The Cramer-Rao lower bound as a criteria for evaluating a large data reduction system such as LACIE
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Guseman, L. F., Jr.↗
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Cramer-Rao bounds for signal-to-noise ratio and combiner weight estimation
Cramer-Rao lower bounds on estimator variance are calculated for arbitrary unbiased estimates of signal-to-noise ratio and combiner weight parameters. Estimates are assumed to be based on a discrete set of observables obtained by matched filtering of a biphase modulated signal. The bounds are developed first for a problem model based on one observable per channel symbol period, and then extended to a more general problem in which subperiod observables are also available.
Dolinar, S. J.↗
Efficient computation of parameter confidence intervals
An important step in system identification of aircraft is the estimation of stability and control derivatives from flight data along with an assessment of parameter accuracy. When the maximum likelihood estimation technique is used, parameter accuracy is commonly assessed by the Cramer-Rao lower bound. It is known, however, that in some cases the lower bound can be substantially different from the parameter variance. Under these circumstances the Cramer-Rao bounds may be misleading as an accuracy measure. This paper discusses the confidence interval estimation problem based on likelihood ratios, which offers a more general estimate of the error bounds. Four approaches are considered for computing confidence intervals of maximum likelihood parameter estimates. Each approach is applied to real flight data and compared.
Murphy, Patrick C.↗
Maximum likelihood algorithm using an efficient scheme for computing sensitivities and parameter confidence intervals
Improved techniques for estimating airplane stability and control derivatives and their standard errors are presented. A maximum likelihood estimation algorithm is developed which relies on an optimization scheme referred to as a modified Newton-Raphson scheme with estimated sensitivities (MNRES). MNRES determines sensitivities by using slope information from local surface approximations of each output variable in parameter space. The fitted surface allows sensitivity information to be updated at each iteration with a significant reduction in computational effort compared to integrating the analytically-determined sensitivity equations or using a finite difference scheme. An aircraft estimation problem is solved using real flight data to compare MNRES with the commonly used modified Newton-Raphson technique; MNRES is found to be faster and more generally applicable. Parameter standard errors are determined using a random search technique. The confidence intervals obtained are compared with Cramer-Rao lower bounds at the same confidence level. It is observed that the nonlinearity of the cost function is an important factor in the relationship between Cramer-Rao bounds and the error bounds determined by the search technique.
Murphy, P. C.↗
Status of input design for aircraft parameter identification
Results are presented on the design of aircraft inputs (i.e. elevator, rudder and aeleron deflection time histories) to identify aircraft stability and control derivatives from flight test data. The problem is first reduced to an optimization problem with differential and integral constraints. The criteria used are either expressed in terms of the Cramer-Rao lower bound on the covariance matrix of the parameter estimates or in terms of the maximum prediction error variance. Both time-domain longitudinal and lateral dynamics of C-8 and Jet Star aircrafts and comparison with doublet type inputs are made.
Mehra, R. K.↗
Flight test design for efficient extraction of aircraft parameters
This paper considers the design of flight control inputs which result in a minimization of the nonuniqueness problem of parameter identification due to statistical correlation. The performance index for the control design is taken as a linear weighted sum of the squares of the correlation coefficients of the aerodynamic stability and control derivatives as computed from the Cramer-Rao lower bound matrix. An optimal input design is demonstrated for an F-8 aircraft with supercritical wing. Reduced correlations are noted and corresponding estimates and confidence levels presented.
Wells, W. R.↗
Optimum data utilization for parameter identification with application to lifting rotors
The work is concerned with determining the minimum quantity of data needed for achieving best possible accuracy of identified parameters in transient testing of a lifting rotor blade. Specifically, the problem is that of determining from blade flapping transients caused by blade pitch inputs the equivalent Lock number and the equivalent collective pitch setting. For a given time-dependent blade pitch input function, the running time for the test less than which insufficient parameter accuracy is obtained and more than which little accuracy improvement is achieved was calculated with the aid of the Cramer-Rao lower bound for the parameter covariance matrix.
