The subject here is generalized (i.e., non-Gaussian) noise models, and specifically their first-order probability density functions (PDFs). Attention is focused primarily on the author's canonical statistical-physical Class A and Class B models. In particular, Class A noise describes the type of electromagnetic interference (EMI) often encountered in telecommunication applications, where this ambient noise is largely due to other, "intelligent" telecommunication operations. On the other hand, ambient Class B noise usually represents man-made or natural "nonintelligent"-i.e., nonmessage-bearing noise-and is highly impulsive. Class A noise is not an /spl alpha/-stable process, nor is it reducible to such, except in the limiting Gaussian cases of high-density noise (by the central limit theorem). Class B noise is also asymptotically normal (before model approximation). Under rather broad conditions, principally governed by the source propagation and distribution scenarios, the PDF of Class B noise alone (no Gaussian component) can usually be approximated by (1) a symmetric Gaussian /spl alpha/-stable (S/spl alpha/S) model in the case of narrowband reception, or when the PDF /spl omega//sub 1/(/spl alpha/) of the amplitude is symmetric; and (2) a nonsymmetric /spl alpha/-stable (NS/spl alpha/S) model (no Gaussian component) can be constructed in broadband regimes. New results here include: (i) counting functional methods for constructing the general qth-order characteristic functions (CFs) of Class A and Class B noise, from which (all) moments and (in principle), the PDFs follow; (ii) the first-order CFs, PDFs, and cumulative probabilities (APDs) of nonsymmetric broadband Class B noise, extended to include additive Gauss noise (AGN); (iii) proof of the existence of all moments in the basic Class A and Class B models; (iv) the key physical role of AGN and the fact that AGN removes /spl alpha/-stability; (v) the explicit roles of the propagation and distribution scenarios; and (vi) extension to noise fields. Although telecommunication applications are emphasized, Class A and Class B noise models apply selectively, but equally well, to other physical regimes, e.g., underwater acoustics and EM (radar, optics, etc.). Supportive empirical data are included.

Classical threshold detection theory for arbitrary noise and signals, based on independent noise samples, i.e., using only the first-order probability density of the noise, is generalized to include the critical additional statistical information contained in the (first-order) covariances of the noise. This is accomplished by replacing the actual, generalized noise by a "quasi-equivalent" (QE-)model employing both the first-order PDF and covariance. The result is a "near-optimum" approach, which is the best available to date incorporating these fundamental statistical data. Space-time noise and signal fields are specifically considered throughout. Even with additive white Gaussian noise (AWGN) worthwhile processing gains per sample (/spl Gamma//sup (c)/) are attainable, often O(10-20 dB), over the usual independent sampling procedures, with corresponding reductions in the minimum detectable signal. The earlier moving average (MA) noise model, while not realistic, is included because it reduces in the Gaussian noise cases to the threshold optimum results of previous analyses, while the QE-model remains suboptimum here because of the necessary constraints imposed in combining the PDF and covariance information into the detector structure. Full space-time formulation is provided in general, with the important special cases of adaptive and preformed beams in reception. The needed (first-order) PDF here is given by the canonical Class A and Class B noise models. The general analysis, including the canonical threshold algorithms, correlation gain factors /spl Gamma//sup (c)/, detection parameters for the QE-model, along with some representative numerical results for both coherent and incoherent detection, based on four representative Toeplitz covariance models is presented.>

A short personal account is given of the author's long association with S.O. Rice and their common interests in noise modelling and nonlinear processing. This association started during World War II and continued until Rice's death. Some personal history is provided, and the early and continuing influence of Rice's work, particularly his Mathematical Analysis of Random Noise (see Bell Syst. Tech. J., vol.23, p.282-332, July 1944; vol.24, p.46-156, Jan. 1945) on the author's research in noise modelling, scattering, and signal processing is traced. Selected references, recent and older, are included to illustrate this stimulation and influence.>

The problem of determining the waveform distortion produced by a (symmetrical) narrow-band linear filter when an off-tune, simultaneously amplitude- and angle-modulated carrier is introduced, is analytically solved by an extension of the Bedrosian-Rice approach [1]. Here, both the instantaneous angle and envelope of the resulting narrowband output are determined. These results provide a basis for numerical calculations and extension to the more complex problem of interference effects produced by multiple inputs of the above type.

