Kullback-Leibler (KL) divergence plays a central role in hypothesis testing. It gives a measure of the statistical distance between two probability distributions. In the distributed detection problem, it is used as a design criterion in the absence of the information regarding the fusion center's (FC) decision rule: The local sensor decision rules are designed to maximize the KL divergence between the distributions of quantized messages sent to the FC under alternative and null hypotheses. In decision making tasks involving humans, subjective perception of probability values due to behavioral biases needs to be taken into account. In this letter, the notion of behavioral KL divergence is proposed. The statistical distance between two distributions is computed based on the perceived values of the probabilities, which are obtained from the actual probabilities using the probability weighting function employed in prospect theory. It is proved that the behavioral KL divergence between the distributions of the quantized decision at the output of a detector under both hypotheses is maximized by either the Neyman-Pearson (NP) rule or flipped Neyman-Pearson (FNP) rule for any fixed false alarm probability. Based on this result, it is also established that under a constraint on the average perceived false alarm probability, the average behavioral KL divergence is maximized by time-sharing between at most two single-threshold likelihood-ratio tests, each of which is either an NP or an FNP rule. The theoretical results are supported by numerical examples.

A prospect theoretic variant of the Neyman-Pearson (NP) detection problem is proposed for a behavioral (human) decision maker, who perceives distorted versions of the detection and false alarm probabilities. The perceived probabilities are obtained as strictly monotonic transformations of the true probabilities via a probability weighting function employed in prospect theory. It is assumed that the the decision maker can employ time-sharing among a number of detectors in order to maximize the average perceived detection probability subject to a constraint on the average perceived false alarm probability. It is shown that time-sharing between at most two distinct NP decision rules is optimal. A sufficient condition for improvability of the detection performance via time-sharing is presented based on the convexity properties of the transformed version of the receiver operating characteristic curve corresponding to the likelihood ratio detector. Numerical examples are provided to corroborate the theoretical results.

The problem of optimal signal design for coherent detection of binary signals in Gaussian noise is revisited under power and secrecy constraints. In particular, the aim is to select the binary transmitted signals in an optimal manner so that the probability of error is minimized at an intended receiver while the probability of error at an eavesdropper is maintained above a threshold value and the signal powers are limited. It is shown that an optimal solution exists in the form of antipodal signaling along the eigenvector corresponding to the solution of a maximum (possibly generalized) eigenvalue problem, which is specified explicitly based on the channel coefficient matrices and the noise covariance matrices at the intended receiver and the eavesdropper. Furthermore, optimal signal design can be performed in an efficient manner by solving a semidefinite programming (SDP) relaxation followed by a matrix rank-one decomposition. Numerical examples are provided to illustrate optimal solutions for three different but exhaustive cases.

A binary ensemble classification method that sequentially processes the data collected from multiple decision agents in the presence of parameter uncertainties is proposed. Agents are assumed to form correlated groups whose decisions are modeled as multivariate Bernoulli random vectors. The prior probabilities of the binary hypotheses and the corresponding probabilities of the outcomes under each hypothesis are treated as unknown deterministic parameters. Cappé's online Expectation-Maximization algorithm is employed to estimate the parameter values, which are then fed into the ensemble classifier. The proposed technique is shown the reduce the computational complexity while delivering performance close to its offline counterpart, which requires multiple passes over the data. Numerical examples are presented to corroborate the results.

The problem of simple M-ary hypothesis testing under a generic performance criterion that depends on arbitrary functions of error probabilities is considered. Using results from convex analysis, it is proved that an optimal decision rule can be characterized as a randomization among at most two deterministic decision rules, each of the form reminiscent to Bayes rule, if the boundary points corresponding to each rule have zero probability under each hypothesis. Otherwise, a randomization among at most M(M-1)+1 deterministic decision rules is sufficient. The form of the deterministic decision rules are explicitly specified. Likelihood ratios are shown to be sufficient statistics. Classical performance measures including Bayesian, minimax, Neyman-Pearson, generalized Neyman-Pearson, restricted Bayesian, and prospect theory based approaches are all covered under the proposed formulation. A numerical example is presented for prospect theory based binary hypothesis testing.

In this paper, modulation classification and symbol decoding problems are jointly considered and optimal strategies are proposed under various settings. In the considered framework, there exist a number of candidate modulation formats and the aim is to decode a sequence of received signals with an unknown modulation scheme. To that aim, two different formulations are proposed. In the first formulation, the prior probabilities of the modulation schemes are assumed to be known and a formulation is proposed under the Bayesian framework. This formulation takes a constrained approach in which the objective function is related to symbol decoding performance whereas the constraint is related to modulation classification performance. The second formulation, on the other hand, addresses the case in which the prior probabilities of the modulation schemes are unknown, and provides a method under the minimax framework. In this case, a constrained approach is employed as well; however, the introduced performance metrics differ from those in the first formulation due to the absence of the prior probabilities of the modulation schemes. Finally, the performance of the proposed methods is illustrated through simulations. It is demonstrated that the proposed techniques improve the introduced symbol detection performance metrics via relaxing the constraint(s) on the modulation classification performance compared with the conventional techniques in a variety of system configurations.

