Hybrid Cramer-Rao Bound Research Articles

On the Unique Channel Identification in Ambient Backscatter Communication Systems

Channel coefficient estimation is a challenging problem in ambient backscatter communication systems. In this paper, unique identification of channel coefficients using the Expectation-Maximization (EM) method is investigated. It is proven that the EM method is not able to always yield unique solutions to channel coefficients and therefore the tag symbol estimation accuracy can be compromised. A correct derivation of the hybrid Cramer-Rao Bound (HCRB) is also provided.

Hybrid Cramer-Rao Bound on Carrier and Sampling Frequency Offset Estimation for OFDM Systems in Rayleigh Fading Channels

For the carrier frequency offset (CFO) and sampling (clock) frequency offset (SFO) estimation, the hybrid Cramer-Rao bound (HCRB) is developed when the CFO, SFO, information-bearing symbols are deterministic and channel coefficients are random. Both noise and channel coefficients are complex Gaussian. The HCRB is a lower bound on the mean squared estimation error for any unbiased estimator of a parameter. For the HCRB to be applicable, it is necessary for deterministic parameters to be identifiable (uniquely determined). Some necessary identifiability conditions of some deterministic parameters are found and presented. The HCRB is dependent on the initial time instant. The HCRB is used to assess the performances of some existing methods via simulation. Our results demonstrate that even the best performance is still around 10 dB higher than the HCRB. Further effort is needed to develop more accurate methods.

Hybrid Cramer-Rao bound for near-field source localization using a spatially spread acoustic vector sensor

This presentation characterizes the source-localization precision obtainable via received signal strength indication (RSSI) based on data from a tri-axial velocity sensor and a spatially separated pressure sensor. That scheme was proposed originally by Y. I. Wu and K. T. Wong in January 2012 in the IEEE Transactions on Aerospace and Electronic Systems. That source-localization scheme depends on the acoustic propagation path-loss exponent, which is typically not precisely known a priori but could be modeled stochastically. That exponent may also differ in value for the propagation path to the tri-axial velocity sensor and for the propagation path to the pressure sensor. This presentation accounts for these two practical considerations in characterizing the scheme's source-localization precision, through the metric of the hybrid Cramer-Rao bound (HCRB), the correctness of which is here validated by Monte Carlo simulations of the corresponding maximum a posteriori (MAP) estimator.

Realistic Lower Bound on Elevation Estimation for Tomographic SAR

The noise in a tomographic synthetic aperture radar (Tomo-SAR) model is normally assumed to be independent and identically distributed (i.i.d.) Gaussian. In this paper, the correlated Tomo-SAR model is introduced by studying the effect of random residual phase and correlated additive Gaussian noise, and a realistic and general hybrid Cramer–Rao bound (HCRB) on elevation estimation is derived for such a model. Then, a simplified calculation of the HCRB is proposed when the bound of elevation is the main focus. Computer simulations are performed to analyze the proposed HCRB for elevation estimation. The results obtained from estimators based on compressive sensing and distributed compressive sensing show that the proposed HCRB can provide a more realistic bound than the CRB derived with the white additive noise and perfect phase compensation assumption. This is also validated through processing results on real data acquired by TerraSAR-X/Tandem-X sensors.

Joint Estimation of MIMO-OTH Radar Measurements and Ionospheric Parameters

To avoid the disadvantage of the traditional over-the-horizon (OTH) radar, which uses only the measurements of ionospheric parameters obtained from an ionosonde to estimate the target parameters, this paper proposes a new multiple-input multiple-output skywave OTH radar, which estimates the ionospheric and target parameters jointly by exploiting the data received by both the ionosonde and the radar receivers. First, we consider the scenario where the prior distribution of ionospheric parameters is unknown, the joint maximum likelihood estimator is developed and the Cramer–Rao bound (CRB) is derived to analyze the estimation performance. Then, the case with known Gaussian ionospheric prior distribution is analyzed, we develop the hybrid maximum likelihood and maximum a posteriori estimator, and the hybrid CRB (HCRB) is derived for analyzing the estimation performance. Finally, we show that the proposed joint estimation approaches lead to better accuracy for the ionospheric and target parameters estimation. We prove that both the CRB and the HCRB vary with the ionosonde measurement error covariance, and the corresponding upper and lower bounds are derived.

