fast coupled adaptation for sparse impulse responses using a partial haar transform this work presents a novel scheme for identifying the impulse response of a sparse channel the scheme consists of two adaptive filters operating sequentially the first adaptive filter adapts using a partial haar transform of the input and yields an estimate of the location of the peak of the sparse impulse response the second adaptive filter is then centered about this estimate both filters are short in comparison to the delay uncertainty of the unknown channel the principle advantage of this scheme is that two short adaptive filters can be used instead of one long adaptive filter resulting in faster overall convergence and reduced computational complexity and storage the scheme is analyzed in detail for a least mean squares lms lms lms type of structure although it can be implemented using any combination of adaptive algorithms monte carlo simulations are shown to be in good agreement with the theoretical model for the behavior of the peak estimating filter as well as for the mean square error mse behavior of the second filter

discrete choquet integral and some of its symmetric extensions classical extensions of the choquet integral defined on 0 1 to 1 1 are the asymmetric and the symmetric choquet integral the second one being called also the sipos integral recently the balancing choquet integral was introduced as another kind of a symmetric extension of the discrete choquet integral we introduce and discuss a new type of such extension the fusion choquet integral and discuss its properties and relationship to the balancing and the symmetric choquet integral the symmetric maximum introduced by grabisch is shown to be a special case of the fusion and the balancing choquet integral several extensions of owa operators are also discussed

on the optimum gain parameter in lms adaptation in this correspondence the optimum selection of the adaptation parameter in lms adaptive estimation is discussed for a white data covarianee matrix it is shown via analysis and numerical evaluation that the optimum selection of is a complicated function of the number of filter taps the initial weight setting the wiener weights and the number of learning samples is not chosen in general to yield the most rapid transient response for most cases of interest a smaller value of will be selected for slower adaptation and smaller misadjustment error at the end of the learning phase

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

time correlation statistics of the lms adaptive algorithm weights the transient and steady state weight correlation statistics of both the real and complex lms adaptive filters are obtained when the inputs are independent samples from real and circularly gaussian processes respectively a matrix relationship is derived between the covariance matrix of the weight vector at two different times and the covariance matrix of the weights at one time these expressions show that the weight fluctuations have the same time constants as the mean behavior of the lms algorithm itself i e the weights are correlated over the same number of iterations that it takes for the algorithm to converge to the wiener weights for stationary inputs

optimal gain derivation for the lms algorithm using a visual fidelity criterion this paper analytically derives the optimal feedback gain parameter 0 for the least mean squares lms algorithm used in a video compression application and subject to a visual fidelity criteria experimental results are presented which show these analytically derived values of 0 to be in excellent agreement with experimentally derived 0 over a wide range of filter and data parameters

steady state mse convergence of lms adaptive filters with deterministic reference inputs with applications to biomedical signals we analyze the steady state mean square error mse convergence of the lms algorithm when deterministic functions are used as reference inputs a particular adaptive linear combiner is presented where the reference inputs are any set of orthogonal basis functions the adaptive orthogonal linear combiner aolc several authors have applied this structure always considering in the analysis a time average behavior over one signal occurrence we make a more precise analysis using the deterministic nature of the reference inputs and their time variant correlation matrix two different situations are considered in the analysis orthogonal complete expansions and incomplete expansions the steady state misadjustment is calculated using two different procedures with equivalent results the classical one analyzing the transient behavior of the mse and as the residual noise at the output of the equivalent time variant transfer function of the system the latter procedure allows a very simple formalism being valid for colored noise as well the derived expressions for steady state misadjustment are contrasted with experimental results in electrocardiographic ecg signals giving exact concordance for any value of the step size

the complex lms adaptive algorithm transient weight mean and covariance with applications to the ale the transient and steady state mean and covariance of the complex valued lms adaptive element weights are investigated when the inputs are samples from circularly normal processes it is shown that the data covariance diagonalizing matrix also diagonalizes the weight covariance matrix this result permits describing the complete transient behavior of the adaptive line enhancer ale weight covariance matrix in closed form for the case of multiple equal power narrow band statistically independent orthogonal rayleigh fading sinusoids in broad band noise based on these transient covariance results it is shown that for any stage of adaptation there exists an optimum ale gain constant that minimizes the excess mean squared prediction error this result is of particular significance when processing time limited data

