on the statistical efficiency of the lms algorithm with nonstationary inputs a fundamental relationship exists between the quality of an adaptive solution and the amount of data used in obtaining it quality is defined here in terms of misadjustment the ratio of the excess mean square error mse in an adaptive solution to the minimum possible mse the higher the misadjustment the lower the quality is the quality of the exact least squares solution is compared with the quality of the solutions obtained by the orthogonalized and the conventional least mean square lms algorithms with stationary and nonstationary input data when adapting with noisy observations a filter trained with a finite data sample using an exact least squares algorithms will have a misadjustment given by m frac n n frac number of weights number of training samples if the same adaptive filter were trained with a steady flow of data using an ideal orthogonalized lms algorithm the misadjustment would be m frac n 4 tau mse frac number of weights number of training samples thus for a given time constant tau mse of the learning process the ideal orthogonalized lms algorithm will have about as low a misadjustment as can be achieved since this algorithm performs essentially as an exact least squares algorithm with exponential data weighting it is well known that when rapid convergence with stationary data is required exact least squares algorithms can in certain cases outperform the conventional widrow hoff lms algorithm it is shown here however that for an important class of nonstationary problems the misadjustment of conventional lms is the same as that of orthogonalized lms which in the stationary case is shown to perform essentially as an exact least squares algorithm

darboux transformations and supersymmetry for the generalized schrodinger equations in 1 1 dimensions we study intertwining relations supersymmetry and darboux transformations for generalized schrodinger equations in 1 1 dimensions we obtain intertwiners in an explicit form by means of which we construct arbitrary order darboux transformations for our class of equations we develop a corresponding supersymmetric formulation and prove equivalence of the darboux transformations with the supersymmetry formalism finally we show that our darboux transformations can also be constructed by means of point transformations avoiding the use of intertwiners

paper error propagation properties of recursive least squares adaptation algorithms the numerical properties of implementations of the recursive least squares identification algorithm are of great importance for their continuous use in various adaptive schemes here we investigate how an error that is introduced at an arbitrary point in the algorithm propagates it is shown that conventional ls algorithms including bierman s ud factorization algorithm are exponentially stable with respect to such errors i e the effect of the error decays exponentially the base of the decay is equal to the forgetting factor the same is true for fast lattice algorithms the fast least squares algorithm sometimes known as the fast kalman algorithm is however shown to be unstable with respect to such errors

performance advantage of complex lms for controlling narrow band adaptive arrays in narrow band adaptive array applications the mean square convergence of the discrete time real least mean square lms algorithm is slowed by image frequency noises generated in the lms loops the complex lms algorithm proposed by widrow et al is shown to eliminate these noises yielding convergence of the mean squared error mse at slightly over twice the rate this paper includes a comprehensive analysis of the mse of adaptation for lms the analysis is based upon the method developed in the 1968 dissertation by k d senne and it represents the most complete treatment of the subject published to date

exponential convergence of recursive least squares with exponential forgetting factor this paper demonstrates that provided the system input is persistently exciting the recursive least squares estimation algorithm with exponential forgetting factor is exponentially convergent further it is shown that the incorporation of the exponential forgetting factor is necessary to attain this convergence and that the persistence of excitation is virtually necessary the result holds for stable finite dimensional linear time invariant systems but has its chief implications to the robustness of the parameter estimator when these conditions fail

unitary representations of the two dimensional euclidean group in the heisenberg algebra e 2 is studied as the automorphism group of the heisenberg algebra h the basis in the hilbert space k of functions on h on which the unitary irreducible representations of the group are realized is explicitly constructed the addition theorem for the kummer functions is derived

representations of su 1 1 in non commutative space generated by the heisenberg algebra su 1 1 is considered as the automorphism group of the heisenberg algebra h the basis in the hilbert space k of functions on h on which the irreducible representations of the group are realized is explicitly constructed from group theoretical considerations summation formulae for the product of two three and four hypergeometric functions are derived

projection based regularized dual averaging for stochastic optimization we propose a novel stochastic optimization framework based on the regularized dual averaging rda method the proposed approach differs from the previous studies of rda in three major aspects first the squared distance loss function to a random closed convex set is employed for stability second a sparsity promoting metric used implicitly by a certain proportionate type adaptive filtering algorithm and a quadratically weighted ell 1 regularizer are used simultaneously third the step size and regularization parameters are both constant due to the smoothness of the loss function these three differences yield an excellent sparsity seeking property high estimation accuracy and insensitivity to the choice of the regularization parameter numerical examples show the remarkable advantages of the proposed method over the existing methods including adagrad and the adaptive proximal forward backward splitting method in applications to regression and classification with real synthetic data

