Noise benefits in joint detection and estimation problems

Abstract

a b s t r a c tAdding noise to inputs of some suboptimal detectors or estimators can improve their performance under certain conditions. In the literature, noise benefits have been studied for detection and estimation systems separately. In this study, noise benefits are investigated for joint detection and estimation systems. The analysis is performed under the Neyman-Pearson (NP) and Bayesian detection frameworks and according to the Bayesian estimation criterion. The maximization of the system performance is formulated as an optimization problem. The optimal additive noise is shown to have a specific form, which is derived under both NP and Bayesian detection frameworks. In addition, the proposed optimization problem is approximated as a linear programming (LP) problem, and conditions under which the performance of the system can or cannot be improved via additive noise are obtained. With an illustrative numerical example, performance comparison between the noise enhanced system and the original system is presented to support the theoretical analysis.

Exploring $Bona~Fide$ Optimal Noise for Bayesian Parameter Estimation

“…In this paper, we will theoretically provide the solutions to aforementioned crucial questions, and elucidate the possibility of exploiting the noise benefits in some easily implemented suboptimal estimators. We argue that the noise-enhanced Bayesian estimators proposed by [22]- [43], [45]- [49] in recent years can be mainly classified into four categories: (i) the noise-modified estimator established on a single sensor [32], (ii) a linear minimum MSE (LMMSE) estimator based on a single sensor, (iii) the noise-enhanced Bayesian estimator as the average of outputs of an ensemble of identical sensors [42] and (iv) the linear combination estimator executing the LMMSE transform on an array of identical or nonidentical sensors [48].…”

“…Proof of Theorem 2 is presented in Appendix A. Although this theorem leads to a negative aspect of the added noise to the optimal MMSE estimatorθ ms (x), it also indicates the possibility of noise benefits in some suboptimal estimators beyond the restricted conditions of [12], [20], [22]- [25], [27], [30], [31], [33], [36]- [43], [45]- [49]. In practice, the MMSE estimatorθ ms (x) is usually too computationally intensive to implement [51], [54], thus we will exploit the optimal added noise in some easily implementable suboptimal estimators as follows.…”

“…For the first situation, the performances of some easily implemented suboptimal estimators were shown to be substantially improved by exploiting the benefits of added noise [22], [32], [34]- [38], [42], [46]- [49]. In the second situation, rich results from utilizing various kinds of noise have been reported for quantized observations [1], [2], [12]- [14], [20], [21], [23]- [25], [27], [30], [31], [33], [36]- [43], [45]- [49]. For instance, Papadopoulos et al developed a methodology of additive control input before signal quantization at the sensor to achieve the maximum possible performance for quantizer-based networks [23].…”

“…Since the addition of noise can be artificially designed, then finding the optimal probability density function (PDF) of added noise becomes an interesting question [26], [28], [29], [32], [33], [36]- [43], [45]- [49]. Especially, Chen et al considered all possible PDFs of added noise to optimize an arbitrary fixed or variable estimator and proved that the optimal noise, if it exists, is just a finite number of (no more than two) constant vectors by using the properties of convex hull and Caratheodory theorem [28], [29], [32], [38], [39].…”

“…Especially, Chen et al considered all possible PDFs of added noise to optimize an arbitrary fixed or variable estimator and proved that the optimal noise, if it exists, is just a finite number of (no more than two) constant vectors by using the properties of convex hull and Caratheodory theorem [28], [29], [32], [38], [39]. Then, this kind of optimal noise PDFs inspired a series of theoretical improvability of estimation under various estimation criteria [26], [33], [37], [40]- [43], [45]- [49]. An interesting question is whether optimal bona fide noise, rather than a constant bias, exists for enhancing the estimator performance or not.…”