Recursive filters

Infinite impulse response filter -> recursive filter... 

Infinite impulse response filter recursive filter Infrared spectroscopy spectroscopy... 

With a new data point Zk+, the updated state yk+ k+ and its covariance matrix Xyjc + iiit +1 can be obtained by using the following Kalman filter recursive formulae (Kalman and Bucy 1961) ... 

N. B. a has the inverse role of a in the first derivative of a Gaussian. Deriche proposes the following recursive implementation of the filter/in two dimensions. Deriche retains the same solution as Canny, that is ... 

The script file kalfild.m solves, in forward-time, the discrete solution of the Kalman filter equations, using equations (9.74), (9.75) and (9.76) in a recursive process. The MATLAB command Iqed gives the same result. 

Raman, V. and Pitsch, H., Large-eddy simulation of a bluff-body-stabilized non-premixed flame using a recursive filter-refinement procedure. Combust. Flame, 142, 329, 2005. 

In this chapter we discuss the principles of the Kalman filter with reference to a few examples from analytical chemistry. The discussion is divided into three parts. First, recursive regression is applied to estimate the parameters of a measurement equation without considering a systems equation. In the second part a systems equation is introduced making it necessary to extend the recursive regression to a Kalman filter, and finally the adaptive Kalman filter is discussed. In the concluding section, the features of the Kalman filter are demonstrated on a few applications. 

Before we introduce the Kalman filter, we reformulate the least-squares algorithm discussed in Chapter 8 in a recursive way. By way of illustration, we consider a simple straight line model which is estimated by recursive regression. Firstly, the measurement model has to be specified, which describes the relationship between the independent variable x, e.g., the concentrations of a series of standard solutions, and the dependent variable, y, the measured response. If we assume a straight line model, any response is described by ... 

The recursive property of the Kalman filter allows the detection of such model deviations, and offers the possibility of disregarding the measurements in the region where the model is invalid. This filter is the so-called adaptive Kalman filter. 

Historically, treatment of measurement noise has been addressed through two distinct avenues. For steady-state data and processes, Kuehn and Davidson (1961) presented the seminal paper describing the data reconciliation problem based on least squares optimization. For dynamic data and processes, Kalman filtering (Gelb, 1974) has been successfully used to recursively smooth measurement data and estimate parameters. Both techniques were developed for linear systems and weighted least squares objective functions. 

As was previously shown, Kalman filtering techniques can be, and have been, successfully used on dynamic process data, to smooth measurement data recursively and... 

In this chapter different aspects of data processing and reconciliation in a dynamic environment were briefly discussed. Application of the least square formulation in a recursive way was shown to lead to the classical Kalman filter formulation. A simpler situation, assuming quasi-steady-state behavior of the process, allows application of these ideas to practical problems, without the need of a complete dynamic model of the process. 

Friedland, B. (1969). Treatment of bias in recursive filtering. IEEE Trans. Autom. Control AC-14,359-367. Gelb, A. (1974). Applied Optimal Estimation. MIT Academic Press, Cambridge, MA. 

However, there is a price to pay in a spectral transform Lanczos algorithm At each recursion step, the action of the filter operator onto the Lanczos vectors has to be evaluated. In the original version, Ericsson and Ruhe update the Lanczos vectors by solving the following linear equation ... 

Interestingly, the spectral transform Lanczos algorithm can be made more efficient if the filtering is not executed to the fullest extent. This can be achieved by truncating the Chebyshev expansion of the filter,76,81 or by terminating the recursive linear equation solver prematurely.82 In doing so, the number of vector-matrix multiplications can be reduced substantially. 

PIST distinguishes itself from other spectral transform Lanczos methods by using two important innovations. First, the linear equation Eq. [38] is solved by QMR but not to a high degree of accuracy. In practice, the QMR recursion is terminated once a prespecified (and relatively large) tolerance is reached. Consequently, the resulting Lanczos vectors are only approximately filtered. This inexact spectral transform is efficient because many less matrix-vector multiplications are needed, and its deficiencies can subsequently... 

The major shortcoming of the spectral method is the rate of convergence. Its ability to resolve eigenvalues is restricted by the width of the filter, which in turn is inversely proportional to the length of the Fourier series (the uncertainty principle). Thus, to accurately characterize an eigenpair in a dense spectrum, one might have to use a very long Chebyshev recursion. 

The scaling laws of FD are dominated by the recursion because diagonalization of small matrices is relatively inexpensive. However, because one must store multiple filtered vectors along the recursion, FD could be a burden for large systems. 

The filtering, namely the construction of energy local bases, can also be carried out using the Lanczos recursion or similar recursive methods. However, filtered vectors at E/ can only be obtained using the Green filter ... 

HOC1,309,310 HArF,311 and C1HC1.71 Most of these calculations were carried out using either the complex-symmetric Lanczos algorithm or filter-diagonali-zation based on the damped Chebyshev recursion. The convergence behavior of these two algorithms is typically much less favorable than in Hermitian cases because the matrix is complex symmetric. 

Recursion Polynomial Expansion of the Green s Function with Absorbing Boundary Conditions Calculations of Resonances of HCO by Filter Diagonalization. 

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