on the role of input signals in deterministic linear filtering this correspondence shows that typical frequency domain design formulations of deterministic linear filters without explicit formal relationship with input signals can be interpreted as worst case designs with respect to normed spaces of input signals filter design formulations optimal only with respect to subsets of input signals are discussed the interesting similarities with the minimax formulations of the robust wiener filtering literature are pointed out

stochastic analysis of the lms algorithm with a saturation nonlinearity following the adaptive filter output this paper presents a statistical analysis of the least mean square lms algorithm with a zero memory scaled error function nonlinearity following the adaptive filter output this structure models saturation effects in active noise and active vibration control systems when the acoustic transducers are driven by large amplitude signals the problem is first defined as a nonlinear signal estimation problem and the mean square error mse performance surface is studied analytical expressions are obtained for the optimum weight vector and the minimum achievable mse as functions of the saturation these results are useful for adaptive algorithm design and evaluation the lms algorithm behavior with saturation is analyzed for gaussian inputs and slow adaptation deterministic nonlinear recursions are obtained for the time varying mean weight and mse behavior simplified results are derived for white inputs and small step sizes monte carlo simulations display excellent agreement with the theoretical predictions even for relatively large step sizes the new analytical results accurately predict the effect of saturation on the lms adaptive filter behavior

a tight upper bound of the average absolute error in a constant step size sign algorithm a direct performance index of the adaptive filtering sign algorithm sa is the average absolute error aae at the output of the filter adopting this performance index an easy analysis of sa is achieved under a weak assumption it is proved for both deterministic and random inputs to the filter that the aae has a tight upper bound that exceeds the minimum aae by half the product of the step size and power of the filter input the assumption used is existence of average squared and average absolute values of filter input signals a practical interest of the result is that it provides a formula for the biggest step size as a function of tolerable adaptation noise to desired signal ratio

adaptive filter performance with nonlinearities in the correlation multiplier it is well known that the correlation multiplier used to form the update signal in adaptive transversal filters need not be a true multiplier by using an approximate multiplier such as a sign bit only multiplier implementation can be made less costly in this paper we derive expressions for the convergence rate and misalignment noise of adaptive transversal filters with arbitrary correlation multiplier non linearities the analysis shows any such nonlinearities to always harm performance and quantifies the penalty suffered sign bit only multiplication suffers a significant penalty and power of two multiplication is suggested as a better alternative its performance is almost equal to that of a true multiplier and in a digital implementation at least it is only slightly more difficult to realize than a sign bit only multiplier the theory is developed in a way emphasizing physical understanding and has been verified to be in good agreement with simulation

saturation effects in lms adaptive echo cancellation for binary data the effect of a saturation type error nonlinearity in the weight update equation in least mean square lms adaptive echo cancellation is investigated for an independent binary data model a nonlinear difference equation is derived for the mean norm of the difference between the estimate and the unknown filter to be estimated by the algorithm the difference equation is evaluated numerically it is shown that far end binary data interference is much more deleterious to algorithm transient behavior than far end gaussian data interference the number of additional bits for the same cancellation convergence rates for binary versus gaussian interference of the same power is studied as a function of various system parameters algorithm convergence rates are studied as a function of an arbitrary probability density function pdf for the far end data it is shown that a binary pdf causes the worst degradation and a gaussian shaped pdf causes the least degradation

algebraic properties of separated power series we study the commutative algebra of rings of separated power series over a ring e and that of their extensions rings of separated and more specifically convergent power series from a field k with a separated e analytic structure both of these collections of rings already play an important role in the model theory of non archimedean valued fields and we establish their algebraic properties this will make a study of the analytic geometry over such fields through the classical methods of algebraic geometry possible

adaptive weighted myriad filter algorithms for robust signal processing in spl alpha stable noise environments stochastic gradient based adaptive algorithms are developed for the optimization of weighted myriad filters wmyfs wmyfs form a class of nonlinear filters motivated by the properties of spl alpha stable distributions that have been proposed for robust non gaussian signal processing in impulsive noise environments the weighted myriad for an n long data window is described by a set of nonnegative weights w sub i sub i l sup n and the so called linearity parameter k 0 in the limit as k spl rarr spl infin the filter reduces to the familiar weighted mean filter which is a constrained linear fir filter necessary conditions are obtained for optimality of the filter weights under the mean absolute error criterion an implicit formulation of the filter output is used to find an expression for the gradient of the cost function using instantaneous gradient estimates an adaptive steepest descent algorithm is then derived to optimize the weights this algorithm involves a very simple update term that is computationally comparable to the update in the classical lms algorithm the robust performance of this adaptive algorithm is demonstrated through a computer simulation example involving lowpass filtering of a one dimensional chirp type signal in impulsive noise