analysis of conjugate gradient algorithms for adaptive filtering the paper presents and analyzes two approaches to the implementation of the conjugate gradient cg algorithm for filtering where several modifications to the original cg method are proposed the convergence rates and misadjustments for the two approaches are compared an analysis in the z domain is used in order to find the asymptotic performance and stability bounds are established the behavior of the algorithms in finite word length computation are described and dynamic range considerations are discussed it is shown that in finite word length computation and close to steady state the algorithms behaviors are similar to the steepest descent algorithm where the stalling phenomenon is observed using 16 bit fixed point number representation our simulations show that the algorithms are numerically stable

the block processing ftf adaptive algorithm a new block type adaptive filtering algorithm is presented this new block adaptive filter differs from the frequency domain block adaptive filters of ferrara 1980 and of clark mitra and parker 1980 in that the new method applies a deterministic time domain least squares criteria within each of the data blocks information is carried from block to block via a weighted initial condition this new block fast transversal filters bftf algorithm is a numerically stable algorithm and can also be used to perform efficient least squares system identification on any one data block in which case it shows a moderate computational advantage over the previous most efficient single data block algorithms of morf et al 1977 of marple 1981 and of kalouptsidis manolakis and carayannis 1984 1985 mechanisms for tracking and varying block length from block to block are also presented and evaluated finally we indicate how the new algorithm could be pipelined for maximum throughput with delay proportional to the number of parameters after computation of the sample correlation lags

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

a fast squares power inversion array for tracking nonstationary environments this correspondence presents and investigates a fast algorithm tracking of nonstationary environments in power inversion the algorithm is based on the use of iterative least squares techniques and the introduction of an additional gradient vector in the objective function so that both the optimal weight vector and its rate of change are estimated recursively although the algorithm is computationally more complex than conventional least squares methods it has better performance in moving sources as the velocities of the latter are also being estimated

green function on the quantum plane a green function which can be called q analogous of the hankel function on the quantum plane eq2 eq 2 u 1 is constructed

reduced complexity self tuning adaptive algorithms in application to channel estimation in this letter reduced complexity self tuning algorithms are proposed using simplified parameter updating procedures convergence analysis based on the independence assumption and the ordinary differential equation ode method shows that the tuning parameter of the proposed algorithm attains the same limit as the conventional self tuning adaptive algorithm simulations are carried out for channel estimation to support the analysis and performance of the proposed algorithms

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

optimal variable step size for the lms newton algorithm with application to subband adaptive filtering an efficient approach for the computation of the optimum convergence factor for the lms least mean square newton algorithm applied to a transversal fir structure is proposed the approach leads to a variable step size algorithm that results in a dramatic reduction in convergence time the algorithm is evaluated in system identification applications where two alternative implementations of the adaptive filter are considered the conventional transversal fir realization and adaptive filtering in subbands

adaptive step size algorithms for blind interference suppression in ds cdma systems this paper develops adaptive step size blind lms algorithms and adaptive forgetting factor blind rls algorithms for code aided suppression of multiple access interference mai and narrowband interference nbi in ds cdma systems these algorithms optimally adapt both the step size forgetting factor and the weight vector of the blind linear multiuser detector using the received measurements simulations are provided to compare the proposed algorithms with previously studied blind rls and blind lms algorithms they show that the adaptive step size blind lms algorithm and adaptive forgetting factor blind rls algorithm field significant improvements over the standard blind lms algorithm and blind rls algorithm in dynamic environments where the number of interferers are time varying

improvement of the fast recursive least squares algorithms via normalization a comparative study this paper deals with the derivation and the properties of fast optimal least squares algorithms and particularly with their normalization it is shown how the well known fast kalman algorithm written in the most general form can be normalized through a purely algebraic point of view leading to the normalized least squares transversal filter derived by cioffi kailath and lev ari from the geometric approach an improved form of the algorithm is presented the different algorithms have been compared from a practical point of view as regards their convergence initialization procedures complexity and numerical properties normalized transversal algorithms are shown to be interesting because of their nice structured form simplicity of conception and improved good numerical behavior

