a new approach to multipath correction of constant modulus signals an adaptive digital filtering algorithm that can compensate for both frequency selective multipath and interference on constant envelope modulated signals is presented the method exploits the fact that multipath reception and various interference sources generate incidental amplitude modulation on the received signal a class of so called constant modulus performance functions is developed which sense this am term but are insensitive to the angle modulation simple adaptive algorithms for finite impulse response fir digital filters are developed which employ a gradient search of the performance function one of the resulting algorithms is simulated for the example of an fm signal degraded by specular multipath propagation substantial improvements in noise power ratio npr are observed e g 25 db with moderately rapid convergence time these results are then extended to include tonal interference on a fm signal and intersymbol interference on a qpsk data signal

tauberian theorems and the central limit theorem 1 results let xn be independent and identically distributed random variables with law p mean 0 and variance a2 00 s 1 xk write p qn 1 for the laws of sn sn nyi a n and the standard normal n o 1 thus qn 1d weak convergence by the global central limit theorem we shall need the local central limit theorem see e g ibragimov and linnik 1971 theorems 4 4 1 4 2 1 4 2 2 we shall assume either llt i some pk has an absolutely continuous component in which case qn 11 3 0 as n x oo variation norm or llt ii p is lattice supported by some arithmetic progression a hk with k integer and h maximal we shall suppose for simplicity that a 0 and h 1 when writing pnk for pn k

asymptotic properties of the algebraic constant modulus algorithm the algebraic constant modulus algorithm acma is a noniterative blind source separation algorithm it computes jointly beamforming vectors for all constant modulus sources as the solution of a joint diagonalization problem we analyze its asymptotic properties and show that unlike cma it converges to the wiener beamformer when the number of samples or the signal to noise ratio snr goes to infinity we also sketch its connection to the related jade algorithm and derive a version of acma that converges to a zero forcing beamformer this gives improved performance in applications that use the estimated mixing matrix such as in direction finding

on the non existence of undesirable equilibria of godard blind equalizers existing results in the literature have proved that particular blind equalization algorithms including godard algorithms are globally convergent in an ideal and nonimplementable setting where a doubly infinite dimensional equalizer is available for adaptation contrary to popular conjectures it is shown that implementable finite dimensional equalizers which attempt to approximate the ideal setting generally fail to have global convergence to acceptable equalizer parameter settings without the use of special remedial measures a theory based on the channel convolution matrix nullspace is proposed to explain the failure of godard algorithms for such practical blind equalization situations this nullspace theory is supported by a simple example showing ill convergence of the godard algorithm

functional limit theorems for weighted sums of i i d random variables on etudie des theoremes limites fonctionnels pour des sommes ponderees specialement pour des poids concernant les methodes de sommabilite classiques telles que celles de cesaro abel borel et euler

blind equalization for an application of unitary space time modulation in isi channels transmit diversity schemes have gained attention due to the promise of increased capacity and improved performance among these schemes unitary space time modulation and differentially encoded unitary space time modulation allow for simple noncoherent decoding for flat fading channels in this paper a new blind equalization algorithm for these transmission schemes in intersymbol interference isi channels is proposed a matrix type constant modulus algorithm that exploits the unitary structure of the space time codes is developed the equalizer is paired with a noncoherent decoder resulting in a completely blind low complexity method for decoding in the presence of isi a noiseless convergence analysis is conducted and verified via simulation in both noiseless and noisy cases the performance of the overall system is evaluated via simulation and semi analytically and the achieved performance is between that of the ideal zero forcing and the minimum mean squared error equalizers

stochastic random walk summability on etudie les proprietes limites des sommes partielles ponderees lorsque les poids sont donnes par une procedure de sommation en marche aleatoire

a modified constrained constant modulus approach to blind adaptive multiuser detection an alternative blind adaptive multiuser detection is investigated based on modified constrained constant modulus cm criterion it is shown that the performance of a cm based receiver is limited by the received power of the desired user in this paper we show that the limitation can be avoided using the noncanonical constraint cm criterion and that in the presence of channel noise the modified cm criterion function is strictly convex by properly selecting some constant with analyzing the extrema of the cost function we point out how to select the constant moreover a simple stochastic gradient algorithm for implementing our scheme is presented and the convergence properties of the algorithm are analyzed simulation examples are given to demonstrate the performance of the proposed scheme

