Cramer-Rao Lower Bound for Parameter Estimation of Multiexponential Signals (original) (raw)
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Cramer-Rao lower bound for parameter estimation in nonlinear systems
IEEE Signal Processing Letters, 2000
Calculation of the Cramer-Rao lower bound, i.e., the inverse of the Fisher information matrix, for output data sets of a general nonlinear system is a challenging problem and is considered in this letter. It is shown that the Fisher information matrix for a data set generated by a nonlinear system with additive Gaussian measurement noise can be expressed in terms of the outputs of its derivative system that is also a nonlinear system. An example is considered arising from surface plasmon resonance experiments to determine the dynamic parameters of molecular interactions.
The stability of nonlinear least squares problems and the Cramer-Rao bound
IEEE Transactions on Signal Processing, 2000
A number of problems of interest in signal processing can be reduced to nonlinear parameter estimation problems. The traditional approach to studying the stability of these estimation problems is to demonstrate finiteness of the Cramér-Rao bound (CRB) for a given noise distribution. We review an alternate, determinstic notion of stability for the associated nonlinear least squares (NLS) problem from the realm of nonlinear programming (i.e., that the global minimizer of the least squares problem exists and varies smoothly with the noise). Furthermore, we show that under mild conditions, identifiability of the parameters along with a finite CRB for the case of Gaussian noise is equivalent to deterministic stability of the NLS problem. Finally, we demonstrate the application of our result, which is general, to the problems of multichannel blind deconvolution and sinusoid retrieval to generate new stability results for these problems with little additional effort.
New Results on Deterministic Cramér–Rao Bounds for Real and Complex Parameters
IEEE Transactions on Signal Processing, 2012
The Cramér-Rao bounds (CRB) is a lower bound of great interest for system analysis and design in the asymptotic region [high signal-to-noise ratio (SNR) and/or large number of snapshots], as it is simple to calculate and it is usually possible to obtain closed form expressions. The first part of the paper is a generalization to complex parameters of the Barankin rationale for deriving MSE lower bounds, that is the minimization of a norm under a set of linear constraints. With the norm minimization approach the study of Fisher information matrix (FIM) singularity, constrained CRB and regularity conditions become straightforward corollaries of the derivation. The second part provides new results useful for system analysis and design: a general reparameterization inequality, the equivalence between reparameterization and equality constraints, and an explicit relationship between parameters unidentifiability and FIM singularity.
A Comparison of Two CramÉr–Rao Bounds for Nonlinear Filtering with<tex>$rm P_d ≪ 1$</tex>
IEEE Transactions on Signal Processing, 2004
The paper presents a comparative study of two recently reported Cramér-Rao lower bounds (CRLBs) for nonlinear filtering, both applicable when the probability of detection is less than unity. The first bound is the information reduction factor CRLB; the second is the enumeration method CRLB. The enumeration method is accurate but computationally expensive. We prove in the paper that the information reduction factor bound is overoptimistic, being always less than the enumeration CRLB. The theory is illustrated by two target tracking applications: ballistic object tracking and bearings-only tracking. The simulations studies confirm the theory and reveal that the information reduction factor CRLB rapidly approaches the enumeration CRLB as the scan number increases.
Journal of Magnetic Resonance, 2000
We have derived analytical expressions of the Cramer-Rao lower bounds on spectral parameters for singlet, doublet, and triplet peaks in noise. We considered exponential damping (Lorentzian lineshape) and white Gaussian noise. The expressions, valid if a sufficiently large number of samples is used, were derived in the time domain for algebraic convenience. They enable one to judge the precision of any unbiased estimator as a function of the spectral and experimental parameters, which is useful for quantitation objectives and experimental design. The influence of constraints (chemical prior knowledge) on parameters of the peaks of doublets and triplets is demonstrated both analytically and numerically and the inherent benefits for quantitation are shown. Our expressions also enable analysis of spectra comprising many peaks. Copyright 2000 Academic Press.
Fisher Information Lower Bounds with Applications in Hardware-Aware Nonlinear Signal Processing
arXiv: Information Theory, 2015
We discuss the problem of deriving compact and tractable lower bounds for the Fisher information matrix. To motivate our particular approach towards such expressions, we first examine the structure of the exact Fisher information matrix in the context of exponential family models. Then, by replacing an arbitrary data model by an equivalent distribution within the exponential family of distributions, we derive a lower bound for the Fisher information measure of probabilistic models with multivariate output and multiple parameters. The pessimistic information matrix allows a tractable quantitative analysis of the parameter-specific information flow through nonlinear random systems. Therefore, the technique is exploited for the performance analysis concerning direction-of-arrival estimation of wireless source signals with a binary radio sensor array. Further, by the example of a sensing device exhibiting amplifier saturation, we outline how the information bound can be used to learn co...
In this study, a generic analysis of sensor impulse response effects on linearly filtered channel noise is presented to illustrate the resulting variations to the Cramèr–Rao lower bounds (CRLBs) of signal parameter estimators in signal processing and communication applications. The authors start by deriving the density function of a filtered signal, which is shown to be a mixture density, and hence the exact expressions for the mean and variance. Simulation results are used to confirm the derivations, which are then used to investigate the effects of impulse response length and variance, as well as channel noise length and variance effects on the resulting CRLBs. Results indicate that for non-zero mean channel noise and impulse responses, the resulting mean of filtered noise can be relatively large causing adverse deviations to parameter estimations. The filtered noise variance is shown to be proportional to the impulse response energy, where for long duration of signal capture the ...
The hybrid Cramér-Rao bound and the generalized Gaussian linear estimation problem
2008
This paper explores the Hybrid Cramér-Rao Lowerbound (HCRLB) for a Gaussian generalized linear estimation problem in which some of the unknown parameters are deterministic while the other are random. In general, the HCRLB on the non-Bayesian parameters is not asymptotically tight. However, we show that for the generalized Gaussian linear estimation problem, the HCRLB of the deterministic parameters coincides with the CRLB, so it is an asymptotically tight bound. In addition, we show that the ML/MAP estimator [1] is asymptotically efficient for the non-Bayesian parameters while providing optimal estimate of the Bayesian parameters. The results are demonstrated on a signal processing example. It is shown the Hybrid estimation can increase spectral resolution if some prior knowledge is available only on a subset of the parameters.