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

postsmoothing for the lms algorithm and a fixed point roundoff error analysis the effect of post algorithm smoothing on digital implementations of the least mean square lms algorithm is studied an expression is derived for the mean square error mse with post algorithm pa smoothing but without finite wordlength effects it is shown that the mse can be reduced from that of lms without pa smoothing pa smoothing results in reduced convergence speed the effect of fixed point finite precision on pa smoothing is studied an expression for the mse is derived monte carlo simulation results that support the result that digital lms with pa smoothing performs better than digital lms without pa smoothing are presented

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

efficient subspace based algorithm for adaptive bearing estimation and tracking in some practical applications of array processing the directions of the incident signals should be estimated adaptively and or the time varying directions should be tracked promptly in this paper an adaptive bearing estimation and tracking abest algorithm is investigated for estimating and tracking the uncorrelated and correlated narrow band signals impinging on a uniform linear array ula based on the subspace based method without eigendecomposition sumwe where a linear operator is obtained from the array data to form a basis for the null space by exploiting the array geometry and its shift invariance property specifically the null space is estimated using the least mean square lms or normalized lms nlms algorithm and the directions are updated using the approximate newton method the transient analyses of the lms and nlms algorithms are studied where the weight i e the linear operator is in the form of a matrix and there is a correlation between the additive noise and input data that involve the instantaneous correlations of the received array data in the updating equation and the step size stability conditions are derived explicitly in addition the analytical expressions for the mean square error mse and mean square deviation msd learning curves of the lms algorithm are clarified the effectiveness of the abest algorithm is verified and the theoretical analyses are corroborated through numerical examples simulation results show that the abest algorithm is computationally simple and has good adaptation and tracking abilities

an efficient recursive total least squares algorithm for fir adaptive filtering an algorithm for recursively computing the total least squares tls solution to the adaptive filtering problem is described this algorithm requires o n multiplications per iteration to effectively track the n dimensional eigenvector associated with the minimum eigenvalue of an augmented sample covariance matrix it is shown that the recursive least squares rls algorithm generates biased adaptive filter coefficients when the filter input vector contains additive noise the tls solution on the other hand is seen to produce unbiased solutions examples of standard adaptive filtering applications that result in noise being added to the adaptive filter input vector are cited computer simulations comparing the relative performance of rls and recursive tls are described

fixed point error analysis of stochastic gradient adaptive lattice filters a theoretical analysis is presented of the stochastic gradient adaptive lattice filter used as a linear one step predictor when the effects of finite precision arithmetic are taken into account only the fixed point implementation is considered both the unnormalized and normalized adaptation algorithms are analyzed expressions of the steady state mean squared values of the accumulated numerical errors in the computation of the reflection coefficients and the prediction errors of different orders have been developed the results show that the dominant term in the expressions for the mean squared values of the numerical errors is inversely proportional to the convergence parameter furthermore they indicate that the quantization errors associated with the reflection coefficients are more critical than those associated with representing the prediction error sequences signals with high correlation among samples produce larger numerical errors in the adaptive lattice filter the authors present several simulation examples that show close agreement with the theoretical results they also show that the gradient adaptive lattice filters have better numerical properties than their transversal counterparts

ale behavior for two sinusoidal signal models the adaptive line enhancer ale weight covariance matrix is determined for a complex monochromatic sine wave signal in an independent circular gaussian noise background the weight covariance matrix is shown to differ significantly from that obtained when the signal is a narrow band sine wave process in spite of this result the misadjustment factors which characterize the mean norm squared error behavior of the ale are essentially indistinguishable for small values of the adaptation parameter noise power product 2

fluctuation analysis of a two layer backpropagation algorithm used for modeling nonlinear memoryless channels neural networks have been used to model the nonlinear characteristics of memoryless nonlinear channels using the backpropagation learning bp with experimental training data the mean transient and convergence behavior of a simplified two layer neural network has been studied previously in order to better understand this neural network application the network was trained with zero mean gaussian data this paper extends these results to include the effects of the weight fluctuations on the mean square error mse a new methodology is presented that can be extended to other nonlinear learning problems the new mathematical model is able to predict the mse learning behavior as a function of the algorithm step size spl mu the performance analysis is based on the derivation of linear recursions for the variance and covariance of the weights that depend nonlinearly on the mean weights these linear recursions can be used to predict the local mean square stability of the weights as in linear gradient search problems lms etc it is shown that there exists an optimum p minimizing the mse which is the result of the tradeoff between fast learning and small weight fluctuations monte carlo simulations display excellent agreement between the actual behavior and the predictions of the theoretical model over a wide range of spl mu values

