In wireless communications, including cellular communication systems, spread spectrum overlay systems, and signals intelligence applications, the degradation caused by rapidly time-varying multipath and unknown co-channel interference can be reduced by adaptive spatial filtering using antenna arrays. In this paper we propose a flexible framework for adapting a spatial filter to extract signals of interest (SOIs) while rejecting signals not of interest (SNOIs) without using a training signal, array calibration data, or knowledge of spatial characteristics of the desired or interfering signals. The generic method within this framework exploits user-selected properties of the SOIs and/or SNOIs to transform the received data set into a new set in which the SOIs are more highly correlated between the two sets than are the SNOIs. The framework then applies the method of canonical correlation analysis to the original and transformed data sets to obtain a linear combining weight vector that performs the spatial filtering. Simulation results illustrate the performance of algorithms developed within the new framework.>

An algorithm for the estimation of time difference of arrival (TDOA) of cyclostationary signals is presented. The algorithm is derived as a generalization of the SPECCOA algorithm. Simulation results indicate that the performance of the new algorithm can greatly exceed that of SPECCOA.>

The problem of detecting the presence of direct sequence spread-spectrum signals in noise is considered, and the detector consisting of a filter followed by a delay-and-multiply device is optimized with respect to the filter bandwidth and the delay to maximize robustness to errors in assumed values of pulse rate and carrier frequency. Output SNR is used for performance optimization, and the results are corroborated by evaluating receiver operating characteristics.>

Conventional time and space filtering of stationary random signals, which amounts to forming linear combinations of time translates and space translates, exploits the temporal and spatial coherence of the signals. By including frequency translates as well, the spectral coherence that is characteristic of cyclostationary signals can also be exploited. Some of the theoretical concepts underlying this generalized type of filtering, called frequency-shift (FRESH) filtering, are developed. The theory of optimum FRESH filtering, which is a generalization of Wiener filtering called cyclic Wiener filtering, is summarized, and the theory is illustrated with specific examples of separating temporally and spectrally overlapping communications signals, including AM, BPSK, and QPSK. The structures and performances of optimum FRESH filters are presented, and adaptive adjustment of the weights in these structures is discussed. Also, specific results on the number of digital QAM signals that can be separated, as a function of excess bandwidth, are obtained.>

The problem of dual-receiver interception of low-SNR signals, which includes detection of the presence of a particular signal type and location of its source, is considered. In particular, source-location based on time-difference-of-arrival (TDOA) measurements is examined, and the low-SNR maximum-likelihood receiver for joint detection and TDOA estimation is taken as a starting point. By explicitly revealing the way in which this receiver exploits the spectral correlation properties of the cyclostationary signal, several partial implementations with optimality properties of their own are proposed. These greatly simplified implementations, which require only a one-dimensional search over the TDOA parameter, are shown by simulation to perform competitively with the relatively complicated maximum-likelihood receiver which requires a two- or three-dimensional search for phase-shift keyed (PSK) signals.>

A class of random time-series inputs for nonlinear time-invariant systems that permit the analytical specification of a set of operators on the input that are orthonormal over all time to the Volterra operators for all orders and all lag sets is introduced. The time series in this class are cyclostationary and complex valued. The orthonormal operators are used to obtain an input-output type of cross-correlation formula for identifying the individual Volterra kernels of arbitrary order for a nonlinear system of possibly infinite order and possibly infinite memory. The real parts of the complex-valued inputs in this class comprise a class of real-valued inputs for which the same sets of specified operators apply. However, the orthogonality for different orders holds for these real inputs only for Volterra operators of order less than the order of the specified operator. Thus, these real inputs can be used to identify Volterra kernels only for finite-order systems. Frequency-domain counterparts of the time-domain methods that can utilize an FFT algorithm are developed.>

The problem of detecting the presence of spread-spectrum phase-shift-keyed signals in variable noise and interference backgrounds is considered, and the performances of four detectors are evaluated and compared. The detectors include the optimum radiometer, the optimum modified radiometer that jointly estimates the noise level and detects the signal, and the maximum-SNR spectral-line regenerator for spectral-line frequencies equal to the chip rate and the doubled carrier frequency. It is concluded that the spectral-line regenerators can outperform both types of radiometers by a wide margin. The performance advantages are quantified in terms of receiver operating characteristics for several noise and interference environments and receiver collection times.>