tracking of time varying mobile radio channels 1 the wiener lms algorithm adaptation algorithms with constant gains are designed for tracking smoothly time varying parameters of linear regression models in particular channel models occurring in mobile radio communications in a companion paper an application to channel tracking in the is 136 tdma system is discussed the proposed algorithms are based on two key concepts first the design is transformed into a wiener filtering problem second the parameters are modeled as correlated arima processes with known dynamics this leads to a new framework for systematic and optimal design of simple adaptation laws based on prior information the algorithms can be realized as wiener filters called learning filters or as lms newton updates complemented by filters that provide predictions or smoothing estimates the simplest algorithm named the wiener lms is presented all parameters are here assumed governed by the same dynamics and the covariance matrix of the regressors is assumed known the computational complexity is of the same order of magnitude as that of lms for regressors which are either white or have autoregressive statistics the tracking performance is however substantially improved

fixed point roundoff error analysis of the exponentially windowed rls algorithm for time varying systems a fixed point roundoff error analysis of the exponentially windowed rls algorithm is presented it is shown that a tradeoff exists in the choice of the forgetting factor in order to reduce the sensitivity of the algorithm to additive noise must be chosen close to one on the other hand the roundoff error increases as lambda rightarrow 1 it is shown that the algorithm is stabilized with lambda rightarrow 1 to derive the theoretical results it is assumed that the input signal is a white gaussian random process finally simulations are presented which confirm the theoretical findings of the paper

discrete time model reference adaptive control for continuous time systems using generalized sampled data hold functions the generalized sampled data hold functions approach of p t kabamba 1987 is extended to the control of linear time invariant systems with unknown parameters the idea of kabamba s approach is to periodically sample the plant output and define the control as the sampled output plus a discrete time reference each multiplied by an individual modulating function such a control makes it possible to assign an arbitrary discrete time transfer function for the sampled closed loop system and does not make assumptions on the plant other than controllability and observability the authors propose an indirect adaptive controller which is based on this approach and estimates the modulating functions online in particular the control is modified so that persistent excitation of the continuous time plant is ensured without making an assumption on the reference signal and discrete time asymptotic model following is nevertheless obtained the only assumptions on the plant are minimality for the continuous and sampled plant and known order

unified large system analysis of mmse and adaptive least squares receivers for a class of random matrix channels we present a unified large system analysis of linear receivers for a class of random matrix channels the technique unifies the analysis of both the minimum mean squared error mmse receiver and the adaptive least squares als receiver and also uses a common approach for both random independent identically distributed i i d and random orthogonal precoding we derive expressions for the asymptotic signal to interference plus noise ratio sinr of the mmse receiver and both the transient and steady state sinr of the als receiver trained using either i i d data sequences or orthogonal training sequences the results are in terms of key system parameters and allow for arbitrary distributions of the power of each of the data streams and the eigenvalues of the channel correlation matrix in the case of the als receiver we allow a diagonal loading constant and an arbitrary data windowing function for i i d training sequences and no diagonal loading we give a fundamental relationship between the transient steady state sinr of the als and the mmse receivers we demonstrate that for a particular ratio of receive to transmit dimensions and window shape all channels which have the same mmse sinr have an identical transient als sinr response we demonstrate several applications of the results including an optimization of information throughput with respect to training sequence length in coded block transmission

susy transformation of the green function and a trace formula an integral relation is established between the green functions corresponding to two hamiltonians which are supersymmetric susy partners and in general may possess both discrete and continuous spectra it is shown that when the continuous spectrum is present the trace of the difference of the green functions for susy partners is a finite quantity which may or may not be equal to zero despite the divergence of the traces of each green function our findings are illustrated by using the free particle example considered both on the whole real line and on a half line