blind source separation the location of local minima in the case of finitely many samples cost functions used in blind source separation are often defined in terms of expectations i e an infinite number of samples is assumed an open question is whether the local minima of finite sample approximations to such cost functions are close to the minima in the infinite sample case to answer this question we develop a new methodology of analyzing the finite sample behavior of general blind source separation cost functions in particular we derive a new probabilistic analysis of the rate of convergence as a function of the number of samples and the conditioning of the mixing matrix the method gives a connection between the number of available samples and the probability of obtaining a local minimum of the finite sample approximation within a given sphere around the local minimum of the infinite sample cost function this shows the convergence in probability of the nearest local minima of the finite sample approximation to the local minima of the infinite sample cost function we also answer a long standing problem of the mean squared error mse behavior of the finite sample least squares constant modulus algorithm ls cma namely whether there exist ls cma receivers with good mse performance we demonstrate how the proposed techniques can be used to determine the required number of samples for ls cma to exceed a specified performance the paper concludes with simulations that validate the results

length and cost dependent local minima of unconstrained blind channel equalizers baud rate linear blind equalizers may converge to undesirable stable equilibria due to different mechanisms one such mechanism is the use of linear fir filters as equalizers it is shown that this type of local minima exist for all unconstrained blind equalizers whose cost functions satisfy two general conditions the local minima generated by this mechanism are thus called length dependent local minima another mechanism is generated by the cost function adopted by the blind algorithm itself this type of local minima are called cost dependent local minima it is shown that several well designed algorithms do not have cost dependent local minimum whereas other algorithms such as the decision directed equalizer and the stop and go algorithm sga do unlike many existing convergence analysis the convergence of the godard 1980 algorithms gas and standard cumulant algorithms scas under gaussian noise is also presented computer simulations are used to verify the analytical results

relationships between the constant modulus and wiener receivers the godard 1980 or the constant modulus algorithm cma is an effective technique for blind receiver design in communications however due to the complexity of the constant modulus cm cost function the performance of the cm receivers has primarily been evaluated using simulations theoretical analysis is typically based on either the noiseless case or approximations of the cost function the following question while resolvable numerically for a specific example remains unanswered in a generic manner in the presence of channel noise where are the cm local minima and what are their mean squared errors mse in this paper a geometrical approach is presented that relates the cm to wiener or minimum mse receivers given the mse and the intersymbol user interference of a wiener receiver a sufficient condition is given for the existence of a cm local minimum in the neighborhood of the wiener receiver the mse bounds on cm receiver performance are derived and shown to be tight in simulations the analysis shows that while in some cases the cm receiver performs almost as well as the nonblind wiener receiver it is also possible that due to its blind nature the cm receiver may perform considerably worse than a nonblind wiener receiver

convergence and performance analysis of godard family and multimodulus algorithms for blind equalization we obtain the convergence of the godard family including the sato and constant modulus cm algorithms and the multimodulus algorithms mma in a unified way our analysis also works for cma fractionally spaced equalizer fse our assumptions are quite realistic the channel input can be asymptotically stationary and ergodic the channel impulse response is finite and can be stationary and ergodic this models fading channels and the equalizer length is finite the noise is independent and identically distributed i i d the channel input can be discrete or continuous our approach allows us to approximate the whole trajectory of the equalizer coefficients this provides estimates of the rate of convergence and the system performance symbol error rate can be evaluated under transience and steady state

parabolic equations for curves on surfaces 2 intersections blow up and generalized solutions we describe a theory for parabolic equations for immersed curves on surfaces which generalizes the curve shortening or flow by mean curvature problem as well as several models in the theory of phase transitions in two dimensions a class of equations is described for which the initial value problem is well posed for rough initial data for which one can give a description of the way a smooth solution becomes singular and for which one can define generalized solutions i e solutions which are smooth except at a discrete set of times the methods which are used in this paper are more geometrical than those of part i by comparing arbitrary solutions with certain special solutions and by considering the way they intersect we derive estimates for the curvature and the tangent which allow one to study the initial value problem and the way solutions become singular

