Cramer-Rao Lower Bound Expressions Research Articles

Reduced-Complexity Decimeter-Level Bluetooth Ranging in Multipath Environments

In this paper, we investigate enhanced super-resolution range estimators with decimeter-level accuracy for multi-antenna and multipath Bluetooth systems. To enhance the traditional MUSIC range estimator for a two-way frequency-hopping Bluetooth channel model, we apply forward-backward averaging and bandwidth extrapolation using Burg’s algorithm to improve ranging accuracy, which is limited by the used Bluetooth bandwidth and the quality of the estimated sample covariance matrix. For the multi-antenna case, we compare the Summed Antenna Processing and Individual Antenna Processing methods to process multiple-antenna Bluetooth channel measurements and enhance the range estimation accuracy compared to the single-antenna case. In addition, we investigate a sparsity-aware range estimator which exploits the sparsity of Bluetooth channel impulse response and achieves comparable ranging accuracy to the enhanced MUSIC estimator but at a much lower computational complexity. We apply the greedy Orthogonal Matching Pursuit algorithm to heuristically solve the sparsity-constrained optimization problem for Bluetooth ranging. Furthermore, we evaluate the computational complexity of our investigated Bluetooth range estimators with two complexity-reduction techniques to further reduce the complexity of MUSIC range estimator. Moreover, we analyze the Cramer-Rao Lower Bound (CRLB) on unbiased range estimation using the frequency-hopping Bluetooth channel model and derive a new insightful CRLB expression for a two-path channel model. Finally, we evaluate the Root-Mean-Square Error and Empirical Cumulative Distribution Function performance of our investigated range estimators both on simulated and real-world Bluetooth data that we collected in line-of-sight (LOS) and non-line-of-sight (NLOS) multipath scenarios. Our proposed enhancements on the range estimators improved the ranging accuracy by 58% for our collected Bluetooth data.

Determination of Cramer-Rao Lower Bound (CRLB) and Minimum Variance Unbiased Estimator of a DC Signal in AWGN Using Laplace Transform

This paper presents an alternative approach for the determination of Cramer-Rao Lower Bound (CRLB) and Minimum Variance Unbiased Estimator (MVUE) using Laplace transformation. In this work, a DC signal in Additive White Gaussian Noise (AWGN) was considered. During the investigation, a number of experiments were conducted to analyze different possible outputs under different conditions, and then the patterns of the outcomes were studied. Finally closed-form expressions for the CRLB and MVUE were deduced employing the Laplace transformation. The resulting expressions showed that the proposed method has almost the same number of steps as the existing method. However, the latter requires only the knowledge of algebra to arrive at the CRLB expressions contrary to the existing approach where a strong mathematical background is required and hence making it superior over the existing method, in that sense.
 Keywords: Additive White Gaussian Noise, Cramer-Rao Lower Bound, DC-Value, Laplace transform, and Minimum Variance Unbiased Estimator (MVUE).

Determination of CRLB and Minimum Variance Unbiased Estimator of a DC Signal in AWGN Using Laplace Transform

This paper presents an alternative approach for the determination of Cramer-Rao Lower Bound (CRLB) and Minimum Variance Unbiased Estimator (MVUE) using Laplace transformation. In this work, a DC signal in Additive White Gaussian Noise (AWGN) was considered. During the investigation, a number of experiments were conducted to analyze different possible outputs under different conditions, and then the patterns of the outcomes were studied. Finally closed-form expressions for the CRLB and MVUE were deduced employing the Laplace transformation. The resulting expressions show that the proposed method has almost the same number of steps as the existing method. However, the later requires only the knowledge of algebra to arrive at the CRLB expressions contrary to the existing approach where a strong mathematical background is required and hence making it superior over the existing method, in that sense.

Performance Bounds for Target Parameter Estimation From Frequency-Domain Electromagnetic Induction Data

We derive the Cramer–Rao lower bounds (CRBs) for all target parameters associated with noisy measurements of a specific class of targets using a frequency-domain electromagnetic induction (EMI) system. The target parameters include the target tensors, the relaxation frequencies and their corresponding amplitudes, as well as the target location. We validate the derivation through Monte Carlo simulation. We then derive approximate CRB (ACRB) expressions based on a new low-rank model perspective for EMI data. These approximate expressions are significant simplifications over the full CRB expressions. They also facilitate the analysis and improve the understanding of the factors impacting the lower bounds on the target parameters. We demonstrate the utility and accuracy of the ACRB expressions for two example targets.