convergence analysis of a data echo canceller with a stochastic gradient adaptive fir filter using the sign algorithm under the assumption that the errors at the taps are gaussian distributed a new set of recurrence formulae is derived for calculating theoretical convergence process of a data echo canceller equipped with an fir filter that is adaptively controlled by using the stochastic gradient sign algorithm with a binary and white process as the filter input convergence curves for the mean squared residual echo based on the recurrence formulae show an excellent agreement with those obtained by simulation approximate recurrence formulae that yield a useful though less accurate estimation of the residual echo convergence are also proposed furthermore a closed form solution to the approximate recurrence formulae is derived and finally probability density function of the residual echo is depicted as it changes its shape as the echo canceller converges

operator integrals and sesquilinear forms abstract we consider various systematic ways of defining unbounded operator valued integrals of complex functions with respect to mostly positive operator measures and positive sesquilinear form measures and investigate their relationships to each other in view of the extension theory of symmetric operators we demonstrate the associated mathematical subtleties with a physically relevant example involving moment operators of the momentum observable of a particle confined to move on a bounded interval

convergence of the sign algorithm for adaptive filtering with correlated data convergence of a decreasing gain sign algorithm sa for adaptive filtering is analyzed the presence of the hard limiter in the algorithm makes a rigorous analysis difficult therefore there are few results available such results normally include restrictive assumptions such as the assumptions that successive observation vectors are independent and the new error signal of the adaptive filter has a time invariant probability density function the former assumption is not valid in the context of adaptive filtering since two successive observation vectors share most of their components while the latter assumption is a restriction on the adaptive weights whose evolution is a priori unknown in lieu of using these assumptions an almost sure convergence of the sa is proved under the assumption that the sequence of observation vectors is m dependent this assumption allows strong correlation between successive observations

locally optimum adaptive signal processing algorithms we propose a new analytic method for comparing constant gain adaptive signal processing algorithms specifically estimates of the convergence speed of the algorithms allow for the definition of a local measure of performance called the efficacy that can be theoretically evaluated by definition the efficacy is consistent with the fair comparison techniques used in signal processing applications using the efficacy as a performance measure we prove that the lms newton algorithm is optimum and is thus the fastest algorithm within a very rich algorithmic class furthermore we prove that regular lms is better than any of its variants that apply the same nonlinear transformation on the elements of the regression vector such as signed regressor quantized regressor etc for an important class of input signals simulations support all our theoretical conclusions

on error saturation nonlinearities in lms adaptation the effect of a saturation type error nonlinearity in the weight update equation in least mean squares lms adaptation is investigated for a white gaussian data model nonlinear difference equations are derived for the eight first and second moments which include the effect of an error function erf saturation type nonlinearity on the error sequence driving the algorithm a nonlinear difference equation for the mean norm is explicitly solved using a differential equation approximation and integration by quadratures the steady state second moment weight behavior is evaluated exactly for the erf nonlinearity using the above results the tradeoff between the extent of error saturation steady state excess mean square error and rate of algorithm convergence is studied the tradeoff shows that 1 starting with a sign detector the convergence rate is increased by nearly a factor of two for each additional bit and 2 as the number of bits is increased further the additional bit by very little in convergence speed asymptotically approaching the behavior of the linear algorithm

analysis of a sign algorithm with delayed data an adaptive filter whose weights are adapted using a sign algorithm with a delayed error signal is analyzed for stationary environments it is proved that the excess average absolute estimation error is bounded for all values of the error signal delay and the algorithm step size for the nonstationary case when the optimal filter weights are time varying the optimum step size which minimizes the excess average absolute error is derived it is shown that the optimum step size does not depend on the additive noise power the analytical results are supported by computer simulations