tracking properties and steady state performance of rls adaptive filter algorithms adaptive signal processing algorithms derived from ls least squares cost functions are known to converge extremely fast and have excellent capabilities to track an unknown parameter vector this paper treats analytically and experimentally the steady state operation of rls recursive least squares adaptive filters with exponential windows for stationary and nonstationary inputs a new formula for the estimation noise has been derived involving second and fourth order statistics of the filter input as well as the exponential windowing factor and filter length furthermore it is shown that the adaptation process associated with lag effects depends solely on the exponential weighting parameter in addition the calculation of the excess mean square error due to the lag for an assumed markov channel provides the necessary information about tradeoffs between speed of adaptation and steady state error it is also the basis for comparison to the simple lms algorithm in a simple case of channel identification it is shown that the lms and rls adaptive filters have the same tracking behavior finally in the last part we present new rls restart procedures applied to transversal structures for mitigating the disastrous results of the third source of noise namely finite precision arithmetic

tracking improvements in fast rls algorithms using a variable forgetting factor the concept of a variable forgetting factor vff is incorporated into fast recursive least squares frls algorithms compromises in the data matrix that are needed to do this are examined both prewindowed and growing memory covariance algorithms are presented in transversal and lattice structures forgetting factor adaptation schemes which improve tracking performance over conventional frls algorithms are suggested finally the bias introduced by the use of the vff is analyzed

quantumsu 2 ande 2 groups contraction procedure in 3 it was shown in the framework of deformed enveloping algebras that quantumsu 2 ande 2 groups are related by the contraction procedure we consider the same problem on thec level as a result we find a number of formulae coupling the comultiplications in quantumsu 2 ande 2 in particular we show that the comultiplications in both groups are implemented by partial isometries an unexpected feature of quantume 2 is discovered and the corresponding strange behavior of quantumsu 2 is described

operator equalities related to the quantum e 2 group the paper deals with normal operatorsr ands satisfying simple commutation relations that we encounter investigating the quantum deformation of thee 2 group we show thatr s admits a normal extension if and only ifr 1s satisfies a certain spectral condition a number of related formulae are derived in particular all the functionsf satisfying the character equationf r s f r f s are found

round off error propagation in four generally applicable recursive least squares estimation schemes abstract the numerical robustness of four generally applicable recursive least squares estimation schemes is analysed by means of a theoretical round off propagation study this study highlights a number of practical interesting insights into the widely used recursive least squares schemes these insights have been confirmed in an experimental verification study

first order tracking properties of weighted least squares estimators the first order tracking properties evolution of the mean tracking error of weighted least squares wls estimators applied to nonstationary system identification are investigated the expected path of the parameter estimates is evaluated the concept of the impulse response associated with the wls estimator is introduced

on the numerical stability and accuracy of the conventional recursive least squares algorithm we study the nonlinear round off error accumulation system of the conventional recursive least squares algorithm and we derive bounds for the relative precision of the computations in terms of the conditioning of the problem and the exponential forgetting factor which guarantee the numerical stability of the finite precision implementation of the algorithm the positive definiteness of the finite precision inverse data covariance matrix is also guaranteed bounds for the accumulated round off errors in the inverse data covariance matrix are also derived in our simulations the measured accumulated roundoffs satisfied in steady state the analytically predicted bounds we consider the phenomenon of explosive divergence using a simplified approach we identify the situations that are likely to lead to this phenomenon simulations confirm our findings

gradient based variable forgetting factor rls algorithm in time varying environments in this paper a new control mechanism for the variable forgetting factor vff of the recursive least square rls adaptive algorithm is presented the control algorithm is basically a gradient based method of which the gradient is derived from an improved mean square error analysis of rls the new mean square error analysis exploits the correlation of the inverse of the correlation matrix with itself that yields improved theoretical results especially in the transient and steady state mean square error it is shown that the theoretical analysis is close to simulation results for different forgetting factors and different model orders the analysis yields a dynamic equation of mean square error that can be used to derive a dynamic equation of the gradient of mean square error to control the forgetting factor the dynamic equation can produce a positive gradient when the error is large and a negative gradient when the error is in the steady state compared with other variable forgetting factor algorithms the new control algorithm gives fast tracking and small mean square model error for different signal to noise ratios snrs

convergence analysis of the binormalized data reusing lms algorithm normalized least mean squares algorithms for fir adaptive filtering with or without the reuse of past information are known to converge often faster than the conventional least mean squares lms algorithm this correspondence analyzes an lms like algorithm the binormalized data reusing least mean squares bndr lms algorithm this algorithm which corresponds to the affine projection algorithm for the case of two projections compares favorably with other normalized lms like algorithms when the input signal is correlated convergence analyses in the mean and in the mean squared are presented and a closed form formula for the mean squared error is provided for white input signals as well as its extension to the case of a colored input signal a simple model for the input signal vector that imparts simplicity and tractability to the analysis of second order statistics is fully described the methodology is readily applicable to other adaptation algorithms of difficult analysis simulation results validate the analysis and ensuing assumptions