Eavesdropper and Jammer Selection in Wireless Source Localization Networks

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 first 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 Cramer-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.

Exact and Closed-Form CRLBS for High-Order Kinematic Parameters Estimation Using LFM Coherent Pulse Train

In radar signal processing, the detection and parameter estimation of high-speed maneuvering targets, which often utilize a coherent pulse train signal with linear frequency modulation, have been receiving increasing attention. Fundamentally and significantly, the Cramer–Rao lower bound (CRLB), as a cornerstone for evaluating the estimation performance of high-order kinematic parameters, has been derived and investigated here. In this paper, a 2-D echo signal model expressed in the fast-frequency and slow-time domains is adopted. The scaled orthogonal Legendre polynomials are deliberately introduced to solve the inverse problem of the Fisher information matrix, and then, a linear mapping relationship between different polynomial parameters can be used to obtain the analytical CRLB expressions. The main contributions included are: 1) the CRLBs, which are exact and closed form, have been extended to arbitrary motion model orders and reference time instants; 2) the influences of the motion model order, the reference time instant, as well as the radar parameters on the CRLBs are exploited comprehensively; and 3) some specific cases, including four low-order motion models and two preferred reference times, are also presented to better demonstrate the CRLB performance relationships. It highlights the fact that the reference time instant corresponding to the middle of the pulse train is a reasonable and compromised choice for parameter estimation, although it is not necessarily optimal for the kinematic parameters of all models and orders. The above research results are illustrated with numerical simulations and further verified using the maximum likelihood estimation method combined with Monte Carlo experiments.

Theoretical Limits for Time Delay Based Location Estimation in Cooperative Relay Networks

Node localization in wireless networks is crucial for supporting advanced location-based services and improving the performance of network algorithms such as routing schemes. In this paper, we study the fundamental limits for time delay based location estimation in cooperative relay networks. The theoretical limits are investigated by obtaining Cramer–Rao Lower Bound (CRLB) expressions for the unknown source location under different relaying strategies when the location of the destination is known and unknown. More specifically, the effects of amplify-and-forward and decode-and-forward relaying strategies on the location estimation accuracy are studied. Furthermore, the CRLB expressions are derived for the cases where the location of only source as well as both source and destination nodes are unknown considering the relays as reference nodes. In addition, the effects of the node topology on the location estimation accuracy of the source node are investigated. The results reveal that the relaying strategy at relay nodes, the number of relays, and the node topology can have significant impacts on the location accuracy of the source node. Additionally, knowing the location of the destination node is crucial for achieving accurate source localization in cooperative relay networks.

Likelihood-Based Modulation Classification for Multiple-Antenna Receiver

Likelihood-based algorithms for the classification of linear digital modulations are systematically investigated for a multiple receive antennas configuration. Existing modulation classification (MC) algorithms are first extended to the case of multiple receive antennas and then a critical problem is identified that the overall performance of the multiple antenna systems is dominated by the worst channel estimate of a particular antenna. To address the performance degradation issue, we propose a new MC algorithm by optimally combining the log likelihood functions (LLFs). Furthermore, to analyze the upper-bound performance of the existing and the proposed MC algorithms, the exact Cramer-Rao Lower Bound (CRLB) expressions of non-data-aided joint estimates of amplitude, phase, and noise variance are derived for general rectangular quadrature amplitude modulation (QAM). Numerical results demonstrate the accuracy of the CRLB expressions and verify that the results reported in the literature for quadrature phase-shift keying (QPSK) and 16-QAM are special cases of our derived expressions. Also, it is demonstrated that the probability of correct classification of the new algorithm approaches the theoretical bounds and a substantial performance improvement is achieved compared to the existing MC algorithm.

