The integration (or fusion) of two or more detection-localization systems often results in performance improvement, especially if they are complementary to each other. On the other hand, the cost of integration in additional signal processing and overhead may be considerable. As a figure of merit for this tradeoff, the authors propose the dB-gain-to-cost ratio and illustrate its evaluation by using a simple example. The example consists of a pair of detection-localization systems which have complementary features. One has higher resolution that the other, but is more susceptible to random disturbances. The author models the first as a system which detects a signal in multiplicative and additive random disturbances and additive white noise. Thus, one main result is the derivation of the maximum likelihood detector under these disturbances. In the second system, the author assumes the random disturbances to be negligible. Hence, the maximum likelihood detector is the matched filter. Input data to both systems are two-dimensional arrays of real numbers corresponding to the same physical area under observation where the second data array has a coarser grid than the first, thus yielding poorer resolution. The detection probabilities, in terms of which the dB-gains are computed, are evaluated by Monte Carlo simulation.>

Data compression has been suggested as a nonparametric way of discriminating between message sources. Compressions obtained from a Lempel-Ziv algorithm for relatively short messages, such as those encountered in practice, are examined. The intuitive notion of message complexity has less connection with compression than one might expect from known asymptotic results about infinite messages. Nevertheless, discrimination by compression remains an interesting possibility.>

Detection of a deterministic signal is considered Because of the presence of the transient disturbance, the traditional totally probabilistic criteria are no longer appropriate, since they cannot be related to the physical reality, as in the case of stationary ergodic noise only. A mixed strategy is proposed for the detection where the disturbance is treated as a semideterministic object while white Gaussian noise is treated as a genuinely random object. Specifically, the author uses as the detection statistic the log-likelihood ratio of two conditional probability measures (corresponding to the signal-presence and the signal-absence hypotheses), given the disturbance, in which the disturbance is replaced by its maximum a posteriori likelihood estimate. The detection statistic takes the generalized (in function space) form of a soft-limiter or a blanker followed by the matched filter. It is an explicit detection algorithm, though improvement in computational efficiency is much desired.>

An approximate log-likelihood ratio for detecting a deterministic signal in linear and nonlinear Gaussian noise is derived. The new aspect of this detection problem is the inclusion of a nonlinear form of Gaussian noise and a certain interaction between the signal and the linear noise, such as modulation of the signal by the noise. The derived log-likelihood ratio consists of a modified version of the classical correlation detector and a new processor where the latter is an order of magnitude smaller than the former. In the absence of the signal-noise interaction, the modification consists of subtracting the estimate of the nonlinear noise from the input (to the correlation detector); the nonlinear processor is a quadratic correlation processor where the correlation is not against the (transformed) signal but against the difference of two estimates of the nonlinear noise under the two hypotheses (signal-presence and signal-absence). Presence of the signal-noise interaction introduces further modification and complication on both processors.>

For pt.I see ibid., vol.36, no.2, p.347-57 (1990). Simple numerical examples are presented to illustrate the effect of the previously derived nonlinear filters for combating nonlinear Gaussian noise in detecting deterministic signals. The nonlinear Gaussian noise is expressed as a quadratic form in stationary Gaussian noise that is also present in the data, together with white Gaussian noise. Thus the nonlinear noise is referred to as the quadratic noise and the stationary noise as the linear noise. The former is assumed to be an order of magnitude smaller than the latter. When the signal overlaps with both the linear and the quadratic noise, use of both nonlinear filters for the small quadratic-noise region improves the detection performance well beyond the optimum level achievable in the absence of the quadratic noise. As the quadratic noise increases, this improvement diminishes and the performance eventually deteriorates below the level achievable by the linear and the first nonlinear filter combination.>

An approximate likelihood ratio is derived for detecting a signal in linear stationary and quadratic Gaussian noise. The novel aspect is the inclusion of the quadratic noise. Such a problem can arise in radar when secondary as well as primary reflection by clutter takes place. Monte Carlo simulation using simple examples is run to illustrate the effects of the nonlinear processors on detection performance.>