nonlinear quantization effects in the frequency domain complex scalar lms adaptive algorithm nonlinear quantization effects in the frequency domain complex scalar lms adaptive algorithm are analyzed by using conditional expectations the probability density function of the quantizer input conditioned on the weight is derived the density is applied to finding the conditional characteristic function and the mth conditional moment at the quantizer output the first and second conditional moments of the quantizer output are used to derive difference equations that approximate the dynamical behavior of the first and second weight moments these difference equations are solved numerically and compare favorably to simulation results a model of the quantizer as an additive noise source is of no analytical value since the quantizatian noise has negligible effect on the mean square error when the model is valid finally a design approach is proposed for selecting the number of bits in the weight accumulator the moment equations are also used to determine the algorithm mean square error for different quantizer step sizes and the optimum algorithm step size when a fixed amount of input data is available for adaptation

tracking analysis of the sign algorithm in nonstationary environments a tracking analysis of the adaptive filters equipped with the sign algorithm and operating in nonstationary environments is presented under the assumption that the nonstationary can be modeled using a random disturbance it is shown that the long term time average of the mean absolute error is bounded and that there exists an optimal choice of the convergence constant mu which minimizes this quality applying the commonly used independence assumption and under the assumption that the nonstationarity is solely due to the time varying behavior of the optimal coefficients it is shown that the distributions of the successive coefficient misalignment vectors converge to a limiting distribution when the adaptive filter is used in the system identification mode finally under the additional assumption that the signals involved are zero mean and gaussian a set of nonlinear difference equations is derived that characterizes the mean and mean squared behavior of the filter coefficients and the mean squared estimation error during adaptation and tracking results of several experiments that show very good correlation with the theoretical analyses are presented

the lms algorithm with delayed coefficient adaptation the behavior of the delayed least mean square dlms algorithm is studied it is found that the step size in the coefficient update plays a key role in the convergence and stability of the algorithm an upper bound for the step size is derived that ensures the stability of the dlms the relationship between the step size and the convergence speed and the effect of the delay on the convergence speed are also studied the analytical results are supported by computer simulations

convergence analysis of the sign algorithm without the independence and gaussian assumptions the paper is concerned with rigorous convergence analysis of the sign algorithm sa in the context of adaptive plant identification asymptotic time averaged convergence for the mean absolute weight misalignment is proved for all values of the algorithm step size and initial weight vector the paper has three main contributions with respect to available convergence results of the sa the first is the deletion of the gaussian assumption which is important when covering the case of discrete valued data no assumption about the distribution of the regressor sequence is used except for the usual assumption of positive definite covariance matrix the assumptions used about the noise allow nonexistence unboundedness and vanishing of the noise probability density function for arguments strictly different from zero the second contribution is the deletion of the assumption of independent successive regressors this deletion is important since in applications two successive regressors usually share all their components except two hence they are strongly dependent even for white plant input the case of colored noise is also analyzed finally the third contribution is the extension of the above results to the nonstationary case the used assumptions allow nonstationarity of the plant input plant noise and plant parameters

noise cancellation with improved residuals an effective adaptive scheme for noise cancellation when the signal to be recovered has known autocorrelation is presented two algorithms that exploit a special form of prior information are investigated in this approach the desired signal is removed from the output feedback by linear prediction the prior information used is the desired signal s autocorrelation knowing this one can find a filter that whitens the desired signal screening the error feedback through this filter removes most of the desired signal energy reducing its interference with the coefficient update this is the basis for the first algorithm discussed namely the least mean square algorithm with augmented predictor lms ap proposed by orgren et al 1986 in many applications the whitening filter may not be strictly positive real spr in such cases a different algorithm is needed one which is assuredly convergent regardless of the satisfaction of the spr condition a modified lms algorithm with augmented predictor mlms ap which provides such an alternative is proposed

adaptive algorithm based on least mean p power error criterion for fourier analysis in additive noise this abstract presents a novel adaptive algorithm for the estimation of discrete fourier coefficients dfc of sinusoidal and or quasiperiodic signals in additive noise the algorithm is derived using a least mean p power error criterion it reduces to the conventional lms algorithm when p takes on 2 it is revealed by both analytical results and extensive simulations that the new algorithm for p 3 4 generates much improved dfc estimates in moderate and high snr environments compared with the lms algorithm whereas both have similar degrees of complexity assuming the gaussian property of the estimation error the proposed algorithm including the lms algorithm is analyzed in detail elegant dynamic equations and closed form noise misadjustment expressions are derived and clarified

