Statistics of the MLE and Approximate Upper and Lower Bounds - Part 1: Application to TOA Estimation

In nonlinear deterministic parameter estimation, the maximum likelihood estimator (MLE) is unable to attain the Cramer-Rao lower bound at low and medium signal-to-noise ratios (SNR) due the threshold and ambiguity phenomena. In order to evaluate the achieved mean-squared-error (MSE) at those SNR levels, we propose new MSE approximations (MSEA) and an approximate upper bound by using the method of interval estimation (MIE). The mean and the distribution of the MLE are approximated as well. The MIE consists in splitting the a priori domain of the unknown parameter into intervals and computing the statistics of the estimator in each interval. Also, we derive an approximate lower bound (ALB) based on the Taylor series expansion of noise and an ALB family by employing the binary detection principle. The accurateness of the proposed MSEAs and the tightness of the derived approximate bounds are validated by considering the example of time-of-arrival estimation

Purposeful Co-Design of OFDM Signals for Ranging and Communications

This paper analyzes the fundamental trade-offs that occur in the co-design of orthogonal frequency-division multiplexing signals for both ranging (via time-of-arrival estimation) and communications. These trade-offs are quantified through the Shannon capacity bound, probability of outage, and the Ziv-Zakai bound on range estimation variance. Bounds are derived for signals experiencing frequency-selective Rayleigh block fading, accounting for the impact of limited channel knowledge and multi-antenna reception. Uncompensated carrier frequency offset and phase errors are also factored into the capacity bounds. Analysis based on the derived bounds demonstrates how Pareto-optimal design choices can be made to optimize the communication throughput, probability of outage, and ranging variance. Different signal design strategies are then analyzed, showing how Pareto-optimal design choices change depending on the channel

Ziv-Zakai Lower Bound for Impulse Radio Ultra-WideBand Ranging Error Correlation Matrix

We derive the Ziv-Zakai lower bound for Impulse Radio (IR) Ultra-WideBand (UWB) ranging error correlation matrix under a classical IR-UWB positioning system. We present the numerical evaluations with IEEE 802.15.4a channel model and the geometry of indoor environments of interest. As our derived bound depends on the geometry of the indoor environments, our bound can be used in real environments with the channel measurements from real environments

Statistics of the MLE and Approximate Upper and Lower Bounds - Part 2: Threshold Computation and Optimal Signal Design

Threshold and ambiguity phenomena are studied in Part 1 of this work where approximations for the mean-squared-error (MSE) of the maximum likelihood estimator are proposed using the method of interval estimation (MIE), and where approximate upper and lower bounds are derived. In this part we consider time-of-arrival estimation and we employ the MIE to derive closed-form expressions of the begin-ambiguity, end-ambiguity and asymptotic signal-to-noise ratio (SNR) thresholds with respect to some features of the transmitted signal. Both baseband and passband pulses are considered. We prove that the begin-ambiguity threshold depends only on the shape of the envelope of the ACR, whereas the end-ambiguity and asymptotic thresholds only on the shape of the ACR. We exploit the results on the begin-ambiguity and asymptotic thresholds to optimize, with respect to the available SNR, the pulse that achieves the minimum attainable MSE. The results of this paper are valid for various estimation problems

Bandwidth scaling behavior in wireless systems : theory, experimentation, and performance analysis

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2010.Cataloged from PDF version of thesis.Includes bibliographical references (p. 167-174).The need for ubiquitous wireless services has prompted the exploration of using increasingly larger transmission bandwidths often in environments with harsh propagation conditions. However, present analyses do not capture the behavior of systems in these channels as the bandwidth changes. This thesis: describes the development of an automated measurement apparatus capable of characterizing wideband channels up to 16 GHz; formulates a framework for evaluating the performance of wireless systems in realistic propagation environments; and applies this framework to sets of channel realizations collected during a comprehensive measurement campaign. In particular, the symbol error probability of realistic wideband subset diversity (SSD) systems, as well as improved lower bounds on time-of-arrival (TOA) estimation are derived and evaluated using experimental data at a variety of bandwidths. These results provide insights into how the performance of wireless systems scales as a function of bandwidth. Experimental data is used to quantify the behavior of channel resolvability as a function of bandwidth. The results show that there are significant differences in the amount of energy captured by a wideband SSD combiner under different propagation conditions. In particular, changes in the number of combined paths affect system performance more significantly in non-line-of-sight conditions than in line-of-sight conditions. Results also indicate that, for a fixed number of combined paths, lower bandwidths may provide better performance because a larger portion of the available energy is captured at those bandwidths. The expressions for lower bounds on TOA estimation, developed based on the Ziv-Zakai bound (ZZB), are able to account for the a priori information about the TOA as well as statistical information regarding the multipath phenomena. The ZZB, evaluated using measured channel realizations, shows the presence of an ambiguity region for moderate signal-to-noise ratios (SNRs). It is shown that in a variety of propagation conditions, this ambiguity region diminishes as bandwidth increases. Results indicate that decreases in the root mean square error for TOA estimation were significant for bandwidths up to approximately 8 GHz for SNRs in this region.by Wesley M. Gifford.Ph.D

