In blind equalization (BE) a cost function based on the fit between the equalizer outputs and the signaling constellation is generally defined. To minimize such a cost function, standard gradient descent learning is commonly used. We exploit the idea of relative gradient (RG) learning to modify such standard Bussgang-type algorithms. Instead of one output each time, our method uses a sliding block of outputs. Our RG-based block Bussgang algorithms have faster convergence than corresponding Bussgang algorithms based on the standard gradient.

Two modified blind equalization algorithms are analyzed for performance. These algorithms add a constellation-matched error term to the cost functions of the generalized Sato and multimodulus algorithms. The dynamic convergence behavior and steady-state performance of these algorithms, and of a related version of the constant modulus algorithm, are characterized. The analysis establishes the improved performance of the proposed algorithms.

Performance of diversity systems is often evaluated under the assumption of perfect interleaving and characterised in terms of long-term parameters such as the average bit-error rate, which does not capture the dynamics of fading channels. Statistical characteristics (static and dynamic) of the envelope of two correlated Rayleigh fading channels are explored using a physical model. For two popular diversity-combining schemes, maximal ratio combining and selection combining, both static and dynamic (level-crossing rate) properties of correlated fading channels are derived. These results are very useful for performance evaluation of diversity systems without bit-level simulations. The results can also provide very useful characteristics such as average duration of fades, fading rate and outage probability for two-channel diversity systems and can be extended to multiple fading channels.

Blind source separation (BSS) techniques allow recovery of individual source signals from observed mixtures, exploiting only the assumption of mutual independence of sources. Generally, complex signals are recovered with an arbitrary phase rotation. In this letter, we propose two BSS algorithms to separate complex sources that have independent in-phase and quadrature (I/Q) parts. The proposed algorithms allow source phase recovery as well as separation. Simulation results demonstrate that the algorithms are effective in recovering source phases without affecting source separation.

A coarray-based synthetic aperture beamformer using stepped-frequency signal synthesis and post-data acquisition processing is presented for wideband imaging of near-field scenes. The proposed beamformer formulation and implementation finds key applications in through-the-wall microwave imaging and landmine detection problems. While coarray techniques offer significant reduction in array elements for a given angular resolution, stepped-frequency realization of wideband systems simplifies implementation and offers flexibility in beamforming. Proof of concept is provided using real data collected in an anechoic chamber.

We derive the maximum a posteriori probability (MAP) receiver for orthogonal frequency-division multiplexed signals in a fading channel. As the complexity of the MAP receiver is high, we obtain a low-complexity, suboptimal receiver and evaluate its performance.

We propose and analyze a hybrid automatic repeat request (ARQ) with a selective combining scheme using rate-compatible punctured convolutional (RCPC) codes for fading channels. A finite-state Markov channel model is used to represent the Rayleigh fading channels. We show that the hybrid ARQ with selective combining yields better performance than the generalized type-II ARQ scheme for fading channels. Furthermore, simulation results of real-time video time division multiple access (TDMA) transmission system are given. Better video quality can be obtained by our proposed scheme, with a bounded delay. Analytical results of throughput and packet error rate (PER) are compared to the simulated results. Our analysis based on a finite-state Markov channel model, is shown to give good agreement with simulations.

We form a finite-state Markov channel model to represent Rayleigh fading channels. We develop and analyze a methodology to partition the received signal-to-noise ratio (SNR) into a finite number of states according to the time duration of each state. Each state corresponds to a different channel quality indicated by the bit-error rate (BER). The number of states and SNR partitions are determined by the fading speed of the channel. Computer simulations are performed to verify the accuracy of the model.

Pilot symbol assisted modulation is a promising scheme to mitigate the effect of fading in a wireless channel. Analytical results for the performance of this scheme are available. Although the use of diversity is known to improve the performance of receivers used in fading channels, pilot symbol assisted diversity reception has not been studied. In this paper, we derive an exact probability of error expression for such a receiver as a function of the channel estimation error variance and the number of diversity channels. An upper bound for the probability of error which illustrates the advantage of using diversity, is also obtained. A numerical example is provided.

Interest in OFDM has renewed with the standardization of OFDM for digital audio broadcasting in Europe. In this paper, we consider an adaptive, diversity receiver for OFDM signals in a Rayleigh fading channel. The diversity receiver has L branches with each branch receiving the signal from L independently fading diversity channels. We model the fading process as a vector auto-regressive process and use the Kalman filter to obtain the MMSE optimum channel estimates for each branch. The channel estimates and the signals are combined using the maximal ratio combining rule to obtain the decision variables. We analyze the performance of this receiver and provide a numerical example to highlight the advantage of using diversity.

We investigate hybrid ARQ schemes for low-bit-rate video transmission over wireless channels. We propose and analyze a hybrid ARQ with a selective combining scheme using rate-compatible punctured convolutional (RCPC) codes on Rayleigh fading channels. It is shown that our scheme performs better than the generalized type-II ARQ for bursty error channels. Simulations of a real-time video TDMA transmission system are also performed. A better video quality can be obtained by our proposed scheme with a bounded delay. Analytical results of the throughput and packet error rate are compared to the simulation results. Our analysis based on a finite-state Markov channel model is shown to give very good agreement with simulations.