Because communications systems are seldom significantly interfered with by classical white Gaussian noise, it is necessary to consider other, appropriate (and tractable) interference models, if realistic estimates of system performance are to be obtained for the general spectral-use environment. For this purpose, Middleton's recently developed canonical statistical-physical model of "impulsive" interference is applied to real-world communication channels. The principal features of this model are first summarized, including the statistical relations required for the solution of signal detection problems. [Excellent agreement of these model statistics with correspondingly measured statistics is also noted.] The model for narrow-band impulsive interference (Class A noise, a subset of the overall model) is next specifically applied to an important class of coherent signal detection problems. Algorithms for error probabilities in optimum detection are then obtained, along with performance bounds, for the same error probabilities. Since it is known that in order to gain significant improvement over current receivers, the number of (essentially) independent samples of the received interference waveform must be enlarged (i.e., large "processing gains"), the performance results here are given parametrically in the number of samples, or equivalently, in the time-bandwidth product. Performance of current suboptimum receivers is then obtained and compared to the optimum performance. It is shown that very substantial savings in signal power and/or spectrum space can usually be achieved by using the indicated optimal algorithms. Since physical realization of the completely optimum detection algorithms cannot, in general, be economically realized, the somewhat more conservative, corresponding locally optimum Bayes detection (LOBD) receivers are derived. In general, these LOBD structures require adaptive, highly non-linear filters, preceding the conventional correlation detector elements characteristic of optimum receivers for Gaussian interference. Performance for these non-linear, optimum threshold systems is then determined, specifically in Part I for coherent reception.

In Part I, the relevant statistical properties of the recently developed statistical-physical model of generalized impulsive interference have been briefly reviewed (for sub Class A noise) and then applied specifically to optimum coherent detection. It is shown that by using optimum and (locally optimum) detection algorithms (canonically and explicitly derived), substantial savings in signal power and/or spectrum space can be achieved for operation in these highly nonGaussian interference environments. This paper (Part II) extends the preceding analysis to cover various important cases of incoherent reception. The same general model for narrow-band (Class A) impulsive interference and interference examples used in Part I are again employed here. In addition to providing both canonical LOBD structures and expressions for performance, this permits explicit quantitative comparisons between coherent and incoherent reception for common classes of specific digital signal waveforms.

The treatment in the above papers^1concerning the effects of a nonvanishing velocity gradient on the waveform and statistical properties of a scattered field and the resulting received reverberation return is modified to include possible focusing phenomena. These, in turn, can become quite pronounced in the neighborhood of caustics and foci in various practical situations vis-á-vis the common unfocused or weakly focused cases.^{2,3}The modification of earlier results is achieved by inserting an appropriate "focusing factor"\beta_0along with the regular absorption factor(Y_{path})for each leg in the propagation path: source\rightarrowscatter\rightarrowreceiver. Extensions to the "singular" cases, treated more recently by the so-called "modified ray theory^4, can also be carried out.

Analytically tractable statistical-physical models of manmade noise environments have been constructed [1]-[3]. These permit quantitative description of the various types of electromagnetic interference appearing in typical radio receivers and, in particular here, for the communication links employed in mobile transportation systems and urban environments generally. This paper presents a summary of some of the principal analytical results obtained to date [1], [4], and includes some suggested next steps for joint theoretical and experimental study of these increasingly important phenomena. Firstorder probability density functions (pdf's) and probability distributions (pd's) are obtained explicitly; (higher order pdf's and pd's may also be found by similar methods) [2]. These models are based on a Poisson distribution of sources in space. The approach is canonical, in that the results are, in form, independent of particular emitted waveforms, propagation conditions, source distributions, beam patterns, and specific system parameters, as long as the interference is narrow-band following (at least) the aperture and/or the RF stages of a typical receiver. Considered here only are the cases of communication interference, where source and receiver bandwidths are comparable. The paper concludes with a short discussion of some features of suggested future interaction between theory and experiment.

