Browsing by Subject "Signal detection."
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Item Open Access Alternative approaches and noise benefits in hypothesis-testing problems in the presence of partial information(2011) Bayram, SuatPerformance of some suboptimal detectors can be enhanced by adding independent noise to their observations. In the first part of the dissertation, the effects of additive noise are studied according to the restricted Bayes criterion, which provides a generalization of the Bayes and minimax criteria. Based on a generic M-ary composite hypothesis-testing formulation, the optimal probability distribution of additive noise is investigated. Also, sufficient conditions under which the performance of a detector can or cannot be improved via additive noise are derived. In addition, simple hypothesis-testing problems are studied in more detail, and additional improvability conditions that are specific to simple hypotheses are obtained. Furthermore, the optimal probability distribution of the additive noise is shown to include at most M mass points in a simple M-ary hypothesis-testing problem under certain conditions. Then, global optimization, analytical and convex relaxation approaches are considered to obtain the optimal noise distribution. Finally, detection examples are presented to investigate the theoretical results. In the second part of the dissertation, the effects of additive noise are studied for M-ary composite hypothesis-testing problems in the presence of partial prior information. Optimal additive noise is obtained according to two criteria, which assume a uniform distribution (Criterion 1) or the least-favorable distribution (Criterion 2) for the unknown priors. The statistical characterization of the optimal noise is obtained for each criterion. Specifically, it is shown that the optimal noise can be represented by a constant signal level or by a randomization of a finite number of signal levels according to Criterion 1 and Criterion 2, respectively. In addition, the cases of unknown parameter distributions under some composite hypotheses are considered, and upper bounds on the risks are obtained. Finally, a detection example is provided to illustrate the theoretical results. In the third part of the dissertation, the effects of additive noise are studied for binary composite hypothesis-testing problems. A Neyman-Pearson (NP) framework is considered, and the maximization of detection performance under a constraint on the maximum probability of false-alarm is studied. The detection performance is quantified in terms of the sum, the minimum and the maximum of the detection probabilities corresponding to possible parameter values under the alternative hypothesis. Sufficient conditions under which detection performance can or cannot be improved are derived for each case. Also, statistical characterization of optimal additive noise is provided, and the resulting false-alarm probabilities and bounds on detection performance are investigated. In addition, optimization theoretic approaches for obtaining the probability distribution of optimal additive noise are discussed. Finally, a detection example is presented to investigate the theoretical results. Finally, the restricted NP approach is studied for composite hypothesistesting problems in the presence of uncertainty in the prior probability distribution under the alternative hypothesis. A restricted NP decision rule aims to maximize the average detection probability under the constraints on the worstcase detection and false-alarm probabilities, and adjusts the constraint on the worst-case detection probability according to the amount of uncertainty in the prior probability distribution. Optimal decision rules according to the restricted NP criterion are investigated, and an algorithm is provided to calculate the optimal restricted NP decision rule. In addition, it is observed that the average detection probability is a strictly decreasing and concave function of the constraint on the minimum detection probability. Finally, a detection example is presented, and extensions to more generic scenarios are discussed.Item Open Access Noise enhanced detection(2009) Bayram, SuatPerformance of some suboptimal detectors can be improved by adding independent noise to their measurements. Improving the performance of a detector by adding a stochastic signal to the measurement can be considered in the framework of stochastic resonance (SR), which can be regarded as the observation of “noise benefits” related to signal transmission in nonlinear systems. Such noise benefits can be in various forms, such as a decrease in probability of error, or an increase in probability of detection under a false-alarm rate constraint. The main focus of this thesis is to investigate noise benefits in the Bayesian, minimax and Neyman-Pearson frameworks, and characterize optimal additional noise components, and quantify their effects. In the first part of the thesis, a Bayesian framework is considered, and the previous results on optimal additional noise components for simple binary hypothesis-testing problems are extended to M-ary composite hypothesis-testing problems. In addition, a practical detection problem is considered in the Bayesian framework. Namely, binary hypothesis-testing via a sign detector is studied for antipodal signals under symmetric Gaussian mixture noise, and the effects of shifting the measurements (observations) used by the sign detector are investigated. First, a sufficient condition is obtained to specify when the sign detectorbased on the modified measurements (called the “modified” sign