Following the foundational work that established basic ideas for optimum distributed defection schemes using multiple sensors (as reviewed in Part I of this two-part review), further work on distributed detection has developed many useful and interesting extensions of the basic concepts. These more recent developments parallel those that arose from the early work on centralized, classical signal detection, resulting in new ideas of asymptotically optimum nonparametric, robust, and sequential centralized detection. Recent developments on these topics in the setting of distributed signal detection are reviewed in the present paper. Results in these directions are important in practice because they allow cases of modeling uncertainty to be addressed, and they provide more efficient detection schemes by optimizing more general performance criteria.

We derive an analytic expression for the point spread function, using the concept of coarrays, for wideband pulse-echo imaging of targets. We analyze the performance of the imaging system under finite and infinite range conditions. The results provide a better understanding of the interplay between bandwidth, element locations, weighting of the elements, pulse shape within the bandwidth, and imaging performance. This leads to new insights on array and pulse design. We support our results through computer simulations.

We investigate a nonlinear technique for restoration of images distorted in color and space, using a class of extended radial basis function networks called the Gaussian mixture basis function networks (GMBFN). We report computer simulation results that indicate usefulness of this new approach.

In this paper, covariance estimation and consequently reduced ordering in multivariate order statistic (OS) filters are studied. The robustness of covariance estimators is characterized by plotting the sensitivity surfaces that describe the change caused by outliers in the condition number of the covariance matrix. The efficiency of the estimators under nominal noise distribution is studied. The estimation of correlations and component variances are addressed separately through eigendecomposition of the covariance matrix. The results indicate that correlations and ratios of component variances are estimated rather accurately using robust estimators whereas a constant correction factor is often necessary to get consistent estimates, The minimum volume ellipsoid (MVE) and minimum covariance determinant (MCD) algorithms based on random sampling do not perform reliably for small sampled or when too few elemental subsets are drawn. The qualitative comparison is performed in a RGB color image filtering task. The filter employing the iterative minimum covariance determinant (IMCD) estimate preserves the edges the best whereas the M-estimator smooths out noise effectively on homogeneous regions. The robustness of the IMCD filter and the efficiency of an M-estimator can be combined using a final refinement step as in the case of S-IMCD filters.

We investigate a technique for image restoration using nonlinear networks based on radial basis functions. The technique is also based on the concept of training or learning by examples. When trained properly, these networks are used as spatially invariant feedforward nonlinear filters that can perform restoration of images degraded by nonlinear degradation mechanisms. We examine a number of network structures including the Gaussian radial basis function network (RBFN) and some extensions of it, as well as a number of training algorithms including the stochastic gradient (SG) algorithm that we have proposed earlier. We also propose a modified structure based on the Gaussian-mixture model and a learning algorithm for the modified network. Experimental results indicate that the radial basis function network and its extensions can be very useful in restoring images degraded by nonlinear distortion and noise.

Active imaging arrays are used to image scenes composed of reflectors of transmitted radiation, and in many such applications, line arrays are employed. In this paper, we discuss scanned active line arrays for imaging based on image synthesis. We define the novel concept of array redundancy for active arrays, analogous to the well-known concept of redundancy applied to passive arrays, and we define and give examples of minimum redundancy and reduced redundancy line arrays composed of transmit/receive elements. Such arrays differ from their passive imaging counterparts both in geometry and in element count.

We consider imaging arrays and apertures from the point of view of their performance in the imaging of spatially incoherent source distributions. The concept of coarray is used to provide an interesting insight into the aperture synthesis process using frequency diversity imaging. Simulation results are also provided. The proposed methods are applicable to imaging of coherent distribution of reflectors as well.

This paper addresses a nonlinear restoration technique based on generalized multiresolution decomposition. The input signal corrupted by impulsive and nonimpulsive noise is decomposed into multiresolution bands using order statistic (OS) filters. Then nonlinear filtering is performed on these bands using denoising techniques. Simulations show that this scheme has better edge and sharp feature preservation, and better noise reduction properties compared to other OS filter based techniques.

The modified trimmed mean (MTM) filters are known to possess desirable robustness and detail preservation properties. They combine the averaging operation with the median operation which implies that the filters also attenuate noise efficiently. We investigate multivariate extensions of the MTM filters. In the multivariate case, there is no unique way to define the MTM filters and hence there are several design options. A few of the most promising definitions for these filters and various design options are studied. The robustness of multivariate MTM filters is analyzed using the influence function approach. Filtering examples are also given using RGB color images.

