is described and illustrated by simulation results showing its effectiveness for system identification with severely corrupted input output measurements a comparison of wiener s method to the speccorr method shows the superior performance of the latter

exact solutions for ising chains in a random field the authors present an exact solution for the one dimensional ising model in a random field the distribution of field strengths is site independent symmetric and has a three peak structure the free energy is obtained for all finite temperatures the low temperature behaviour is studied in detail they find an expansion for the free energy in integer powers of temperature the ground state energy and the zero point entropy are calculated explicitly the specific heat is linear for low temperatures in contradiction with mean field theory the origin of an additional exponential parameter is discussed

cyclo period estimation for discrete time cyclo stationary signals this paper presents a new method named as the variability method to estimate the cyclo period of a discrete time cyclo stationary signal the method is essentially based on the time varying correlation and or the time varying mean whose estimators are associated with some statistics of blocked signals a plot of variability of these statistics as a function of the blocking operator index visually reveals a periodic pattern from which the cyclo period is obtained properties of the variability method are analyzed and compared with three existing cyclo period estimation methods via simulation and real life examples

exact electronic spectra and inverse localization lengths in one dimensional random systems i random alloy liquid metal and liquid alloy analytic continuations into the complex energy plane of dyson schmidt type of equations for the calculation of the density of states are constructed for a random alloy model a liquid metal and for a liquid alloy in all these models the characteristic function follows from the solution of this equation its imaginary part yields the accumulated density of states and its real part is a measure for the inverse of the localization length of the eigenfunctions the equations have been solved exactly for some distributions of the random variables in the random alloy case the strengths of the delta potentials have an exponential distribution they may also have finite exponentially distributed values with probability 0 p 1 and be infinite with probability q 1 p in the liquid metal the liquid particles are assumed to behave like hard rods this implies an exponential distribution of the distances between the particles the common electronic potential may be arbitrary but is assumed to vanish outside the rods in the one dimensional liquid alloy there is apart from positional randomness of the liquid particles a distribution of the strengths of the electronic delta potentials for cauchy distributions an argument of lloyd is extended to obtain the characteristic function from the one in the model with equal strengths for the case of a liquid of point particles a three parameter class of distributions of the strengths is shown to yield a solution in the form of known functions of the equation mentioned above for several cases numerical calculations of the density of states and the inverse localization length of the eigenfunctions are presented and discussed new results are found exponential decay of the density of states near special energies in the random alloy and liquid metal divergence of the density of states at certain energies with non classical exponent 13 in the random alloy if the average of the potential strengths vanishes exponentially small broadening of the bound state levels for low concentrations of the liquid particles peak in the localization length at the bound state energies which becomes exponentially narrow for low concentrations different exponent in the decay of the inverse localization length at large energies for delta potentials and square well potentials further an expression for the grand potential is given involving a sum over the characteristic function at certain points and divergence of the zero point energy is found for cauchy distributions of the delta potential strengths

exact solutions for spectra and green s functions in random one dimensional systems for chains of harmonic oscillators with random masses a set of equations is derived which determine the spatial fourier components of the average one particle green s function these equations are valid for complex values of the frequency a relation between the spectral density and functions introduced by schmidt is discussed exact solutions for this green s function and the less complicated characteristics function the analytic continuation into the complex frequency plane of the accumulated spectral density and the inverse localization length of the eigenfunctions are derived for exponential distributions of the masses for some cases the characteristic function is calculated numerically for gamma distributions the equations are cast in the form of ordinary higher order differential equations these have been solved numerically for determining the characteristic function for arbitrary mass distributions a cumulant expansion and a peculiar symmetry of the green s function are discussed the method is also applied to chains where the spring constants and or the masses have random values also for these systems exact solutions are discussed for exponential distributions e g of both masses and spring constants the characteristic function is expressed in bessel functions the relation with certain random relaxation models is shown finally x y hamiltonians with random exchange constants and or magnetic fields or equivalently tight binding electron models with diagonal and or off diagonal disorder are considered here the green s function does not depend on the wave number if the distribution of exchange constants is symmetric around the origin new solutions for the characteristic function and green s function are derived for a number of cases including exponentially distributed magnetic fields and power law distributed exchange constants

a new approach to the problem of disordered harmonic chains abstract in the problem of harmonic chains with random masses the exponential growth rate and the spectral distribution function are shown to be the real and imaginary part of dyson s characteristic function along its cut in the complex plane a new function satisfying a certain integral equation is introduced from which the characteristic function follows immediately the functions introduced by dyson schmidt and matsuda and ishii are recovered as special cases and it is shown that schmidt s distributions w n converge as n relations between dyson s and schmidt s functions are obtained finally the same method is applied to the problem of harmonic chains with random variables k n m n and k n 1 m n and to the tight binding electron model with random site energies

