Showing papers on "Kalman filter published in 1986"

Adaptive Filter Theory

Abstract: Background and Overview. 1. Stochastic Processes and Models. 2. Wiener Filters. 3. Linear Prediction. 4. Method of Steepest Descent. 5. Least-Mean-Square Adaptive Filters. 6. Normalized Least-Mean-Square Adaptive Filters. 7. Transform-Domain and Sub-Band Adaptive Filters. 8. Method of Least Squares. 9. Recursive Least-Square Adaptive Filters. 10. Kalman Filters as the Unifying Bases for RLS Filters. 11. Square-Root Adaptive Filters. 12. Order-Recursive Adaptive Filters. 13. Finite-Precision Effects. 14. Tracking of Time-Varying Systems. 15. Adaptive Filters Using Infinite-Duration Impulse Response Structures. 16. Blind Deconvolution. 17. Back-Propagation Learning. Epilogue. Appendix A. Complex Variables. Appendix B. Differentiation with Respect to a Vector. Appendix C. Method of Lagrange Multipliers. Appendix D. Estimation Theory. Appendix E. Eigenanalysis. Appendix F. Rotations and Reflections. Appendix G. Complex Wishart Distribution. Glossary. Abbreviations. Principal Symbols. Bibliography. Index.

Estimation of Object Motion Parameters from Noisy Images

Abstract: An approach is presented for the estimation of object motion parameters based on a sequence of noisy images. The problem considered is that of a rigid body undergoing unknown rotational and translational motion. The measurement data consists of a sequence of noisy image coordinates of two or more object correspondence points. By modeling the object dynamics as a function of time, estimates of the model parameters (including motion parameters) can be extracted from the data using recursive and/or batch techniques. This permits a desired degree of smoothing to be achieved through the use of an arbitrarily large number of images. Some assumptions regarding object structure are presently made. Results are presented for a recursive estimation procedure: the case considered here is that of a sequence of one dimensional images of a two dimensional object. Thus, the object moves in one transverse dimension, and in depth, preserving the fundamental ambiguity of the central projection image model (loss of depth information). An iterated extended Kalman filter is used for the recursive solution. Noise levels of 5-10 percent of the object image size are used. Approximate Cramer-Rao lower bounds are derived for the model parameter estimates as a function of object trajectory and noise level. This approach may be of use in situations where it is difficult to resolve large numbers of object match points, but relatively long sequences of images (10 to 20 or more) are available.

State-space models

Abstract: This chapter reviews the usefulness of the Kalman filter for parameter estimation and inference about unobserved variables in linear dynamic systems. Applications include exact maximum likelihood estimation of regressions with ARMA disturbances, time-varying parameters, missing observations, forming an inference about the public's expectations about inflation, and specification of business cycle dynamics. The chapter also reviews models of changes in regime and develops the parallel between such models and linear state-space models. The chapter concludes with a brief discussion of alternative approaches to nonlinear filtering.

Estimation, Prediction, and Interpolation for ARIMA Models with Missing Data

Abstract: We show how to define and then compute efficiently the marginal likelihood of an ARIMA model with missing observations. The computation is carried out by using the univariate version of the modified Kalman filter introduced by Ansley and Kohn (1985a), which allows a partially diffuse initial state vector. We also show how to predict and interpolate missing observations and obtain the mean squared error of the estimate.

An Introduction to Kalman Filters

Abstract: Kalman filters are a powerful tool for reducing the effects of noise in measurements. This paper gives a no-nonsense introduction to the subject for people with A-level maths. The basic ideas of getting better estimates from many measurements are simply introduced. Thereafter, the basics are gradually developed, using worked examples, to a full Kalman filter. Wherever possible, variations, simplifications, and applications are given in the hope that the reader will be encouraged to use Kalman filter techniques

A recursive modified Gram-Schmidt algorithm for least- squares estimation

Abstract: This paper presents a recursive form of the modified Gram-Schmidt algorithm (RMGS). This new recursive least-squares (RLS) estimation algorithm has a computational complexity similar to the conventional RLS algorithm, but is more robust to roundoff errors and has a highly modular structure, suitable for VLSI implementation. Its properties and features are discussed and compared to other LS estimation algorithms.