It is generally recognized that digital channel equalization can be interpreted as a problem of nonlinear classification. Networks capable of approximating nonlinear mappings can be quite useful in such applications. The radial basis function network (RBFN) is one such network. We consider an extension of the RBFN for complex-valued signals (the complex RBFN or CRBFN). We also propose a stochastic-gradient (SG) training algorithm that adapts all free parameters of the network. We then consider the problem of equalization of complex nonlinear channels using the CRBFN as part of an equalizer. Results of simulations we have carried out show that the CRBFN with the SG algorithm can be quite effective in channel equalization.>

In this paper we investigate radial basis function networks (RBFNs) for application in adaptive interference cancellation. We study the problem from the perspective of optimal signal estimation. Optimum interference cancellation usually requires nonlinear processing of signals. RBFNs, owing to their nonlinear function approximation capability, can be expected to be able to implement or approximate the operation of optimum interference cancellation with appropriate network configuration and training. We examine a number of different RBFN structures as well as training algorithms. In particular, we show that in some special cases the optimum interference canceler can be exactly implemented by a class of normalized RBFNs that we have recently proposed. Through a number of simulation examples, we demonstrate that the various neural networks based on radial basis functions can be very useful for interference cancellation problems in which traditional linear cancelers may fail badly. We also discuss issues related to network performance and learning algorithms, and some practical considerations. Wir untersuchen Netzwerke mit radialen Basisfunktionen (RBFNs) in ihrer Anwendung zur adaptiven Störkompensation. Wir gehen an das Problem vom Blickwinkel optimaler Signalschätzung aus heran. Optimale Interferenzlöschung erfordert normalerweise eine nichtlineare Signalverarbeitung. Von RBFNs kann man erwarten, daβ sie aufgrund ihrer Fähigkeit zur Nachbildung nichtlinearer Funktionen in der Lage sind, die Operation optimaler Störbeseitigung mit einer passenden Netzwerkkonfiguration und geeignetem Training zumindest näherungsweise zu realisieren. Wir prüfen eine Reihe verschiedener RBFN-Strukturen sowie Trainingsalgorithmen. Insbesondere zeigen wir, daβ der optimale Inter-ferenzkompensator in einigen Sonderfällen durch eine Klasse normalisierter RBFNs exakt realisiert werden kann, die wir kürzlich vorgeschlagen haben. Anhand einer Reihe von Simulationsbeispielen demonstrieren wir, daβ die verschiedenen Netzwerke auf der Grundlage radialer Basisfunktionen sehr nützlich für Störlöschungsprobleme sein können, in denen herkömmliche lineare Entstörer manchmal völlig versagen. Wir sprechen auch Fragen von Netzwerkleistungsfähigkeit und Lernverfahren an sowie einige praktische Überlegungen. Nous étudions dans cet article les réseaux de fonctions radiales (RBFN) dans le cadre d'application à l'annulation adaptative d'interférences. Nous envisageons ce problème dans la perspective de l'estimation optimale des signaux. L'annulation d'interférence optimale requiert habituellement un traitement non linéaire des signaux. Les RBFN, du fait de leur capacité d'approximation de fonction non linéaires, devraient permettre d'implanter ou d'approximer l'opération de suppression d'interférence optimale à l'aide d'une configuration appropriée du réseau et d'un apprentissage. Nous examinons un certain nombre de structures de RBFN différentes ainsi que des algorithmes d'apprentissage. En particulier, nous montrons que dans certains cas spéciaux Pannulateur d'interférence optimal peut être exactement implanté par une classe de RBFN normalisés que nous avons récemment introduite. A l'aide d'un certain nombre d'exemples de simulation, nous montrons que les différents réseaux neuronaux basés sur les fonctions radiales peuvent être très utiles pour les problèmes d'annulation d'interférences dans lesquels les annulateurs linéaires traditionnels peuvent échouer. Nous discutons également les questions touchant aux performances des réseaux et des algorithmes d'apprentissage, et certaines considérations pratiques.