Ultra-wideband range estimation: Theoretical limits and practical algorithms

The high time resolution of ultra-wideband (UWB) signals enables wireless devices to perform accurate range estimation. In order to realize UWB systems with accurate ranging capabilities, both theoretical limits on range estimation and practical algorithms that approach those limits should be investigated. This paper provides a survey of various UWB ranging algorithms and discusses their performance and complexity tradeoffs. In addition, theoretical limits on range estimation are discussed in terms of Cramer-Rao and Ziv-Zakai lower bounds. Index Terms- Ultra-wideband (UWB), time-of-arrival (TOA) estimation, ranging, Cramer-Rao lower bound (CRLB), Ziv-Zakai lower bound (ZZLB). ©2008 IEEE

Target localization in passive and active systems : performance bonds

The main goal of this dissertation is to improve the understanding and to develop ways to predict the performance of localization techniques as a function of signal-to-noise ratio (SNR) and of system parameters. To this end, lower bounds on the maximum likelihood estimator (MLE) performance are studied. The Cramer-Rao lower bound (CRLB) for coherent passive localization of a near-field source is derived. It is shown through the Cramer-Rao bound that, the coherent localization systems can provide high accuracies in localization, to the order of carrier frequency of the observed signal. High accuracies come to a price of having a highly multimodal estimation metric which can lead to sidelobes competing with the mainlobe and engendering ambiguity in the selection of the correct peak. The effect of the sidelobes over the estimator performance at different SNR levels is analyzed and predicted with the use of Ziv-Zakai lower bound (ZZB). Through simulations it is shown that ZZB is tight to the MLEs performance over the whole SNR range. Moreover, the ZZB is a convenient tool to assess the coherent localization performance as a function of various system parameters. The ZZB was also used to derive a lower bound on the MSE of estimating the range and the range rate of a target in active systems. From the expression of the derived lower bound it was noted that, the ZZB is determined by SNR and by the ambiguity function (AF). Thus, the ZZB can serve as an alternative to the ambiguity function (AF) as a tool for radar design. Furthermore, the derivation is extended to the problem of estimating target’s location and velocity in a distributed multiple input multiple output (MIMO) radar system. The derived bound is determined by SNR, by the product between the number of transmitting antennas and the number of receiving antennas from the radar system, and by all the ambiguity functions and the cross-ambiguity functions corresponding to all pairs transmitter-target-receiver. Similar to the coherent localization, the ZZB can be applied to study the performance of the estimator as a function of different system parameters. Comparison between the ZZB and the MSE of the MLE obtained through simulations demonstrate that the bound is tight in all SNR regions

Soft information for localization-of-things

Location awareness is vital for emerging Internetof- Things applications and opens a new era for Localizationof- Things. This paper first reviews the classical localization techniques based on single-value metrics, such as range and angle estimates, and on fixed measurement models, such as Gaussian distributions with mean equal to the true value of the metric. Then, it presents a new localization approach based on soft information (SI) extracted from intra- and inter-node measurements, as well as from contextual data. In particular, efficient techniques for learning and fusing different kinds of SI are described. Case studies are presented for two scenarios in which sensing measurements are based on: 1) noisy features and non-line-of-sight detector outputs and 2) IEEE 802.15.4a standard. The results show that SI-based localization is highly efficient, can significantly outperform classical techniques, and provides robustness to harsh propagation conditions.RYC-2016-1938

Cramer-Rao bounds in the estimation of time of arrival in fading channels

This paper computes the Cramer-Rao bounds for the time of arrival estimation in a multipath Rice and Rayleigh fading scenario, conditioned to the previous estimation of a set of propagation channels, since these channel estimates (correlation between received signal and the pilot sequence) are sufficient statistics in the estimation of delays. Furthermore, channel estimation is a constitutive block in receivers, so we can take advantage of this information to improve timing estimation by using time and space diversity. The received signal is modeled as coming from a scattering environment that disperses the signal both in space and time. Spatial scattering is modeled with a Gaussian distribution and temporal dispersion as an exponential random variable. The impact of the sampling rate, the roll-off factor, the spatial and temporal correlation among channel estimates, the number of channel estimates, and the use of multiple sensors in the antenna at the receiver is studied and related to the mobile subscriber positioning issue. To our knowledge, this model is the only one of its kind as a result of the relationship between the space-time diversity and the accuracy of the timing estimation.Peer ReviewedPostprint (published version