Simultaneous detection and estimation under multiple hypotheses when data from only one observation interval are available, are treated on the basis of statistical decision theory. Estimation is carried out under the assumption that the signal of interest is not present with probability 1, which is necessary if detection is to be a meaningful operation. Also, we consider the case where the operations of detection and estimation are coupled. Specific detector and estimator structures are determined for the case of strong coupling when the cost of estimation error is given by a quadratic function. The detector structures are in general complex nonlinear functions of the received data. However, a detailed analysis of the Gaussian case resulted in a type of correlation detector, which correlates the received data with the least square estimators of the possible signals in the absence of uncertainty. The associated optimum estimator structure is found to be a weighted sum of least square estimators in the absence of uncertainty. Also, joint detection and estimation under multiple hypotheses is discussed for the case of a simple cost function. The estimators that result can be interpreted as generalized maximum-likelihood estimators. Finally, optimum prediction and filtering are briefly considered.

A theory of reverberation and related first-order scattered fields, developed earlier by the author, is now extended to include the often critical effects of absorption in the medium, multiple specular reflections, and nonzero velocity gradients(\nabla_c \neq 0). As before, a combination of field and ray theory is used to obtain typical scattered and reflected waveforms; a spatial and temporal ensemble of random point scatterers is embedded in the now inhomogeneous medium(\nabla_c \neq 0)and the handling of complex geometries is emphasized. Also as before, the effects of the scattering mechanism are described in generality by linear time-varying filter responses. A general vertical velocity gradient is postulated, which may be approximated by contiguous linear velocity profiles. Propagation characteristics for the latter are obtained in detail, including expressions for ray path lengths and delays for arbitrary beam orientations. A useful operator formalism is developed, which assists in determining the principal contributions to the received return when different modes of propagation e.g., convergence zone (CVZ), bottom bounce (BB), surface duct (SD), etc., are employed. Platform and scatterer Doppler are included, as well as arbitrary apertures and beam patterns. Specific waveform results for the important cases of narrow-band signal excitation are given, in addition to the general broad-band theory developed for both bistatic and monostatic reception. A number of models of possible scatterer response are also included, for both physical and biological applications in oceanography.

The earlier theory (Parts I and II) is extended to include the determination of various important statistics of the reverberation process under the broadened conditions of and using the results for the scattered and reflected fields developed in Part III. The new conditions involve nonzero velocity gradients, absorbing media, and multiple reflections, as well as the usual Doppler, general signals and apertures, and mono- and bistatic operation. Of principal interest are the mean return (specular reflection), the mean intensity, and the covariance of these generally nonstationary processes, following reception by the receiving aperture. These we determine under the basic Poisson assumptions of primarily independent single scattering, as before. Also of major importance are the auto- and cointensity spectra associated with the various second-order statistics. General relations for these and higher order moments are indicated. The quasi-stationary cases of "short" narrow-band signals occur in many practical operational situations and are treated here in detail, under a variety of general and special conditions, appropriate to both oceanographic and general communication applications (e.g., detection and classification). The development of our quasi-phenomenological model (Parts I-IV) concludes with a critique of the role of the Poisson assumption of independent scatterers in both volume and surface scatter and a short account of possible and needed extensions of the present "macroscopic" theory to one that is fully physical at the "microscopic" level as well.