A wide variety of techniques have been developed to deal with coherent signals at a sensor array. Many of these techniques have the structure of a pre-processor followed by a standard adaptive beamforming or angle of arrival estimation algorithm. The pre-processor is designed to reduce the cross-correlations between the arriving signals. A general decorrelation technique of this type called autocorrelation matrix smoothing (AMS) is described in this paper. The processing in AMS consists of a two-dimensional linear filtering operation on the correlation matrix of the array measurements. A number of the previously proposed decorrelation techniques can be interpreted as special cases of AMS, corresponding to different choices for the mask used to filter the correlation matrix. Although the previous methods were not originally formulated in terms of AMS, the unified interpretation points out relations between the techniques and suggests improved techniques. This paper summarizes the basic properties of AMS, explains the unified interpretation of previous decorrelation methods, and describes some extensions for improved decorrelation. Eine groβe Vielfalt von Verfahren wurden zur Behandlung kohärenter Signale in einer Sensorgruppe entwickelt. Viele dieser Verfahren besitzen die Struktur einer Vorverarbeitungsstufe, auf die ein Standardalgorithmus zu adaptiven Keulenformung oder zur Schätzung des Einfallswinkels folgt. Die Vorverarbeitungsstufe wird ausgelegt, um die Kreuzkorrelationen zwischen den einfallenden Signalen zu verringern. In diesem Aufsatz wird ein deratiges allgemeines Dekorrelationsverfahren beschrieben, das als Glättung der Autokorrelationsmatrix (autocorrelation matrix smoothing, AMS) bezeichnet wird. Die Verarbeitung bei AMS besteht in einer zweidimensionalen linearen Filterung, angewendet auf die Korrelationsmatrix der Gruppe von Meβsignalen. Eine Anzahl früher vorgeschlagener Dekorrelationsverfahren können als Spezialfälle von AMS interpretiert werden, die einer unterschiedlichen Auswahl der Maske zur Filterung der Korrelationsmatrix entsprechen. Obwohl die früheren Methoden ursprünglich nicht im Rahmen von AMS formuliert wurden, zeigt die vereinheitlichte Interpretation Beziehungen zwischen den Verfahren auf und liefert Hinweise auf verbesserte Verfahren. Dieser Aufsatz faβt die grundlegenden Eigenschaften von AMS zusammen, erklärt die vereinheitlichte Interpretation früherer Dekorrelationsmethoden und beschreibt einige Erweiterungen für eine verbesserte Dekorrelation. Une grande variété de techniques a été développée pour le traitement de signaux cohérents par un réseau de capteurs. Beaucoup de ces techniques ont la structure d'un pré-processeur suivi d'un algorithme standard de formatage de voie ou d'estimation d'angle d'arrivée. Le pré-processeur est conçu pour réduire les intercorrélations entre les signaux arrivants. Une technique générale de décorrélation de ce type, appelée adoucissement de la matrice d'autocorrélation (AMS) est décrite dans cet article. Le traitement AMS consiste en une opération de filtrage linéaire bi-dimensionnel sur la matrice de corrélation des mesures sur le réseau. Un certain nombre de techniques de décorrélation déjà proposées peuvent être interprétées comme des cas spéciaux d'AMS, correspondant à différents choix du masque utilisé pour filtrer la matrice de corrélation. Bien que ces méthodes n'aient pas été formulées originellement en termes d'AMS, l'interprétation unifiée met en lumière les relations entre les techniques et suggère des techniques améliorées. Cet article résume les propriétés de base de l'AMS, explique l'interprétation unifiée des méthodes de corrélation déjà existantes, et décrit certaines extensions à des fins d'amélioration de la décorrélation.

Constant false alarm rate detection is considered in a decentralized, two-sensor context. Cases with observations which are dependent from sensor to sensor are investigated, for which results have been lacking. The in-phase and quadrature components of the received narrowband observation at each sensor consist of a common weak random signal in additive Gaussian combined clutter and noise. The sensors use cell-averaging CFAR tests to each generate binary decisions which are sent to a fusion center. Optimal sensor thresholds are found and the performance of the best schemes using AND and OR fusion rules are compared. The ability of these schemes to maintain constant false alarm probability in the presence of clutter edges is also studied.>

The asymptotic relative efficiency (ARE) of two centralized detection schemes has proved useful in large-sample-size and weak-signal performance analysis. In the present paper ARE is applied to some distributed detection cases which use counting fusion rules. In such cases one finds that ARE generally depends on the power of the tests which can make its application difficult. This dependence turns out to be relatively weak in the cases considered and the ARE is reasonably well approximated by the limit of the ARE as the detection probability approaches the false alarm probability. This approximation should be useful for distributed cases. Some specific results provide the best counting (k-out-of-N) fusion rules for cases with identical sensor detectors if one uses asymptotically large observation sample sizes at each sensor. These results indicate that for false alarm probabilities of less than 0.5, OR rules are generally never optimum.>

We investigate a nonlinear technique for image restoration using radial basis function networks. Computer simulation results indicate this new technique can be very useful.>

This paper addresses the problems of image enhancement and restoration in the case of multivariate image data. Multivariate generalizations of median filtering are studied. The robustness properties of two such techniques are investigated using the influence function approach. Robust polynomial filters are introduced for applications where the original image has to be restored with high fidelity. Filtering examples are given using multivariate noise processes with equal component variances, unequal component variances, correlated noise components, and in the presence of outliers. Both simulated data and multivariate data from a range imaging sensor are used.>

Adaptive channel equalization accomplished without resorting to a training sequence is known as blind equalization. The Godard algorithm and the generalized Sato algorithm are two widely referenced algorithms for blind equalization of a QAM system. These algorithms exhibit very slow convergence rates when compared to algorithms employed in conventional data-aided equalization schemes. In order to speed up the convergence process, these algorithms may be switched over to a decision-directed equalization scheme once the error level is reasonably low. The authors present a scheme which is capable of operating in two modes: blind equalization mode and a mode similar to the decision-directed equalization mode. In this proposed scheme, the dominant mode of operation changes from the blind equalization mode at higher error levels to the mode similar to the decision-directed equalization mode at lower error levels. Manual switch-over to the decision-directed mode from the blind equalization mode, or vice-versa, is not necessary since transitions between the two modes take place smoothly and automatically.>