linear fuzzy game with coalition interaction and its coincident solutions abstract in this study we consider the solution concepts for fuzzy coalition games i e cooperative games with fuzzy coalition under a certain participation level in general cooperative games with fuzzy coalition are based on the assumption that all the fuzzy coalition values for different fuzzy coalitions must be represented by the same formula which may omit the coalition interaction under different participation ratios for players considering these conditions we propose the coincident fuzziness form for games with fuzzy coalition which are represented by a mapping from the characteristic function of the crisp game to that of the fuzzy coalition game the proposed fuzzy coalition games admit the differences in coalition interactions for different fuzzy coalitions where the coalition interactions are represented by the fuzzy coalition values in different ways or formulas for a fixed fuzzy coalition the maximum fuzzy coalition game is proven to be the choquet integral form on the condition that the associate crisp game is convex in order to seek appropriate solutions for the proposed games based on a certain participation level the fuzzy shapley axioms are defined and the explicit shapley value is represented by the shapley value of the associated crisp games moreover the fuzzy core of this proposed fuzzy coalition game is the stable solution set for the given fuzzy coalition which is also denoted by the crisp cores furthermore we study the relationship between the fuzzy shapley value and the fuzzy core by adding restrictions on the fuzzy core we propose a strong fuzzy core which is a more stable solution than the fuzzy core in addition the fuzzy core is equivalent to the strong fuzzy core under some condition

on the probability density function of the lms adaptive filter weights the joint probability density function of the weight vector in least mean square lms adaptation is studied for gaussian data models an exact expression is derived for the characteristic function of the weight vector at time n 1 conditioned on the weight vector at time n the conditional characteristic function is expanded in a taylor series and averaged over the unknown weight density to yield a first order partial differential difference equation in the unconditioned characteristic function of the weight vector the equation is approximately solved for small values of the adaption parameter and the weights are shown to be jointly gaussian with time varying mean vector and covariance matrix given as the solution to well known difference equations for the weight vector mean and covariance matrix the theoretical results are applied to analyzing the use of the weights in detection and time delay estimation simulations that support the theoretical results are also presented

behavior of the e normalized lms algorithm with gaussian inputs the first and second moment behavior of one variation of the normalized lms e nlms algorithm is investigated for a white covariance matrix and gaussian statistics for the data for this model it is shown that the e nlms algorithm has neither behavior independent of the input data power nor a performance significantly better than the lms algorithm for which the input power level also must be known a priori hence based upon the results of the analysis it is recommended that the algorithm not be used in place of the lms for known input power levels or in place of the nlms for unknown input power levels

lms adaptation with correlated data a scalar example in this note the statistical behavior of the complex scalar lms adaptation algorithm is investigated when the input data stream consists of statistically dependent inputs the asymptotic time constant of the mean behavior of the algorithm is evaluated taking into account the time correlation properties of the input data sequence it is shown that the behavior of the correlated data case does not degrade significantly from the uncorrelated case until the input sequence is very highly correlated 0 99

fast adaptive microwave beamforming using array signal estimation a new perturbation technique is proposed which enables adaptive beamforming abf in microwave domain using a single port beamformer in this technique for a system with l antennas the weight vector is independently perturbed l times to obtain l correlated outputs these outputs are then used to find a noisy approximation of the antenna array signal for the gradient vector estimation the process of weight perturbations is performed faster than the nyquist rate mainly to increase the temporal correlation of the consecutive antenna array signal samples and to lower the perturbation error performance of the proposed perturbation technique with the adaptive unconstrained least mean square ulms algorithm is investigated for different channel scenarios the ulms algorithm with single port beamformer converges in very high noise and interference levels and with a convergence speed close to that of the multi port receiver after convergence both the single port and the multiport beamformer algorithms achieve the same steady state signal to interference plus noise ratio sinr gain effects of weight quantization are also investigated for the single port beamformer with the proposed perturbation technique