The problem of simultaneous detection and estimation of signals in noise is formulated in the language of statistical decision theory. Optimum structures and corresponding general measures of system performance are derived under the Bayes criterion of minimum average risk for the detectors and estimators appropriate to this type of joint operation: It is shown that whereas the structures of the resulting optimum detectors require a class of modified likelihood ratios, the structures of the optimum estimators, which act on the data when there is uncertainty as to the presence of a signal[p(H_{1}) < 1 ], have a common canonical form for a wide variety of operating strategies. This form is identical with that obtained for estimation alone[p(H_{1}) < 1 ], even though there is generally mutual coupling between detector and extractor. A simple structure is obtained for the estimation of amplitude and waveform in the case of a quadratic cost function (least mean-square error), where it is found that the estimator which is optimum here[p(H_{1}) < 1 ]is the product of the corresponding Bayes estimator in the "classical" case [P(H1) = 1 ] and a simple algebraic function of the generalized likelihood ratio. In this case, one can also show that estimators that are unbiased in the classical sense remain unbiased. In parallel with the classical theory, a generalized version of unconditional maximum likelihood estimation is obtained for the "simple" cost function whenp(H_{1} < 1. It is found that estimators that are linear in the classical case[p(H_{1}) = 1 ]are nonlinear in the more general situation[p(H_{1})<1], where increased structural complexity is always the rule for both detectors and estimators. A specific example involving the coherent estimation of signal amplitude illustrates the approach. Analogous extensions to prediction and filtering are formulated, making it evident that a broad area for further generalizations of classical Bayes detection and extraction theory is available for systematic investigation.

A theory of reverberation and related first-order scattered fields is developed, based on the assumption of weak inhomogeneities (i.e., primary scattering only), and a consequent representation in terms of Poisson point processes in space and time. Both surface and volume reverberation are included, separately and together, for general geometries, source and receiver at the same and different locations, and arbitrary transmitting and receiving apertures. A combination of field- and ray-theory is employed to obtain a characteristic scattered waveform, where the inhomogeneous medium is replaced by a homogeneous and isotropic one in which a spatially and temporally random ensemble of point scatterers is embedded. The effects of the scattering mechanism are described generally by a linear, time-varying filter response. The medium itself is seen to be dispersive and is represented by a set of linear (statistical) space-time operators, by which the signal source and the receiver are coupled to one another, as well as to the point scatterers. Broadband as well as narrowband signals and reverberation are included in the model, which is capable of handling general apertures, illuminating signals, doppler of the scatterers, multiple sources and receivers (overlapping beams), and a characteristic time-varying scatter mechanism, that reveals in detail the inherent nonstationarity of the reverberation. Shadowing effects of "rough" surfaces are included, and a variety of important special results, such as the case of narrowband excitation, and simple (time- and frequency-independent) scattering, are also described. The emphasis is on broadband (frequency-dependent) structures, and their associated space-time operators, by which the system as a whole is represented, and with the help of which one can apply the general methods of statistical communication theory to the central problems of signal processing for detection, communication, and classification, in an environment dominated by reverberation or clutter and analogous signal-dependent noise processes.