The Cramer-Rao lower bound on the variance of estimates of directions of arrival, signal strengths, and carrier frequencies and phases of cyclostationary signals arriving at an array is considered. Numerical evaluations of the bound for the estimates of directions of arrival show that it can be orders of magnitude lower than the corresponding bound for stationary signals, despite the great number of unknown parameters in the former case. Comparisons with the mean squared error of direction estimates obtained by the cyclic least squares algorithm show that the bound is not overly optimistic.>

A method of channel identification is proposed that exploits the spectral correlation properties of pulse- and carrier-modulated signals to identify channels in the presence of arbitrary noise and nearly arbitrary interference. Although a pilot or training signal is required, no replica of the transmitted pilot/training signal is needed at the receiver. The price paid for this simplicity and the tolerance to extreme channel corruption from noise or interference is that the method is slow. That is, relatively long averaging times are needed for measurement of the spectral correlation of the received signal.>

Two alternative philosophical frameworks for the two problems of estimating the time-invariant or time-variant autocorrelation function and its Fourier transform for stationary and cyclostationary time-series are compared. One is based on the stochastic process model, and the other is based on the nonstochastic time-series model. It is then explained that results on estimator bias and variance for these two problems couched within the stochastic process framework have analogs within the nonstochastic framework. The bias and variance results for cyclostationary time-series that are available within these two frameworks are then summarized.>

A recent paper by E. Biglieri et al. (see ibid., vol.35, p.620-5, May 1989) shows that the matched-filter/tapped-delay-line structure is optimum not only for linear pulse-modulated signals and linear channel distortion, but also for nonlinear finite-alphabet pulse-modulation and some nonlinear channel distortion. This has important practical applications. Therefore, its connection with other work reported in the literature is presented.>

An algorithm that combines the signal selectivity of cyclic MUSIC with the ability to function well in the presence of fully correlated multipath is presented. Analytical results and computer simulations are presented to support the claims, which include the ability to estimate directions of arrival of desired signals in the presence of multipath and arbitrary numbers of arbitrarily closely spaced interfering signals. Furthermore, the algorithm can estimate the number of spectrally correlated signals with any particular cycle frequency, and it implicitly obtains maximum signal-to-interference-plus-noise ratio (SINR) estimates of those signals.>

A new approach to blind adaptive signal extraction using narrowband antenna arrays is presented. The approach has the capability to extract communication signals from cochannel interference environments using only known spectral correlation properties of those signals, i.e. without using knowledge of the content or direction of arrival of the transmitted signal, or the array manifold or background noise covariance of the receiver, to train the antenna array. The class of spectral self-coherence restoral (SCORE) objective functions is introduced, and algorithms for adapting antenna arrays to optimize these objective functions are developed. Using the theory of spectral correlation, it is shown by analysis and simulation that these algorithms maximize the signal-to-interference-and-noise ratio at the output of the narrowband antenna array when a single communication signal with spectral self-coherence at a known value of frequency separation, along with an arbitrary number of interferers without spectral self-coherence at that frequency separation, are impinging on the array.

Signal-selective direction-finding algorithms that overcome many of the limitations of existing techniques are presented. The algorithms automatically classify signals as desired or undesired on the basis of their known spectral correlation properties and estimate only the desired signals' directions of arrival. The signal-selective nature of the techniques eliminates the need for knowledge of the characteristics of the noise or interference in the environment and makes it possible to resolve a number of desired signals not exceeding the number of sensors in the presence of arbitrary noise and a virtually unlimited number of unknown interferers. For example, the interferers can exhibit an arbitrarily high degree of correlation among themselves and can arrive from directions arbitrarily close to those of the desired signals.>