moment operators of the cartesian margins of the phase space observables the theory of operator integrals is used to determine the moment operators of the cartesian margins of the phase space observables generated by the mixtures of the number states the moments of the x margin are polynomials of the position operator and those of the y margin are polynomials of the momentum operator

a comparison of two quantized state adaptive algorithms quantized state qs adaptive algorithms reduce the numerical complexity and dynamic range requirements of least mean squares lms adaptation by replacing multiplications with shifts bit comparisons or table lookups a theoretical foundation with which to distinguish two primary qs algorithm forms and to predict which algorithm is most appropriate in a given context is presented an extended lyapunov approach is used to derive a persistence of excitation pe condition which guarantees linear stability of the quantized error qe form averaging theory is then used to derive pe condition which guarantees exponential stability of the quantized regressor qreg form failure to meet this latter condition which is not equivalent to the spectral richness pe condition for lms can result in exponential instability the qe and qreg algorithms are then compared in terms of conditions for stability convergence properties of the prediction and parameter errors convergence rates and steady state errors

operator integrals and phase space observables the operator integral f de of a complex valued measurable function f with respect to a positive operator measure e is considered if f is a neumark dilation of e into a projection measure then the projected operator integral pr fdf is a restriction of the operator f de necessary and sufficient conditions for the equality pr f df f de are obtained the results are applied to determine the moment operators of the phase space observables generated by the number states

characterizations of the canonical phase observable in this paper we investigate various properties of phase observables which could serve to determine the canonical phase observable among the family of all phase observables we also show that any contractive weighted shift operator defines a unique phase observable and we characterize phase observables that give the most accurate phase distribution in coherent states in the classical limit

adaptive equalizer using finite bit power of two quantizer the stochastic gradient algorithm using a simplified arithmetic is analyzed in this paper a power of two quantizer is used for the input of the multiplier to reduce the multiplication to at most a simple shift in spite of its simple implementation the performance is shown to be comparable to the classical lms algorithm a linearized approximation to the quantizer is first derived followed by the analysis of an exact nonlinear model the derivation is based on the gaussian assumption and the effects of removing the gaussian assumption are later considered the roundoff error due to the finite bit computation is calculated computer simulation results are provided to support the analysis

effect of delay on the performance of the leaky lms adaptive algorithm this correspondence studies the effect of a nonzero delay on the performance of the leaky lms algorithm the stability bound on the step size is derived for error convergence it is shown that leakage allows for larger bounds on the step size of the delayed lms algorithm theoretical bounds are verified by simulations

phase space quantization and the operator moment problem we consider questions related to a quantization scheme in which a classical variable f r on a phase space is associated with a preferably unique semispectral measure ef such that the moment operators of ef are required to be of the form fk with a suitable mapping from the set of classical variables to the set of not necessarily bounded operators in the hilbert space of the quantum system in particular we investigate the situation where the map is implemented by the operator integral with respect to some fixed positive operator measure the phase space is first taken to be an abstract measurable space then a locally compact unimodular group and finally r2 where we determine explicitly the relevant operators fk for certain variables f in the case where the quantization map is implemented by a translation covariant positive operator measure in addition we consider the question under what conditions a positive operator measure is projection valued

almost sure convergence of a decreasing gain sign algorithm for adaptive filtering the convergence analysis of a sign algorithm with decreasing gain when governing the weights of an adaptive filter is presented the analysis is done in a noiseless case a rigorous proof of almost sure convergence to the optimal filter is attained under a weak ergodicity assumption that includes the case important in practice of correlated observations

alpha stable signals and adaptive filtering let d sub k x sub k be jointly stationary symmetric spl alpha stable processes with index 1 spl alpha 2 we first consider the problem of optimal fir linear and regression filters we then consider the signed error and sign sign adaptive filtering algorithms for the estimation of the desired signal d sub j on the basis of the input vector x sub j various convergence results are established for the signal and tap weights estimation errors

adaptive threshold nonlinear algorithm for adaptive filters with robustness against impulse noise a new adaptation algorithm for adaptive filters is proposed introducing adaptive threshold in the nonlinear correlation function the adaptive threshold controlled by the long term average of the error signal power makes the filtering system highly robust against impulse noise which is demonstrated by the results of simulation and theoretical calculation