detector) can have smaller probability of error than the original sign detector. Also, two suf- ficient conditions under which the original sign detector cannot be improved by measurement modification are derived in terms of desired signal and Gaussian mixture noise parameters. Then, for equal variances of the Gaussian components in the mixture noise, it is shown that the probability of error for the modified detector is a monotone increasing function of the variance parameter, which is not always true for the original detector. In addition, the maximum improvement, specified as the ratio between the probabilities of error for the original and the modified detectors, is specified as 2 for infinitesimally small variances of the Gaussian components in the mixture noise. Finally, numerical examples are presented to support the theoretical results, and some extensions to the case of asymmetric Gaussian mixture noise are explained. In the second part of the thesis, the effects of adding independent noise to measurements are studied for M-ary hypothesis-testing problems according to the minimax criterion. It is shown that the optimal additional noise can be represented by a randomization of at most M signal values. In addition, a convex relaxation approach is proposed to obtain an accurate approximation to the noise probability distribution in polynomial time. Furthermore, sufficient conditions are presented to determine when additional noise can or cannot improve the performance of a given detector. Finally, a numerical example is presented. Finally, the effects of additional independent noise are investigated in the Neyman-Pearson framework, and various sufficient conditions on the improvability and the non-improvability of a suboptimal detector are derived. First, a sufficient condition under which the performance of a suboptimal detector cannot be enhanced by additional independent noise is obtained according to the Neyman-Pearson criterion. Then, sufficient conditions are obtained to specifywhen the detector performance can be improved. In addition to a generic condition, various explicit sufficient conditions are proposed for easy evaluation of improvability. Finally, a numerical example is presented and the practicality of the proposed conditions is discussed.Item Open Access Noise enhanced detection in restricted Neyman-Pearson framework(2013) Gültekin, ŞanHypothesis tests frequently arise in many different engineering problems. Among the most frequently used tests are Bayesian, minimax, and Neyman-Pearson. Even though these tests are capable of addressing many real-life problems, they can be insufficient in certain scenarios. For this reason, developing new hypothesis tests is an important objective. One such developed test is the restricted NeymanPearson test, where one tries to maximize the average detection probability while keeping the worst-case detection and false-alarm probabilities bounded. Finding the best hypothesis testing approach for a problem-at-hand is an important point. Another important one is to employ a detector with an acceptable performance. In particular, if the employed detector is suboptimal, it is crucial that it meets the performance requirements. Previous research has proven that performance of some suboptimal detectors can be improved by adding noise to their inputs, which is known as noise enhancement. In this thesis we investigate noise enhancement according to the restricted Neyman-Pearson framework. To that aim, we formulate an optimization problem for optimal additive noise. Then, generic improvability and nonimprovability conditions are derived, which specify if additive noise can result in performance improvements. We then analyze the special case in which the parameter space is discrete and finite, and show that the optimal noise probability density function is discrete with a certain number of point masses. The improvability results are also extended and more precise conditions are derived. Finally, a numerical example is provided which illustrates the theoretical results and shows the benefits of applying noise enhancement to a suboptimal detector.Item Open Access Optimal detector randomization in cognitive radio receivers in the presence of imperfect sensing decisions(2013) Sezer, Ahmet DündarIn cognitive radio systems, spectrum sensing is one of the crucial tasks to be performed by secondary users in order to limit the interference to primary users. Therefore various spectrum sensing methods have been proposed in the literature. Once secondary users make a sensing decision, they adapt their communication parameters accordingly, which means that they perform communications when the channel is sensed as idle whereas they either do not transmit at all or transmit at a reduced power when the channel is sensed as busy. However, in practical systems, sensing decisions of secondary users are never perfect; hence, there can be cases in which the sensing decision is idle (busy) but primary user activity actually exists (does not exist). Therefore, the optimal design of secondary systems requires the consideration of imperfect sensing decisions. In this thesis, optimal detector randomization is developed for secondary users in a cognitive radio system in the presence of imperfect spectrum sensing decisions. Also, suboptimal detector randomization is proposed under the assumption of perfect sensing decisions. It is shown that the minimum average probability of error can be achieved by employing no more than four maximum a-posteriori probability (MAP) detectors at the secondary receiver. Optimal and suboptimal MAP detectors and generic expressions for