Array processing involves manipulation of signals induced on various antenna elements. Its capabilities of steering nulls to reduce cochannel interferences and pointing independent beams toward various mobiles, as well as its ability to provide estimates of directions of radiating sources, make it attractive to a mobile communications system designer. Array processing is expected to play an important role in fulfilling the increased demands of various mobile communications services. Part I of this paper showed how an array could be utilized in different configurations to improve the performance of mobile communications systems, with references to various studies where feasibility of apt array system for mobile communications is considered. This paper provides a comprehensive and detailed treatment of different beam-forming schemes, adaptive algorithms to adjust the required weighting on antennas, direction-of-arrival estimation methods-including their performance comparison-and effects of errors on the performance of an array system, as well as schemes to alleviate them. This paper brings together almost all aspects of array signal processing.

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Application of antenna arrays to mobile communications. II. Beam-forming and direction-of-arrival considerations

From the Publisher:
Adaptive Filtering: Algorithms and Practical Implementation is a concise presentation of adaptive filtering, covering as many algorithms as possible while avoiding adapting notations and derivations related to the different algorithms. Furthermore, the book points out the algorithms which really work in a finite-precision implementation, and provides easy access to the working algorithms for the practicing engineer. Adaptive Filtering: Algorithms and Practical Implementation may be used as the principle text for courses on the subject, and serves as an excellent reference for professional engineers and researchers in the field.

Adaptive Filtering: Algorithms and Practical Implementation

A least-mean-square (LMS) adaptive filter with a variable step size is introduced. The step size increases or decreases as the mean-square error increases or decreases, allowing the adaptive filter to track changes in the system as well as produce a small steady state error. The convergence and steady-state behavior of the algorithm are analyzed. The results reduce to well-known results when specialized to the constant-step-size case. Simulation results are presented to support the analysis and to compare the performance of the algorithm with the usual LMS algorithm and another variable-step-size algorithm. They show that its performance compares favorably with these existing algorithms. >

A variable step size LMS algorithm

An overview is presented of several frequency-domain adaptive filters that efficiently process discrete-time signals using block and multirate filtering techniques. These algorithms implement a linear convolution that is equivalent to a block time-domain adaptive filter, or they generate a circular convolution that is an approximation. Both approaches exploit the computational advantages of the FFT. Subband adaptive filtering is also briefly described. Here the input data are first processed by a bank of narrowband bandpass filters that are approximately nonoverlapping. The transformed signals are then decimated by a factor that depends on the degree of aliasing that can be tolerated, resulting in a large computational savings. Several performance issues are considered, including convergence properties and computational complexities of the adaptive algorithms and the effects of circular convolution and aliasing on the converged filter coefficients. >

Frequency-domain and multirate adaptive filtering

Industrial applications of model based predictive control

The LMS adaptive filter algorithm requires a priori knowledge of the input power level to select the algorithm gain parameter μ 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 μ 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 μ 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 μ. 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.

Analysis of the normalized LMS algorithm with Gaussian inputs

A new application of the LMS adaptive filter, that of determining the time delay in a signal between two split-array outputs, is described. In a split array sonar, this time delay can be converted to the bearing of the target radiating the signal. The performance of such a tracker is analyzed for stationary broad-band targets. It is shown that a continuous adaptive tracker performs within 0.5 dB of the Cramer-Rao lower bound. Further, performance predictions are developed for a discrete adaptive tracker which demonstrates excellent agreement with simulations. It is shown that the adaptive tracker can have significantly less sensitivity to changing input spectra than a conventional tracker using a fixed input filter.

Time delay estimation using the LMS adaptive filter--Dynamic behavior

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.

The complex LMS adaptive algorithm--Transient weight mean and covariance with applications to the ALE

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. >

On error-saturation nonlinearities in LMS adaptation

The transient behavior of the LMS adaptive filter is studied when configured as an adaptive line enhancer operating in the presence of a fixed or variable complex frequency sine-wave signal buried in white noise. For a fixed frequency signal, the mean weights are shown to respond to signal more rapidly than to noise alone. For a chirped signal, a fixed parameter matrix first-order difference equation is derived for the mean weights and a closed-form steady-state solution obtained. The transient response is obtained as a function of the eigenvectors and eigenvalues of the input covariance matrix. Sufficient conditions for the stability of the transient response are derived and an upper bound on the eigenvalues obtained. Finally, the mean-square error is evaluated when responding to a chirped signal. A gain coefficient of the LMS algorithm is determined which minimizes the mean-square error for chirped signals as a function of chirp rate and signal and noise powers.

N. Bershad

Papers

Analysis of the normalized LMS algorithm with Gaussian inputs

Time delay estimation using the LMS adaptive filter--Dynamic behavior

The complex LMS adaptive algorithm--Transient weight mean and covariance with applications to the ALE

On error-saturation nonlinearities in LMS adaptation

Tracking characteristics of the LMS adaptive line enhancer-response to a linear chirp signal in noise