subnormal operators regarded as generalized observables and compound system type normal extension related to 1 1 in this paper subnormal operators not necessarily bounded are discussed as generalized observables in order to describe not only the information about the probability distribution of the output data of their measurement but also the framework of their implementations we introduce a new concept of a compound system type normal extension and we derive the compound system type normal extension of a subnormal operator which is defined from an irreducible unitary representation of the algebra 1 1 the squeezed states are characterized as the eigenvectors of an operator from this viewpoint and the squeezed states in multi particle systems are shown to be the eigenvectors of the adjoints of these subnormal operators under a representation the affine coherent states are discussed in the same context as well

optimization of adaptive identification for time varying filters adaptive identification in a time varying context is studied when controlled by the lms algorithm with constant gain under the assumption of correlated successive input vectors it is well known by experience that the tracking mean square error mse epsilon mu results from the tradeoff between the gradient part which is increasing and the lag contribution which is decreasing in this note we clarify the relative roles of the gradient and lag errors by proving their decoupled character this property relies upon independence between the additive noise at the output of the plant to be identified and the information vector at the plant input convergence of the mse is established rather than assumed quantitative evaluations of upper and lower bounds allow an approximate optimization of the gain in two important cases the optimum is exact one of these cases is slow variations it is defined in a quantitative manner thanks to the ratio of the variation noise to the output additive noise

constructible motivic functions and motivic integration 1 1 in this paper intended to be the first in a series we lay new general foundations for motivic integration and give answers to some important issues in the subject since its creation by maxim kontsevich 23 motivic integration developed quickly and has spread out in many directions in a nutshell in motivic integration numbers are replaced by geometric objects like virtual varieties or motives but classicaly not only numbers are defined using integrals but also interesting classes of functions the previous constructions of motivic integration were all quite geometric and it was quite unclear how they could be generalized to handle integrals depending on parameters the new approach we present here based on cell decomposition allows us to develop a very general theory of motivic integration taking parameters into account more precisely we define a natural class of functions constructible motivic functions which is stable under integration the basic idea underlying our approach is to construct more generally push forward morphisms f which are functorial they satisfy f g f g so that performing motivic integration corresponds to taking the push forward to the point this strategy has many technical advantages in essence it allows to reduce the construction of f to the case of closed immersions and projections and in the latter case we can perform induction on the relative dimension the basic case being that of relative dimension 1

analysis of the normalized lms algorithm with gaussian inputs the lms adaptive filter algorithm requires a priori knowledge of the input power level to select the algorithm gain parameter for stability and convergence since the input power level is usually one of the statistical unknowns it is normally estimated from the data prior to beginning the adaptation process it is then assumed that the estimate is perfect in any subsequent analysis of the lms algorithm behavior in this paper the effects of the power level estimate are incorporated in a data dependent that appears explicitly within the algorithm the transient mean and second moment behavior of the modified lms nlms algorithm are evaluated taking into account the explicit statistical dependence of upon the input data the mean behavior of the algorithm is shown to converge to the wiener weight a constant coefficient matrix difference equation is derived for the weight fluctuations about the wiener weight the equation is solved for a white data covariance matrix and for the adaptive line enhancer with a single frequency input in steady state for small expressions for the misadjustment error are also presented it is shown for the white data covariance matrix case that the averaging of about ten data samples causes negligible degradation as compared to the lms algorithm in the ale application the steady state weight fluctuations are shown to be mode dependent being largest at the frequency of the input

stochastic approximation with correlated data new almost sure convergence results for a special form of the multidimensional robbins monro stochastic approximation procedure are developed the results are applicable to cases where the training data is heavily correlated no conditional expectation properties or boundedness assumptions are required to apply the new results for example when the data sequence is normal and i m dependent ii autoregressive moving average or iii band limited the results can be used to establish the almost sure convergence of each algorithm treated the special form of the robbins monro procedure considered is motivated by a consideration of several algorithms that have been proposed for discrete time adaptive signal processing applications most of these algorithms can also be viewed as stochastic gradient following algorithm the ease with which the new results can be applied is illustrated

comparison of the convergence of two algorithms for adaptive fir digital filters the convergence properties of two different algorithms for the updating of the coefficients of an adaptive fir digital filter are investigated and compared with one another these algorithms are the stochastic iteration algorithm and the sign algorithm in this latter algorithm a one bit gradient estimation is used which makes its implementation very simple the convergence is characterized by the residual echo variance after convergence and a parameter that indicates the speed of the convergence it is shown that the convergence of the sign algorithm can always be assured but is much slower than that of the stochastic iteration algorithm if the same variance of the residual echo is to be obtained