In applications such as smoothing and enhancement of images, adaptive filtering techniques offer the flexibility needed for good performance with non-stationary observations. Many adaptive schemes can be based on the idea of determining the local statistics of the signal through appropriate tests on the data, to aid in the selection of a filtering procedure that is suited to the data. In the paper, the authors consider decision-directed or data-dependent adaptive filtering schemes that are based on order statistics. A general formulation for such a class of adaptive order statistics filters is presented. Approximate statistical performance analysis, especially in the presence of edges, may be carried out for this entire class of filters. The authors give examples of some existing filters that fit into this framework. The formulation also accommodates filters that employ multiple windows in their operation. To illustrate the potential of this class of multiple window (MW) filters, they construct and analyze simple filters, like the triple window median (TW-MED) and the triple window median of means (TW-MOM) filters, that are shown to yield useful performance. The class of mean-median hybrid (MMH) filters is also presented as a simple example which may be extended to give interesting performance.

The cell averaging constant false alarm rate detector has been assumed to be optimal for detecting Swerling I targets embedded in exponential clutter and noise of unknown power. This is because the detector uses a minimum variance unbiased estimate (which is also a maximum likelihood estimate) of the unknown clutter-plus-noise power to set the threshold. The authors prove, using a result concerning least favorable distributions in composite hypotheses testing, that the cell averaging detector is indeed optimal in that it is a uniformly most powerful detector.>

The effectiveness of a square boundary array in finite-range, pulse-echo imaging is investigated. The images produced by such an array are quite poor when no additional signal processing is used. It is demonstrated through simulations that a synthetic-aperture signal processing technique called image addition can be used to reduce the sidelobes associated with the square boundary array, thereby improving the image quality. Image addition was originally proposed for narrowband imaging of far-field scenes, but it is also useful for finite-range, pulse-echo imaging. The conclusions are expected to apply to other, nonsquare boundary array geometries.

High-resolution acoustical imaging requires the use of short-duration (wideband) pulses for good resolution in range, and two-dimensional (2-D) transducer arrays for good resolution and scanning in both lateral directions. However, 2-D arrays often contain an impractically large number of elements, so it is important to find sparse 2-D arrays which can perform as well as densely populated 2-D arrays. Boundary arrays are a particular class of sparse arrays which have elements located only on the boundary of some planar region. Signal processing techniques have been developed that effectively fill in the missing elements in the interior of the boundary array. These techniques were originally proposed in the context of narrowband imaging of far-field scenes. The techniques are applied to near-field imaging with wideband pulses. Simulations indicate that the methods are effective in this context also, and thus 2-D boundary arrays can produce high-quality images in acoustical imaging applications.>

A general approach to super resolution imaging of point sources using active arrays of transmit/receive elements is presented. The usual techniques of high resolution imaging using single transmitters and passive receive arrays fail in the presence of sets of coherent point sources, which often arise due to coherent multipath. However, data obtained from transmit/receive arrays may be arranged into matrices to which eigenspace direction of arrival estimation may be successfully applied, even int he presence of coherent sources. Each such matrix may be thought of as corresponding to a different transmit/receive array; this may be either the actual transmit/receive array or a virtual transmit/receive array whose effect is synthesized. This approach provides great flexibility, since a large number of different synthetic or virtual arrays may be available for a given transmit/receive array, and each can provide a different tradeoff between the total number of resolvable targets and the largest number of mutually coherent targets which can be resolved.

An elliptical boundary aperture is a collection of points lying on an ellipse from which energy is transmitted and/or received. An important special case is the circular boundary aperture. When these apertures are used with beamforming to produce a narrowband image of a far-field source, the corresponding point spread function (PSF) is characterized by high sidelobes. The concept of the coarray of an imaging system is used here to develop techniques which synthesize the effect of a more desirable PSF with an elliptical boundary aperture. Techniques are given for use in active imaging of spatially coherent sources, as well as passive imaging of spatially incoherent sources. Discrete arrays and continuous apertures are considered separately. The approach shows that the PSF synthesis problem can be solved in many more ways than previously recognized, and this fact is exploited to develop procedures which have a least-squares optimality property.