It is shown that for the problem of multiple-source location using antenna arrays, the algorithms MUSIC and ESPRIT can be simplified by use of the property called cyclostationarity, which is exhibited by modulated signals. The advantages of the modified algorithms are reduction in the required number of array elements and associated reduction in the SVD (singular-value-decomposition) computations. The disadvantages are the requirement of either knowing or measuring frequency parameters, such as carrier frequency or baud rate, and the requirement of longer integration time for correlation measurement, as well as the requirement of measuring a different correlation matrix for each signal of interest.>

The unifying framework of the spectral-correlation theory of cyclostationary signals is used to present a broad treatment of weak, random signal detection for interception purposes. The relationships among a variety of previously proposed ad hoc detectors, optimum detectors, and newly proposed detectors are established. The spectral-correlation-plane approach to the interception problem is put forth as especially promising for detection, classification, and estimation in particularly difficult environments involving unknown and changing noise levels and interference activity. A fundamental drawback of the popular radiometric methods in such environments is explained.>

As a continuation of Part I, the spectral correlation function is presented for a variety of types of digitally modulated signals. These include digital pulse-amplitude, pulse-width, and pulse-position modulation, and various types of phase-shift keying and frequency-shift keying. The magnitudes of the spectral correlation functions are graphed as the heights of surfaces above a bifrequency plane, and these graphs are used as visual aids for comparison and contrast of the spectral correlation properties of different modulation types.

Although the magnitude of the discrete Fourier transform of a maximal-length shift-register sequence is flat, except for its value at zero frequency, the higher resolution spectral content given by the Fourier-series transform is highly erratic. This little-known fact is described, and its ramifications on fast Fourier transforms of one-digit-extended pseudo noise and zero-padded pseudo noise are explained.

A new method for despreading direct-sequence spreadspectrum signals, without use of the spreading code, is evaluated by analysis and simulation. The new method exploits the cyclic autocorrelation in place of the conventional autocorrelation, which was used in a prior method. Both methods require that the period of the code be equal to a multiple of the data symbol interval. Broad-band noise and narrowband interference rejection capabilities of both methods are compared. It is shown that the new method, unlike the prior method, can provide substantial immunity to strong narrow-band interference.

The popular class of synchronizers that consist of a quadratic nonlinearity followed by a phase-lock loop is investigated, and it is shown that the optimum design of the quadratic transformation is characterized in terms of a spectral correlation function for the signal to be synchronized to. It is also shown that the SNR performance of this quadratic transformation, and the mean-square phase jitter of the phaselock loop are both characterized in terms of spectral correlation functions. The conditions under which the optimum quadratic transformations, for symbol synchronization of BPSK, QPSK, SQPSK, and MSK, and for carrier synchronization of BPSK, reduce to the well-known matched-filter-squarer are identified. In addition, the well-known zeromean-square-phase-jitter condition is generalized from PAM to all synchronizable signals, and is characterized in terms of the spectral correlation function. The low-SNR maximum-likelihood synchronizer for all quadratically synchronizable signals is characterized in terms of a multiplicity of maximum-SNR quadratic spectral-line generators. A closed form implementation in terms of a matched filter, squarer, and symbol-rate-synchronized averager is obtained for BPSK and QPSK signals.

It is shown that a large class of nonstationary discrete parameter stochastic processes possess novel ergodic properties, which are referred to as {\em cycloergodic} properties. Specifically, it is shown that periodic components of time-varying probabilistic parameters can be consistently estimated from time averages on one sample path. The cycloergodic theory developed herein extends and generalizes existing ergodic theory for asymptotically mean stationary andN-stationary (cyclostationary) processes, and is presented in both wide-sense and strict-sense contexts.

Explicit formulas for locally (SNR\rightarrow 0)optimum (MMSE) signal estimators (smoother, filter, and predictor) for discrete-time observations of a random signal in additive random noise are derived and used to characterize the locally optimum (likelihood ratio) signal detector for on-off signaling. The characterizations are canonical (distribution-free) detector structures involving estimator-correlators. These structural characterizations provide new interpretations of known detectors for various special cases. If the one-step signal predictor is recursive and the noise is white (possibly non-Gaussian or nonstationary), there is a canonical structure that admits recursive computation. The primary motivation for these structural characterizations is to render the estimator-correlator design philosophy applicable for the purpose of simplifying implementations and enhancing adaptability. Unlike the known esfimator-correlator structural characterizations for continuous-time globally optimum detectors, the new characterizations apply for non-Gaussian as well as Gaussian noise, and the estimators are explicit rather than implicit.