adaptive filtering with binary reinforcement recently there has been increased interest in high speed adaptive filtering where the usual stochastic gradient or least mean square lms algorithm is replaced with the simpler algorithm where adaptation is guided only by the polarity of the error signal in this paper the convergence of this binary reinforcement br algorithm is proved under the usual independence assumption and the surprising observation is made that unlike the lms algorithm convergence occurs for any positive value of the step size parameter while the stochastic gradient algorithm attempts to minimize a mean square error cost function the binary reinforcement algorithm in fact attempts to minimize a mean absolute error cost function it is proved for the binary reinforcement algorithm that the tap weight vector converges in distribution to a random vector that is suitably concentrated about the optimal value based on a least mean absolute error cost function for a sufficiently small step size the expected cost of the asymptotic weight vector can be made as close as desired to thc minimum cost attain ed by the optimum weight vector

square integrable imprimitivity systems we give a definition of square integrability for imprimitivity systems and we prove that the square integrable ones share most of the properties of square integrable representations of groups

improved convergence analysis of stochastic gradient adaptive filters using the sign algorithm convergence analysis of stochastic gradient adaptive filters using the sign algorithm is presented in this paper the methods of analysis currently available in literature assume that the input signals to the filter are white this restriction is removed for gaussian signals in our analysis expressions for the second moment of the coefficient vector and the steady state error power are also derived simulation results are presented and the theoretical and empirical curves show a very good match

definable sets motives and p adic integrals 0 1 let x be a scheme reduced and separated of finite type over z for p a prime number one may consider the set x zp of its zp rational points for every n in n there is a natural map n x zp x z p assigning to a zp rational point its class modulo p the image yn p of x zp by n is exactly the set of z p rational points which can be lifted to zp rational points denote by nn p the cardinality of the finite set yn p by a result of the first author 7 the poincare series

convergence analysis of adaptive filtering algorithms with singular data covariance matrix the paper provides a rigorous analysis of the behavior of adaptive filtering algorithms when the covariance matrix of the filter input is singular the analysis is done in the context of adaptive plant identification the considered algorithms are lms rls sign sa and signed regressor sra algorithms both the signal and weight behavior of the algorithms are considered the signal behavior is evaluated in terms of the moments of the excess output error of the filter the weight behavior is evaluated in terms of the moments of the filter weight misalignment vector it is found that the rls and sra diverge when the input covariance matrix is singular the steady state signal behavior of the lms and sa can be made arbitrarily fine by using sufficiently small step sizes of the algorithms indeed the long term average of the mean square excess error of the lms is proportional to the algorithm step size the long term average of the mean absolute excess error of the sa is proportional to the square root of the algorithm step size on the other hand the steady state weight behavior of both the lms and sa have biases that depend on the weight initialization the analytical results of the paper are supported by simulations

uniform properties of rigid subanalytic sets in the context of rigid analytic spaces over a non archimedean valued field a rigid subanalytic set is a boolean combination of images of rigid analytic maps we give an analytic quantifier elimination theorem for complete algebraically closed valued fields that is independent of the field in particular the analytic quantifier elimination is independent of the valued field s characteristic residue field and value group in close analogy to the algebraic case this provides uniformity results about rigid subanalytic sets we obtain uniform versions of smooth stratification for subanalytic sets and the lojasiewicz inequalities as well as a unfiorm description of the closure of a rigid semianalytic set

transient analysis of adaptive filters with error nonlinearities the paper develops a unified approach to the transient analysis of adaptive filters with error nonlinearities in addition to deriving earlier results in a unified manner the approach also leads to new performance results without restricting the regression data to being gaussian or white the framework is based on energy conservation arguments and avoids the need for explicit recursions for the covariance matrix of the weight error vector

on the optimum data nonlinearity in lms adaptation the effect of an arbitrary nonlinear operation on the data input to the weight update equation in the lms adaptive algorithm is investigated for a gaussian data model exact difference equations are derived for the weight first and second moments that include the effects of an arbitrary nonlinear operation on the data sequence the difference equations are used to obtain expressions for the transient behavior of the mean square error the mean square error is minimized over the choice of nonlinearity for a fixed transient behavior the best choice of nonlinearity is shown exactly to be of the form x 1 bx2 when the input data are white for an arbitrary data covariance matrix the optimum nonlinearity is shown to be linear when the product of the algorithm step size and input power is much less than unity

