An array of sensor elements has long been an attractive solution for severe reception problems that commonly involve signal detection and estimation. The basic reason for this attractiveness is that an array offers a means of overcoming the directivity and sensitivity limitations of a single sensor, offering higher gain and narrower beamwidth than that experienced with a single element. In addition, an array has the ability to control its response based on changing conditions of the signal environment, such as direction of arrival, polarization, power level, and frequency. The advent of highly compact, inexpensive digital computers has made it possible to exploit well-known results from signal processing and control theory to provide optimization algorithms that automatically adjust the response of an adaptive array and has given rise to a new domain called 'smart arrays.' This self-adjusting capability renders the operation of such systems more flexible and reliable and (more importantly) offers improved reception performance that would be difficult to achieve in any other way. This revised edition acquaints the reader with the historical background of the field and presents important new developments that have occurred over the last quarter century that have improved the utility and applicability of this exciting field.

This book section discusses the following topics: signal environment; array element spacing; array perfomance; nulling limitations due to miscellaneous array effects; narrowband and broadband signal processing; and adaptive array performance measure-coverage improvement factor (CIF).

Optimum array processing is an optimum multichannel filtering problem. The objective of array processing is to enhance the reception (or detection) of a desired signal that may be either random or deterministic in a signal environment containing numerous interference signals. The desired signal may also contain one or several uncertain parameters (e.g., spatial location, signal energy, phase) that it may be advantageous to estimate. Optimum array processing techniques are broadly classified as processing appropriate for ideal propagation conditions and processing appropriate for perturbed propagation conditions. Ideal propagation implies an ideal nonrandom, nondispersive medium where the desired signal is a plane (or spherical) wave and the receiving sensors are distortionless. In this case the optimum processor is said to be matched to a plane wave signal. Any performance degradation resulting from deviation of the actual operating conditions from the assumed ideal conditions is minimized by the use of complementary methods, such as the introduction of constraints. When operating under the aforementioned ideal conditions, vector weighting of the input data succeeds in matching the desired signal.

Gradient algorithms are popular because they are simple, easy to understand, and solve a large class of problems. The performance and adaptive weights determine the nature of the performance surface. When performance is a quadratic function of the weight settings, then it is a bowl-shaped surface with a minimum at the 'bottom of the bowl.' In this case, local optimization methods, such as gradient methods, can find the bottom. In the event that the performance surface is irregular, having several relative optima or saddle points, then the transient response of the gradient-based minimum-seeking algorithms get stuck in a local minimum. The gradient-based algorithms considered in this chapter are as follows: least mean square (LMS); Howells-Applebaum loop; differential steepest descent (DSD); accelerated gradient (AG); and steepest descent for power minimization.

The usefulness of an adaptive array often depends on its convergence rate. For example, when adaptive radars simultaneously reject jamming and clutter while providing automatic platform motion compensation, then rapid convergence to steady-state solutions is essential. Convergence of adaptive sensor arrays using the popular maximum signal-to-noise ratio (SNR) or least mean squares (LMS) algorithms depend on the eigenvalues of the noise covariance matrix. When the covariance matrix eigenvalues differ by orders of magnitude, then convergence is exceedingly long and highly example dependent. One way to speed convergence and circumvent the convergence rate dependence on eigenvalue distribution is to directly compute the adaptive weights using the sample covariance matrix of the signal environment.

This chapter discusses the weighted least squares error processor, updated covariance matrix inverse, Kalman filter methods for adaptive array processing, minimum variance processor and simulation results. The specific form selected for a recursive processor should reflect the data weight scheme that is appropriate for the desired application. The various recursive algorithms may be developed by applying the matrix inversion lemma to the same basic weight update equation. Since the recursive algorithms are different from a DMI algorithm, primarily because the required matrix inversion is accomplished in a recursive manner, it is hardly surprising that many of the desirable properties found to apply to DMI algorithms also hold for recursive algorithms. Rapid convergence rates and insensitivity to eigenvalue spread are characteristics that make recursive processors attractive algorithm candidates provided sufficient computational power and accuracy are available to carry out the required calculations.

The least mean squares (LMS) and maximum signal-to-noise ratio (SNR) algorithms converge slowly whenever there is a wide spread in the eigenvalues of the input signal correlation matrix. A wide eigenvalue spread occurs if the signal environment includes a very strong source of interference together with other weaker but nevertheless potent interference sources. This condition also happens when two or more very strong interference sources arrive at the array from closely spaced but not identical directions.

Random searches are classified as either guided or unguided, depending on whether information is retained whenever the outcome of a trial step is learned. Furthermore, both the guided and unguided varieties of random search are given accelerated convergence by increasing the adopted step size in a successful search direction. Four representative examples of random search algorithms used for adaptive array applications are considered in this chapter: linear random search (LRS), accelerated random search (ARS), guided accelerated random search (GARS), and genetic algorithm (GA).