correlation function of random field ising chains is it lorentzian or not the two point connected correlation function or wavevector dependent susceptibility of the ferromagnetic ising chain in a random field is calculated exactly at any temperature for a two parameter family of diluted symmetric exponential distributions of the magnetic fields thermodynamic properties of this model have been derived in a previous work by the authors besides the correlation function itself the solution provides exact results for the usual susceptibility the correlation length and the edwards anderson parameter the low temperature regime is examined in full detail the authors obtain in closed form the limit value at zero temperature of various quantities and the first correction to this limit which behaves linearly with temperature the correlation length is discontinuous at zero temperature they also derive the scaling form of the correlation function for small p where the dilution p is the probability for a spin to be subjected to a non zero random field even in this limit the correlation function is more complicated than the simple lorentzian predicted e g by mean field theory

conditions for optimality of naive quantized finite horizon control this paper presents properties of a control law which quantizes the unconstrained solution to a unitary horizon quadratic programme this naive quantized control law underlies many popular algorithms such as converters and decision feedback equalities and is easily shown to be globally optimal for horizon one however the question arises as to whether it is also globally optimal for horizons greater than one i e whether it solves a finite horizon quadratic programme where decision variables are restricted to belonging to a quantized set by using dynamic programming we develop sufficient conditions for this to hold the present analysis is restricted to first order plants however this case already raises a number of highly non trivial issues the results can be applied to arbitrary horizons and quantized sets which may contain a finite or an infinite though countable number of elements

low temperature thermodynamics of random field ising chains exact results the authors consider the random field ferromagnetic ising chain for a class of continuous symmetric probability distributions of the random magnetic fields of a diluted power times exponential type this class of distribution is exactly solvable in the sense that the disorder can be integrated explicitly at any temperature a detailed analysis of the low temperature thermodynamics is presented exact expressions are obtained for the ground state energy the zero temperature entropy and the amplitude of the specific heat which vanishes linearly at low temperature the diluted symmetric binary distribution where the magnetic fields can only assume the values or hb or zero can be viewed as a limiting case of the class of exactly solvable distributions the low temperature physics of this discrete model is investigated in detail including the exponential fall off of the specific heat

nonuniform multirate filter banks analysis and design with an spl hscr sub spl infin performance measure a general multirate filter bank system with multiple channels and nonuniform bands is studied with the reconstruction performance measured by the worst case energy gain spl hscr sub spl infin norm of the error system between the multirate system and a pure time delay system using blocking and polyphase decomposition we associate with the multirate system an equivalent linear time invariant lti system whose transfer matrix can be computed by a simple procedure based on this lti system the optimal design problem can be reduced to one of spl hscr sub spl infin optimization which can be solved by ready made software in many cases for illustration optimal synthesis filters are designed for a two channel example and their properties are studied in detail

second order statistical characterization of the filter bank and its elements this correspondence proposes an alternative scheme for characterizing the wide sense cyclostationary wscs processes and applies it for the analysis of the filter banks whose output is wscs for wide sense stationary and wscs inputs the proposed representation clearly illustrates the kind of transformation taking place in multiscale stochastic systems these relationships are extended to the filter bank to make clear the effect of the various stages of the filter bank on wscs processes

survey paper set invariance in control the properties of positively invariant sets are involved in many different problems in control theory such as constrained control robustness analysis synthesis and optimization in this paper we provide an overview of the literature concerning positively invariant sets and their application to the analysis and synthesis of control systems

design of multichannel nonuniform transmultiplexers using general building blocks the paper considers design of multichannel nonuniform band transmultiplexers it is well known that using traditional building blocks up and downsamplers and linear time invariant lti causal filters nonuniform transmultiplexers typically do not achieve perfect reconstruction to alleviate this we propose to build nonuniform transmultiplexers using general dual rate structures that provide more design freedom and hence perfect reconstruction can be achieved such general transmultiplexers have a new source of error called aliasing distortion in addition to the traditional cross talk magnitude and phase distortions we propose a composite error criterion that captures all four distortions in one using this error criterion as reconstruction performance measure we develop an optimal design procedure and apply it to a three channel nonuniform example yielding an fir transmultiplexer that has good frequency limiting properties in the synthesis end and is very close to perfect reconstruction

control of constrained discrete time linear systems using quantized controls abstract the theory of control of continuous time systems with control constraints is extended to the case where the controls are of the form u i n i s where n i is an integer and s is a scaling factor these results permit the analysis of the controllability of digital control systems with quantized controls they also provide the theoretical framework for recently suggested real time suboptimal controllers based on the application of artificial intelligence techniques such an application is presented at the end of the paper