The Cramer-Rao bound for pole and amplitude coefficient estimatesof damped exponential signals in noise

A complete Cramer-Rao bound (CRB) derivation is provided for the case in which signals consist of arbitrary exponential terms in noise. Expressions for the CRBs of the parameters of a damped exponential model with one set of poles and multiple sets of amplitude coefficients are derived. CRBs for the poles and amplitude coefficients are derived in terms of rectangular and polar coordinate parameters. For rectangular parameters it is shown that CRBs for the real and imaginary parts of poles and amplitude coefficients are equal and uncorrelated. In polar coordinates, the angle and magnitude CRBs are also uncorrelated. Furthermore, the CRBs of the pole angles and relative magnitudes are equal and are logarithmically symmetric about the unit circle     

On the cramer-rao bound for carrier frequency estimation in the presence of phase noise

We consider the carrier frequency offset estimation in a digital burst-mode satellite transmission affected by phase noise. The corresponding Cramer-Rao lower bound is analyzed for linear modulations under a Wiener phase noise model and in the hypothesis of knowledge of the transmitted data. Even if we resort to a Monte Carlo average, from a computational point of view the evaluation of the Cramer-Rao bound is very hard. We introduce a simple but very accurate approximation that allows to carry out this task in a very easy way. As it will be shown, the presence of the phase noise produces a remarkable performance degradation of. the frequency estimation accuracy. In addition, we provide asymptotic expressions of the Cramer-Rao bound, from which the effect of the phase noise and the dependence on the system parameters of the frequency offset estimation accuracy clearly result. Finally, as a by-product of our derivations and approximations, we derive a couple of estimators specifically tailored for the phase noise channel that will be compared with the classical Rife and Boorstyn algorithm, gaining in this way some important hints on the estimators to be used in this scenario     

Maximum likelihood and lower bounds in system identification withnon-Gaussian inputs

We consider the problem of estimating the parameters of an unknown discrete linear system driven by a sequence of independent identically distributed (i.i.d.) random variables whose probability density function (PDF) may be non-Gaussian. We assume a general system structure that may contain causal and noncausal poles and zeros. The parameters characterizing the input PDF may also be unknown. We derive an asymptotic expression for the Cramer-Rao lower bound, and show that it is the highest (worst) in the Gaussian case, indicating that the estimation accuracy can only be improved when the input PDF is non-Gaussian. It is further shown that the asymptotic error variance in estimating the system parameters is unaffected by lack of knowledge of the PDF parameters, and vice verse. Computationally efficient gradient-based algorithms for finding the maximum likelihood estimate of the unknown system and PDF parameters, which incorporate backward filtering for the identification of non-causal parameters, are presented. The dual problem of blind deconvolution/equalization is considered, and asymptotically attainable lower bounds on the equalization performance are derived. These bounds imply that it is preferable to work with compact equalizer structures characterized by a small number of parameters as the attainable performance depend only on the total number of equalizer parameters     

Realizable lower bounds for time delay estimation

The accuracy of time delay estimates obtainable in active localization systems is studied, focusing on the effect of ambiguities in the time delay estimates. Such ambiguities occur when the transmitted signal has small relative bandwidth. Then, for signal to noise ratios below a certain threshold, the commonly used Cramer-Rao lower bound is not realizable. The study concentrates on the region of intermediate SNR values, where the Cramer-Rao bound is no longer achievable, but useful information on time delays can still be obtained from the measurement. Realizable bounds for the single and two echo cases are obtained by deriving a new form of the Barankin (1949) bound for active time delay estimation. This form maintains the realizability property of the most general form, but is of reasonable complexity. New bounds are derived for the multiple echo case. Examples are presented to illustrate the dependence of the bound on parameters such as SNR, relative bandwidth, and echo separation     

Two-Dimensional Polynomial Phase Signals: Parameter Estimation and Bounds

This paper considers the problem of parametric modeling and estimation of nonhomogeneous two-dimensional (2-D) signals. In particular, we focus our study on the class of constant modulus polynomial-phase 2-D nonhomogeneous signals. We present two different phase models and develop computationally efficient estimation algorithms for the parameters of these models. Both algorithms are based on phase differencing operators. The basic properties of the operators are analyzed and used to develop the estimation algorithms. The Cramer-Rao lower bound on the accuracy of jointly estimating the model parameters is derived, for both models. To get further insight on the problem we also derive the asymptotic Cramer-Rao bounds. The performance of the algorithms in the presence of additive white Gaussian noise is illustrated by numerical examples, and compared with the corresponding exact and asymptotic Cramer-Rao bounds. The algorithms are shown to be robust in the presence of noise, and their performance close to the CRB, even at moderate signal to noise ratios.     