statistical performance analysis of the algebraic constant modulus algorithm this paper presents a large sample analysis of the covariance of the beamformers computed by the analytical constant modulus algorithm acma method for blindly separating constant modulus sources this can be used to predict the signal to interference plus noise ratio sinr performance of these beamformers as well as their deviation from the nonblind wiener receivers to which they asymptotically converge the analysis is based on viewing acma as a subspace fitting optimization where the subspace is spanned by the eigenvectors of a fourth order covariance matrix the theoretical performance is illustrated by numerical simulations and shows a good match

bounds for the mse performance of constant modulus estimators the constant modulus cm criterion has become popular in the design of blind linear estimators of sub gaussian independent and identically distributed i i d processes transmitted through unknown linear channels in the presence of unknown additive interference in this paper we present an upper bound for the conditionally unbiased mean squared error umse of cm minimizing estimators that depends only on the source kurtoses and the umse of wiener estimators further analysis reveals that the extra umse of cm estimators can be upper bounded by approximately the square of the wiener i e minimum umse since our results hold for vector valued finite impulse response infinite impulse response fir iir linear channels vector valued fir iir estimators with a possibly constrained number of adjustable parameters and multiple interferers with arbitrary distribution they confirm the longstanding conjecture regarding the general mean square error mse robustness of cm estimators

low complexity variable forgetting factor mechanism for blind adaptive constrained constant modulus algorithms in this paper we propose a novel low complexity variable forgetting factor vff mechanism for blind adaptive constrained constant modulus ccm recursive least squares rls algorithms applied to linear interference suppression in direct sequence code division multiple access ds cdma systems the proposed variable forgetting factor mechanism employs an updated component related to the time average of the constant modulus cm cost function to automatically adjust the forgetting factor in order to ensure good tracking of the interference and the channel convergence and tracking analyses are carried out analytical expressions for predicting the mean squared error of the proposed adaptation technique are obtained simulation results are presented for nonstationary environments and show that the proposed variable forgetting factor mechanism achieves superior performance to previously reported methods at a reduced complexity

avoiding divergence in the shalvi weinstein algorithm the most popular algorithms for blind equalization are the constant modulus algorithm cma and the shalvi weinstein algorithm swa it is well known that swa presents a higher convergence rate than cma at the expense of higher computational complexity if the forgetting factor is not sufficiently close to one if the initialization is distant from the optimal solution or if the signal to noise ratio is low swa can converge to undesirable local minima or even diverge in this paper we show that divergence can be caused by an inconsistency in the nonlinear estimate of the transmitted signal or when the algorithm is implemented in finite precision by the loss of positiveness of the estimate of the autocorrelation matrix or by a combination of both in order to avoid the first cause of divergence we propose a dual mode swa in the first mode of operation the new algorithm works as swa in the second mode it rejects inconsistent estimates of the transmitted signal assuming the persistence of excitation condition we present a deterministic stability analysis of the new algorithm to avoid the second cause of divergence we propose a dual mode lattice swa which is stable even in finite precision arithmetic and has a computational complexity that increases linearly with the number of adjustable equalizer coefficients the good performance of the proposed algorithms is confirmed through numerical simulations

the heat equation shrinks embedded plane curves to round points soit c 0 s 1 r 2 une courbe lisse plongee dans le plan alors c s 1 0 t r 2 existe en satisfaisant c t k n ou k est la courbure de c et n est son vecteur unite normal entrant c t est lisse pour tout t il converge vers un point quand t t et sa forme limite quand t t est un cercle rond avec convergence dans norme c

monotonic convergence of fixed point algorithms for ica we re examine a fixed point algorithm proposed by hyvarinen for independent component analysis wherein local convergence is proved subject to an ideal signal model using a square invertible mixing matrix here we derive step size bounds which ensure monotonic convergence to a local extremum for any initial condition our analysis does not assume an ideal signal model but appeals rather to properties of the contrast function itself and so applies even with noisy data and or more sources than sensors the results help alleviate the guesswork that often surrounds step size selection when the observed signal does not fit an idealized model

an analysis of constant modulus receivers this paper investigates connections between non blind wiener receivers and blind receivers designed by minimizing the constant modulus cm cost applicable to both t spaced and fractionally spaced fir equalization the main results include 1 a test for the existence of cm local minima near wiener receivers 2 an analytical description of cm receivers in the neighborhood of wiener receivers 3 mean square error mse bounds for cm receivers when the channel matrix is invertible we also show that the cm receiver is approximately colinear with the wiener receiver and provide a quantitative measure of the size of neighborhoods that contain the cm receivers and the accuracy of the mse bounds