In Part I a general, first-order scattering model for reverberation has been constructed, from which the characteristic aperture and medium operators have been obtained for general geometries, signals, and arrays. Here, in Part II, the second-order statistics (intensities, covariances, and spectra) of such reverberation processes are determined in detail. In addition to the fluctuation, or purely random component of the received scatter process, an average or purely deterministic component often exists and may be significant, particularly for scattering from surfaces. Since the medium is dispersive (and time varying), reverberation is a non-stationary process (as is well known). Except for purely sinusoidal signals, these processes are (approximately) stationary only for signals of short duration (i.e., at fixed ranges), and only then can be said to possess an intensity spectrum in the usual sense. In this latter instance the familiar reverberation laws (intensity\siminput signal, and\simrange^{-3}or range^{-2}, respectively, for surface or volume scatter) hold for the fluctuation component, while different laws (intensity\simrange^{-2}, or range^{0}) govern the deterministic components. With the composite process no such simple behavior usually appears, unless the geometry insures the dominance of a surface or volume effect. Moreover, in the general case these laws are only approximations, which may often be poor ones. Particular attention is also given to narrowband signals. A variety of expressions for the covariances of the slowly-varying components of the received reverberation is developed in detail, for both surface and volume scatter. These results are very simply combined (additively in the means, variances, etc.) to yield analogous results for the complete, or composite reverberation process. Conditions for normality are given and a number of new results for the probability densities of such nonstationary, narrowband Gaussian processes are obtained. The first-order statistics of the envelope and phase of the received scattered return are also derived, for general narrowband signals, extending the earlier work of Rayleigh, Rice, Hoyt, Nakagami, Middleton, and Beckmann. Many special cases for arbitrary illuminating signals are considered, including an approach to complex (distributed) targets, uniform beams, multiple receivers, transmitter and receiver at the same location, large and small doppler, etc., where the general formulation (as in Part I) is broadband, given in terms of frequency-selective apertures, time-varying dynamic cross sections, and arbitrary input signal waveforms. A number of second-order statistical properties of the medium response is defined and evaluated, which are needed for the combined theoretical and experimental exploration of the medium itself. It is found that the present model of the dispersive reverberation channel cannot be fully described by the statistical structures postulated earlier (and successfully) in radar astronomy: a considerably more involved second-order statistic is required (and obtained) here. To what extent it may provide an adequate description of reality remains to be studied. Part II concludes with a short summary of the principal results and assumptions. Many special cases for arbitrary illuminating signals are considered, including an approach to complex (distributed) targets, uniform beams, multiple receivers, transmitter and receiver at the same location, large and small doppler, etc., where the general formulation (as in Part I) is broadband, given in terms of frequency-selective apertures, time-varying dynamic cross sections, and arbitrary input signal waveforms. A number of second-order statistical properties of the medium response is defined and evaluated,which are needed for the combined theoretical and experimental exploration of the medium itself. It is found that the present model of the dispersive reverberation channel cannot be fully described by the statistical structures postulated earlier (and successfully) in radar astronomy: a considerably more involved second-order statistic is required (and obtained) here. To what extent it may provide an adequate description of reality remains to be studied. Part II concludes with a short summary of the principal results and assumptions.

This paper deals with the forms and properties of a detector operating in an environment in which little statistical information about either the signal or noise field is or can be available. An adaptive detector based on the theory of nonparametric statistics has been designed for this purpose. The detector uses amplitude samples taken from two distinct receivers, only one of which may contain signals from a target. The detector maintains a constant false alarm rate despite any nonstationarity of the noise. The theory of nonparametric statistics suggests that the data from the two sources be ranked in order of amplitude, that a linear weighting (a correlator) be applied to the ranking, and the result be applied to a threshold. In this study, we have considered an adaptive detector of this form. The adaptation mechanism selects the weight (correlating) function on the basis of the past observed data. This class of adaptive detectors is shown to have excellent performance in terms of signal detectability for signals of adequate strength. However, for a given detector design, signals weaker than a certain critical threshold cannot be detected regardless of the amount of data available. Thus, in contrast to the kind of signal suppression effect which characterizes conventional passive detection schemes, the adaptive detector has a sharp minimum detectable signal threshold.