Direction finding with the sensors' gains suffering bayesian uncertainty —hybrid CRB and MAP estimation

The paper analyzes how a sensor array's direction-finding accuracy may be degraded by any stochastic uncertainty in the sensors' complex value gains, modeled here as complex value Gaussian random variables. This analysis is via the derivation of the hybrid Cramer-Rao bound (HCRB) of the azimuth-elevation direction-of-arrival estimates. This HCRB is analytically shown to be inversely proportional to a multiplicative factor equal to one plus the variance of the sensors' gain uncertainty. This finding applies to any array grid geometry. The maximum a posteriori (MAP) estimator corresponding to this uncertain gain data model is also derived. Monte Carlo simulations demonstrate that this estimator approaches the lower bound derived.

Bayesian Estimation in the Presence of Deterministic Nuisance Parameters—Part I: Performance Bounds

How accurately can one estimate a random parameter subject to unknown deterministic nuisance parameters? The hybrid Cramer-Rao bound (HCRB) provides an answer to this question for a restricted class of estimators. The HCRB is the most popular performance bound on the mean-square-error (MSE) for random parameter estimation problems which involve deterministic parameters. The HCRB is useful when one is interested in both the random and the deterministic parameters and in the coupling between their estimation errors. This bound refers to locally weak-sense unbiased estimators with respect to (w.r.t.) the deterministic parameters. However, if these parameters are nuisance, it is unnecessary to restrict their estimation as unbiased. This paper is the first of a two-part study of Bayesian parameter estimation in the presence of deterministic nuisance parameters. It begins with a study on order relations between existing Cramer-Rao (CR)-type bounds of mean-unbiased Bayesian estimators. Then, a new CR-type bound is developed with no assumption of unbiasedness on the nuisance parameters. Alternatively, Lehmann’s concept of unbiasedness is employed rather than conventional mean-unbiasedness. It is imposed on a risk that measures the distance between the estimator and the minimum MSE (MMSE) estimator which assumes perfect knowledge of the nuisance parameters. In the succeeding paper, asymptotic performances of some Bayesian estimators with maximum likelihood based estimates for the nuisance parameters are investigated. The proposed risk-unbiased bound (RUB) is proved to be asymptotically achieved by the MMSE estimator with maximum likelihood estimates for the nuisance parameters, while the existing CR-type bounds are not necessarily achievable.

基于相位同步的分布式全相参雷达参数估计及相参性能

For distributed aperture coherent radar with a general architecture, a signal model for coherent parameter estimation and coherent processing based on system phase synchronization is proposed in this paper, and the sensitivity of parameter estimation and coherent processing performance to synchronization errors is analyzed in detail. A closed-form hybrid Cramer-Rao bound (HCRB) of the estimation of time delay difference is first derived in multiple-input and multiple output (MIMO) work mode. Following this, in fully coherent work mode, the analytical formula of output signal-to-noise ratio gain (oSNRg) is addressed, and a numerical solution of the analytical formula is obtained by using high-order polynomial fitting. The principle of antenna configuration is then studied. It is concluded that the estimation accuracy of time delay difference or coherent processing performance is determined by both synchronization error and input SNR. Compared with existing methods, the method based on phase synchronization can yield higher estimation accuracy of time delay differences and better coherent processing performance, given the assumption that the phase synchronization error is small and the input SNR is low.

Identifiability Analysis for Array Shape Self-Calibration Based on Hybrid Cramér-Rao Bound

In array shape self-calibration where the sensor position errors and source locations are both unknown, the identifiability of the unknowns is a fundamental problem. Previously, using an approximate hybrid Cramer-Rao bound (HCRB), it was found that under the assumption of small random errors, a nominally linear array is impossible to self-calibrate, but a nominally non-linear array is possible to self-calibrate with three noncollinear far-field sources. In this letter, both small and large random errors are considered, thus the crucial small-error approximation has to be dropped. An accurate HCRB is then derived using fully the prior information about the errors. The accurate HCRB proves that if it is tight, a perturbed nominally linear array is possible to self-calibrate. The larger the sensor position errors, the easier the self-calibration. This is important because of the wide application of linear array. Moreover, two noncollinear far-field sources are sufficient to self-calibrate an array of arbitrary nominal shape.