Banerjee, D.↗
Effects of control inputs on the estimation of stability and control parameters of a light airplane
The maximum likelihood parameter estimation technique was used to determine the values of stability and control derivatives from flight test data for a low-wing, single-engine, light airplane. Several input forms were used during the tests to investigate the consistency of parameter estimates as it relates to inputs. These consistencies were compared by using the ensemble variance and estimated Cramer-Rao lower bound. In addition, the relationship between inputs and parameter correlations was investigated. Results from the stabilator inputs are inconclusive but the sequence of rudder input followed by aileron input or aileron followed by rudder gave more consistent estimates than did rudder or ailerons individually. Also, square-wave inputs appeared to provide slightly improved consistency in the parameter estimates when compared to sine-wave inputs.
Cannaday, R. L.↗
Extreme Mean and Its Applications
Extreme value statistics obtained from normally distributed data are considered. An extreme mean is defined as the mean of p-th probability truncated normal distribution. An unbiased estimate of this extreme mean and its large sample distribution are derived. The distribution of this estimate even for very large samples is found to be nonnormal. Further, as the sample size increases, the variance of the unbiased estimate converges to the Cramer-Rao lower bound. The computer program used to obtain the density and distribution functions of the standardized unbiased estimate, and the confidence intervals of the extreme mean for any data are included for ready application. An example is included to demonstrate the usefulness of extreme mean application.
Swaroop, R.↗
On identifiability of flexible structure parameters
This report investigates the identifiability of modal parameters of flexible structures. Expressions are derived for Cramer-Rao lower bounds for the modal parameters, that is, frequencies, damping ratios and mode shapes or slopes. The optimal initial state, which maximizes the trace of the Fisher information matrix in the absence of persistent input, is obtained. The concepts discussed are applied to a finite-element model of the 122 meter hoop/column antenna. The numerical results show that the identifiability of the structural frequencies is excellent, followed by that of the damping ratios and the mode-slopes.
Joshi, S. M.↗
Controller design and parameter identifiability studies for a large space antenna
The problem of control systems synthesis and parameter identifiability are considered for a large, space-based antenna. Two methods are considered for control system synthesis, the first of which uses torque actuators and collocated attitude and rate sensors, and the second method is based on the linear-quadratic-Gaussian (LQG) control theory. The predicted performance obtained by computing variances of pointing, surface and feed misalignment errors in the presence of sensor noise indicates that the LQG-based controller yields superior results. Since controller design requires the knowledge of the system parameters, the identifiability of the structural parameters is investigated by obtaining Cramer-Rao lower bounds. The modal frequencies are found to have the best identifiability, followed by damping ratios, and mode-slopes.
Joshi, S. M.↗
Gas temperature and density measurements based on spectrally resolved Rayleigh-Brillouin scattering
The use of molecular Rayleigh scattering for measurements of gas density and temperature is evaluated. The technique used is based on the measurement of the spectrum of the scattered light, where both temperature and density are determined from the spectral shape. Planar imaging of Rayleigh scattering from air using a laser light sheet is evaluated for ambient conditions. The Cramer-Rao lower bounds for the shot-noise limited density and temperature measurement uncertainties are calculated for an ideal optical spectrum analyzer and for a planar mirror Fabry-Perot interferometer used in a static, imaging mode. With this technique, a single image of the Rayleigh scattered light can be analyzed to obtain density (or pressure) and temperature. Experimental results are presented for planar measurements taken in a heated air stream.
Seasholtz, Richard G.↗
2D velocity and temperature measurements in high speed flows based on spectrally resolved Rayleigh scattering
The use of molecular Rayleigh scattering for measurements of gas velocity and temperature is evaluated. Molecular scattering avoids problems associated with the seeding required by conventional laser anemometry and particle image velocimetry. The technique considered herein is based on the measurement of the spectrum of the scattered light. Planar imaging of Rayleigh scattering using a laser light sheet is evaluated for conditions at 30 km altitude (typical hypersonic flow conditions). The Cramer-Rao lower bounds for velocity and temperature measurement uncertainties are calculated for an ideal optical spectrum analyzer and for a planar mirror Fabry-Perot interferometer used in a static, imaging mode. With this technique, a single image of the Rayleigh scattered light from clean flows can be analyzed to obtain temperature and one component of velocity. Experimental results are presented for planar velocity measurements in a Mach 1.3 air jet.