Theoretical Limits for Estimation of Periodic Movements in Pulse-Based UWB Systems

In this paper, Cramer-Rao lower bounds (CRLBs) for estimation of signal parameters related to periodically moving objects in pulse-based ultra-wideband (UWB) systems are presented. The results also apply to estimation of vital parameters, such as respiration rate, using UWB signals. In addition to obtaining the CRLBs, suboptimal estimation algorithms are also presented. First, a single-path channel with additive white Gaussian noise is considered, and closed-form CRLB expressions are obtained for sinusoidal object movements. Also, a two-step suboptimal algorithm is proposed, which is based on time delay estimation via matched filtering followed by least-squares estimation, and its asymptotic optimality property is shown in the limit of certain system parameters. Then, a multipath environment is considered, and exact and approximate CRLB expressions are derived. Moreover, suboptimal schemes for parameter estimation are studied. Simulation studies are performed for the estimation of respiration rates in order to evaluate the lower bounds and performance of the suboptimal algorithms for realistic system parameters.

The Multitarget Monopulse CRLB for Matched Filter Samples

It has recently been found that by jointly processing multiple (sum, azimuth-, and elevation-difference) matched filter samples, it is possible to extract and localize several (more than two) targets spaced more closely than the classical interpretation of radar resolution. This paper derives the Cramer-Rao lower bound (CRLB) for sampled monopulse radar data. It is worthwhile to know the limits of such procedures; and in addition to its role in delivering the measurement accuracies required by a target tracker, the CRLB reveals an estimator's efficiency. We interrogate the CRLB expressions for cases of interest. Of particular interest are the CRLB's implications on the number of targets localizable: assuming a sampling-period equal to a rectangular pulse's length, five targets can be isolated between two matched filter samples given that the target's signal-to-noise ratios (SNRs) are known. This reduced to three targets when the SNRs are not known, but the number of targets increases back to five (and beyond) when a dithered boresight strategy is used. Further insight to the impact of pulse shape and of the benefits of oversampling are given.

Computation of the exact cramer-rao lower bound for 2-D ARMA parameter Estimation-I: the quarter-plane case

A closed-form expression for computing the exact Cramer-Rao lower bound (CRLB) on unbiased estimates of the parameters of a two-dimensional (2-D) autoregressive moving average (ARMA) model is developed. The formulation is based on a matrix representation of 2-D homogeneous Gaussian random process that is generated uniformly from the related 2-D ARMA model. The formulas derived for the exact Fisher information matrix (FIM) are an explicit function of the 2-D ARMA parameters and are valid for real-valued homogeneous quarter-plane (QP) 2-D ARMA random fields, where data are propagated using only the past values. It is noteworthy that our approach is practical especially for quantifying the accuracy of 2-D ARMA parameter estimates realized with short data records. Computer simulations display the behavior of the derived CRLB expression for some QP causal 2-D ARMA processes, as a function of the number of data points. The extension of this algorithm for the nonsymmetric half-plane (NSHP) case will be presented in a subsequent paper

The combined effect of signal decorrelation and random noise on the variance of time delay estimation

Tissue motion and elasticity imaging techniques commonly use time delay estimation (TDE) for the assessment of tissue displacement. The performance of these techniques is limited because the signals are corrupted by various factors including electronic noise, quantization, and speckle decorrelation. Speckle decorrelation is caused by changes in the coherent interference among scatterers when the tissue moves relative to the ultrasound beam. In time delay estimation, the effect of noise is usually addressed through the signal-to-noise ratio (SNR) term. Decorrelation, often a significant source of error in medical ultrasound, is commonly described in terms of the correlation coefficient. A relationship between the correlation coefficient and the SNR was previously derived in the literature, for identical signals corrupted by uncorrelated random noise. In this paper, we derive the relationship between the peak of the correlation coefficient function and the SNR for two jointly stationary signals when a delay is present between the signals. Recently, an expression for the Cramer-Rao lower bound (CRLB) has been derived in the literature for partially decorrelated signals in terms of the SNR and the correlation coefficient. Since the applicability of the CRLB is determined not only by the SNR, but also by the correlation coefficient, it is important to unify the expression for the CRLB for partially correlated signals. In this paper, we derive an expression for the CRLB in term of an equivalent SNR converted from the correlation coefficient using an SNR-p relationship, and show this expression to be equivalent to the expression for CRLB. We also corroborate the validity of the SNR-p expression with a simulation. Using this formulation, correlation measurements can be converted to SNR to obtain a composite SNR. The use of this composite SNR in lieu of those in the CRLB expression in the literature allows the extension of the literature results to the solution of the common TDE problems that involve signal decorrelation.