A general canonical theory is developed for the systematic approximation of optimum, or Bayes, detection procedures in the critical limiting threshold mode of operation. The approximations to Bayes detectors introduced here are called locally optimum Bayes detectors (LOBD's) and are defined by the condition that they produce the same value of average risk and its derivative for vanishingly small input signals(\theta \rightarrow 0)as do the corresponding Bayes systems. The LOBD,x̂, in the binary (i.e., two-alternative cases:H_{1}, signal and noise, vs.H_{0}, noise alone) is found to be the expansion of the logarithm of the optimum or likelihood ratio (functional) form\Lambda, including a suitable bias term, e.g.,x̂ = \log \mu + B̂_{n}(\theta)+\theta \left( \frac{\delta \log \Lambda_{n}}{\delta \theta} \right)_{\theta = 0} \mbox{(1)}Suitable choices of biasBcan frequently be made so thatx̂is asymptotically as efficient asx^{\ast}, the corresponding Bayes detector. The principal advantages of the LOBD are that 1) its comparative simplicity vis-à-vis the Bayes system,x^{\ast}; 2) its structure can always be found, even when the structure of the Bayes system cannot be obtained explicitly; and 3) expected performance (average risk and error probabilities) can often be determined for the LOBD when such is not possible for the Bayes system. In addition, the LOBD is canonically derived, i.e., the form ofx̂, (1), is independent of the particular noise statistics and signal, structure. New results include 1) the fact that the logarithm of the likelihood ratio is the function of the received data to be approximated, 2) correlated samples (including continuous sampling on an interval), 3) structures analogous to (1) which hold for the LOBD in binary sequential detection, and 4) similar expansions of the logarithm of the likelihood ratios, etc., which are required for the LOBD in multiple alternative cases. Moreover, in the binary situation of testing for signals(S_{1})of one class against signals(S_{2})of another in noise, the LOBD is seen to be a linearly weighted sum of characteristics like (1). To account for the effects of correlated samples and limiting distributions that do not obey the Central Limit Theorem, i.e., are not asymptotically normal (as illustrated in several of the examples), a criterion of asymptotic relative efficiency (ARE)_{\theta \geq 0is developed. Sufficient, and necessary and sufficient conditions on the bias and on\^{\theta}itself are established, which when satisfied insure that the LOBD is at least asymptotically as efficient, or equivalent to, the corresponding Bayes detector in the limit.

The maximum improvement that can be expected from amplitude and/or phase adaptivity in the performance of an optimum receiver is evaluated in detail for a general class of broad- and narrow-band signals subject to slow Rayleigh fading in the "on-off" mode of operation. The pertinent Bayes risks are evaluated and compared for different states of a priori information. It is found that amplitude adaptivity yields an advantage increasing without bounds with the SNR, whereas phase adaptivity is a decreasing function of the SNR ratio. For the interesting region of moderately low error probabilities, and with equal a priori error risks, amplitude adaptivity yields an improvement in performance equivalent to3-6dB in SNR, whereas phase adaptivity yields an improvement of2or1dB depending on whether the amplitude also is known or not. The effect of asymmetry of a priori error risks is discussed. It is also pointed out that the optimum test for deciding on the presence or absence of the signal is uniformly most powerful with respect to the amplitude, so that no advantage can be expected from amplitude adaptivity in the Neyman-Pearson mode of operation.

Formulas are derived for linear (least-square) reconstruction of multidimensional ({\em e.g.}, space-time) random fields from sample measurements taken over a limited region of observation. Data may or may not be contaminated with additive noise, and the sampling points may or may not be constrained to lie on a periodic The solution of the optimum filter problem in wave-number space is possible under certain restrictive conditions: 1) that the sampling locations be periodic and occupy a sector of the Euclidean sampling space, and 2) that the wave-number spectrum be factorable into two components, one of which represents a function nonzero only within the data space, the other only within the sector imaging the data space through the origin. If the values of the continuous field are accessible before sampling, a prefiltering operation can, in general, reduce the subsequent error of reconstruction. However, the determination of the optimum filter functions is exceedingly difficult, except under very special circumstances. A one-dimensional second-order Butterworth process is used to illustrate the effects of various postulated constraints on the sampling and filtering configuration.

The necessary and sufficient (n. ands.) conditions for the nonsingularity, i.e., regularity, and for the singularity of optimum tests for the presence of one Gaussian process vs another on a finite sample are established, for both nonstationary and stationary processes, including those with nonrational spectra. In the stationary cases, the condition may be expressed alternatively in terms of an integral of suitable spectral ratios when the random processes possess rational spectra and for certain classes of nonrational spectra as well. Equivalently, for rational spectra then. ands. condition for nonsingularity is that the spectral ratio approach unity as frequency becomes infinite and that the spectral ratio be finite and nonzero for all frequencies, while for singularity then. ands. condition requires that this ratio differ from unity in the limit or if unity in the limit, that this ratio vanish or be unbounded at some one (or more) finite frequencies. Some of the implications of these results in applications to signal detection are considered, and a method of solution of an associated class of integral equations, of the type \begin{equation} \int_{0-}^{T+} L(\tau, u)K(\bar u - t \bar) du = G(t, \tau), O -Kis a rational kernel andGis suitably specified, is briefly outlined. Specific results in the case of RC and LRC noise kernels, withGcorrespondingly the difference of two (different) RC or LRC covariance functions, are also given.