Locally optimum (LO) distributed detection is considered for observations that are dependent from sensor to sensor. The necessary conditions are presented for LO distributed sensor detector designs. and a locally optimum fusion rule for an N-sensor parallel distributed detection system with dependent sensor observations is given. Specific solutions are obtained for a random signal additive noise detection problem with two sensors. These solutions indicate that the LO sensor detector nonlinearities, in general, contain a term proportional to f'/f, where f is the noise probability density function (pdf). For some non-Gaussian pdf's, the new term is significant and causes the LO sensor detector nonlinearities to be nonsymmetric even for symmetric pdfs. LO solutions are presented for finite sample sizes, and the solutions for the asymptotic case are discussed. These results are extended to yield the form of the solutions for the N-sensor LO random signal distributed detection problem that generalize the two-sensor results.>

Nonparametric detection of a zero-mean random signal in additive noise is considered. The locally optimum detector based on signs and ranks of observations is derived, for good weak-signal detection performance under any specified noise probability density function. This detector is shown to have interesting similarities to the locally optimum detector for random signals. It may also be viewed as a generalization of the locally optimum rank detector for known signals. Examples of the test statistic of the detector are given for some specific noise probability density functions. Asymptotic and finite sample-size performance of the locally optimum rank detector is also considered.>

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.>

The limitations of the asymptotic relative efficiency (ARE) in predicting finite sample size detector performance have been noted in several previous studies. It has been observed that the finite-sample-size relative efficiency (RE) may converge very slowly to its asymptotic limit. In addition, in some cases the RE approaches the ARE from below, while in other cases the RE either starts above or overshoots the ARE and approaches it from above. Results indicate that both of these effects can be predicted for a useful class of detectors for known signals in additive noise. The prediction formula is developed for the generalized correlation detector structure. The result is then used to analyze some specific detectors. Among these are the sign detector, the dead-zone detector, the soft-limiter and the noise blanker. Calculations and simulations giving the relative efficiency as a function of sample size have been used to verify predictions.>

The nonlinear equations whose solution determines the locally optimum detection quantizer design are derived for a general parametric detection problem where the breakpoints are constrained to be time invariant. These quantizers maximize the efficacy of a test based on quantized data. Some specific optimum detection quantizer problems for the case of time-invariant breakpoints have been solved in the past, but only for the special case when the locally optimum nonlinearity factors in a certain way. Examples of observation models that do not satisfy these conditions are given. It is demonstrated that the locally optimum quantizer design for the time-invariant breakpoint constraint is the same as that quantizer design that minimizes the time-average mean-square difference between the quantizer and the locally optimum time-varying nonlinearity. A specific result shows that the optimum quantizer is not symmetric for the quadratic detector for random signals in Gaussian noise.>

The resolution of a linear imaging system is determined by the size of the aperture on which it measures data. For a fixed number of sensor elements, large 2-D apertures are formed when the elements are placed on the boundary of some planar region. It is proved that it is possible, in principle, to use only the boundary of some convex region as the aperture, while synthesizing the effect of having placed sensors throughout the interior of that region. This allows sidelobe levels to be controlled. Such a possibility is clearly of interest to designers of large, distributed sensor arrays for use in both active, transmit/receive imaging and passive, receive-only imaging. The central idea that is exploited in the work is that of the coarray of an aperture. Signal processing techniques have been developed that effectively fill in the missing elements from the interior of the aperture for the cases of rectangular, hexagonal, and triangular boundary apertures.>

Systems of two-dimensional (2-D) imaging arrays and apertures are considered from the point of view of their performance in the imaging of spatially incoherent as well as coherent source distributions. Such systems find applications in radar, sonar, and ultrasound imaging, as well as in applications such as seismology and radio astronomy. For linear imaging techniques related to beamforming and based on the Fourier transform relationship between the source distribution and the aperture plane measurements, the point spread function of the system completely characterizes its performance. This function is determined by the geometry of the physical aperture or array as well as the weighting that can be applied to measurements. It is shown that the introduction of the concept of coarray, both for receive apertures in incoherent imaging and for transmit/receive systems in reflection-mode coherent imaging, provides a convenient and elegant framework within which many apparently isolated techniques for point-spread function or aperture synthesis can be understood. In addition to this unifying role, coarray concept gives new insight into the aperture synthesis process, which allows interesting new imaging techniques to be developed, especially in coherent imaging.

A generalized observation model for signal detection problems is proposed. The model allows consideration of several interesting special cases including additive noise, multiplicative noise, and signal-dependent noise, and includes both deterministic and random signals. Locally optimum (LO) detectors are interesting generalizations of those for the additive noise model. Under the proposed observation model the performance of the LO detectors is compared with that of other common detectors.>

Locally optimum detectors for weak random signals are derived for a generalized observation model incorporating signal-dependent and multiplicative noise. It is shown that the locally optimum random-signal detectors in the generalized observation model are interesting generalizations of those that would be obtained in the purely additive noisy signal model. Examples of explicit results for the locally optimum detector test statistics are given for some typical cases. Both asymptotic and finite sample-size performances of the locally optimum detectors are considered and compared with those of other standard detector structures.>

A high resolution pattern diversity method is presented for resolving multiple coherent plane waves impinging on a linear array. In this method, use is made of several different choices for the common pattern of array elements forming the array, allowing the rank of the resulting signal-only covariance matrix to be equal to the number of plane wave arrivals even if multiple coherent sources are presented. The underlying method does not require a uniform array, nor does it require an array in motion. The theoretical development of the method and simulation results are provided.>