their average probability of error are derived in the presence of possible sensing errors. Numerical results are presented and the importance of taking possible sensing errors into account is illustrated in terms of average probability of error optimization.Item Open Access Optimal stochastic approaches for signal detection and estimation under inequality constraints(2012) Dülek, BerkanFundamental to the study of signal detection and estimation is the design of optimal procedures that operate on the noisy observations of some random phenomenon. For detection problems, the aim is to decide among a number of statistical hypotheses, whereas estimating certain parameters of the statistical model is required in estimation problems. In both cases, the solution depends on some goodness criterion by which detection (or estimation) performance is measured. Despite being a well-established field, the advances over the last several decades in hardware and digital signal processing have fostered a renewed interest in designing optimal procedures that take more into account the practical considerations. For example, in the detection of binary-valued scalar signals corrupted with additive noise, an analysis on the convexity properties of the error probability with respect to the transmit signal power has suggested that the error performance cannot be improved via signal power randomization/sharing under an average transmit power constraint when the noise has a unimodal distribution (such as the Gaussian distribution). On the contrary, it is demonstrated that performance enhancement is possible in the case of multimodal noise distributions and even under Gaussian noise for three or higher dimensional signal constellations. Motivated by these results, in this dissertation we adopt a structured approach built on concepts called stochastic signaling and detector randomization, and devise optimal detection procedures for power constrained communications systems operating over channels with arbitrary noise distributions. First, we study the problem of jointly designing the transmitted signals, decision rules, and detector randomization factors for an M-ary communications system with multiple detectors at the receiver. For each detector employed at the receiver, it is assumed that the transmitter can randomize its signal constellation (i.e., transmitter can employ stochastic signaling) according to some probability density function (PDF) under an average transmit power constraint. We show 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 when optimal maximum a-posteriori probability (MAP) detectors are employed in both cases. Next, we prove that a randomization between at most two MAP detectors corresponding to two deterministic signal vectors results in the optimal performance. Sufficient conditions are also provided to conclude ahead of time whether the correct decision performance can or cannot be improved by detector randomization. In the literature, the discussions on the benefits of stochastic signaling and detector randomization are severely limited to the Bayesian criterion. Therefore, we study the convexity/concavity properties for the problem of detecting the presence of a signal emitted from a power constrained transmitter in the presence of additive Gaussian noise under the Neyman-Pearson (NP) framework. First, it is proved that the detection probability corresponding to the α−level likelihood ratio test (LRT) is either concave or has two inflection points such that the function is concave, convex and finally concave with respect to increasing values of the signal power. Based on this result, optimal and near-optimal power sharing/randomization strategies are proposed for average and/or peak power constrained transmitters. Using a similar approach, the convexity/concavity properties of the detection probability are also investigated with respect to the jammer power. The results indicate that a weak Gaussian jammer should employ on-off time sharing to degrade the detection performance. Next, the previous analysis for the NP criterion is generalized to channels with arbitrary noise PDFs. Specifically, we address the problem of jointly designing the signaling scheme and the decision rule so that the detection probability is maximized under constraints on the average false alarm probability and average transmit power. In the case of a single detector at the receiver, it is shown that the optimal solution can be obtained by employing randomization between at most two signal values for the on-signal and using the corresponding NP-type LRT at the receiver. When multiple detectors are available at the receiver, the optimal solution involves a randomization among no more than three NP decision rules corresponding to three deterministic signal vectors. Up to this point, we have focused on signal detection problems. In the following, the trade-offs between parameter estimation accuracy and measurement device cost are investigateed under the influence of noise. First, we seek to determine the most favorable allocation of the total cost to measurement devices so that the average Fisher information of the resulting measurements is maximized for arbitrary observation and measurement statistics. Based on a recently proposed measurement device cost model, we present a generic optimization problem without assuming any specific estimator structure. Closed form expressions are obtained in the case of Gaussian observations and measurement noise. Finally, a more elaborate analysis of the relationship between parameter estimation accuracy and measurement device cost is presented. More specifically, novel