H.L. Van Trees

A Bayesian approach to robust adaptive beamforming

An adaptive beamformer that is robust to uncertainty in source direction-of-arrival (DOA) is derived using a Bayesian approach. The DOA is assumed to be a discrete random variable with a known a priori probability density function (PDF) that reflects the level of uncertainty in the source DOA. The resulting beamformer is a weighted sum of minimum variance distortionless response (MVDR) beamformers pointed at a set of candidate DOAs, where the relative contribution of each MVDR beamformer is determined from the a posteriori PDF of the DOA conditioned on previously observed data. A simple approximation to the a posteriori PDF results in a straightforward implementation. Performance of the approximate Bayesian beamformer is compared with linearly constrained minimum variance (LCMV) beamformers and data-driven approaches that attempt to estimate signal characteristics or the steering vector from the data.

A signal subspace approach for speech enhancement

A perceptually based linear signal estimator for enhancing speech signals degraded by uncorrelated additive noise is developed. The estimator is designed by minimizing the signal distortion while maintaining the residual noise level below some given threshold. The estimator is shown to be a Wiener filter with adjustable input noise level. This level is determined by the threshold of the permissible residual noise. The estimator is implemented using the signal subspace approach. The vector space of the noisy signal is decomposed into a signal subspace and complementary orthogonal noise subspace. Estimation is performed from vectors in the signal subspace only, since the orthogonal subspace does not contain signal information. The proposed estimator is shown to be a refinement of a version of the spectral subtraction signal estimator. The latter estimator is shown to be asymptotically optimal for stationary signal and noise in the linear minimum mean square error sense.<<ETX>>

A recursive least squares implementation for LCMP beamforming under quadratic constraint

Quadratic constraints on the weight vector of an adaptive linearly constrained minimum power (LCMP) beamformer can improve robustness to pointing errors and to random perturbations in sensor parameters. We propose a technique for implementing a quadratic inequality constraint with recursive least squares (RLS) updating. A variable diagonal loading term is added at each step, where the amount of loading has a closed-form solution. Simulations under different scenarios demonstrate that this algorithm has better interference suppression than both the RLS beamformer with no quadratic constraint and the RLS beamformer using the scaled projection technique, as well as faster convergence than LMS beamformers.

Extended Ziv-Zakai lower bound for vector parameter estimation

The Bayesian Ziv-Zakai bound on the mean square error (MSE) in estimating a uniformly distributed continuous random variable is extended for arbitrarily distributed continuous random vectors and for distortion functions other than MSE. The extended bound is evaluated for some representative problems in time-delay and bearing estimation. The resulting bounds have simple closed-form expressions, and closely predict the simulated performance of the maximum-likelihood estimator in all regions of operation.

Functional techniques for the analysis of the nonlinear behavior of phase&#8211;locked loops

In this paper we consider the analysis of a nonlinear feedback system. The purpose of the paper is twofold. The first objective is to demonstrate the efficiency of the Volterra functional expansion technique as a method of analyzing nonlinear feedback systems. The techniques we demonstrate are valid for a large class of nonlinear systems. Several important advantages of the functional approach are as follows: 1) Random and deterministic inputs and disturbances are included. 2) All input-output relationships are explicit. One does not have to solve complicated differential equations. 3) Once one becomes facile with the properties of the expansion, the analysis of any particular nonlinear system is rapid and straightforward. The second objective is to obtain some new and useful results for a device of practical importance. The particular nonlinear system that we will use as an example represents a phase-locked loop whose input signal is a phase-modulated sinewave which has been corrupted by additive noise. Two interesting cases of phase modulation are considered. In the first case the phase θ 1 (t) is a deterministic function. In the second case the phase θ 1 (t) is a sample function from a random process. The results are presented as closed form analytic expressions. Several interesting cases are plotted as a function of the significant parameters.