on the joint characteristic function of the complex scalar lms adaptive weight in this paper the joint characteristic function of the single weight complex scalar lms adaptive algorithm is studied an integral equation is derived for the joint characteristic function of the real and imaginary parts of the weight this integral equation is used to obtain the weight moments and is approximately solved for a sufficiently small adaptation parameter it is shown that for small and in steady state the real and imaginary parts of the weights are statistically independent gaussian random variables with means equal to the wiener weight these results are applied to a detection problem using the weight magnitude square as the detection statistic assuming the weight is also gaussian during the transient phase of adaptation the detection performance is optimized over for a fixed number of data samples and a known observation interval the optimum selection of approaches zero so that the adaptation process reduces to a cross correlation operation when the observation interval is not known a priori a bounded away from zero is required this detector is shown to be 3 db degraded from the optimum narrow band envelope detector using the same number of data samples

convergence analysis of an algorithm for blind equalization the authors consider a key algorithm for blind equalization and derive expressions for the evolution of the equalizer output error trajectory the authors develop a model to examine the convergence behavior of this algorithm suitable approximations are incorporated into the model to facilitate analysis the validity of these approximations is demonstrated for a typical communication channel it is shown that even for a channel for which the assumption of gaussianity of the equalizer input data may not be very good the analysis presented predicts the convergence behavior reasonably well

correlation function for the weight sequence of the frequency domain adaptive filter the statistical behavior of the single complex adaptive weight in the frequency domain adaptive filter is further investigated in a previous note 8 an expression was evaluated for the expected value of the magnitude square of the weight sequence at any time in this note the covariance of the weight at different times is evaluated and in steady state shown to be geometric with the same time constant as the frequency domain adaptive filter algorithm itself using these results the reduction in the fluctuations of the weight values are computed for an exponentially decaying post algorithm filter

weight modulation effects in the adaptive line enhancer the steady state behavior of the adaptive line enhancer is modeled by the sum of two filters the wiener weight filter and a weight misadjustment filter the effects of randomly time varying coefficients in the weight misadjustment filter are modeled so as to display line broadening of single frequency ale inputs

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

the composite regressor algorithm for iir adaptive systems an adaptive iir algorithm called the composite regressor algorithm cra is developed the algorithm is a generalization of the common equation error a priori output error and a posteriori output error adaptive iir algorithms the cra is analyzed for convergence in a noiseless environment and for bias in a stochastic setting it is determined that by using a parameter called the regressor composition parameter a tradeoff can be obtained between the automatic convergence but large bias results of the equation error algorithm and the difficult convergence condition but small bias results of the output error algorithms in proving results for the cra it is shown that the a posteriori output error algorithm produces estimates with nonzero bias when the adaptive gain is small but bounded away from zero a convergence condition for the a priori output error algorithm is derived for the first time

fuzzy bi cooperative games in multilinear extension form in this paper we introduce the notion of a fuzzy bi cooperative game in multilinear extension form an lg value as a possible solution concept is obtained using standard fuzzy game theoretic axioms

nonvanishing of l series associated to cubic twists of elliptic curves has a rational solution an extensive compilation of the older history of the problem is given in dickson 6 within the past century researchers have tried to exploit either explicitly or unknowingly the fact that the curve 0 1 is in fact an elliptic curve during the nineteenth century lucas and later sylvester used a descent argument to prove that 0 1 had no solution for infinitely many d in certain congruence classes mod 9 and 18 see 6 ch xxi zagier and kramarz 19 have produced a great deal of numerical evidence about the l series of the curves based on these computations they have argued heuristically that for

an approximate statistical analysis of the widrow lms algorithm with application to narrow band interference rejection the widrow lms algorithm is considered for the implementation of an adaptive prewhitening filter in a direct sequence ds spread spectrum receiver exact expressions for the steady state tapweight covariance matrix and resulting average excess mean square error are developed for the real lms algorithm when the input contains a random binary sequence used to model a pseudonoise spreading sequence it is shown here that the output samples of the adaptive filter possess approximately gaussian statistics under the conditions of slow convergence and a large number of filter taps using this approximation expressions for the resulting bit error rate ber when the adaptive algorithm is used to suppress a fading gone jammer are developed and numerical results obtained from these expressions are compared to simulation results for the ds receiver

statistical analysis of initialization methods for rls adaptive filters theoretical analysis is used to evaluate the mean and second moment properties of recursive least squares algorithms incorporating the fast exact initialization and soft constrained initialization methods during the initialization period it is shown that the weight vector mean and covariance produced by fast exact initialization are undefined for this period theoretical results are derived for soft constrained initialization that show that the weight vector mean and covariance are finite and expressions are given for these quantities simulations for various cases are presented to support the accuracy of these theoretical results