An approximate but explicit likelihood ratio is derived for detecting deterministic signals in Gaussian and compound Poisson noise. The approximation in the derivation is based on the assumption that the localized noise elements rarely overlap each other. The derived log-likelihood ratio consists of two distinct parts. One is the conventional correlation detector for detecting deterministic signals in Gaussian noise. The other is a nonlinear processor which compensates for the degradation of the correlation detector caused by the localized noise, and is activated only by the presence of the localized noise. As such, it involves covariance operators of both the Gaussian and the localized noise, and is obtained by using the simultaneous diagonalization and orthogonalization of quadratic forms in function space involving eigenfunctions of certain composite operators.>

We study a model of dynamic storage allocation in which requests for single units of memory arrive in a Poisson stream at rate $\lambda $ and are accommodated by the first available location found in a linear scan of memory. Immediately after this first-fit assignment, an occupied location commences an exponential delay with rate parameter $\mu $, after which the location again becomes available. The set of occupied locations (identified by their numbers) at time t forms a random subset $S_t $ of $\{ 1,2, \cdots \}$. The extent of the fragmentation in $S_t $, i.e. the alternating holes and occupied regions of memory, is measured by $\max (S_t ) - |S_t |$. In equilibrium, the number of occupied locations, $|S|$, is known to be Poisson distributed with mean $\rho = \lambda/ \mu$. We obtain an explicit formula for the stationary distribution of max $(S)$, the last occupied location, and by independent arguments we show that $(E\max (S) - E|S|) /E|S| \to 0$ as the traffic intensity $\rho \to \infty $. Moreover, we verify numerically that for any $\rho $ the expected number of wasted locations in equilibrium is never more than $\frac{1}{3}$ the expected number of occupied locations. Our model applies to studies of fragmentation in paged computer systems, and to containerization problems in industrial storage applications. Finally, our model can be regarded as a simple concrete model of interacting particles [Adv. Math., 5 (1970), pp. 246–290].

Suppose requests to store files arrive at a storage facility in a Poisson stream at rate 1. Each file is allocated storage space on arrival and each remains independently for an exponential time with mean l/p. The lengths of the files are assumed to be independent with common distribution F. Each file is placed in the lowest addressed contiguous sequence of locations large enough to accommodate the fre at its arrival time. This is the so-called first-fit storage discipline. We conjecture that first-fit is asymptotically optimal in the sense that the ratio of expected empty space to expected occupied space tends to zero as p → 0, i.e., as the occupied space tends to ∞. This conjecture seems very hard to prove, but it has been proved for constant file lengths [1], i.e., when F degenerates. We are unable to prove the conjecture but give a graphic display of the results of a Monte Carlo simulation which makes it very convincing.

A piecewise linear random path from a sequence of line processes on a plane is constructed. It is shown that the entropy of such a path is maximized if the sequence of line processes are independent identically distributed, Poisson line processes. We also establish that the properly scaled (contracted) coordinates of a constantly moving object on a piecewise linear random path become asymptotically normal as time goes on if the sequence of line processes is independent identically distributed and satisfies certain conditions.

An object is moving on a piecewise linear random path with a constant speed. Given the initial location and direction, the probability distribution of the object's location is calculated as a function of time, and the result is presented graphically. Then the probability of detecting such an object is calculated by using circular and line sensors.

A mathematical treatment of the random field which we have used as a stochastic model for wind pressure exciting the ocean surface is presented. It is a composite stochastic process consisting of an infinite series of ordinary processes and point processes. It is shown that this is a well-defined homogeneous fourth-order random field.

