Browsing by Subject "Calculus of variations"

Now showing 1 - 3 of 3

Open Access
Entropy, invertibility and variational calculus of adapted shifts on Wiener space
(2009) Üstünel, A.S.
In this work we study the necessary and sufficient conditions for a positive random variable whose expectation under the Wiener measure is one, to be represented as the Radon-Nikodym derivative of the image of the Wiener measure under an adapted perturbation of identity with the help of the associated innovation process. We prove that the innovation conjecture holds if and only if the original process is almost surely invertible. We also give variational characterizations of the invertibility of the perturbations of identity and the representability of a positive random variable whose total mass is equal to unity. We prove in particular that an adapted perturbation of identity U = IW + u satisfying the Girsanov theorem, is invertible if and only if the kinetic energy of u is equal to the entropy of the measure induced with the action of U on the Wiener measure μ, in other words U is invertible ifffrac(1, 2) under(∫, W) | u |H 2 d μ = under(∫, W) frac(d U μ, d μ) log frac(d U μ, d μ) d μ . The relations with the Monge-Kantorovitch measure transportation are also studied. An application of these results to a variational problem related to large deviations is also given. © 2009 Elsevier Inc. All rights reserved.
Open Access
Estimation theoretic analyses of location secrecy and ris-aided localization under hardware impairments
(2022-06) Öztürk, Cüneyd
In this thesis, we present estimation theoretic analyses of location secrecy and reconfigurable intelligent surface (RIS) aided localization under hardware impairments. First, we consider a wireless source localization network in which a target node emits localization signals that are used by anchor nodes to estimate the target node position. In addition to target and anchor nodes, there can also exist eavesdropper nodes and jammer nodes which aim to estimate the position of the target node and to degrade the accuracy of localization, respectively. We propose the problem of eavesdropper selection with the goal of optimally placing a given number of eavesdropper nodes to a subset of possible positions in the network to estimate the target node position as accurately as possible. As the performance metric, the Cramér-Rao lower bound (CRLB) related to the estimation of the target node position by eavesdropper nodes is derived, and its convexity and monotonicity properties are investigated. By relaxing the integer constraints, the eavesdropper selection problem is approximated by a convex optimization problem and algorithms are proposed for eavesdropper selection. Moreover, in the presence of parameter uncertainty, a robust version of the eavesdropper selection problem is developed. Then, the problem of jammer selection is proposed where the aim is to optimally place a given number of jammer nodes to a subset of possible positions for degrading the localization accuracy of the network as much as possible. A CRLB expression from the literature is used as the performance metric, and its concavity and monotonicity properties are derived. Also, a convex optimization problem and its robust version are derived after relaxation. Moreover, the joint eavesdropper and jammer selection problem is proposed with the goal of placing certain numbers of eavesdropper and jammer nodes to a subset of possible positions. Simulation results are presented to illustrate performance of the proposed algorithms. Second, a wireless source localization network consisting of synchronized target and anchor nodes is considered. An anchor placement problem is formulated to minimize the CRLB on estimation of target node positions by anchor nodes. It is shown that the anchor placement problem can be approximated as a minimization problem of the ratio of two supermodular functions. Due to the lack of a polynomial time algorithm for such problems, an anchor selection problem is proposed to solve the anchor placement problem. Via relaxation of integer constraints, the anchor selection problem is approximated by a convex optimization problem, which is used to propose two algorithms for anchor selection. Furthermore, extensions to quasi-synchronous wireless localization networks are discussed. To examine the performance of the proposed algorithms, various simulation results are presented. Third, we investigate the problem of RIS-aided near-field localization of a user equipment (UE) served by a base station (BS) under phase-dependent amplitude variations at each RIS element. Through a misspecified Cramér -Rao bound (MCRB) analysis and a resulting lower bound (LB) on localization, we show that when the UE is unaware of amplitude variations (i.e., assumes unit-amplitude responses), severe performance penalties can arise, especially at high signal-to-noise ratios (SNRs). Leveraging Jacobi-Anger expansion to decouple range-azimuth-elevation dimensions, we develop a low-complexity approximated mismatched maximum likelihood (AMML) estimator, which is asymptotically tight to the LB. To mitigate performance loss due to model mismatch, we propose to jointly estimate the UE location and the RIS amplitude model parameters. The corresponding Cramér -Rao bound (CRB) is derived, as well as an iterative refinement algorithm, which employs the AMML method as a subroutine and alternatingly updates individual parameters of the RIS amplitude model. Simulation results indicate fast convergence and performance close to the CRB. The proposed method can successfully recover the performance loss of the AMML under a wide range of RIS parameters and effectively calibrate the RIS amplitude model online with the help of a user that has an a-priori unknown location. Fourth, we consider RIS-aided localization scenarios with RIS pixel failures, where individual RIS elements can become faulty due to hardware imperfections. We explore the impact of such failures on the localization performance. To that aim, an MCRB analysis is conducted and numerical results indicate that performance loss for estimating the UE position can be significant in the presence of pixel failures. To remedy this issue, we develop two different diagnosis strategies to determine which pixels are failing, and design robust methods to perform localization in the presence of faulty elements. One strategy is based on the l_1-regularization method, the second one employs a successive approach. Both methods significantly reduce the performance loss due to pixel failures. The successive one performs very close to the theoretical bounds at high SNRs even though it has a higher computational cost than the l_1-regularization based method. In the final part of the dissertation, the optimal encoding strategy of a scalar parameter is performed in the presence of jamming based on an estimation theoretic criterion. Namely, the aim is to obtain the optimal encoding function at the transmitter that minimizes the expectation of the conditional Cramér -Rao bound (ECRB) at the receiver when the jammer has access to the parameter and alters the received signal by sending an encoded version of the parameter. Via calculus of variations, the optimal encoding function at the transmitter is characterized explicitly, and an algorithm is proposed to calculate it. Numerical examples demonstrate benefits of the proposed optimal encoding approach.
Open Access
NVIS HF signal propagation in ionosphere using calculus of variations
(KeAi Communications Co., 2018) Sezen, U.; Arikan, F.; Arıkan, Orhan
Modeling Near Vertical Incidence Sounding (NVIS) High Frequency (HF) signal propagation in the ionosphere is important. Because, ionosondes which are special types of radars probing the ionosphere with certain HF frequencies (between 2 and 30 MHz), work mostly in NVIS mode (where elevation angle is between 89 and 90°). In this work, we are going to propose a new method for NVIS wave propagation in the ionosphere by discretizing the NVIS wave propagation path into mediums in which the refractive index changes linearly, where we solve the ray propagation in each medium analytically using calculus of variations and use Snell's Law at medium changes. The main advantage of the proposed solution is the reduced computational complexity and time. This algorithm can be used to simulate and compare the behavior of vertical ionosondes together with other ray tracing algorithms.