The mean-squared-error measure of quality is used as a basis for a general nearest prototype signal-classification methodology. Canonical signal features for this methodology are identified. A consistency requirement is proposed and used to develop a general approach for determining appropriate class prototypes in discriminant space. It is shown that the class indicator, which is a commonly used class prototype in pattern recognition applications, will often violate the consistency requirement. The general results are used to obtain a solution to a previously posed complexity-performance trade-off problem for matched-filter-tapped-delay-line receivers for serial signal classification in anM-ary data transmission system.

The major purpose of this paper is to promote interchange between the fields of pattern recognition and communications, in the realm of statistical classification. The general class of second-order measures of quality for statistical classification is defined. The variety of members in this class that have been used by practitioners or proposed by theorists for numerical pattern-classification and signal waveform-classification are compared and contrasted. The several measures that are the most generally applicable are shown to be either equivalent to each other or characterizable in terms of each other, thereby revealing an inherent unity. For example, the ratio of between-class-scatter to within-class-scatter used in pattern recognition and the ratio of signal-energy to noise-energy used in communications are unified through an identification of signal with between-class-scatter and noise with within-class-scatter. Results on equivalences are stated and proved for waveform classification rather than numerical classification in order to complement the extensive literature on the latter, and to emphasize applicability to communications. This entails introduction of a scatter ratio for waveforms. In a companion paper, second-order measures of quality are used as a basis for a general nearestprototype signal-classification methodology; canonical signal features for this methodology are identified, and a general approach for determining appropriate class prototypes is given. These two papers provide an integrated approach to the design of a complete signal classifier, i.e., feature extraction and discriminant-functional design tailored to fit a minimumdistance discrimination rule.

A novel two-stage adaptive signal extractor for intermittent signal applications is presented. If the presence and absence of the signal can be detected, the first stage will adapt only while the signal is absent and thereby effect a reduction in noise, whereas the second stage will adapt only when the signal is present and thereby effect a reduction in the distortion caused by the gust stage. Bounds on performance are derived, and performance improvement relative to a conventional one-stage adaptive noise canceller is assessed.

The basic premise in a recent paper that the matchedfilter square-law envelope detector is generally appropriate for fastfading as well as slow-fading incoherent channels is questioned. Its inappropriateness for a fast-fading incoherent Gaussian channel is demonstrated by evaluation of probability of error for this receiver and for the optimum receiver. As a result, modifications to some of the conclusions drawn regarding attainable performance gain with increasing observation time are proposed. In addition, it is pointed out that the performance predictor adopted, viz., deflection, exhibits anomalous behavior (for at least some incoherent fading channels), thereby rendering the other conclusions questionable.

The Cramér-Rao bound on the error variance of an unbiased estimator of all unknown parameter is given an intuitively pleasing interpretation in terms of the sensitivity of the likelihood function to changes in parameter value.

A generalized measure of receiver output signal-to-noise ratio (SNR) that includes several conventional measures of SNR as special eases is presented. The potential for anomalous behavior of the new measure when used to predict signal-detection performance is demonstrated through detailed performance analysis of quadratic receivers for incoherent slow- and fast-fading Gaussian channels.

The familiar notion of inducing stationarity into a cyclostationary process by random translation is extended through characterization of the class of all second-order continuous-parameter processes (with autocorrelation functions that possess a generalized Fourier transform) that are {\em stationarizable} in the wide sense by random translation. This class includes the nested set of proper subclasses: {\em almost cyclostationary} processes, {\em quasi-cyclostationary} processes, and {\em cyclostationary} processes. The random translations that induce stationarity are also characterized. The concept of stationarizability is extended to the concept of asymptotic stationarizability, and the class of {\em asymptotically stationarizable} processes is characterized. These characterizations are employed to derive characterizations of optimum linear and nonlinear time-invariant filters for nonstationary processes. Relative to optimum time-varying filters, these time-invariant filters offer advantages of implementational simplicity and computational efficiency, but at the expense of increased filtering error which in some applications is quite modest. The uses of a random translation for inducing stationarity-of-order-n, for increasing the degree of local stationarity, and for inducing stationarity into discrete-parameter processes are briefly described.