In this paper it is shown how the earlier concepts of the matched filter may be generalized by recognizing explicitly the decision-making character of most reception systems. Accordingly, when an approach making use of statistical decision theory is applied for both signal detection and extraction, a variety of new classes of matched filters (Bayes matched filters) can be defined. These can be described specifically in the critical situation of threshold reception, where system optimality is at a premium. It is shown, for incoherent reception in some important special instances, that matched filters based on maximizing output signal-to-noise ratio (theS/Nmatched filters of the earlier theory) are also optimum from the broader, decision viewpoint. The required optimum filters are themselves time-varying and nonunique, and thus permit a measure of design freedom. In all instances, realizable filters are possible, and it is shown how their weighting functions may be determined. Both discrete and continuous filtering on a finite interval,(0, T), are considered.

The problem of estimating signal waveform from received data that is corrupted by noise is briefly considered from the viewpoint of decision theory, in extension of some earlier work. The noise is assumed to be a Gauss process, which may or may not be stationary. Here, however, nothing is known about the signal process except that it may be deterministic, entirely random, or a mixed process. Two new features in the present application are the representation of the signal process as a linear expansion (M. S.) in complete orthonormal sets, and suitable choices of these sets. Examples involving discrete and continuous sampling on a finite interval, with various choices of a priori distributions of signal parameters are described, including calculations of Bayes and Minimax risks.

Thershold coherent and incoherent detection of signals in normal noise is examined when random data sampling is employed at the receiver. The resulting optimum system and its thershold performance are compared with the corresponding cases where periodic sampling is used. Expressions for the szstem structure, the error probabilities, and the Bayes risk are obtained; for most of the specific examples examined it is found that periodic sampling gives better performance than random sampling. A discussion of the generality of these results is given in the concluding section. (Author)

The problem of optimum and suboptimum detection of normal signals in additive normal noise backgrounds is examined by the methods of statistical decision theory. Some general results for optimum receiver structure, error probabilities, and average risk are obtained for the case of colored noise backgrounds. A detailed study of threshold reception in white-noise backgrounds is included, along with calculations of Bayes risk, bias terms, and minimum detectable signals for broad-band RC-noise signals and narrow-band, i.e., high-Q, LRC-noise signal processes. Optimum detector structures for signal processes with rational intensity spectra are also determined for the white noise case, and particular attention is paid to optimum receiver design in terms of physically realizable elements. Suboptimum receiver structure and performance are considered briefly, as well as a number of limiting cases of more special interest. General methods of attack are illustrated, with details given in Appendixes I-V. Application of the results to a variety of communication problems is indicated.

A device which performs a sequential test on a mixture of signal and noise is called a Sequential Detector. With such a device, two thresholds are introduced, each of which is associated with a terminal decision. The length of the detection process (integration time) is not fixed in advance of the experiment but is a random variable, depending on the progress of the test. An optimum form of such a test exists and is characterized by the fact that detection is performed on the average faster than with conventional; i.e., fixed sample size (optimum or non-optimum), devices. The sequential analysis developed by A. Wald is fully applied in this paper, but an important new feature is the treatment of correlated samples and its application to continuous sampling processes. In the introduction, the problem is presented within the framework of Wald's Statistical Decision Theory, and the optimum properties of sequential detectors are discussed accordingly. It is pointed out that a sequential detector is defined in terms of conditional probabilities and hence its operation is essentially independent of a priori information, although the average risk or cost of detection necessarily depends on the a priori signal data. The general theory is illustrated with some cases of special interest. The simplest example of detection involves independent, discrete observations; e.g., the case of a pulsed carrier in normal noise. Here the optimum detector still has the well-known\log I_0structure, but it is shown that the square law approximation for weak signals requires a bias correction due to the fourth order term. Coherent sequential detection of causal signals in normal noise provides another illustration of the theory. An interesting result is that the probabilities of error do not depend on the shape of the filter, provided the proper computer is used. The use of RC-filtered noise illustrates the treatment of continuous detection processes. Finally, the reduction in minimum detectable signal level resulting from the use of a sequential detector is computed. A third example is the sequential detection of random signals in normal noise. It is shown that, although the optimum computer involves the knowledge of the inverted correlation matrix, the average length of the test does not. Hence a curious result is obtained that in this instance detection can be performed in an arbitrarily short time. The paper concludes with a discussion of the practical necessity of truncating the detection process and exact expressions for the error probabilities of such truncated tests are derived and compared with Wald's original approximations.