A Kalman filtering approach to short-time Fourier analysis

Abstract: The problem of estimating time-varying harmonic components of a signal measured in noise is considered. The approach used is via state estimation. Two methods are proposed, one involving pole-placement of a state observer, the other using quadratic optimization techniques. The result is the development of a new class of filters, akin to recursive frequency-sampling filters, for inclusion in a parallel bank to produce sliding harmonic estimates. Kalman filtering theory is applied to effect the good performance in noise, and the class of filters is parameterized by the design tradeoff between noise rejection and convergence rate. These filters can be seen as generalizing the DFT.

Toward the Application of the Kalman Filter to Regional Open Ocean Modeling

Abstract: A partial differential equation model is defined for ocean meteorological prediction and synoptic analysis. The Kalman filter used for data assimilation is described and applied to the one-dimensional linear barotropic quasi-geostrophic model with periodic and open boundary conditions. The model accounts for eddy scale dynamics in the ocean. The assumptions made in the forecast model are discussed, along with comparisons of the error variances expected with the filter and from an objective analysis method. The effectiveness of the Kalman filter is demonstrated and subsequent efforts to extend the filter to two dimensions are indicated.

Improvement of the fast recursive least-squares algorithms via normalization: A comparative study

Abstract: This paper deals with the derivation and the properties of fast optimal least-squares algorithms, and particularly with their normalization It is shown how the well-known fast Kalman algorithm, written in the most general form, can be normalized through a purely algebraic point of view, leading to the normalized least-squares transversal filter derived by Cioffi, Kailath, and Lev-Ari from the geometric approach An improved form of the algorithm is presented The different algorithms have been compared from a practical point of view as regards their convergence, initialization procedures, complexity, and numerical properties Normalized transversal algorithms are shown to be interesting because of their nice structured form, simplicity of conception, and improved good numerical behavior

Map-aided navigation system employing TERCOM-SITAN signal processing

Abstract: A navigation system contains a modified Kalman filter processor which continuously receives both TERCOM and SITAN control information so that the operation of the SITAN processing is effectively continuously optimized. The system employs an over flight terrain data storage map to which position and altitude signals are coupled for extracting both elevation and slope information from the stored map. The extracted elevation and slope data are coupled, together with the outputs of baromatic and radar altimeter sensors and estimated altitude and position data outputs from the navigation unit, to a correlation/modified Kalman filter processor. Elevation profile data accessed from the map is correlated with elevation profile signals derived from aircraft on-board sensors, in order to find that flight path on the stored map, parallel to, but displaced from the flight path indicated by the navigation unit, for which successive elevation data values correspond most closely to the elevations measured by the altimeter sensors. In this correlation processing operation the "most likely" path is selected by defining a performance index associated with each path and selecting that path with the best performance index. The result of the correlation processing provides a position fix to be combined with the position estimates in the modified Kalman filter.

Time-Series Methods and Recursive Estimation in Hydrological Systems Analysis

Abstract: Previous chapters have shown how models of hydrological systems can be formulated in many different ways and with various levels of complexity. In this chapter, we will see how these kinds of models can be considered within a unified stochastic setting and how it is then possible to treat model calibration as a problem of time-series analysis. In this manner, powerful time-series techniques, such as recursive estimation (Young 1984) can be used in the identification, estimation and validation of the models. And, because ot their inherently stochastic nature, such models can subsequently provide a natural vehicle for real-time flow forecasting. Moreover, the recursive approach to estimation allows for continuous updating of the model parameter estimates and the possibility of more advanced “self-adaptive” forecasting and control procedures.