Chapters 4 through 9 considered the transient response characteristics and implementation considerations associated with different classes of algorithms that are widely used for adaptive array applications. This chapter summarizes the principal characteristics of each algorithm class before considering some practical problems associated with adaptive array system design. In each chapter of Part 2 the convergence speed of an algorithm representing a distinct adaptation philosophy was compared with the convergence speed of the least mean squares (LMS) algorithm. The convergence speeds of the various algorithms are compared for a selected example in this chapter. Since the misadjustment versus rate of adaptation tradeoffs for the random search algorithms-linear random search (LRS), accelerated random search (ARS), and guided accelerated random search (GARS) - and for the differential steepest descent (DSD) algorithm of Chapter 4 are unfavorable compared with the LMS algorithm, recourse to these methods would be taken only if the meager instrumentation required was regarded as a cardinal advantage or nonunimodal performance surfaces were of concern. Furthermore, the Howells-Applebaum maximum signal-to-noise ratio (SNR) algorithm has a misadjustment versus convergence speed trade-off that is nearly identical with the LMS algorithm.

Array errors due to manufacturing tolerances distort the array pattern. To minimize these errors, the array must be calibrated at the factory and at regular intervals once deployed. The transversal filter consisting of a sequence of weighted taps with intertap delay spacing offers a practical means for achieving the variable amplitude and phase weighting as a function of frequency that is required if an adaptive array system is to perform well against wideband interference signal sources. The distortionless channel transfer functions for a two-element array were derived. It was found that to ensure distortion-free response to a broadband signal the channel phase is a linear function of frequency, whereas the channel amplitude function is nearly flat over a 40% bandwidth. Quadrature hybrid processing provides adequate broadband signal response for signals having as much as 20% bandwidth. Tapped delay line processing is a practical necessity for 20% or more bandwidth signals. A transversal filter provides an attractive means of compensating the system auxiliary channels for the undesirable effects of the following: multipath interference; interchannel mismatch; and propagation delay across the array.

In this chapter, we apply maximum likelihood (ML) estimation methods to estimate the direction of arrival (DOA), or angle of arrival (AOA), of one or more signal sources, using data received by the elements of an N-element antenna array. The Cramer-Rao (CR) lower bound on angle estimation error is derived under several different signal assumptions. The CR bound helps determine system performance versus signal-to-noise ratio (SNR) and array size.

This chapter presents several innovations that have taken place since the first edition of this book. Wireless communication applications often resort to very simple beam switching, in which multiple beams simultaneously exist and the one with the best signal reception is selected. Moving radars or sonars must deal with clutter as well as interfering signals. Space-time adaptive processing (STAP) combines a spatial adaptive array with a temporal adaptive array to improve clutter cancellation and null placement. Another relatively recent development is multiple input, multiple output (MIMO) antenna array systems where an adaptive array is used for both transmit and receive to increase channel capacity. Reconfigurable antennas change their physical layout using switches to adapt for example the pattern, frequency response, and polarization response to match the desired signal. Partial adaptivity is of interest when only a portion of the total number of elements is controlled, thereby reducing the number of processors required to achieve an acceptable level of adaptive array performance.

Since the impulse response of the transversal filter consists of a sequence of weighted impulse functions, it is convenient to adopt the z-transform description for the filter transfer function instead of the Laplace transform description. It has been seen that the frequency response H(jω) is periodic with period determined by the signal bandwidth 1/Δ and that the number of zeros that can occur across the signal bandwidth is equal to the number of delay elements in the tapped-delay line. It remains to show that the resolution associated with each of the zeros of H(jω) is approximately 1/NΔ, where N = number of taps in the tapped-delay line.

Complex envelope notation provides a convenient means of characterizing the information-carrying component of modulated carrier signals so the carrier frequency component of the signal (which does not carry information) does not appear in the signal description. The properties of complex signal descriptions outlined in this appendix are discussed in some detail in references 1 and 2.

The scalar function trace [AB^T] has all the properties of an inner product, so the formulas for differentiation of the trace of various matrix products that appear in reference 1 are of interest for computing the gradients of certain inner products that appear in adaptive array optimization problems. For many problems, the extension of vector and matrix derivative operations to matrix functions of vectors and matrices is important; this generalization is described in reference 2.

This appendix summarizes some matrix properties that are especially useful for solving optimization problems arising in adaptive array processing. Derivations for these relations as well as a treatment of the basic properties of matrixes may be found in the references 1-3 for this appendix.

Useful properties that apply to real Gaussian random vectors are briefly discussed and extension to complex Gaussian random vectors are presented in this appendix.

For purposes of interpreting array gain expressions, which initially appear to be quite complicated, it is very convenient to introduce certain geometric aspects of various complex vector relationships and thereby obtain quite simple explanations of the results. The discussion in this appendix is based on the development given by Cox.

Appendix G discusses eigenvalues and eigenvectors.

This book section comprises answers and solutions to problems in adaptive arrays.