For restoration of nonstationary signals in impulsive noise the combination or C-filter uses rank-order dependent coefficients in an FIR (finite-impulse response) filter structure. This allows frequency selectivity as well as edge-preservation and impulse rejection. The nonlinear C-filter design is facilitated by the use of an LMS (least-mean-squares) algorithm. This approach is explained and examples are given of C-filter performance for one- and two-dimensional signals.>

An algorithm is presented for order-statistic determination which operates in a number of steps fewer than or equal to the number of bits in the binary representation of the input. The algorithm uses a bit-level thresholding operation on input data in binary representation with both adaptive thresholds and adaptive binary weights, and it does not require any auxiliary data structure. A bit-level pipelined implementation for this scheme is also presented whose hardware complexity is bounded by N(log/sub 2/N) where N is the window size; this leads to a pipelined structure for L-filtering.>

The beam pattern of an aperture of sensor elements determines the quality of the image of far-field sources produced by it. Resolution is limited by the physical size of the aperture in linear beamforming, and sidelobe characteristics are determined by the aperture weighting function or aperture apodization. For a given number of elements the largest two-dimensional physical apertures are formed if elements are distributed only around their boundaries. Results developed recently are extended to show that in general, for any convex boundary aperture in a plane, it is possible to synthesize any beam pattern which is obtainable from a filled aperture with the same boundary. Specific results for elliptical and rectangular boundaries are briefly reviewed, and a general synthesis theorem which provides the basis of the two-dimensional pattern synthesis scheme is established. This is applicable for both incoherent and coherent imaging, the latter implemented using a transmit-receive scheme.>

Several generalizations of order statistic filtering are examined and proposed for nonlinear edge preservation and impulsive noise suppression. The Double Window Modified Trimmed Mean filter's performance is characterized. Recursive generalizations are considered for enhanced smoothing capabilities. A structure combining temporal and rank order statistic processing is proposed. While exposition and examples are given for one dimensional input sequences, the techniques and results are applicable to multidimensional filtering.

In recent years there has been much interest in robustness issues in general and in robust signal processing schemes in particular. Robust schemes are useful in situations where imprecise a priori knowledge of input characteristics makes the sensitivity of performance to deviations from assumed conditions an important factor in the design of good signal processing schemes. In this survey we discuss the minimax approach for the design of robust methods for signal processing. This has proven to be a very useful approach because it leads to constructive procedures for designing robust schemes. Our emphasis is on the contributions which have been made in robust signal processing, although key results of other robust statistical procedures are also considered. Most of the results we survey have been obtained in the past fifteen years, although some interesting earlier ideas for minimax signal processing are also mentioned. This survey is organized into five main parts, which deal separately with robust linear filters for signal estimation, robust linear filters for signal detection and related applications, nonlinear methods for robust signal detection, nonlinear methods for robust estimation, and robust data quantization. The interrelationships among many of these results are also discussed in the survey.

The design of a fixed filter is considered for equalization of an imprecisely known channel. The channel frequency response is assumed to have amplitude and phase characteristics lying within specified bounds at each frequency, and a minimax filter optimizing worst case mean-squared error (MSE) performance is derived. The general result is illustrated by considering a two-path channel model with an uncertain secondary path delay characteristic.

Robust matched filtering is considered for multiple-input systems, with problems formulated in both frequency and time domains being given specific treatment. Robust solutions are found for bounded classes containing well-defined but not exactly specified input characteristics. Several properties for some special cases of interest are examined, and an example is given to illustrate the usefulness of the results. Finally, the results are applied to a narrowband spatial array system, for which the robust scheme has an interesting and useful characteristic.

Robust Wiener filtering has previously been considered for the single-input (scalar) case where there is no channel distortion and where the signal to be estimated is the source signal itself. Here, these results are extended to the multiple-input (vector) case where linear channel distortion is allowed and the signal to be estimated is a linear-filtered version of the source signal. The results are obtained from those for the single-input ease by modifying the constraints on signal and noise characteristics. Such a modification is motivated by examining the expression of the mean-squared error for the optimum filter.

L-smoothers and M-smoothers are introduced as generalizations of the median filter for nonlinear smoothing of noisy data, and their properties are derived. In addition, a double-window smoothing algorithm which is shown to be a data-dependent modification of L- and M-smoothers is proposed for filtering noisy signals with sharp edges. Simulation results are given to demonstrate the performance characteristics of these smoothing algorithms.

Minimax robust Wiener filtering is considered for the case in which the signal and noise spectral-density matrix is not completely specified. Results are obtained for spectral-density matrix classes which are defined by upper and lower bounds on the components of the matrix. These results form an extension of earlier results on robust Wiener filtering for the case of uncorrelated signals and noise.

The detection of signals in noise with possibly asymmetric probability density functions is considered. The noise density model allows a symmetric contaminated-nominal central part and an arbitrary tail behavior. For detection of known signals, the robust nonlinear-correlator (NC) detector is obtained based on detector efficacy as performance criterion. The robustM-detector structure for constant-signal detection is also explicitly obtained.