green s functions sum rules and matrix elements for susy partners it is now well known that the solutions in the schrodinger equations for two potentials which are supersymmetric partners are linked by intertwining relations involving first order differential operators in this paper we explore the consequence of this linkage for the green s functions corresponding to susy partners and the relation between the sums over the inverses of the eigenvalues for the two potentials we also establish some relations between the matrix elements of certain operators evaluated between the eigenstates of two partner potentials we show that there can be circumstances where some matrix elements vanish as a consequence of the presence of supersymmetry

green function on the q symmetric space invariant distance on the non commutative algebra is constructed and the generalized functions on the q symmetric space are introduced the green function and the kernel on m are derived a path integration is formulated the green function for the free massive scalar field on the non commutative einstein space is presented

a trace formula for dirac green s functions related by darboux transformations we construct green s functions and a trace formula for pairs of stationary dirac equations under sturm liouville boundary conditions where the equations are related to each other by a darboux transformation our findings generalize former results pozdeeva e 2008 j phys a math theor at press

fixed point error analysis of the normalized ladder algorithm an attempt is made to analyze the fixed point error performance of the normalized ladder algorithm for autoregressive system identification assuming rounding arithmetic a preliminary simulation study of this algorithm has shown that the bias in the estimated reflection coefficients is much more predominant than the variance of the error in the estimate the study therefore is directed to find a model for predicting the bias in the estimated reflection coefficients the analysis shows that the roundoff errors associated with the square root operations in one of the algorithm equations are mainly responsible for the bias in the estimated reflection coefficients these errors arise because of the normalization procedure that makes the quantities under the square root operations very close to one two main results are presented in the paper 1 a simplified theoretical expression for predicting the average bias in the estimated reflection coefficients at any stage is derived 2 a recursive relation for the average error arising from the finite precision arithmetic in the squared residuals is derived this relation illustrates how the errors made in one stage affect the errors in the succeeding stages simulations are performed to check the theoretical models the experimental results agree very closely with the theoretical predictions

a note on the error propagation analysis of recursive least squares algorithms the article comments on the error propagation analysis of recursive least squares algorithms presented by ljung and ljung 1985 verhaegen 1989 and haykin 1991 in particular it corrects some incorrect statements about the error propagation given in the literature it considers the problem of spatial adaptive filtering using the rls minimization

on the statistics of fixed point roundoff error roundoff error after fixed point multiplication is commonly modeled as uniformly distributed white noise that is uncorrelated with the signal this paper presents a statistical analysis of fixed point roundoff error that identifies the conditions under which this model is valid and examines the statistical behavior of roundoff error when these conditions are not satisfied the results show that if the multiplier coefficient is expressed as a n m where m is a positive integral power of two and n is an odd integer then the errors generated by roundoff after multiplication can generally be modeled as uniformly distributed white and uncorrelated with the signal if the signal has sufficiently wide bandwidth and has a dynamic range that extends over approximately m quantum steps for narrow band low level signals the roundoff error statistics can differ significantly from the uniform white uncorrelated model in addition these results show that statistical behavior of roundoff error can differ significantly from that of the quantization error that is generated when a continuous random variable is quantized

mlse for an unknown channel i optimality considerations the problem of performing joint maximum likelihood ml estimation of a digital sequence and unknown dispersive channel impulse response is considered starting from a continuous time ct model previous investigations of this problem have not considered the front end fe processing in detail rather a discrete time signal model has been assumed we show that a fractionally spaced whitened matched filter matched to the known data pulse provides a set of sufficient statistics when a tapped delay line channel model is assumed and that the problem is ill posed when the channel impulse response is generalized to a ct finite length model practical approximations are considered that circumvent this ill posed condition recursive computation of the joint ml metric is developed together the fe processing and metric recursion provide a receiver structure which may be interpreted as the theoretical foundation for the previously introduced technique of per survivor processing and they lead directly to generalizations several fe processors representative of those suggested in the literature are developed and related to the practically optimal fe

