Cramer–Rao lower bounds for change points in additive and multiplicative noise

The paper addresses the problem of determining the Cramer–Rao lower bounds (CRLBs) for noise and change-point parameters, for steplike signals corrupted by multiplicative and/or additive white noise. Closed-form expressions for the signal and noise CRLBs are first derived for an ideal step with a known change point. For an unknown change-point, the noise-free signal is modeled by a sigmoidal function parametrized by location and step rise parameters. The noise and step change CRLBs corresponding to this model are shown to be well approximated by the more tractable expressions derived for a known change-point. The paper also shows that the step location parameter is asymptotically decoupled from the other parameters, which allows us to derive simple CRLBs for the step location. These bounds are then compared with the corresponding mean square errors of the maximum likelihood estimators in the pure multiplicative case. The comparison illustrates convergence and efficiency of the ML estimator. An extension to colored multiplicative noise is also discussed

Probabilistic state estimation with discrete-time chaotic systems

Includes bibliographical references (p. 89-90).Supported by the Defense Advanced Research Projects Agency, monitored by the U.S. Navy Office of Naval Research. N00014-89-J-1489 Supported by the U.S. Air Force Office of Scientific Research. AFOSR-91-0034-A Supported by a subcontract from Lockheed Sanders, Inc., under U.S. Navy Office of Naval Research grant. N00014-91-C-0125Michael D. Richard

Approximate Gaussian conjugacy: parametric recursive filtering under nonlinearity, multimodality, uncertainty, and constraint, and beyond

Since the landmark work of R. E. Kalman in the 1960s, considerable efforts have been devoted to time series state space models for a large variety of dynamic estimation problems. In particular, parametric filters that seek analytical estimates based on a closed-form Markov–Bayes recursion, e.g., recursion from a Gaussian or Gaussian mixture (GM) prior to a Gaussian/GM posterior (termed ‘Gaussian conjugacy’ in this paper), form the backbone for a general time series filter design. Due to challenges arising from nonlinearity, multimodality (including target maneuver), intractable uncertainties (such as unknown inputs and/or non-Gaussian noises) and constraints (including circular quantities), etc., new theories, algorithms, and technologies have been developed continuously to maintain such a conjugacy, or to approximate it as close as possible. They had contributed in large part to the prospective developments of time series parametric filters in the last six decades. In this paper, we review the state of the art in distinctive categories and highlight some insights that may otherwise be easily overlooked. In particular, specific attention is paid to nonlinear systems with an informative observation, multimodal systems including Gaussian mixture posterior and maneuvers, and intractable unknown inputs and constraints, to fill some gaps in existing reviews and surveys. In addition, we provide some new thoughts on alternatives to the first-order Markov transition model and on filter evaluation with regard to computing complexity

Modified Cramér-Rao lower bound for TOA and symbol width estimation. An application to search and rescue signals

International audienceThis paper focuses on the performance of time of arrival estimators for distress beacon signals which are defined by pulses with smooth transitions. These signals are used in the satellite-based search and rescue Cospas-Sarsat system. We propose a signal model based on sigmoidal functions. Closed-form expressions for the modified Cramér-Rao bounds associated with the parameters of this model are derived. The obtained expressions are easy to interpret since they analytically depend on the system parameters. Simulations conducted on realistic search and rescue signals show good agreement with the theoretical results

Performance analysis of symbol timing estimators for time-varying MIMO channels

The purpose of this thesis is to derive and analyze the theoretical limits for estimatingthe symboltiming delayof Multiple-Input Multiple-Output (MIMO)systems. Two main N X M system models are considered, where N represents the number of transmit antennas and M denotes the number of receive antennas, the 2 X 2 system used by S.-A. Yangand J. Wu and the 4 X 4system used by Y.-C. Wu and E. Serpedin. The second model has been extended to take into account the symbol time-varying fading. The theoretical estimation limits are shown by several bounds: modified Cramer-Rao bound (MCRB), Cramer-Rao bound (CRB) and Barankin bound (BB). BB will be exploited to obtain accurate information regarding the necessary length of data to obtain good estimation. Two scenarios for synchronization are presented: data-aided (DA) and non-data-aided (NDA). Two models for the fading process are considered: block fading and symbol time-varying fading, respectively, the second case being assumed to be Rayleigh distributed. The asymptotic Cramer-Rao bounds for low signal-to-noise ratio (low-SNR) and for high-SNR are derived and the performance of several estimators is presented. The performance variation of bounds and estimators is studied byvarying different parameters, such as the number of antennas, the length of data taken into consideration during the estimation process, the SNR, the oversampling factor, the power and the Doppler frequency shift of the fading

Cramer-Rao bound analysis of multi-frame blind deconvolution

This thesis explores how support constraints and multiple frames affect multi-frame blind deconvolution. Previous research in non-blind deconvolution, which seeks to estimate an object from a blurred and noisy image, characterized how the use of support constraints exploited spatial noise correlations to reduce noise in the estimate of the object. In multi-frame blind deconvolution, the blurring function is unknown and must be estimated along with the object. Applying a support constraint to both the object and the blurring functions, when using blind deconvolution, is one way to ensure a unique solution. The effects on the estimate of the object as a function of the size of the supports are analyzed. Also, the benefit in noise reduction in the estimate of the object from including multiple blurred and noisy images is considered. Cramer-Rao Bound theory is employed to provide an algorithm-independent metric to analyze the effects from these parameters. The Cramer-Rao bound is a lower limit to the variance of any estimate of an unknown parameter. In this research, the unknown parameters are the intensities of the object which is estimated