Robust constrained linear receivers for CDMA wireless systems

For code-division multiple access (CDMA) communication systems, many constrained linear receivers have been developed to suppress multiple access interference. The linearly constrained formulations are generally sensitive to multipath fading and other types of signal mismatch. We develop robust linear receivers by exploring appropriate constraints. Multiple linear constraints are exploited to preserve the output energy that is scattered in multipath channels. In addition, a quadratic inequality constraint on the weight vector norm is used to improve robustness with respect to imprecise signal modeling. These constraints can be applied to the minimum output energy (MOE) detector to mitigate the signal mismatch problem and to the decision directed minimum mean square error (MMSE) detector to prevent error propagation and eliminate the need for training sequences at startup. Adaptive implementations for the proposed detectors are presented using recursive least square (RLS) updating in both the direct form and the partitioned linear interface canceller (PLIC) structure. Simulations are performed in a multipath propagation environment to illustrate the performance of the proposed detectors.

A recursive least squares implementation for adaptive beamforming under quadratic constraint

Quadratic constraints on the weight vector of an adaptive linearly constrained minimum power (LCMP) beamformer can improve robustness to pointing errors and to random perturbations in sensor parameters. In this paper, we propose a technique for implementing a quadratic inequality constraint with recursive least squares (RLS) updating. A variable diagonal loading term is added at each step, where the amount of loading is found from the solution to a quadratic equation. Simulations under different scenarios demonstrate that this algorithm outperforms both the RLS beamformer with no quadratic constraint, and the RLS beamformer using the scaled projection technique.

A unified method for measurement and tracking of contacts from an array of sensors

A multiple target track estimation method that operates directly from array data is presented. The maximum a-posteriori (MAP) estimator for contact states is derived for temporally uncorrelated signals and uncorrelated contact tracks, where the number of contacts is assumed known. This estimator is an iterative algorithm employing a nonlinear programming (NLP) penalty method in conjunction with an expectation-maximization (EM) algorithm. The NLP technique is used to find the MAP track estimate based on the synthetic signal estimates produced by the EM algorithm. This method eliminates the data association step of traditional multitarget tracking approaches by conditioning the measurement process on individual target state distributions. It results in a process similar to the EM algorithm for direction finding, with an additional penalty term imposed by the track distributions. The algorithm is derived as a batch method. An extension to support sequential tracking is also developed. Simulation results for a relevant submarine towed array scenario are presented and discussed.

Adaptive and non-adaptive beampattern control using quadratic beampattern constraints

A general framework for adaptive and non-adaptive beampattern synthesis based on minimum mean-square error (MMSE) beamforming with quadratic beampattern constraints is presented. Main beam and sidelobe pattern control is achieved by imposing a set of inequality constraints on the weighted mean-square error between the adaptive pattern and a desired beampattern over a given angular region. By proper choice of angular regions and desired beampatterns in each region, we can trade off the level of pattern control with algorithmic complexity. The technique is used to synthesize a nearly uniform sidelobe level quiescent pattern for a non-linear array, and to control sidelobe levels for the same array in an adaptive manner.

Analog communication over randomly-time-varying channels

The problem of analog communication over a randomly-time-varying channel is considered. An analog source generates a message which is assumed to be a sample function from a Gaussian random process. The message is passed through a linear realizable system before modulation. (This corresponds to the pre-emphasis network in FM.) The output of this system is the modulating signal for a no-memory modulator which, in general, will be nonlinear. The modulated signal is transmitted over a time-varying channel We restrict ourselves to Gaussian multiplicative channels. At the channel output, noise is added. The specific problem of interest is to find the optimum estimate of the message. The principle results are: \begin{enumeratge} \item An integral equation whose solution is the optimum estimate. \item A feedback demodulator whose output is the optimum estimate over a certain range of signal-to-noise ratios. \item A proof that the optimum demodulator corresponds to a joint channel and message estimator. This result is the continuous analog of the estimator-correlator result in digital systems. Some related problems and possible extensions are discussed briefly. \end{enumerate}