Maximum-likelihood parameter estimation of the harmonic,evanescent, and purely indeterministic components of discretehomogeneous random fields

This paper presents a maximum-likelihood solution to the general problem of fitting a parametric model to observations from a single realization of a two-dimensional (2-D) homogeneous random field with mixed spectral distribution. On the basis of a 2-D Wold-like decomposition, the field is represented as a sum of mutually orthogonal components of three types: purely indeterministic, harmonic, and evanescent. The suggested algorithm involves a two-stage procedure. In the first stage, we obtain a suboptimal initial estimate for the parameters of the spectral support of the evanescent and harmonic components. In the second stage, we refine these initial estimates by iterative maximization of the conditional likelihood of the observed data, which is expressed as a function of only the parameters of the spectral supports of the evanescent and harmonic components. The solution for the unknown spectral supports of the harmonic and evanescent components reduces the problem of solving for the other unknown parameters of the field to a linear least squares. The Cramer-Rao lower bound on the accuracy of jointly estimating the parameters of the different components is derived, and it is shown that the bounds on the purely indeterministic and deterministic components are decoupled. Numerical evaluation of the bounds provides some insight into the effects of various parameters on the achievable estimation accuracy. The performance of the maximum-likelihood algorithm is illustrated by Monte Carlo simulations and is compared with the Cramer-Rao bound     

Generalized pencil-of-function method for extracting poles of an EMsystem from its transient response

A generalized pencil-of-function (GPOF) method is developed for extracting the poles of an electromagnetic system from its transient response. The GPOF method needs the solution of a generalized eigenvalue problem to find the poles. Subspace decomposition is also used to optimize the performance of the GPOF method. The GPOF method has advantages over the Prony method in both computation and noise sensitivity, and approaches the Cramer-Rao bound when the signal-to-noise ratio (SNR) is above threshold. An application of the GPOF method to a thin-wire target is presented     

On the accuracy of estimating the parameters of a regularstationary process

Any regular stationary random processes can be represented as the sum of a purely indeterministic process and a deterministic one. This paper considers the achievable accuracy in the joint estimation of the parameters of these two components, from a single observed realization of the process. An exact form of the Cramer-Rao bound (CRB) is derived, as well as a conditional CRB. The relationships between these bounds, and their relations to the previously derived asymptotic bound, are explored by analysis and numerical examples     

Application of the cramer-rao bound to target motion analysis

Theoretical limits to the accuracy with which target trajectory parameters can be estimated from stochastic data are derived via the Cramer-Rao bound. As examples of their use, the limits are applied to a simple one-dimensional tracking exercise and to a more complicated bearings-only tracking problem.     

2-D moving average models for texture synthesis and analysis

A random field model based on moving average (MA) time-series model is proposed for modeling stochastic and structured textures. A frequency domain algorithm to synthesize MA textures is developed, and maximum likelihood estimators are derived. The Cramer-Rao lower bound is also derived for measuring the estimator accuracy. The estimation algorithm is applied to real textures, and images resembling natural textures are synthesized using estimated parameters.     

Bounds for estimation of complex exponentials in unknown colorednoise

We consider the problem of estimating the parameters of complex exponentials in the presence of complex additive Gaussian noise with unknown covariance. Bounds are derived for the accuracy of jointly estimating the parameters of the exponentials and the noise. We first present an exact Cramer-Rao bound (CRB) for this problem and specialize it for the cases of circular Gaussian processes and autoregressive processes. We also derive an approximate expression for the CRB, which is related to the conditional likelihood function. Numerical evaluation of these bounds provides some insights on the effect of various signal and noise parameters on the achievable estimation accuracy     

Bounds on the accuracy of estimating the parameters of discretehomogeneous random fields with mixed spectral distributions

This paper considers the achievable accuracy in jointly estimating the parameters of a real-valued two-dimensional (2-D) homogeneous random field with mixed spectral distribution, from a single observed realization of it. On the basis of a 2-D Wold-like decomposition, the field is represented as a sum of mutually orthogonal components of three types: purely indeterministic, harmonic, and evanescent. An exact form of the Cramer-Rao lower bound on the error variance in jointly estimating the parameters of the different components is derived. It is shown that the estimation of the harmonic component is decoupled from that of the purely indeterministic and the evanescent components. Moreover, the bound on the parameters of the purely indeterministic and the evanescent components is independent of the harmonic component. Numerical evaluation of the bounds provides some insight into the effects of various parameters on the achievable estimation accuracy     

The asymptotic CRLB for the spectrum of ARMA processes

This paper addresses the issue of quantifying the frequency domain accuracy of autoregressive moving average (ARMA) spectral estimates as dictated by the Cramer-Rao lower bound (CRLB). Classical work in this area has led to expressions that are asymptotically exact as both data length and model order tend to infinity, although they are commonly used in finite model order and finite data length settings as approximations. More recent work has established quantifications that, for AR models, are exact for finite model order. By employing new analysis methods based on rational orthonormal parameterizations, together with the ideas of reproducing kernel Hilbert spaces, this paper develops quantifications that extend this previous work by being exact for finite model order in all of the AR, MA, and ARMA system cases. These quantifications, via their explicit dependence on poles and zeros of the underlying spectral factor, reveal certain fundamental aspects of the accuracy achievable by spectral estimates of ARMA processes.     

