The Sensing Capacity of Sensor Networks

This paper demonstrates fundamental limits of sensor networks for detection problems where the number of hypotheses is exponentially large. Such problems characterize many important applications including detection and classification of targets in a geographical area using a network of sensors, and detecting complex substances with a chemical sensor array. We refer to such applications as largescale detection problems. Using the insight that these problems share fundamental similarities with the problem of communicating over a noisy channel, we define a quantity called the sensing capacity and lower bound it for a number of sensor network models. The sensing capacity expression differs significantly from the channel capacity due to the fact that a fixed sensor configuration encodes all states of the environment. As a result, codewords are dependent and non-identically distributed. The sensing capacity provides a bound on the minimal number of sensors required to detect the state of an environment to within a desired accuracy. The results differ significantly from classical detection theory, and provide an ntriguing connection between sensor networks and communications. In addition, we discuss the insight that sensing capacity provides for the problem of sensor selection.Comment: Submitted to IEEE Transactions on Information Theory, November 200

Channel Detection in Coded Communication

We consider the problem of block-coded communication, where in each block, the channel law belongs to one of two disjoint sets. The decoder is aimed to decode only messages that have undergone a channel from one of the sets, and thus has to detect the set which contains the prevailing channel. We begin with the simplified case where each of the sets is a singleton. For any given code, we derive the optimum detection/decoding rule in the sense of the best trade-off among the probabilities of decoding error, false alarm, and misdetection, and also introduce sub-optimal detection/decoding rules which are simpler to implement. Then, various achievable bounds on the error exponents are derived, including the exact single-letter characterization of the random coding exponents for the optimal detector/decoder. We then extend the random coding analysis to general sets of channels, and show that there exists a universal detector/decoder which performs asymptotically as well as the optimal detector/decoder, when tuned to detect a channel from a specific pair of channels. The case of a pair of binary symmetric channels is discussed in detail.Comment: Submitted to IEEE Transactions on Information Theor

The Recursive Form of Error Bounds for RFS State and Observation with Pd<1

In the target tracking and its engineering applications, recursive state estimation of the target is of fundamental importance. This paper presents a recursive performance bound for dynamic estimation and filtering problem, in the framework of the finite set statistics for the first time. The number of tracking algorithms with set-valued observations and state of targets is increased sharply recently. Nevertheless, the bound for these algorithms has not been fully discussed. Treating the measurement as set, this bound can be applied when the probability of detection is less than unity. Moreover, the state is treated as set, which is singleton or empty with certain probability and accounts for the appearance and the disappearance of the targets. When the existence of the target state is certain, our bound is as same as the most accurate results of the bound with probability of detection is less than unity in the framework of random vector statistics. When the uncertainty is taken into account, both linear and non-linear applications are presented to confirm the theory and reveal this bound is more general than previous bounds in the framework of random vector statistics.In fact, the collection of such measurements could be treated as a random finite set (RFS)

On optimum parameter modulation-estimation from a large deviations perspective

We consider the problem of jointly optimum modulation and estimation of a real-valued random parameter, conveyed over an additive white Gaussian noise (AWGN) channel, where the performance metric is the large deviations behavior of the estimator, namely, the exponential decay rate (as a function of the observation time) of the probability that the estimation error would exceed a certain threshold. Our basic result is in providing an exact characterization of the fastest achievable exponential decay rate, among all possible modulator-estimator (transmitter-receiver) pairs, where the modulator is limited only in the signal power, but not in bandwidth. This exponential rate turns out to be given by the reliability function of the AWGN channel. We also discuss several ways to achieve this optimum performance, and one of them is based on quantization of the parameter, followed by optimum channel coding and modulation, which gives rise to a separation-based transmitter, if one views this setting from the perspective of joint source-channel coding. This is in spite of the fact that, in general, when error exponents are considered, the source-channel separation theorem does not hold true. We also discuss several observations, modifications and extensions of this result in several directions, including other channels, and the case of multidimensional parameter vectors. One of our findings concerning the latter, is that there is an abrupt threshold effect in the dimensionality of the parameter vector: below a certain critical dimension, the probability of excess estimation error may still decay exponentially, but beyond this value, it must converge to unity.Comment: 26 pages; Submitted to the IEEE Transactions on Information Theor

The Sampling Rate-Distortion Tradeoff for Sparsity Pattern Recovery in Compressed Sensing

Recovery of the sparsity pattern (or support) of an unknown sparse vector from a limited number of noisy linear measurements is an important problem in compressed sensing. In the high-dimensional setting, it is known that recovery with a vanishing fraction of errors is impossible if the measurement rate and the per-sample signal-to-noise ratio (SNR) are finite constants, independent of the vector length. In this paper, it is shown that recovery with an arbitrarily small but constant fraction of errors is, however, possible, and that in some cases computationally simple estimators are near-optimal. Bounds on the measurement rate needed to attain a desired fraction of errors are given in terms of the SNR and various key parameters of the unknown vector for several different recovery algorithms. The tightness of the bounds, in a scaling sense, as a function of the SNR and the fraction of errors, is established by comparison with existing information-theoretic necessary bounds. Near optimality is shown for a wide variety of practically motivated signal models

Adaptive sensing performance lower bounds for sparse signal detection and support estimation

This paper gives a precise characterization of the fundamental limits of adaptive sensing for diverse estimation and testing problems concerning sparse signals. We consider in particular the setting introduced in (IEEE Trans. Inform. Theory 57 (2011) 6222-6235) and show necessary conditions on the minimum signal magnitude for both detection and estimation: if ${\mathbf {x}}\in \mathbb{R}^n$ is a sparse vector with $s$ non-zero components then it can be reliably detected in noise provided the magnitude of the non-zero components exceeds $\sqrt{2/s}$. Furthermore, the signal support can be exactly identified provided the minimum magnitude exceeds $\sqrt{2\log s}$. Notably there is no dependence on $n$, the extrinsic signal dimension. These results show that the adaptive sensing methodologies proposed previously in the literature are essentially optimal, and cannot be substantially improved. In addition, these results provide further insights on the limits of adaptive compressive sensing.Comment: Published in at http://dx.doi.org/10.3150/13-BEJ555 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Adaptive channel selection for DOA estimation in MIMO radar

We present adaptive strategies for antenna selection for Direction of Arrival (DoA) estimation of a far-field source using TDM MIMO radar with linear arrays. Our treatment is formulated within a general adaptive sensing framework that uses one-step ahead predictions of the Bayesian MSE using a parametric family of Weiss-Weinstein bounds that depend on previous measurements. We compare in simulations our strategy with adaptive policies that optimize the Bobrovsky- Zaka{\i} bound and the Expected Cram\'er-Rao bound, and show the performance for different levels of measurement noise.Comment: Submitted to the 25th European Signal Processing Conference (EUSIPCO), 201

Frequentist and Bayesian Quantum Phase Estimation

Frequentist and Bayesian phase estimation strategies lead to conceptually different results on the state of knowledge about the true value of the phase shift. We compare the two frameworks and their sensitivity bounds to the estimation of an interferometric phase shift limited by quantum noise, considering both the cases of a fixed and a fluctuating parameter. We point out that frequentist precision bounds, such as the Cram\`er-Rao bound, for instance, do not apply to Bayesian strategies and vice-versa. Similarly, bounds for fluctuating parameters make no statement about the estimation of a fixed parameter.Comment: 4 figure