On the Joint Estimation of the RSS-Based Location and Path-loss Exponent

Due to its straightforward implementation, the received signal strength (RSS) has been an advantageous approach for low cost localization systems. Although the propagation model is difficult to characterize in uncertain environments, the majority of current studies assume to have exact knowledge of the path-loss exponent (PLE). This letter deals with RSS based localization in an unknown path-loss model. First, we derive an analytical expression for the mean square error on location estimates for incorrect PLE assumption and examine, via simulation, the effects of error in the PLE on the location accuracy. Second, we enhance a previously proposed RSS-PLE joint estimator (JE) by reducing its complexity. We also propose a maximum a posteriori (MAP) estimator by considering the PLE as an unknown random variable. Finally, we derive the Hybrid Cramer Rao Bound (HCRB) as a benchmark for the MAP estimator. Error analysis results predict large error due to incorrect PLE assumption which are in agreement with the simulation results. Further simulations show that the MAP estimator exhibits better performance at low signal to noise ratio (SNR) and that the relation between the HCRB and CRB depends on the network geometry.

Antenna Placement Analysis for Coherent MIMO Radar with Phased Synchronization Errors

Phase synchronization errors are practically inevitable, so in this paper the quantitative tool to asses the effect of antenna placement for localization performance with phased synchronization errors is provided. The lower bound of mean-square error (MSE) is set by the hybrid Cramer-Rao bound (HCRB) for the joint estimation of the target location and phase synchronization errors at the receivers. It is shown that HCRB follow up to a lower limit, determined by the synchronization error variance and the number of transmit and receive sensors. For uniform antenna distributing, symmetrical placement is optimal, and all forms give exactly the same performance. The localization performance of nonuniform arrays when the antennas are spread out as much as possible is a little better than symmetrical placement at high SNR. Simulation results verify the correctness of conclusions.

Notes on the Tightness of the Hybrid CramÉr–Rao Lower Bound

In this paper, we study the properties of the hybrid Cramer-Rao bound (HCRB). We first address the problem of estimating unknown deterministic parameters in the presence of nuisance random parameters. We specify a necessary and sufficient condition under which the HCRB of the nonrandom parameters is equal to the Cramer-Rao bound (CRB). In this case, the HCRB is asymptotically tight [in high signal-to-noise ratio (SNR) or in large sample scenarios], and, therefore, useful. This condition can be evaluated even when the CRB cannot be evaluated analytically. If this condition is not satisfied, we show that the HCRB on the nonrandom parameters is always looser than the CRB. We then address the problem in which the random parameters are not nuisance. In this case, both random and nonrandom parameters need to be estimated. We provide a necessary and sufficient condition for the HCRB to be tight. Furthermore, we show that if the HCRB is tight, it is obtained by the maximum likelihood/maximum <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a</i> <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">posteriori</i> probability (ML/MAP) estimator, which is shown to be an unbiased estimator which estimates both random and nonrandom parameters simultaneously optimally (in the minimum mean-square-error sense).

Hybrid CramÉr–Rao Bounds for Crustal Displacement Field Estimators in SAR Interferometry

This letter focuses on the performance achievable by spaceborne synthetic aperture radar interferometry (InSAR) in the estimation of line-of-sight crustal deformations from acquisitions over a distributed scatterer. Our model is suited for exploiting the hybrid Cramer-Rao bound (HCRB), where the unknowns are both deterministic parameters and stochastic variables. We take into account both target decorrelation and atmospheric phase screen (APS). This approach leads to a viable evaluation of InSAR performance as a function of system configuration, target decorrelation, and APS variance.

On the use of Cramer-Rao-like bounds in the presence of random nuisance parameters

We focus on the use of Cramer-Rao-like bounds for the estimation of a deterministic parameter in the presence of random nuisance parameters. As a working example, we consider the problem of estimating the carrier frequency offset affecting a linearly modulated waveform received through a Rician fading channel. We calculate the relevant Cramer-Rao bound (CRB) and also a set of Cramer-Rao-like bounds proposed in the literature, among which we mention the hybrid CRB, the modified CRB and the Miller-Chang (1978) bound. An extension of the latter bound is also provided. The relative merits of these bounds are discussed, both in terms of their tightness and ease of calculation. A few useful inequalities involving the above bounds are also established.