optimality in the choice of the convergence factor for gradient based adaptive algorithms the convergence and the adaptation speed of gradient based adaptive algorithms are controlled by the chosen value for the convergence factor in this paper the existence of an optimal value for this convergence factor is investigated for two classes of algorithms a proof is first presented for the general case of the complex adaptive linear combiner alc the results are applied to the complex and real lms algorithms this is followed by a second proof for algorithms which are linear only in a subset of their adaptive coefficients these cases are found in iir applications such as the hybrid recursive lattice recursive and recursive algorithms using the direct realization iir for each case the optimal value is shown to be generated using instantaneous signal estimates the resulting adaptive algorithms become self optimizing in terms of their convergence factor and dependence on incoming training signal levels is reduced moreover a correction factor is introduced in each case to regulate the adaptation process and accommodate practical applications where additive signals are present with the desired signal

a novel approach for stabilizing recursive least squares filters a novel approach for stabilizing recursive least squares rls filters is presented the approach relies on a detailed fixed point analysis which provides two important benefits the analysis reveals a bias in the error propagation mechanism providing an analytical basis for instability problems the analysis then indicates which specific roundoff errors are causing instability these roundoff errors are then biased in such a way that the overall filter is biased towards stable performance experimental results indicate that stability can be achieved with negligible loss in least squares performance

a robust recursive least squares algorithm a new algorithm is developed which guarantees the normalized bias in the weight vector due to persistent and bounded data perturbations to be bounded robustness analysis for this algorithm has been presented an approximate recursive implementation is also proposed it is termed as the robust recursive least squares rrls algorithm since it resembles the rls algorithm in its structure and is robust with respect to persistent bounded data perturbation simulation results are presented to illustrate the efficacy of the rrls algorithm

fast recursive least squares transversal filters for adaptive filtering fast transversal filter ftf implementations of recursive least squares rls adaptive filtering algorithms are presented in this paper substantial improvements in transient behavior in comparison to stochastic gradient or lms adaptive algorithms are efficiently achieved by the presented algorithms the true not approximate solution of the rls problem is always obtained by the ftf algorithms even during the critical initialization period first n iterations of the adaptive filter this true solution is recursively calculated at a relatively modest increase in computational requirements in comparison to stochastic gradient algorithms factor of 1 6 to 3 5 depending upon application additionally the fast transversal filter algorithms are shown to offer substantial reductions in computational requirements relative to existing fast rls algorithms such as the fast kalman algorithms of morf ljung and falconer 1976 and the fast ladder lattice algorithms of morf and lee 1977 1981 they are further shown to attain steady state unnormalized or improve upon first n initialization steps the very low computational requirements of the efficient rls solutions of carayannis manolakis and kalouptsidis 1983 finally several efficient procedures are presented by which to ensure the numerical stability of the transversal filter algorithms including the incorporation of soft constraints into the performance criteria internal bounding and rescuing procedures and dynamic range increasing square root normalized variations of the transversal filters

green s functions and trace formulas for generalized sturm liouville problems related by darboux transformations we study green s functions of the generalized sturm liouville problems that are related to each other by darboux equivalently supersymmetrical transformations we establish an explicit relation between the corresponding green s functions and derive a simple formula for their trace the class of equations considered here includes the conventional schrodinger equation and generalizations such as for position dependent mass and with linearly energy dependent potential as well as the stationary fokker planck equation

a class of mixed transversal and ladder adaptive filters with pure order updates a class of exact recursive least squares rls adaptive filter structures with pure order updates called mixed transversal and ladder mital filters is presented in which the propagation of numerical errors along in time is limited to at most m sampling periods so the long term numerical stability can be guaranteed the adaptive structures can be treated either as transversal forms or as ladder forms to meet the requirements of different applications the outstanding numerical performances of the adaptive filters are illustrated by computer simulations

explicit darboux transformations of arbitrary order for generalized time dependent schrodinger equations we construct darboux transformations of arbitrary order for a generalized linear time dependent schrodinger equation special cases of which correspond to time dependent hamiltonians coupled to a magnetic field with position dependent mass and with weighted energy our darboux transformation reduces correctly to these known cases and also to new generalized schrodinger equations furthermore fundamental properties of the conventional darboux transformation are maintained such as factorization of the nth order transformation into first order transformations and existence of a reality condition for the transformed potentials

