In this paper, a new semi-definite programming approach is devised for approximating nonlinear estimation problems. The main idea is to include the noise components as parameters of interests, which increases the flexibility in the convex optimization formulation. Using the source localization as an illustration, we develop semi-definite relaxation (SDR) positioning algorithms using angle-of-arrival, time-of-arrival and time-difference-of-arrival measurements. Numerical examples are included to show the effectiveness of the proposed SDR approach.

Frequency estimation of a complex single-tone in additive white Gaussian noise from irregularly-spaced samples is addressed. In this Letter, we study the periodogram and weighted phase averager, which are standard solutions in the uniform sampling scenarios, for tackling the problem. It is shown that the estimation performance of both approaches can attain the optimum benchmark of the Cramer-Rao lower bound, although the former technique has a smaller threshold signal-to-noise ratio.

In this Letter, we explore semi-definite relaxation (SDR) program for finding the real roots of a real polynomial. By utilizing the square of the polynomial, the problem is approximated using the convex optimization framework and a real root is estimated from the corresponding minimum point. When there is only one real root, the proposed SDR method will give the exact solution. In case of multiple real roots, the resultant solution can be employed as an accurate initial guess for the iterative approach to get one of the real roots. Through factorization using the obtained root, the reminding real roots can then be solved in a sequential manner.

In this Letter, the problem of estimating the time-difference-of-arrival between signals received at two spatially separated sensors is addressed. By taking discrete Fourier transform of the sensor outputs, time delay estimation corresponds to finding the frequency of a noisy sinusoid with time-varying amplitude. The generalized weighted linear predictor is utilized to estimate the time delay and it is shown that its estimation accuracy attains Cramer-Rao lower bound.

A conventional approach to mobile positioning is to utilize the time-of-arrival (TOA) measurements between the mobile station (MS) and several receiving base stations (BSs). The TOA information defines a set of circular equations from which the MS position can be calculated with the known BS geometry. However, when the TOA measurements are obtained from the non-line-of-sight (NLOS) paths, the position estimation performance can be very unreliable. Assuming that the NLOS probability and distribution are known and the NLOS-induced error dominates the corresponding TOA measurement, two maximum a posteriori probability (MAP) algorithms for NLOS detection and MS localization are derived in this paper. The first provides a standard MAP solution, while the second is a simplified version based on geometric constraints. It is shown that the former achieves more accurate estimation performance at the expense of higher computational cost.

Finding the position of a radiative source based on time-difference-of-arrival (TDOA) measurements from spatially separated receivers has important applications in sonar, radar, mobile communications and sensor networks. Each TDOA defines a hyperbolic locus on which the source must lie and the position estimate can then be determined with the knowledge of the sensor array geometry. While extensive research works have been performed on algorithm development for TDOA estimation and TDOA-based localization, limited attention has been paid in sensor array geometry design. In this paper, an optimum two-dimensional sensor placement strategy is derived with the use of optimum TDOA measurements, assuming that each sensor receives a white signal source in the presence of additive white noise. The minimum achievable Cramér–Rao lower bound is also produced.

Finding the positions of nodes in an ad hoc wireless sensor network (WSN) with the use of the incomplete and noisy distance measurements between nodes as well as anchor position information is currently an important and challenging research topic. However, most WSN localization studies have considered that the anchor positions and the signal propagation speed are perfectly known which is not a valid assumption in the underwater and underground scenarios. In this paper, semi-definite program- ming (SDP) algorithms are devised for node localization in the presence of these uncertainties. The corresponding Cramer-Rao lower bound (CRLB) is also produced. Computer simulations are included to contrast the performance of the proposed algorithms with the conventional SDP method and CRLB.

A common technique for passive source localization is to utilize the range-difference (RD) measurements between the source and several spatially separated sensors. The RD information defines a set of hyperbolic equations from which the source position can be calculated with the knowledge of the sensor positions. Under the standard assumption of Gaussian distributed RD measurement errors, it is well known that the maximum-likelihood (ML) position estimation is achieved by minimizing a multimodal cost function which corresponds to a difficult task. In this correspondence, we propose to approximate the nonconvex ML optimization by relaxing it to a convex optimization problem using semidefinite programming. A semidefinite relaxation RD-based positioning algorithm, which makes use of the admissible source position information, is proposed and its estimation performance is contrasted with the two-step weighted least squares method and nonlinear least squares estimator as well as Cramer-Rao lower bound.

In this paper, we present a distributed particle fllter (DPF) for target tracking in a sensor network. The proposed DPF consists of two major steps. First, particle compression based on support vector machine is performed to reduce the cost of transmission among sensors. Second, each sensor fuses the compressed information from its neighboring nodes with use of consensus or gossip algorithm to estimate the target track. Computer simulations are included to verify the efiectiveness of the proposed approach.

A conventional approach for source localization is to utilize time delay measurements of the emitted signal received at an array of sensors. The time delay information is then employed to construct a set of hyperbolic equations from which the target position can be determined. In this paper, we utilize semi-definite programming (SDP) technique to derive a passive source localization algorithm which can integrate the available a priori knowledge such as admissible target range and other cues. It is shown that the SDP method is superior to the well-known two-step weighted least squares method at lower signal-to-noise ratio conditions.