Closed-loop properties of the infinite-time linear-quadratic optimal regulator for systems with delays

Abstract: The infinite-time linear-quadratic optimal-control problem for systems with delays is discussed from the viewpoint of closed-loop properties: the Kalman equation and the circle condition for the optimal closed-loop control law are derived, and the optimal closed-loop poles are characterized via the hamiltonian matrix.

Identification of Time Varying Systems and Application of System Identification to Signal Processing

Abstract: Part IA new approach to identification of time varying systems is presented, and evaluated using computer simulations. The new approach is built upon the similarities between recursive least squares identification and Kalman filtering.The parameter variations are modelled as process noise in a state space model and then identified using adaptive Kalman filtering. A method for adaptive Kalman filtering is derived and analysed. The simulations indicate that this new approach is superior to previous methods based on adjusting the forgetting factor. This improvement is however gained at the price of a signification increase in computational complexity.Part IIIn this part we apply parameter estimation to the problem of transmission line protection.One approach based on recursive least squares identification is presented. The method has ben tested using simulated data generated by the program EMTP.Another approach based on the theory of travelling waves is also discussed.Part IIIIn this part a method for input estimation or deconvolution is presented. The basis of the method is to use a parametrized model the input signal. To use the method we should thus be able to express the input signal as a function of some unknown parameters and time. The algorithms simultaneously estimates the parameters of the input signal and the parameters of the system transfer function. The presentation here is restricted to transfer functions of all pole type, i.e. ARX-models. The method can be extended to handle zeros in the transfer function. The computational burden would however increase significantly. The algorithm uses efficient numerical methods, as for instance QR-factorization thorugh Householder transformation.The algorithm is in this paper applied to a problem in speech coding. It has been observed that the quality of synthesized speech can be improved, if a more detailed model than an impulse train is used for the pitch pulses, see Fant (1980). It is here shown how the method presented in this paper can be used to estimate the system parameters of the speech production and the parameters of the glottal pulse simultaneously.

Use of optimal estimation theory, in particular the Kalman filter, in data analysis and signal processing

Abstract: The Kalman filter, a powerful and useful optimal estimation technique, does not seem to be widely known among physicists. Here we outline the derivation of the algorithm, and give three examples of its use: (a) in estimating the value of a constant, with both system and measurement noise, (b) in numerical differentiation of noisy data, and (c) in optimally estimating the amplitude of a signal with arbitrary but known time dependence superimposed on a noisy background.

Robust adapative Kalman filtering for systems with unknown step inputs and non-Gaussian measurement errors

Abstract: Target tracking with Kalman filters is hampered by target maneuvering and unknown process and measurement noises. We show that moving data windows may be used to analyze state and measurement error sequences, determining robust estimates of bias and covariance. For steps in the system forcing functions and non-Gaussian measurement errors, the robust estimators yield improvements over linear bias and covariance estimators. Extensive simulations compare conventional, linear adaptive, and robust adaptive average step responses of a first-order system filter. Quantities examined are state estimate, state error, process and measurement covariance estimates, Kalman gain, and input step estimate.

A Maximum Likelihood Method for Estimating EEG Evoked Potentials

Abstract: The problem of EEG evoked potential (EP) estimation is basically one of estimating a transient signal embedded in nonstationary mostly additive noise; and as such it is well suited to a nonstationary estimation approach utilizing, for example, the Kalman filter. The method presented in this paper is based on a model of the EEG response which is assumed to be the sum of the EP and independent correlated Gaussian noise representing the spontaneous EEG activity. The EP is assumed to vary in both shape and latency; the latter is assumed to be governed by some unspecified probability density; and no assumption on stationarity is needed for the noise. With the model described in state-space form, a Kalman filter is constructed, and the variance of the innovation process is derived; a maximum likelihood solution to the EP estimation problem is then obtained via this innovation process. The method was tested on simulated as well as real EEG data.

