A study is made of three recursive identification algorithms with respect to their tracking capability and disturbance sensitivity. The algorithms are applied to the identification of time-varying systems with finite impulse response. The performance of the algorithms are measured in terms of the mean square error of the estimated transfer function. Using asymptotic methods, explicit expressions for the mean square error are derived, and these expressions are used to evaluate the performance of the algorithms when applied to systems with different behavior.>

The emphasis of this study is on direction-of-arrival (DOA) estimation of narrowband emitter signals. There are two main signal models that have been used in the sensor array problem, namely, stochastic and deterministic. A discussion is presented of the models, their effect on the Cramer-Rao lower bounds (CRB), cases where the maximum likelihood (ML) estimator asymptotically achieves this bound, and how the bounds are related. It is shown that the asymptotic variance of the ML method based on the stochastic model is lower than that of the ML method based on the deterministic model. The stochastic ML variance and CRB are based on the assumption of Gaussian emitter signals, which can be an unrealistic assumption when narrowband signals are assumed. The authors show that the ML estimator based on the Gaussian signal model achieves the stochastic CRB asymptotically, regardless of the signal distribution. Numerical evaluations of the theoretical variances for different scenarios are presented to clarify the relationship between the methods and bounds discussed.>

Parameter estimation problems that can be formulated as linear regressions are quite common in many applications. Recursive (on-line, sequential) estimation of such parameters can be performed using the recursive least squares (RLS) algorithm or a stochastic gradient version of this algorithm. In this paper the convergence properties of the gradient algorithm are analyzed under the assumption that the gain tends to zero. The technique is the same as the so-called ordinary differential equation approach, but the treatment here is self-contained and includes a proof of the boundedness of the estimates. A main result is that the convergence conditions for the gradient algorithm are the same as those for the recursive least squares algorithm.

Some basic results on recursive identification techniques and their properties are reviewed. The link between adaptive algorithms, recursive identification, and off-line identification is stressed. The fundamental character of the prediction and its gradient with respect to the adjustable parameters is pointed out.

Very rapid initial convergence of the equalizer tap coefficients is a requirement of many data communication systems which employ adaptive equalizers to minimize intersymbol interference. As shown in recent papers by Godard, and by Gitlin and Magee, a recursive least squares estimation algorithm, which is a special case of the Kalman estimation algorithm, is applicable to the estimation of the optimal (minimum MSE) set of tap coefficients. It was furthermore shown to yield much faster equalizer convergence than that achieved by the simple estimated gradient algorithm, especially for severely distorted channels. We show how certain "fast recursive estimation" techniques, originally introduced by Morf and Ljung, can be adapted to the equalizer adjustment problem, resulting in the same fast convergence as the conventional Kalman implementation, but with far fewer operations per iteration (proportional to the number of equalizer taps, rather than the square of the number of equalizer taps). These fast algorithms, applicable to both linear and decision feedback equalizers, exploit a certain shift-invariance property of successive equalizer contents. The rapid convergence properties of the "fast Kalman" adaptation algorithm are confirmed by simulation.

A new implementation is presented for the optimum likelihood ratio detector for stationary Gaussian signals in white Gaussian noise that uses only two causal time-invariant filters. This solution also has the advantage that fast algorithms based on the Levinson and Chandrasekhar equations can he used for the determination of these time-invariant filters. By using a notion of "closeness to stationarity,' there is a natural extension of the above results for nonstationary signal processes.

A state-space model of a second-order random process is a representation as a linear combination of a set of state-variables which obey first-order linear differential equations driven by an input process that is both white and uncorrelated with the initial values of the state-variables. Such a representation is often called a Markovian representation. There are applications in which it is useful to consider time running backwards and to obtain corresponding backwards Markovian representations. Except in one very special circumstance, these backwards representations cannot be obtained simply by just reversing the direction of time in a forwards Markovian representation. We show how this problem can be solved, give some examples, and also illustrate how the backwards model can be used to clarify certain least squares smoothing formulas.

The results stated in the above paper1 concerning an approved stochastic approximation method are considered. Formulas for the variances of the estimates are derived, and it is found that, in fact, the new algorithm is inferior to previously suggested ones.