The problem of node localization in a wireless sensor network (WSN) with the use of the incomplete and noisy distance measurements between nodes as well as anchor position information is currently an an important yet challenging research topic. Most WSN localization studies at present have assumed that the anchor positions are perfectly known which is not valid in the underwater and underground scenarios. In this paper, semi-definite programming (SDP) algorithms are devised for finding the localizations of unknown-position nodes in the presence of anchor position uncertainty. Computer simulations are included to contrast the performance of the proposed algorithms with the conventional SDP method and Cramer-Rao lower bound.

By utilizing the second and fourth order linear prediction errors, a novel estimator for a single noisy sinusoid is devised. The frequency estimate is obtained from a solving a cubic equation and a simple root selection procedure is provided. Asymptotical variance of the estimated frequency is derived and confirmed by computer simulations. It is demonstrated that the proposed estimator is superior to the reformed Pisarenko harmonic decomposer, which is the improved version of Pisarenko harmonic decomposer.

In this correspondence, based on an alternative derivation of the Pisarenko harmonic decomposition (PHD) method, a new asymptotically unbiased estimator for the frequency of a single real tone in white noise is devised with the use of novel sample covariance expressions. Furthermore, extension to sample covariances with higher lags for performance enhancement is investigated while a simple and effective scheme is suggested to resolve the corresponding frequency ambiguity problem. The variance of the modified Pisarenko's method is also derived, which is then utilized to find the best estimate among all admissible solutions from various sets of sample covariances. Computer simulations are included to corroborate the theoretical development and to demonstrate that the proposed approach outperforms several existing low-complexity frequency estimators in terms of nearly uniform performance and estimation accuracy.

A two-stage autocorrelation approach is proposed for single-tone frequency estimation of a real sinusoid in white noise. In the first stage, we transform the received data to another noisy sinusoidal sequence of same frequency via an autocorrelation procedure. Autocorrelation functions of the converted sequence are then employed for frequency estimation in the second stage. A simple frequency estimator is first derived, followed by a generalized algorithm. Computer simulations are included to illustrate the accuracy of our proposed approach via comparison with Cramér–Rao lower bound and several conventional frequency estimators.

In this paper, a recursive Gauss-Newton (RGN) algorithm is first developed for adaptive tracking of the amplitude, frequency and phase of a real sinusoid signal in additive white noise. The derived algorithm is then simplified for computational complexity reduction as well as improved with the use of multiple forgetting factor (MFF) technique to provide a flexible way of keeping track of the parameters with different rates. The effectiveness of the simplified MFF-RGN scheme in sinusoidal parameter tracking is demonstrated via computer simulations.

We study the convex optimization approach for parameter estimation of several sinusoidal models, namely, single complex/real tone, multiple complex sinusoids, and single two-dimensional complex tone, in the presence of additive Gaussian noise. The major difficulty for optimally determining the parameters is that the corresponding maximum likelihood (ML) estimators involve finding the global minimum or maximum of multimodal cost functions because the frequencies are nonlinear in the observed signals. By relaxing the nonconvex ML formulations using semidefinite programs, high-fidelity approximate solutions are obtained in a globally optimum fashion. Computer simulations are included to contrast the estimation performance of the proposed semi-definite relaxation methods with the iterative quadratic maximum likelihood technique as well as Cramér-Rao lower bound.

In this paper, a new topological approach for studying a sufficiently long random number sequence is proposed. By segmenting the sequence into groups of digits which represent the node identities while the undirected edges symbolize the adjacency between them, a network is constructed for analysis. In particular, the network constructed from a π sequence is examined in detail and its properties are contrasted with the Erdos–Renyi (ER) random graph model. Based on the observation that there are more nodes with even degrees than the adjacent odd counterparts in the constructed network, a new random graph model named Random Eulerian (RE) model and its extension are finally proposed and analyzed.

A popular strategy for source localization is to utilize the measured differences in arrival times of the source signal at multiple pairs of receivers. Most of the time-difference-of-arrival (TDOA) based algorithms in the literature assume that the signal transmission speed is known which is valid for in-air propagation. However, for in-solid scenarios such as seismic and tangible acoustic interface applications, the signal propagation speed is unknown. In this paper, we exploit the ideas in the two-step weighted least squares method [1] to design a three-step algorithm for joint source position and propagation speed estimation. Simulation results are included to contrast the proposed estimator with the linear least squares scheme as well as Cramér–Rao lower bound.

Time-difference-of-arrival (TDOA) based source localization has been intensively studied and broadly applied in many fields. In this paper, particle swarm optimization (PSO) is employed for positioning with TDOA measurements in the circumstances of known and unknown propagation speed. The optimization criterion is first developed and the PSO technique is then employed to search the global minimum of the cost function. For sufficiently small noise conditions, simulation results show that the PSO approach provides accurate source location estimation for both known and unknown propagation speed, and also gives an efficient speed estimate in the later case.