The optimal stochastic signaling based on the joint design of prior distribution and signal constellation is investigated under an average bit rate and power constraints. First, an optimization problem is formulated to maximize the average probability of correct decision over the set of joint distribution functions for prior probabilities and the corresponding constellation symbols. Next, an alternative problem formulation, for which the optimal joint distribution is characterized by a randomization among at most three mass points, is provided, and it is shown that both formulations share the same solution. Three special cases of the problem are investigated in detail. First, in the absence of randomization, the optimal prior probability distribution is analyzed for a given signal constellation and a closed-form solution is provided. Second, the optimal deterministic pair of prior probabilities and the corresponding signal levels are considered. Third, a binary communication system with scalar observations is investigated in the presence of a zero-mean additive white Gaussian noise, and the optimal solution is obtained under practical assumptions. Finally, numerical examples are presented to illustrate the theoretical results. It is observed that the proposed approach can provide improvements in terms of average symbol error rate over the classical scheme for certain scenarios.

Hybrid likelihood-based approaches equipped with the expectation-maximization (EM) algorithm have received attention in the modulation classification literature during recent years. Despite their superior classification performance, these methods, which rely on iterative batch (offline) processing of the measurements, are not particularly suitable for practical applications due to their high computational demand. In this paper, two online methods that facilitate close to optimal classification performance with reduced computational complexity are employed for modulation classification over unknown nonidentical flat block-fading additive white Gaussian noise channels using multiple sensors. The presented algorithms rely on Titterington's and Cappé's online versions of the EM algorithm that are derived from stochastic approximations of the E- and M-steps of the standard EM algorithm. Unlike the batch method, where the a posteriori probabilities for all the transmitted symbols are updated based on the current set of channel estimates, only the a posteriori probability for the most recently transmitted symbol is computed at each symbol interval and this information is employed to estimate the parameters in a sequential manner. Since the a posteriori probabilities corresponding to the previous symbols are not updated at a given iteration, the computational demand is significantly reduced in comparison with the offline variants. Furthermore, the Polyak-Ruppert averaging is employed to improve the convergence speed. Numerical examples are provided to corroborate the effectiveness of the proposed online EM-based classifiers.

An optimal decision framework is proposed for joint detection and decoding when the prior information is available with some uncertainty. The proposed framework provides tradeoffs between the average inclusive error probability (computed using estimated prior probabilities) and the worst case inclusive error probability according to the amount of uncertainty while satisfying constraints on the probability of false alarm and the maximum probability of miss-detection. Theoretical results that characterize the structure of the optimal decision rule according to the proposed criterion are obtained. The proposed decision rule reduces to some well-known detectors in the case of perfect prior information or when the constraints on the probabilities of miss-detection and false alarm are relaxed. Numerical examples are provided to illustrate the theoretical results.

In this paper, we consider distributed maximum likelihood (ML) classification of digital amplitude-phase modulated signals using multiple sensors that observe the same sequence of unknown symbol transmissions over nonidentical flat blockfading Gaussian noise channels. A variant of the expectation-maximization (EM) algorithm is employed to obtain the ML estimates of the unknown channel parameters and compute the global log-likelihood of the observations received by all the sensors in a distributed manner by means of an average consensus filter. This procedure is repeated for all candidate modulation formats in the reference library, and a classification decision, which is available at any of the sensors in the network, is declared in favor of the modulation with the highest log-likelihood score. The proposed scheme improves the classification accuracy by exploiting the signal-to-noise ratio (SNR) diversity in the network while restricting the communication to a small neighborhood of each sensor. Numerical examples show that the proposed distributed EM-based classifier can achieve the same classification performance as that of a centralized classifier, which has all the sensor measurements, for a wide range of SNR values.

In this paper, we consider the problem of automatic modulation classification with multiple sensors in the presence of unknown time offset, phase offset and received signal amplitude. We develop a novel hybrid maximum likelihood (HML) classification scheme based on a generalized expectation maximization (GEM) algorithm. GEM is capable of finding ML estimates numerically that are extremely hard to obtain otherwise. Assuming a good initialization technique is available for GEM, we show that the classification performance (in terms of the probability of error) can be greatly improved with multiple sensors compared to that with a single sensor, especially when the signal-to-noise ratio (SNR) is low. We further demonstrate the superior performance of our approach when simulated annealing (SA) with uniform as well as nonuniform grids is employed for initialization of GEM in low SNR regions. The proposed GEM based approach employs only a small number of samples (in the order of hundreds) at a given sensor node to perform both time and phase synchronization, signal power estimation, followed by modulation classification. We provide simulation results to show the efficiency and effectiveness of the proposed algorithm.