Polytopes, a novel approach to tracking

Abstract: Standard target tracking techniques such as Kalman filters or maximum liklihood estimators approach nonlinearities from an essentially local point of view; that is, they determine a single solution even though the problem may admit more than one. This lack of uniqueness may be due to the absence of global observability as in Doppler tracking where several isolated solutions can occur, or a result of imperfect measurements producing multiple minima in a cost function. The latter is particulary significant since noisy data often produce situations in which local minima abound, trapping a maximum liklihood steepest descent search or causing an extended Kalman filter to diverge. This paper introduces a novel tracking algorithm based on a global approach utilizing containment regions approximated by four-dimensional polytopes. The algorithm is presented in the context of bearings-only tracking, the various tradeoffs with conventional techniques are examined, and potential applications discussed.

State-deviation-estimation circuit employing a phase-locked-loop phase reference

Abstract: A system (34) for determining deviations in the state of motion of a projectile (10) from its intended state includes a comparison module (38) that receives the outputs S of a sensor array (36). The comparison module (38) converts the sensor outputs to a measurement vector Z m and computes the deviation of this measurement vector from an intended measurement vector Z 0 received from a control system. The comparison module (38) then determines the difference E Z between this measured deviation and the deviation predicted by a Kalman filter (44, 46). In generating the measurement vector Z m from the outputs of the sensor array (36), the comparison module (38) "de-spins" the array outputs in accordance with the output of a phase reference (42), whose purpose is to indicate the phase with respect to gravity of the spin of the projectile (10) about its longitudinal axis. The Kalman filter's state-deviation estimator (46) weights the vector output of the comparison module (36) and adds it to the output of the Kalman filter's state-deviation predictor (44) to provide an updated state-deviation estimation E X (t:t), which the state-deviation predictor (44) then uses as a basis for its next cycle. The output of the comparison module (38) constitutes a measure of the performance of the state-deviation predictor (44) and is used as a correction factor for the phase reference (42) so that the system acts as a phase-locked loop to lock the phase reference (42) onto the projectile spin without the need for an external sensor to determine the direction of gravity.

Time series regression with unequally spaced data

Abstract: Regression analysis with stationary errors is extended to the case when observations are not equally spaced. The errors are modelled as either a discrete-time ARMA process with missing observations, or as a continuous-time autoregression with observational error observed at arbitrary times. Using a state-space representation, a Kalman filter is used to calculate the exact likelihood. The linear regression coefficients are separated out of the likelihood so non-linear optimization is required only with respect to the parameters modelling the error structure.

Analysis of Tree Ring Data Using the Kalman Filter Technique

Abstract: A statistical mcthod is presented to filter the influence of weather variations out of a tree ring chronology. The Kaiman filter technique is introduced to estimate a multiple regression model with stochastically fluctuating weather parameters. It cnables the detection of any change in response of trees to weather. The method is in two ways an improvement upon the frequentIy used method of response functions: I) it is not necessary to assume constant model parameters, and 2) the estimation process is not based on the fitting but on the forecast performance of the model.

Stochastic adaptive control for exponentially convergent time-varying systems

Abstract: This paper shows that the standard stochastic adaptive control algorithms for time-invariant systems have an inherent robustness property which renders them applicable, without modification, to time-varying systems whose parameters converge exponentially. One class of systems satisfying this requirement is those having non-steady-state Kalman filter or innovations representations. This allows the usual assumption of a stationary ARMAX representation to be replaced by a more general state space model.

Assessment of Kalman-filter channel estimators for an HF radio link

Abstract: The paper compares a Kalman-filter channel estimator with an alternative and much simpler estimator, for a synchronous serial data-transmission system operating at 9600 bit/s over a model of a voiceband HF radio link. A 16-level QAM signal is transmitted over the voiceband link, the latter having two independent Rayleigh fading sky waves, with a relative transmission delay of 2 ms and a frequency spread of 1 Hz. Five arrangements of the Kalman estimator and two arrangements of the alternative estimator are studied. Following a description of the various systems, the results of computer simulation tests are presented, comparing the accuracies of the estimates of the sampled impulse response of the channel given by the different estimators. It is shown that the alternative estimator is potentially much more cost-effective than the Kalman estimator.