Two related techniques have been proposed in the past for improving the performance of the sign detector through higherorder data quantization. The fixed-thresholdm-interval detector and the generalized sign detector using a conditional test are both nonparametric detectors which are fairly simple to implement. In this paper we compare the asymptotic and finite-sample, finite-signal performance characteristics of these two detectors, and point out their relative advantages and disadvantages.

Least favorable pairs of probability density functions and robust tests are obtained for classes of density functions specified as bands with upper and lower bounds. The results are shown to extend Huber's solution for the\epsilon-contaminated classes, which is contained as a special case. A minimum-distance property of the least favorable density pair is also made explicit.

A phase modulator with good phase linearity over an extended range of inputs is described. Performance results in terms of signal-to-distortion ratio are given.

Matched and Wiener filters are considered for two-dimensional signal processing applications when the a priori information about signal and noise characteristics are not completely specified. This work is an extension of previous work on robust filters for one-dimensional processing. The approach is to design filters which are saddle-point or mini-max solutions for the criterion functional (mean-squared-error or signal-to-noise ratio) over the classes of allowable signal shapes and signal and noise spectral densities.

Multilevel coincidence correlators (CC's) are considered for detecting a weak random signal common to two or more input channels against a background of independent additive noise. For two-input systems the structure of the optimum detector maximizing the efficacy is derived, and a comparison is made between the optimum CC and some suboptimal CC's of interest. An extension of the two-input CC is considered for general multiinput systems.

Performance criteria for the design of optimum quantizers are considered. A distance criterion for quantizer input and output probability distribution functions is formulated, and its relationship to the usual distortion criteria is established. The criterion based on absolute error is shown to have unique properties, justifying its use as a performance index for optimum quantization. Numerical results are presented, and the implication for adaptive quantization of the use of this criterion is discussed.

Finite-bit representation of the coefficients and quantization of input data are studied in digital matched filters for weak-signal detection. An algorithm for optimum coefficients and equations for the optimum input quantizer are obtained for the known signal in additive noise problem. Some numerical performance results are given.

Detection of random signals in arrays of receivers is considered when the exact distribution of the additive noise is not known. Locally robust correlator-type test statistics are obtained for weak-signal detection, for multivariate array noise densities in classes described by three different extensions of the univariate Tukey-Huber contaminated nominal density model. The robust correlator nonlinearities are shown to he of the form previously derived for known signal detection. In addition to robust correlator-type statistics, a quadratic-limiter-sum detector is investigated for robust performance in contaminated Ganssian noise.

Optimum quantization of data, primarily for signal detection applications, is considered. It is shown that two useful detection criteria lead to quantization which gives the minimum mean-squared error between the quantized output and the locally optimum nonlinear transform for each data sample. This criterion is an extension of the usual minimum distortion criterion for optimum quantizers. Numerical results show that it leads to optimum quantizers which can be considerably better in their performance for non-Guassian inputs than the minimum-distortion quantizers.

A one-input nonparametric detector employing the Wilcoxon signed-rank test and conditional testing is described. The rank test is applied to input samples which are larger in absolute value than a positive constant, so that the number of samples ranked is a random variable. A conditional test allows nonparametric operation; the detector is shown to maintain efficient performance with reduced ranking requirements, especially when used in conjunction with the technique of "mixed" statistical tests.

Generalizations of the simple sign detector are considered for nonparametric detection. Nonparametric operation is obtained with only a symmetry assumption on the noise density functions, by basing the detectors on conditional statistical tests. The performance (both asymptotic and finite-sample, finite-signal) of the generalized sign detector is shown to be very good compared to the Wilcoxon detector and to the linear detector, whereas the detector structure remains simple.

The Tukey-Huber contaminated noise model is used toobtain rain-max detectors in the asymptotic case for known signals inadditive noise. According to this model, the noise densityf(x)is defined byf(x) = (1 - \ )g(x) + \varepsilon h(x)for a given\varepsilonand densityg(x), withh(x)an arbitrary density from a large class. A general theorem is obtainedspecifying the most robust detector for additive contaminated noisewithg(x)satisfying certain regularity conditions. As an example,detector structures are derived by the application of the theorem for the case whereg(x)belongs to the class of generalized Gaussian densities(parameterized by their rates of exponential decay). The sign detector is shown to be the asymptotically most robust detector wheng(x)is a double-exponential density.

A class of nonparametric detectors is formulated for dependent input sequences of sampled data. Detectors in this class are nonadaptive modifications of standard nonparametric detectors designed for independent inputs and operate on multivariate samples derived by grouping the dependent univariate input samples. These groups are transformed to single variables and processed according to the corresponding standard nonparametric detection scheme. Performance analysis is initially based on assumptions implying independence of the multivariate samples, and results are then shown to remain valid under weaker conditions. Two modified one-channel detectors are analyzed to illustrate the improved performance achieved over corresponding standard non-parametric detectors.