Two solutions are presented to the problem of efficiently approximating a family of noises parameterized by a scalar\Upsilon, 0 \leq \Upsilon \leq \infty. The noises are represented in the form of vectors withmrandom components, and their covariance matrices are such that the number of significant eigenvalues increases with\Upsilon. The noise sample vector is to be approximated, within a specified error\epsilon, by a linear combination of vectors taken from a fixed set ofmvectors that are independent of\Upsilon. Furthermore, for each\Upsilonthe number of approximating vectors is to be mlnlmlzed while keeping the error below\epsilon. This number increases with\Upsilonas does the number of significant eigenvalues. The problem is to find a sequence of parameter values\Upsilon{1} < \cdots < \Upsilon_{m},andasetofvectorsu_{1}, \cdots ,u_{m}such that, for eachj, \Upsilon_{j}is the maximum value of\Upsilonfor which the noise can be approximated within the error of\epsilonby using onlyjvectors, andu_{1}, \cdots , u_{j}are the approximatingjvectors corresponding to\Upsilon_{j}The critical constraint is that the set ofmapproximating vectors be independent of\Upsilon. In the first solution, the root-mean-square error is used for the error that is to remain below\epsilon. In the second, the sample error is used but the\epsilon-approximation is limited to only those noise samples which have nonnegligible average power. In both solutions a recursive scheme is given for obtaining\Upsilon_{1}, \cdots , \Upsilon_{m}andu_{1} , \cdots , u_{m}, the resultant\Upsilon-sequence andu-set (orthonormal) are unique. The result is applied to adaptive spatial processing for signal detection in the case where the signal wave, though temporally incoherent, has a known wavefront, the dominant noise ls spatlally localized, and the processor must be nearly opthnum for a wide range of frequencies.

Two methods are presented for simultaneously detecting a random signal field and estimating its Markovian parameters in the qq prurience of a white Gaussian noise field. The first method provides the Neyman-Pearson decision rule for determining the absence-or-presence of the signal and the maximum {\em a posteriori} likelihood estimators of the parameters. The second provides the Bayes decision rule for the absence-or-presence of the signal and the minimum mean-square error estimators of the parameters. Both methods allow continual causal detection and estimation based on the continuous space-time data, and their detection and estimation statistics are given in terms of a single statistic, which is the solution of a certain stochastic integral equation. Furthermore, an approximate scheme is developed for recursively generating this statistic by using spatially and temporally sampled data.

The optimum detector of Gaussian signal fields in the multipath-anisotropic noise environment is derived. The optimum detector consists of a series of matched quadratic filters and a decisionmaker. The filters are spatially and temporally matched to the eigenwaves of the signal-to-noise covariance, and the decisionmaker compares a quadratic form with a preassigned threshold. The optimum detector for single-path signals in isotropic white noise is the well-known delay-and-sum beamformer, which is widely used because of its simple structure. In the case of multipath signals in anisotropic noise, a substantial improvement in detection performance is possible over the beamformer. This is shown by numerically comparing the detection probability of the beamformer with that of the optimum detector for this case. In addition, we numerically evaluate the sensitivity of such an improvement to accurate knowledge of the signal and noise parameters that are required by the optimum detector. We also show the possibility of further improvement by the addition of a vertical segment to the commonly used horizontal linear array to provide sensitivity to the elevation angles of multipath signals.

Previously, a coding theorem and its converse for stationary asymptotically memoryless continuous-time channels were proved, giving the capacityCwith an almost sure constraint on the input cost. For calculation, it is more convenient to find a capacity\bar{C}with the constraint on the expected input cost. Obviously,C \leq \bar{C}, but the coding theorem may not hold for\bar{C}. For a class of input costs typified by the time-average power, we prove the coding theorem for\bar{C}by directly showingC = \bar{C}.

We define continuous incrementally stationary channels with finite incremental input- and output-memories and prove the coding theorem and its converse for such channels. These channels include, as special cases, stationary channels with finite input- and output-memories and incrementally stationary and memoryless channels. The former is defined here and the latter has been defined previously. It is emphasized that, with an elementary measure-theoretic formulation, the standard method of proving the coding theorem for discrete channels becomes directly applicable for continuous channels. Consequently, the tedious step of representing a continuous channel by an infinite series of discrete channels can be avoided entirely.