mmse decision feedback equalizers and coding i equalization results the minimum mean squared error decision feedback equalizer mmse dfe has properties that suggest that it is a canonical equalization structure for systems that combine equalization with coded modulation the structure and performance of the mmse dfe are succinctly derived using linear estimation theoretic principles in this first part of this two part paper the front end of the mmse dfe called the mean square whitened matched filter ms wmf is preferable in some ways to a matched filter or a whitened matched filter as a canonical receiver front end in a coded system the feedback filter of the mmse dfe may be implemented in the transmitter using precoding the mmse dfe can perform significantly better than a zero forcing decision feedback equalizer particularly at moderate to low snr s and on severe isi channels the mmse dfe is biased the optimum unbiased mmse dfe is the mmse dfe with the bias removed removing bias improves error probability but reduces the snr to snr sub mmse dfe u snr sub mmse dfe 1 it is shown that this snr relationship is a particular case of a very general result and that snr sub mmse dfe u gives a more realistic estimate of snr the results are extended to partial response equalization and to equalization with correlated inputs in an appendix

performance analysis of the linearly constrained constant modulus algorithm based multiuser detector this paper quantifies the adaptive performance of a blind adaptive multiuser detector mud based on the linearly constrained constant modulus algorithm lccma in both a stationary and nonstationary channel a framework is developed to apply the feedback analysis method to analyzing adaptive mud schemes a closed form expression for the excess mean square error emse of lccma blind adaptive mud in a cdma communications system is derived for both of the steady state and tracking cases the effects of additive noise and multiple access interference are considered a transient analysis is performed that predicts the learning curve of the adaptive filter computer simulation is used to verify the accuracy of the analysis

effect of channel noise on blind equalization and carrier phase recovery of cma and mma this work investigates the effect of channel noise on the performances of the constant modulus algorithm cma the multimodulus algorithm mma and a phase tracker pt following the cma for carrier phase recovery when various symmetrical two dimensional complex valued source constellations are used the contributions of this work are two fold first a generalized formulation of the relocation of stationary points toward the origin owing to channel noise is derived for both the cma and the mma allowing their sensitivity to channel noise to be explicitly compared when various constellations are used second the curvatures of the error surface at the stationary points in both the radial and the angular directions are reduced in a way that is linearly proportional to the square of the distances between the stationary points and the origin as the stationary points of the cma and the mma relocate toward the origin in the presence of additive channel noise the reduction of the error surface curvature in the radial direction reduces the rate of blind equalization convergence whereas the reduction in the angular direction degrades the capacity for carrier phase recovery

super exponential blind adaptive beamforming the objective of the beamforming with the exploitation of a sensor array is to enhance the signals of the sources from desired directions suppress the noises and the interfering signals from other directions and or simultaneously provide the localization of the associated sources in this paper we present a higher order cumulant based beamforming algorithm namely the super exponential blind adaptive beamforming algorithm which is extended from the super exponential algorithm sea and the inverse filter criteria ifc while both sea and ifc assume noise free conditions this requirement is no longer needed and all the noise components are taken into account in the proposed algorithm two special conditions are derived under which the proposed blind beamforming algorithm achieves the performance of the corresponding optimal nonblind beamformer in the sense of minimum mean square error mmse simulation results show that the proposed algorithm is effective and robust to diverse initial weight vectors its performance with the use of the fourth order cumulants is close to that of the nonblind optimal mmse beamformer

parabolic equations for curves on surfaces part i curves with p integrable curvature this is the first of a two part paper in which we develop a theory of parabolic equations for curves on surfaces which can be applied to the so called curve shortening of flow by mean curvature problem as well as to a number of models for phase transitions in two dimensions we introduce a class of equations for which the initial value problem is solvable for initial data with p integrable curvature and we also give estimates for the rate at which the p norms of the curvature must blow up if the curve becomes singular in finite time a detailed discussion of the way in which solutions can become singular and a method for continuing the solution through a singularity will be the subject of the second part