fast subsampled updating stabilized fast transversal filter fsu sftf rls algorithm for adaptive filtering we present a new doubly fast algorithm for recursive least squares rls adaptive filtering that uses displacement structure and subsampled updating the fast subsampled updating stabilized fast transversal filter fsu sftf algorithm is mathematically equivalent to the classical fast transversal filter ftf algorithm the ftf algorithm exploits the shift invariance that is present in the rls adaptation of an fir filter the ftf algorithm is in essence the application of a rotation matrix to a set of filters and in that respect resembles the levinson 1947 algorithm in the subsampled updating approach we accumulate the rotation matrices over some time interval before applying them to the filters it turns out that the successive rotation matrices themselves can be obtained from a schur type algorithm that once properly initialized does not require inner products the various convolutions that appear in the algorithm are done using the fast fourier transform fft the resulting algorithm is doubly fast since it exploits ftf and ffts the roundoff error propagation in the fsu sftf algorithm is identical to that in the sftf algorithm a numerically stabilized version of the classical ftf algorithm the roundoff error generation on the other hand seems somewhat smaller for relatively long filters the computational complexity of the new algorithm is smaller than that of the well known lms algorithm rendering it especially suitable for applications such as acoustic echo cancellation

summation formulae for the product of the q kummer functions from e q 2 using the representation of eq 2 on the non commutative space zz qz z q 0 summation formulae for the product of two three and four q kummer functions are derived

subspace based adaptive method for estimating direction of arrival with luenberger observer in this paper we propose a computationally simple and efficient subspace based adaptive method for estimating directions of arrival amend for multiple coherent narrowband signals impinging on a uniform linear array ula where the previously proposed qr based method is modified for the number determination a new recursive least squares rls algorithm is proposed for null space updating and a dynamic model and the luenberger state observer are employed to solve the estimate association of directions automatically the statistical performance of the rls algorithm in stationary environment is analyzed in the mean and mean squares senses and the mean square error mse and mean square derivation msd learning curves are derived explicitly furthermore an analytical study of the rls algorithm is carried out to quantitatively compare the performance between the rls and least mean square lms algorithms in the steady state the theoretical analyses and effectiveness of the proposed rls algorithm are substantiated through numerical examples

efficient algorithms for adaptive capon and apes spectral estimation in this paper fast algorithms for adaptive capon and amplitude and phase estimation apes methods for spectral analysis of time varying signals are derived fast stable nonrecursive formulae are derived based on time shifting properties of the pertinent variables as a consequence efficient frequency domain recursive least squares rls based as well as fast rls based algorithms for the adaptive estimation of the power spectra are developed stability issues of the frequency domain estimators are considered and stabilization procedures are proposed the computational complexity of the proposed algorithms is lower than relevant existing methods the performance of the proposed algorithms is demonstrated through extensive simulations

analysis of lms newton adaptive filtering algorithms with variable convergence factor an analysis of two lms newton adaptive filtering algorithms with variable convergence factor is presented the relations of these algorithms with the conventional recursive least squares algorithm are first addressed their performance in stationary and nonstationary environments is then studied and closed form formulas for the excess mean square error mse are derived the paper deals in addition with the effects of roundoff errors for the case of fixed point arithmetic specifically closed form formulas for the excess mse caused by quantization are obtained the paper concludes with experimental results that demonstrate the validity of the analysis presented

stabilizing the fast kalman algorithms the fast kalman algorithms are stabilized by introducing a quantity that measures the accumulation of the roundoff errors this quantity is used to correct the variables of the algorithm at every time step the correction is defined as the solution of a specific minimization problem the resulting algorithm still has the nice complexity properties of the original algorithm linear in the number of parameters to be estimated but it has a much more stable behavior

a fast quasi newton adaptive filtering algorithm the convergence rate of an adaptive system is closely related to its ability to track a time varying optimum basic adaptive filtering algorithms give poor convergence performance when the input to the adaptive system is colored more sophisticated algorithms which converge very rapidly regardless of the input spectrum algorithms typically require o n sup 2 computation where n is the order of the adaptive filter a significant disadvantage for real time applications also many of these algorithms behave poorly in finite precision implementation an adaptive filtering algorithm is introduced which employs a quasi newton approach to give rapid convergence even with colored inputs the algorithm achieves an overall computational requirement of o n and appears to be quite robust in finite precision implementations