A general approach to the problem of designing structurally constrained receivers for signal detection and estimation is proposed. The approach is based on the constrained Bayesian methodology wherein risk-minimizing inference (or decision) rules are modified (constrained) by replacement of true posterior probabilities with estimated posterior probabilities. The estimators are structurally constrained minimum-mean-squared-error (MMSE) estimators for random posterior probabilities. This methodology is, in essence, an extension and generalization of the well-known linear MMSE estimation methodology. The approach is employed to design linearly constrained coherent receivers for signals in additive and multiplicative noise, and quadratically constrained noncoherent receivers for signals in additive noise. An analysis of these receivers shows that they are very similar to those that are optimum for additive Gaussian noise. The methodology provides a unified theory of receiver design based on the constrained MMSE criterion. This unification yields new insight into this old approach, clarifying both strengths and weaknesses of the approach.

An equivalent linear model for minimum mean-squared error (MMSE) linear estimation of marked and filtered doubly stochastic Poisson processes is presented. The equivalence is employed to determine the structure of the MMSE noncausal steady-state linear receiver for synchronousm-ary optical data signals.

A recently presented series solution to smoothing, filtering, and prediction problems is extended to include correlation between the white and colored components of the observations, as required in applications to feedback communications and feedback control.

Many communication and control systems employ signal formats that involve some form of periodic processing operation. Signals produced by samplers, scanners, multiplexors, and modulators are familiar examples. Often these signals are appropriately modeled by random processes that are cyclostationary (CS), i.e., processes with statistical parameters, such as mean and autocorrelation, that fluctuate periodically with time. In this paper we examine two methods for representing nonstationary processes that reveal the special properties possessed by CS processes. These representations are the harmonic series representation (HSR) and the translation series representation (TSR). We show that the HSR is particularly appropriate for characterizing the structural properties of CS processes and that the TSR provides natural models for many types of communication signal formats. The advantages gained by modeling signals as CS processes rather than stationary processes is illustrated by consideration of the optimum filtering problem. We present general solutions for filters that minimize mean-square error for continuous-waveform estimation, and we discuss several specific examples for the particular case of additive noise. These examples demonstrate improvement in performance over that of filter designs based on stationary models for the signal processes.

A simple solution to linear least-mean-squared-error smoothing, filtering, and prediction problems that is based on series representations for continuous-time random processes is presented. A method for arbitrarily closely approximating correlation functions is employed to obtain approximants that lead to a simple but general closed-form solution for the impulse-response function of the estimating system. The system is asymptotically optimum, and is automatically synthesized with a canonical structure that is convenient for implementation and amenable to adjustment. The method of solution is valid when the observations are restricted to a finite interval and contain a white component.

The application of series representations for random processes to problems of causal least-mean-square estimation on infinite intervals has met with some difficulty in the past. This correspondence brings to light two of the more outstanding difficulties in relation to the contribution of Cambanis, which appears in this issue.

Several authors have shown that the structure of the least-mean-square linear estimator of the sequence of random amplitudes in a synchronous pulse-amplitude-modulated signal that suffers intersymbol interference and additive noise is a matched filter whose output is periodically sampled and passed through a transversal filter (tapped delay line). It is our purpose in this paper to generalize this result to synchronousm-ary signals (e.g., FSK, PSK, PPM signals). We prove that the structure of the least-mean-square linear estimator of the sequence of random parameters in a synchronousm-ary signal, which suffers intersymbol interference and additive noise, is a parallel connection ofmmatched filters followed by tapped delay lines. A similar structure is derived for the continuous waveform estimator of a synchronousm-ary signal. Finally, we present a structure for estimation-decision detection of synchronousm-ary signals, which is based on least-mean-suare linear estimates of aposterioriprobabilities.

The well-known sampling theorem for wide-sense stationary random processes is generalized to the class of nonstationary random processes.