The problem of the optimum detection of signals in noise, when a possible signal can arise from more than one class of signal types, is considered from the point of view of decision theory. Specifically, optimization here consists of minimizing the average risk for preassigned costs appropriate to the possible correct and incorrect decisions, when only a single signal, of classk, may occur out ofN + 1mutually exclusive signal classes. The analysis is a generalization of the authors' earlier work on binary, or single alternative detection systems. The present treatment outlines the optimization procedure for additive signals and noise, indicates the general structure of the detector, and gives expressions for the probabilities of error and minimum average (or Bayes) risk. Some explicit results for normal statistics are included, and an example-- coherent detection of similar signals, differing only in amplitude-- illustrates the general approach.

When reception in the theory of communication is recognized as a problem in statistical inference, system design and system analysis appear as the counterparts of designing and evaluating statistical tests. This paper discusses the optimum properties of designs based on statistical decision theory from the risk point of view, and from that of information theory. Connections between risk and information loss are established, which result in a unified theory of system design. This includes Minimax methods capable in principle of handling all degrees of a priori knowledge of signal and noise statistics, new methods for comparing actual and ideal systems for the same purpose, and new interpretations of previously used formulations as special cases of the more general theory. Both detection and extraction of signals in noise are considered, the former as a problem of testing statistical hypotheses and the latter as one of estimating parameters. Formulation of the general reception problem as a decision operation is followed by a summary of statistical decision theory from the risk point of view, with some examples of Bayes and Minimax tests and optimum classes of decision rules. Applications to detection show the optimum nature of likelihood ratio receivers as a class, and indicate methods for defining the minimum detectable signal and for comparing system performance. As an illustration, curves of Bayes and Minimax risk are given for detection of a pulsed carrier in noise. Applications to extraction show the nature of optimum extraction and the roles of the mean square error and maximum likelihood criteria from the more general point of view of risk theory. Conditions under which information loss is an extremum in detection and extraction are established, and information loss itself as a criterion of performance is compared with that of the risk measure.

A complete theory of detection is presented, which is capable of treating general types of signals (e.g. periodic, aperiodic, random) in noise of arbitrary statistical character. By proper formulation of the detection problem as a test of statistical hypotheses, the precise structure of the optimum detector can be specified and minimum detectable signals uniquely determined. For threshold reception (the problem of main interest) two classes of operation arise: if detection is coherent, as far as dependence on the input signal-to-noise ratio is concerned one has a linear system, no matter how weak the signal; on the other hand for incoherent reception one always has a quadratic dependence on this input ratio (modulation suppression). Threshold reception in these two instances requires respectively a suitably weighted cross-correlation of the received data with the a priori known signal, or a suitably weighted autocorrelation of the received data with itself. The optimum detector is in general a computer, involving non-linear operations and terminating in a decision operation, which depends on the type of statistical test (e.g. Neyman-Pearson, Ideal, Sequential, etc.) defining the observer. The threshold of decision is necessarily determined by a suitable betting or cost curve. (Both discrete (digital) and continuous (analog) sampling of the data are considered.) In this way optimum performance, consistent with the external constraints, is specified, and the extent by which actual systems depart from this limiting optimum can be calculated.