Letx_1, \cdots ,x_nandy_1, \cdots, y_nbe input and output sequences of a channel. In the case of memoryless input sources, the following inequality on mutual information is well known: \begin{equation} I((x_1, \cdots ,x_n),(y_1, \cdots ,y_n)) \geq \sum I(x_i, y_i). \end{equation} It is straightforward to show that the inequality sign is reversed if the channel instead of the input source is memoryless. In this paper we establish these inequalities when the input and output are functions instead of sequences.

We consider the class of sequential estimation systems containing additive Gaussian noise channels. Letmbe a random variable representing the source andx(t),y(t)be stochastic processes representing the signal and the channel output, respectively. Denote by\hat{m}an estimate ofmbased on observingy(t)from 0 to\tau, where\tauis a random variable determined by a certain stopping rule of the decoder depending on a realization ofy. Letd(m,\hat{m})be the distortion of\hat{m}relative tomandR(\cdot)be the rate distortion function ofmwith respect to the distortion measured. Denote byP_oandN_othe available average signal power and the noise power level, respectively. We show thatE_{\tau} \geq (2 N_o / P_o)R. (Ed(m,\hat{m}))and henceEd(m,\hat{m}) \geq R ^ {-1} (P_o E_ \tau /2N_o ).. That is, given the average distortionEd(m,\hat{m}), the average transmission time required can be no smaller than(2N_o/P_o)R(Ed(m,\hat{m})). Conversely, given the average transmission timeE_{\tau}, the average distortion can be no smaller thanR ^ {-1} (P_o E _ {\tau} /2N_o).

The following model for the white Gaussian channel with or without feedback is considered: \begin{equation} Y(t) = \int_o ^{t} \phi (s, Y_o ^{s} ,m) ds + W(t) \end{equation} wheremdenotes the message,Y(t)denotes the channel output at timet,Y_o ^ {t}denotes the sample pathY(\theta), 0 \leq \theta \leq t. W(t)is the Brownian motion representing noise, and\phi(s, y_o ^ {s} ,m)is the channel input (modulator output). It is shown that, under some general assumptions, the amount of mutual informationI(Y_o ^{T} ,m)between the messagemand the output pathY_o ^ {T}is directly related to the mean-square causal filtering error of estimating\phi (t, Y_o ^{t} ,m)from the received dataY_o ^{T} , 0 \leq t \leq T. It follows, as a corollary to the result forI(Y_o ^ {T} ,m), that feedback can not increase the capacity of the nonband-limited additive white Gaussian noise channel.

Shannon showed that the capacity of a discrete memoryless channel can not be increased by noiseless feedback. It has been conjectured that this should be true for a continuous memoryless channel, provided such a channel is appropriately defined. We precisely define such a channel from two mathematically different points of view and rigorously prove that its capacity can not be increased by feedback.

This paper examines the question of representing a continuous parameter random process\{ x_t, t \in T \}by a sequence of random variables "without loss of information." The principal result is that such a representation by expansion coefficients relative to a basis\{ \phi_i \}of\mathcal{L}_2(T)is always possible, regardless of the orthogonality of\{ \phi_i \}and of the boundedness of the time intervalT, provided only that the process is continuous in probability and almost every sample path has finite energy.

This paper is concerned with the mathematical aspect of a detection problem (a random signal in white Gaussian noise). Specifically, we obtain a sufficient condition for nonsingular detection and derive a likelihood-ratio expression in terms of least-mean-square estimates. The problem itself is old, and the likelihood-ratio expression is also well known. The contribution of this paper is a relatively elementary and self-contained derivation of the likelihood-ratio expression as well as the nonsingularity condition.

Given a channel and its input probability distribution, the fundamental lemma of Feinstein assures the existence of a code whose maximum error probability is bounded by a certain known constant. The lemma has been established for discrete-time channels and inputs that possess probability densities. We generalize the lemma for the case of continuous-time channels and inputs that do not possess probability densities.