multiaccess interference suppression in orthogonal space time block coded mimo systems by adaptive projected subgradient method this paper introduces adaptive filters that are effective to suppress multiple access interference mai in orthogonal space time block coded multiple input multiple output ostbc mimo systems we define an optimal linear filter that minimizes the mean square error between the filter output and a scaled version of the desired output under a constraint defined by the available channel state information csi the adaptive filters refine a given estimate of the optimal filter by suppressing a sequence of closed convex functions with the adaptive projected subgradient method apsm at each iteration to provide robustness against imperfect csi the adaptive filters use not only the available csi but also estimates of previously transmitted symbols which usually belong to a small finite set in digital communication systems the resulting algorithms employ computationally efficient projections onto hyperplanes or hyperslabs and do not require any matrix inversion an efficient recursive scheme based on such an algorithm is also presented convergence analysis and simulation results show the excellent performance of the proposed schemes

unconstrained frequency domain adaptive filter an unconstrained frequency domain least mean squares uflms algorithm is presented the algorithm is based on the overlap save technique used in frequency domain filtering the proposed algorithm converges to the wiener solution without the time domain constraint on the impulse response as proposed by ferrara for a large number of taps geq32 the uflms algorithm offers a significant reduction in computation another advantage is its fast convergence for highly correlated input signals the performance of the algorithm is illustrated by a simulation

error energy bounds for adaptive gradient algorithms the paper establishes robustness optimality and convergence properties of the widely used class of instantaneous gradient adaptive algorithms the analysis is carried out in a purely deterministic framework and assumes no a priori statistical information it employs the cauchy schwarz inequality for vectors in an euclidean space and derives local and global error energy bounds that are shown to highlight as well as explain relevant aspects of the robust performance of adaptive gradient filters along the lines of h sup spl infin theory

performance analysis of adaptive filters equipped with the dual sign algorithm a convergence analysis for stochastic gradient adaptive filters equipped with the dual sign algorithm dsa is presented expressions for the mean and mean squared values of the coefficient misalignment vector are derived under the assumption that the input signal is gaussian the main differences between the current analysis and previous analyses are 1 the present analysis is valid for arbitrary autocorrelation matrices and 2 the analysis does not approximate the behavior for the dsa as switching between two sign algorithms a certain amount of time after the dsa is initialized a simulation example comparing the analytical results with empirical ones is presented and the two curves show excellent agreement with each other

the stability of adaptive minimum mean square error equalizers using delayed adjustment the necessary and sufficient conditions for the stability and convergence of adaptive minimum mean square error equalizers are developed the analysis includes the effect of delays in the adjustment path as occur for instance when the decoder in a data transmission system delays its output decisions

phase space observables and isotypic spaces we give necessary and sufficient conditions for the set of neumark projections of a countable set of phase space observables to constitute a resolution of the identity and we give a criteria for a phase space observable to be informationally complete the results will be applied to the phase space observables arising from an irreducible representation of the heisenberg group

optimum step size of sign algorithm for nonstationary adaptive filtering the adaptive filtering sign algorithm is analyzed in the case of nonstationary and correlated data the excess time average absolute estimation error eaae is used as a performance index of the algorithm it is proved that the eaae is the sum of two terms one proportional to the algorithm step size mu and the other proportional to 1 mu the optimum step size that minimizes the eaae is found equal to square root q p with q being the power of fluctuations of the optimal filter weights and p being the power of filter input this formula is derived under the assumption that increments of optimal filter weights are white computer simulations are used to examine the validity of the formula for other models of variation of the optimal filter simulation results show a good fit with the obtained optimum step size formula

analysis of adaptive filters using normalized signed regressor lms algorithm in this paper adaptive filters using the normalized signed regressor lms algorithm nsra with gaussian reference inputs are proposed and analyzed to yield difference equations for theoretically calculating expected convergence of the filters a simple difference equation for mean squared error mse is derived when the filter input is a white and gaussian process whereas approximate difference equations for colored gaussian inputs are proposed and tested stability conditions and residual mse after convergence are also obtained agreement of theoretical results with those of simulation in the experiment with some examples of filter convergence shows sufficient accuracy of the theory and assures the usefulness of the difference equations in estimating filter performances thus facilitating the design of adaptive filters using the nsra