analytic behavior of the lms adaptive line enhancer for sinusoids corrupted by multiplicative and additive noise the least mean squares adaptive line enhancer lms ale has been widely used for the enhancement of coherent sinusoids in additive wideband noise this paper studies the behavior of the lms ale when applied to the enhancement of sinusoids that have been corrupted by both colored multiplicative and white additive noise the multiplicative noise decorrelates the sinusoid spreads its power spectrum and acts as an additional corrupting noise closed form expressions are derived for the optimum wiener filter ale output snr as a function of the residual coherent sine wave power the noncoherent sine wave power spectrum and the background additive white noise when the coherent to noncoherent sine wave power ratio is sufficiently small it is shown that a nonlinear e g square law transformation of the ale input results in a larger optimum ale output snr

a full mean square analysis of clms for second order noncircular inputs a full mean square transient and steady state analysis of the complex least mean square clms algorithm is provided for strictly linear estimation of general second order noncircular improper gaussian inputs to this end we also consider the performance assessment in terms of the evolution of the complementary mean square error cmse and the complementary covariance pseudocovariance matrix of the weight error vector of clms this makes it possible to measure the degrees of noncircularity of the output error and the weight error vector which arise due to second order noncircularity improperness of the system input and system noise the recently introduced approximate uncorrelating transform which allows for joint direct diagonalization of both the input covariance and complementary covariance matrices with a single singular value decomposition is then employed in order to derive a unified bound on the step size which guarantees the convergence of both the standard mse and the proposed cmse a joint consideration of the standard mean square performance analysis and the proposed complementary performance analysis is shown to provide full second order closed form statistical descriptions of both the transient and steady state performances of clms for second order noncircular improper gaussian input data simulations in the system identification setting support the analysis

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

fuzzy harsanyi solution for a kind of fuzzy coalition games abstract the fuzzy linear game in ref 32 is a class of fuzzy cooperative games with fuzzy coalition fuzzy coalition game for short which admits the difference of interactions for different fuzzy coalitions the fuzzy shapley value for fuzzy linear game has been studied as shapley value is just one point in the harsanyi solution it is interesting to study the explicit form of harsanyi solution in the fuzzy coalition game this paper defines the fuzzy harsanyi solution which distributes the harsanyi dividends such that the dividend shares of players in each fuzzy coalition are proportional to the corresponding players participation index when the fuzzy coalition payoff is average distributed between players in the fuzzy coalition the fuzzy harsanyi solution also coincides with the fuzzy shapley value just as their relationship in a crisp cooperative game we provide two axiomatic characterizations for the fuzzy harsanyi solution one uses axioms and the other uses distribution matrices the fuzzy harsanyi solution is a unique value on the fuzzy linear game that satisfies fuzzy efficiency fuzzy null player and fuzzy additivity property meanwhile based on probability distribution fuzzy harsanyi solution is seen as an expected marginal contribution of each player in the fuzzy coalition which provides more choices for players in the fuzzy coalition

normalized data nonlinearities for lms adaptation properly designed nonlinearly modified lms algorithms in which various quantities in the stochastic gradient estimate are operated upon by memoryless nonlinearities have been shown to perform better than the lms algorithm in system identification type problems the authors investigate one such algorithm given by w sub k l w sub k spl mu d sub k w sub k sup t x sub k x sub k f x sub k in which the function f x sub k is a scalar function of the sum of the squares of the n elements of the input data vector x sub k this form of algorithm generalizes the so called normalized lms nlms algorithm they evaluate the expected behavior of this nonlinear algorithm for both independent input vectors and correlated gaussian input vectors assuming the system identification model by comparing the nonlinear algorithm s behavior with that of the lms algorithm they then provide a method of optimizing the form of the nonlinearity for the given input statistics in the independent input case they show that the optimum nonlinearity is a single parameter version of the nlms algorithm with an additional constant in the denominator and show that this algorithm achieves a lower excess mean square error mse than the lms algorithm with an equivalent convergence rate additionally they examine the optimum step size sequence for the optimum nonlinear algorithm and show that the resulting algorithm performs better and is less complex to implement than the optimum step size algorithm derived for another form of the nlms algorithm simulations verify the theory and the predicted performance improvements of the optimum normalized data nonlinearity algorithm