Seasholtz, Richard G.↗
Error analysis of combined stereo/optical-flow passive ranging
The motion of an imaging sensor causes each imaged point of the scene to correspondingly describe a time trajectory on the image plane. The trajectories of all imaged points are reminiscent of a flow (e.g., of liquid) which is the source of the term 'optical flow'. Optical-flow ranging is a method by which the stream of two-dimensional images obtained from a forward-looking forward-moving passive sensor is used to compute depth (or range) to points in the field of view. Another well-known ranging method consists of triangulation based on stereo images obtained from at least two stationary sensors. In this paper we analyze the potential accuracies of a combined optical flow and stereo passive-ranging system in the context of helicopter nap-of-the-earth obstacle avoidance. The Cramer-Rao lower bound is developed for the combined system under the assumption of an unknown angular bias error common to both cameras of a stereo pair. It is shown that the depth accuracy degradations caused by a bias error is negligible for a combined optical-flow and stereo system as compared to a monocular optical-flow system.
Barniv, Yair↗
Hubble Space Telescope characterized by using phase-retrieval algorithms
We describe several results characterizing the Hubble Space Telescope from measured point spread functions by using phase-retrieval algorithms. The Cramer-Rao lower bounds show that point spread functions taken well out of focus result in smaller errors when aberrations are estimated and that, for those images, photon noise is not a limiting factor. Reconstruction experiments with both simulated and real data show that the calculation of wave-front propagation by the retrieval algorithms must be performed with a multiple-plane propagation rather than a simple fast Fourier transform to ensure the high accuracy required. Pupil reconstruction was performed and indicates a misalignment of the optical axis of a camera relay telescope relative to the main telescope. After we accounted for measured spherical aberration in the relay telescope, our estimate of the conic constant of the primary mirror of the HST was - 1.0144.
Fienup, J. R.↗
An extended-source spatial acquisition process based on maximum likelihood criterion for planetary optical communications
This paper describes an extended-source spatial acquisition process based on the maximum likelihood criterion for interplanetary optical communications. The objective is to use the sun-lit Earth image as a receiver beacon and point the transmitter laser to the Earth-based receiver to establish a communication path. The process assumes the existence of a reference image. The uncertainties between the reference image and the received image are modeled as additive white Gaussian disturbances. It has been shown that the optimal spatial acquisition requires solving two nonlinear equations to estimate the coordinates of the transceiver from the received camera image in the transformed domain. The optimal solution can be obtained iteratively by solving two linear equations. Numerical results using a sample sun-lit Earth as a reference image demonstrate that sub-pixel resolutions can be achieved in a high disturbance environment. Spatial resolution is quantified by Cramer-Rao lower bounds.
Yan, Tsun-Yee↗
Theoretical and simulated performance for a novel frequency estimation technique
A low complexity, open-loop, discrete-time, delay-multiply-average (DMA) technique for estimating the frequency offset for digitally modulated MPSK signals is investigated. A nonlinearity is used to remove the MPSK modulation and generate the carrier component to be extracted. Theoretical and simulated performance results are presented and compared to the Cramer-Rao lower bound (CRLB) for the variance of the frequency estimation error. For all signal-to-noise ratios (SNR's) above threshold, it is shown that the CRLB can essentially be achieved with linear complexity.
Crozier, Stewart N.↗
Error Analysis of Combined Optical-Flow and Stereo Passive Ranging
The motion of an imaging sensor causes each imaged point of the seem to describe a time trajectory on the image plane. The trajectories of all imaged points are reminiscent of a flow (eg, of liquid) which is the source of the term "optical flow". Optical-flow ranging is a method by which the stream of two-dimensional images obtained from a forward-looking forward-moving passive sensor is used to compute range to points in the field of view. Another well-known ranging method consists of triangulation based on stereo images obtained from at least two stationary sensors. In this paper we analyze the potential accuracies of a combined optical flow and stereo passive-ranging system in the context of helicopter nap-of-the-earth obstacle avoidance. The Cramer-Rao lower bound (CRLB) is developed for the combined system under the assumption of a random angular misalignment common to both cameras of a stereo pair. It is shown that the range accuracy degradations caused by misalignment is negligible for a combined optical-flow and stereo system as compared with a monocular optical-flow system.
Barniv, Yair↗