geometrical characterizations of constant modulus receivers convergence properties of the constant modulus cm and the shalvi weinstein sw algorithms in the presence of noise remain largely unknown a new geometrical approach to the analysis of constant modulus and shalvi weinstein receivers is proposed by considering a special constrained optimization involving norms of the combined channel receiver response this approach provides a unified framework within which various blind and nonblind wiener receivers can all be analyzed by circumscribing an ellipsoid by norm balls of different types a necessary and sufficient condition for the equivalence among constant modulus shalvi weinstein zero forcing and wiener receivers are obtained answers to open questions with regard to cm and sw receivers including their locations and their relationship with wiener receivers are provided for the special orthogonal channel and the general two dimensional 2 d channel receiver impulse response it is also shown that in two dimensions each cm or sw receiver is associated with one and only one wiener receiver

the heat equation shrinking convex plane curves soient m et m des varietes de riemann et f m m une application reguliere si m est une courbe convexe plongee dans le plan r 2 l equation de la chaleur contracte m a un point

on the nonvanishing stability of undesirable equilibria for fir godard blind equalizers the existence of undesirable equilibria of godard 1980 or constant modulus algorithms cma is demonstrated for all finite dimensional fir equalizers sufficient stability conditions are presented for these equilibria it is shown that for the godard 2 algorithm these undesirable equilibria are locally stable as long as the equalizer length remains finite results indicate that merely increasing the equalizer length does not necessarily nullify the potential for local convergence by godard algorithms

ill convergence of godard blind equalizers in data communication systems the existence of stable undesirable equilibria for the godard algorithm is demonstrated through a simple autoregressive ar channel model these undesirable equilibria correspond to local but nonglobal minima of the underlying mean cost function and thus permit the ill convergence of the godard algorithms which are stochastic gradient descent in nature simulation results confirm predicted misbehavior for channel input of constant modulus it is shown that attaining the global minimum of the mean cost necessarily implies correct equalization a criterion is also presented for allowing a decision at the equalizer as to whether a global or nonglobal minimum has been reached

convergence analysis of the constant modulus algorithm we study the global convergence of the stochastic gradient constant modulus algorithm cma in the absence of channel noise as well as in the presence of channel noise the case of fractionally spaced equalizer and or multiple antenna at the receiver is considered for the noiseless case we show that with proper initialization and with small step size the algorithm converges to a zero forcing filter with probability close to one in the presence of channel noise such as additive gaussian noise we prove that the algorithm diverges almost surely on the infinite time horizon however under suitable conditions the algorithm visits a small neighborhood of the wiener filters a large number of times before ultimately diverging

domains of attraction of shalvi weinstein receivers domains of attraction doa of shalvi weinstein 1990 receivers are analyzed it is shown that there is a one to one correspondence between doa in the receiver parameter space and those in the global or combined channel receiver parameter space for general noiseless channels doa of sw receivers in the global response space are the minimum distance decision regions on a unit sphere in the presence of noise and for the class of orthogonal channels doa of sw receivers for independent and identically distributed i i d input signals are the minimum distance decision regions on an ellipsoid determined by the channel coefficients and the noise variance the doa in the receiver parameter space are also characterized for the general nonuniformly distributed sources the size of the doa is shown to be affected by the signal power the signal constellation the noise level and the channel condition it is also demonstrated that although the optima of the shalvi weinstein algorithm and those of the constant modulus algorithm are one to one correspondent their doa are different in general

weighted sums of random variables attracted to stable laws on etudie des sommes ponderees de variables aleatoires en relation avec la loi des grands nombres et la loi du logarithme itere

a class of channels resulting in ill convergence for cma in decision feedback equalizers this paper analyzes the convergence of the constant modulus algorithm cma in a decision feedback equalizer using only a feedback filter several works had already observed that the cma presented a better performance than decision directed algorithm in the adaptation of the decision feedback equalizer but theoretical analysis always showed to be difficult specially due to the analytical difficulties presented by the constant modulus criterion in this paper we surmount such obstacle by using a recent result concerning the cm analysis first obtained in a linear finite impulse response context with the objective of comparing its solutions to the ones obtained through the wiener criterion the theoretical analysis presented here confirms the robustness of the cma when applied to the adaptation of the decision feedback equalizer and also defines a class of channels for which the algorithm will suffer from ill convergence when initialized at the origin