Parameter estimation of two-dimensional moving average randomfields

This paper considers the problem of estimating the parameters of two-dimensional (2-D) moving average random (MA) fields. We first address the problem of expressing the covariance matrix of nonsymmetrical half-plane, noncausal, and quarter-plane MA random fields in terms of the model parameters. Assuming the random field is Gaussian, we derive a closed-form expression for the Cramer-Rao lower bound (CRLB) on the error variance in jointly estimating the model parameters. A computationally efficient algorithm for estimating the parameters of the MA model is developed. The algorithm initially fits a 2-D autoregressive model to the observed field and then uses the estimated parameters to compute the MA model. A maximum-likelihood algorithm for estimating the MA model parameters is also presented. The performance of the proposed algorithms is illustrated by Monte-Carlo simulations and is compared with the Cramer-Rao bound     

A radar application of a modified Cramer-Rao bound: parameterestimation in non-Gaussian clutter

In this paper, we derive a lower bound on the error covariance matrix for any unbiased estimator of the parameters of a signal composed of a mixture of spherically invariant random processes (SIRPs). The proposed approach represents a special case of the global Cramer-Rao bound for hybrid random and deterministic parameters estimation, and it is particularly useful when the data, conditioned on a vector of unwanted random parameters (nuisance parameters) with a priori known probability density function, can be modeled as a Gaussian vector. The case of signal composed of a mixture of K-distributed clutter, Gaussian clutter, and thermal noise belongs to this set, and it is regarded as a realistic radar scenario. In the radar problem considered here, this bound can be numerically computed in closed-form, whereas the computation of the true (marginal) Cramer-Rao bound turns out to be infeasible. The performance of some practical estimators are compared with it for two study cases     

On the computation of an asymptotic bound for estimating autoregressive signals in white noise

The Cramer-Rao lower bound (CRLB) provides a useful reference for evaluating the performance of parameter estimation techniques. This paper considers the problem of estimating the parameters of an autoregressive signal corrupted by white noise. An explicit formula is derived for computing the asymptotic CRLB for the signal and noise parameters. Formulas for the asymptotic CRLB for functions of the signal and noise parameters are also presented. In particular, the center frequency, bandwidth and power of a second order process are considered. Some numerical examples are presented to illustrate the usefulness of these bounds in studying estimation accuracy.     

Bootstrap Resampling for Image Registration Uncertainty Estimation Without Ground Truth

We address the problem of estimating the uncertainty of pixel based image registration algorithms, given just the two images to be registered, for cases when no ground truth data is available. Our novel method uses bootstrap resampling. It is very general, applicable to almost any registration method based on minimizing a pixel-based similarity criterion; we demonstrate it using the SSD, SAD, correlation, and mutual information criteria. We show experimentally that the bootstrap method provides better estimates of the registration accuracy than the state-of-the-art Cramer-Rao bound method. Additionally, we evaluate also a fast registration accuracy estimation (FRAE) method which is based on quadratic sensitivity analysis ideas and has a negligible computational overhead. FRAE mostly works better than the Cramer-Rao bound method but is outperformed by the bootstrap method.     

Estimation of amplitude and phase parameters of multicomponentsignals

This paper considers the problem of estimating signals consisting of one or more components of the form a(t)e/sup jφ(t/), where the amplitude and phase functions are represented by a linear parametric model. The Cramer-Rao bound (CRB) on the accuracy of estimating the phase and amplitude parameters is derived. By analyzing the CRB for the single-component case, if is shown that the estimation of the amplitude and the phase are decoupled. Numerical evaluation of the CRB provides further insight into the dependence of estimation accuracy on signal-to-noise ratio (SNR) and the frequency separation of the signal components. A maximum likelihood algorithm for estimating the phase and amplitude parameters is also presented. Its performance is illustrated by Monte-Carlo simulations, and its statistical efficiency is verified     

A simple derivation of the constrained multiple parameterCramer-Rao bound

J.D. Gorman and A.O. Hero (1988) obtained a remarkable extension to the classical multiple parameter Cramer-Rao (CR) lower bound that accounts for deterministic nonlinear equality constraints on the parameters. The virtue of Gorman and Hero's result is that the constrained CR bound on all of the parameters is obtained by subtracting an easily computed nonnegative definite correction matrix from the unconstrained CR bound matrix. The author presents a new, simple derivation of the constrained CR bound and a new necessary condition for an estimator to satisfy the constrained CR bound with equality