We address the problem of discovering unknown digital amplitude-phase modulations over block-fading additive noise channels. The proposed method uses the iterative Richardson-Lucy algorithm to determine the distribution of the transmitted symbols, which completely characterizes the underlying signal constellation. The decoding of the received signals can then be carried out based on the estimate of the signal constellation. An important application of the proposed method is to construct a modulation dictionary in an offline manner prior to performing any type of real time classification, thereby improving the performance of the automatic modulation classification algorithms proposed in the literature.

It was shown in the above-titled paper by Rajan and Tepedelenlioglu (see ibid., vol. 59, no. 6, p. 3922–31, June 2013) that the symbol error rate (SER) of an arbitrary multidimensional constellation subject to additive white Gaussian noise is characterized as the product of a completely monotone function with a nonnegative power of signal-to-noise ratio (SNR) under minimum distance detection. In this comment, it is proved that the probability of correct decision of an arbitrary constellation admits a similar representation as well. Based on this fact, it is shown that the stochastic ordering$\{\leq_{{\cal G}_{\alpha}},\alpha\geq 0\}$proposed by the authors as an extension of the existing Laplace transform order to compare the average SERs over two different fading channels actually predicts that the average SERs are equal for any constellation of dimensionality smaller than or equal to$2\alpha$. Furthermore, it is noted that there are no positive random variables$X_{1}$and$X_{2}$such that the proposed stochastic ordering is satisfied in the strict sense, i.e.,$X_{1}<_{{\cal G}_{\alpha}}X_{2}$, when$\alpha=N/2$for any positive integer$N$. Additional remarks are noted about the fading scenarios at low SNR and the generalization to additive compound Gaussian noise originally discussed in the subject paper.

Optimal channel switching and detector design is studied for M-ary communication systems in the presence of stochastic signaling, which facilitates randomization of signal values transmitted for each information symbol. Considering the presence of multiple additive noise channels (which can have non-Gaussian distributions in general) between a transmitter and a receiver, the joint optimization of the channel switching (timesharing) strategy, stochastic signals, and detectors is performed in order to achieve the minimum average probability of error. It is proved that the optimal solution to this problem corresponds to either (i) switching between at most two channels with deterministic signaling over each channel, or (ii) time-sharing between at most two different signals over a single channel (i.e., stochastic signaling over a single channel). For both cases, the optimal solutions are shown to employ corresponding maximum a posteriori probability (MAP) detectors at the receiver. Numerical results are presented to investigate the proposed approach.

The optimum power randomization problem is studied to minimize outage probability in flat block-fading Gaussian channels under an average transmit power constraint and in the presence of channel distribution information at the transmitter. When the probability density function of the channel power gain is continuously differentiable with a finite second moment, it is shown that the outage probability curve is a nonincreasing function of the normalized transmit power with at least one inflection point and the total number of inflection points is odd. Based on this result, it is proved that the optimum power transmission strategy involves randomization between at most two power levels. In the case of a single inflection point, the optimum strategy simplifies to on-off signaling for weak transmitters. Through analytical and numerical discussions, it is shown that the proposed framework can be adapted to a wide variety of scenarios including log-normal shadowing, diversity combining over Rayleigh fading channels, Nakagami-m fading, spectrum sharing, and jamming applications. We also show that power randomization does not necessarily improve the outage performance when the finite second moment assumption is violated by the power distribution of the fading.

Power constrained on-off keying communications systems are investigated in the presence of stochastic signaling and detector randomization. The joint optimal design of decision rules, stochastic signals, and detector randomization factors is performed. It is shown that the solution to the most generic optimization problem that employs both stochastic signaling and detector randomization can be obtained as the randomization among no more than three Neyman-Pearson (NP) decision rules corresponding to three deterministic signal vectors. Numerical examples are also presented.

Optimal receiver design is studied for a communications system in which both detector randomization and stochastic signaling can be performed. First, it is proven that stochastic signaling without detector randomization cannot achieve a smaller average probability of error than detector randomization with deterministic signaling for the same average power constraint and noise statistics. Then, it is shown that the optimal receiver design results in a randomization between at most two maximum a-posteriori probability (MAP) detectors corresponding to two deterministic signal vectors. Numerical examples are provided to explain the results.