convex measurement cost minimization problems are proposed based on various estimation accuracy constraints assuming a linear system subject to additive Gaussian noise for the deterministic parameter estimation problem. Robust allocation of the total cost to measurement devices is also considered by assuming a specific uncertainty model on the system matrix. Closed form solutions are obtained in the case of an invertible system matrix for two estimation accuracy criteria. Through numerical examples, various aspects of the proposed optimization problems are compared. Lastly, the discussion is extended to the Bayesian framework assuming that the estimated parameter is Gaussian distributed.Item Open Access Signal and detector randomization for multiuser and multichannel communication systems(2013) Tutay, Mehmet EminRandomization can be considered as a possible approach to enhance error performance of communication systems subject to average power constraints. In the first part of this dissertation, we consider downlink of a multiuser communications system subject to an average power constraint, where randomization is employed at the transmitter and receiver sides by modeling signal levels as random variables (stochastic signals) and employing different sets of detectors via time-sharing (detector randomization), respectively. In the second part, we consider single-user systems, where we assume that there exist multiple channels between the transmitter and receiver with arbitrary noise distributions over each of them and only one of the channels can be employed for transmission at any given time. In this case, randomization is performed by choosing the channel in use according to some probability mass function and employing stochastic signaling at the transmitter. First, the jointly optimal power control with signal constellation randomization is proposed for the downlink of a multiuser communications system. Unlike a conventional system in which a fixed signal constellation is employed for all the bits of a user (for given channel conditions and noise power), power control with signal constellation randomization involves randomization/time-sharing among different signal constellations for each user. A formulation is obtained for the problem of optimal power control with signal constellation randomization, and it is shown that the optimal solution can be represented by a randomization of (K + 1) or fewer distinct signal constellations for each user, where K denotes the number of users. In addition to the original nonconvex formulation, an approximate solution based on convex relaxation is derived. Then, detailed performance analysis is presented when the receivers employ symmetric signaling and sign detectors. Specifically, the maximum asymptotical improvement ratio is shown to be equal to the number of users, and the conditions under which the maximum and minimum asymptotical improvement ratios are achieved are derived. In the literature, it is known that employing different detectors with corresponding deterministic signals via time-sharing may enhance error performance of communications systems subject to average power constraints. Motivated by this result, as a second approach, we study optimal detector randomization for the downlink of a multiuser communications system. A formulation is provided to obtain optimal signal amplitudes, detectors, and detector randomization factors. It is shown that the solution of this joint optimization problem can be calculated in two steps, resulting in significant reduction in computational complexity. It is proved that the optimal solution is achieved via randomization among at most min{K, Nd} detector sets, where K is the number of users and Nd is the number of detectors at each receiver. Lower and upper bounds are derived on the performance of optimal detector randomization, and it is proved that the optimal detector randomization approach can reduce the worst-case average probability of error of the optimal approach that employs a single detector for each user by up to K times. Various sufficient conditions are obtained for the improvability and nonimprovability via detector randomization. In the special case of equal crosscorrelations and noise powers, a simple solution is developed for the optimal detector randomization problem, and necessary and sufficient conditions are presented for the uniqueness of that solution. Next, a single-user M−ary communication system is considered in which the transmitter and the receiver are connected via multiple additive (possibly nonGaussian) noise channels, any one of which can be utilized for a given symbol transmission. Contrary to deterministic signaling (i.e., employing a fixed constellation), a stochastic signaling approach is adopted by treating the signal values transmitted for each information symbol over each channel as random variables. In particular, the joint optimization of the channel switching (i.e., time-sharing among different channels) strategy, stochastic signals, and decision rules at the receiver is performed in order to minimize the average probability of error under an average transmit power constraint. It is proved that the solution to this problem involves either one of the following: (i) deterministic signaling over a single channel, (ii) randomizing (time-sharing) between two different signal constellations over a single channel, or (iii) switching (time-sharing) between two channels with deterministic signaling over each channel. For all cases, the optimal strategies are shown to employ corresponding maximum a posteriori probability (MAP) decision rules at the receiver.