This paper treats the problem of estimating a signals(t)in the presence of noisen(t)wheres(t)andn(t)are independent nonstationary Gaussian processes. Specifically, we present the maximum likelihood and the minimum mean-square estimates ofs(t)for eacht, T_{1} \leq t \leq T_{2}, by observing the entire signal-plus-noise waveformx(\cdot)during the interval[T_{1}, T_{2}]. Under the condition that the signal cannot be detected perfectly in the presence of noise, we explicilyty prove that both estimates, denoted byŝ(t), are given byE\{s(t)|x(\cdot)\}, the conditional expectation ofs(t)givenx(\cdot). With the use of simultaneously orthogonal expansions ofx(t)andn(t), we further obtain explicit expressions in the form of infinite series forŝ(t)and\epsilon(t) \equiv E|s(t) - ŝ(t)|^{2}, the minimum mean-square error. Moreover, if the integral equation\sum_{j=0}^{p} \int H̃_{j}(s, u) \frac{\delta^{j}}{\delta u^{j}}X(u, t) du = S(s, t)admits a formal solution\{H̃_{j}(s,t)\}, thenŝ(t)and\epsilon(t)have the closed-form expressionsŝ(t) = \sum_{j=0}^{p} \int H̃_{j} (t,u)x^{j}(u)du,\epsilon(t) = \sum_{j=0}^{p} \int H̃_{j}(t,u)\frac{\delta^{j}}{\delta u^{j}}N(u,t)du,whereS(s,t), N(s,t)andX(s,t)are the covariances ofs(t), n(t)andx(t), respectively, andpis the largest integer for which the2pth partial derivatives of the covariances are continuous. The last result is a generalization of the classical Wiener filtering theory for stationary processes, and it is valid without the condition of imperfect detection if existence of the formal solution is assumed instead. Finally, we exhibit a general solution of the integral equation in the case where boths(t)andn(t)have rational power spectra.

Simultaneously orthogonal expansion of two processes is one of the major mathematical tools for solving the problem of optimum detection of Gaussian signals in Gaussian noise. This expansion yields two integral equations: a homogeneous equation for the threshold and an inhomogeneous one for the test statistic of an optimum decision rule. After reviewing the optimum detection theory leading to the integral equations, four examples are presented to illustrate techniques of solving these equations and determination of the thresholds and test statistics. These techniques involve only elementary calculus and simple linear algebra. Finally, by way of example, an asymptotic interpretation of "white noise" in the context of optimum detection theory is given.

The first part of this paper is concerned with differentiation of the Karhunen-Loève expansion of a stochastic process. In particular, we establish that the expansion series can be differentiated term by term while retaining the same sense of convergence, ff the covarianceR(s, t)has a continuous second partial derivative and the sample functionx(t)is almost surely differentiable. The result can be generalized to the case of higher-order differentiation. Namely, if(\delta^{2n}/\delta s^{n} \deltat^{n}) R(s, t)is continuous andx(t)has thenth derivativex^{(n)}(t)almost surely, then the series can be differentiated term by termntimes, and the resultant series converges in the stochastic mean tox^{(n)}(t)uniformly int. In the second half, the above result is applied to the problem of optimum reception of binary signals in Gaussian noise. Suppose the binary sure signals arem_{1}(t)andm_{2}(t)and the noise covariance isR(s, t). Then we prove the well-known conjecture that the optimum receiver correlates the observable waveform with the solutiong(t)of the integral equation\int R(s, t)g(s) ds = m_{2}( t) - m_{1}(t)even if the solution contains\delta-functions and their derivatives. This result can be generalized to the case ofM-ary sure signals.

Consider the problem of discriminating two Gaussian signals by using only a finite number of linear observables. How to choose the set of n observables to minimize the error probabilityP_{e}, is a difficult problem. BecauseH, the Hellinger integral, andH^{2}form an upper and a lower bound forP_{e}, we minimizeHinstead. We find that the set of observables that minimizesHis a set of coefficients of the simultaneously orthogonal expansions of the two signals. The same set of observables maximizes the HájekJ-divergence as well.