Linear time-invariant space-variant filters and the parabolic equation approximation

Wave propagation in a random, inhomogeneous ocean is treated as transmission through a linear, time-invariant, space-variant, random communication channel. Using the parabolic equation approximation of the Helmholtz wave equation, a random transfer function of the ocean volume is derived. The ocean volume is characterized by a three-dimensional random index of refraction which is decomposed into deterministic and random components. Two additional calculations are performed using the transfer function. The first involves the derivation of the equations for the random, output electrical signals at each element in a receive planar array of complex weighted point sources in terms of the frequency spectrum of the transmitted electrical signal, the transmit and receive arrays, and the transfer function of the ocean medium. The second involves the derivation of the coherence function.     

Robust matched filters

The problem of designing robust matched filters for situations in which there is uncertainty in the signal structure or noise statistics is considered. Two general aspects of this problem are treated. First, maximin robust designs are characterized for a general Hilbert-space formulation of the matched filtering problem and explicit solutions are given for two intuitively appealing models for uncertainty. These results are seen to generalize earlier results for specific matched filtering problems. The second general aspect treated is the application of the theoretical maximin results to the particular problem of designing filters to combat uncertain nonlinear channel distortion. Channel distortion is modeled by considering an unknown received signal which may differ inL_{2}-norm from the transmitted or nominal signal by no more than a fixed amount. The effect of the channel distortion is seen to be equivalent to that of adding a white noise to the channel whose spectral height depends on the degree of distortion. General expressions are developed for the determination of the robust filter and its performance, and numerical results are presented for the case of baseband detection in Gauss-Markov noise.     

Analysis of stability and performance of adaptation algorithms with time-invariant gains

Adaptation laws that track parameters of linear regression models are investigated. The considered class of algorithms apply linear time-invariant filtering on the instantaneous gradient vector and includes least mean squares (LMS) as its simplest member. The asymptotic stability and steady-state tracking performance for prediction and smoothing estimators is analyzed for parameter variations described by stochastic processes with time-invariant statistics. The analysis is based on a novel technique that decomposes the inherent feedback of adaptation algorithms into one time-invariant loop and one time-varying loop. The impact of the time-varying feedback on the tracking error covariance can be neglected under certain conditions, and the performance analysis then becomes straightforward. Performance analysis in the presence of a non-negligible time-varying feedback is performed for algorithms that use scalar measurements. Convergence in mean square error (MSE) and the MSE tracking performance is investigated, assuming independent consecutive regression vectors. Closed-form expressions for the tracking MSE are thereafter derived without this independence assumption for a subclass of algorithms applied to finite impulse response (FIR) models with white inputs. This class includes Wiener LMS adaptation.     

Robust microphone arrays using subband adaptive filters

A new adaptive beamformer which combines the idea of subband processing and the generalised sidelobe canceller structure is presented. The proposed subband beamformer has a blocking matrix that uses coefficient-constrained subband adaptive filters to limit target cancellation within an allowable range of direction of arrival. Simulations comparing the fullband and subband adaptive beamformers show that the subband beamformer has better performance than the fullband beamformer when the target and/or interfering signals are coloured. In reverberant environments, the proposed subband beamformer also performs better than its fullband counterpart     

Losslessness and stability of LDI ladders and filters

The stability of digital ladder filters close in form, via the lossless discrete integrator (LDI) transformation, to doubly terminatedLC Cauer ladder low-pass filters is studied. The LDI transformation does not necessarily map stable analogue into stable digital filters. Necessary and sufficient conditions for these digital filter to be lossless and for a corresponding doubly terminated filter to be stable are given. LDIs are often used in the design of switched capacitor (SC) filters. For this case we provide a threshold sampling rate above which the SC LDI filter retains the stability of the analogue filter. In spite of the stability problem with the LDI mapping, it has been observed that when applied to simulate analogue filters the resulting filter is stable. It can now be argued that, in these cases, other factors in the determination of the sampling rate leads typically to a choice that is also sufficiently above the threshold rate for stability. The theory described is also useful to derive more general digital filters and to perform the design of stable LDI filters exclusively in the Z-plane without reference to analogue prototypes.This work was partly supported by the U.S.-Israel Binational Science Foundation under Grant No. 88-425.     

On the stability of bilinear filters

We consider stability properties of discrete-time bilinear filters. Simple sufficient conditions are given for bounded-input bounded-output stability (with not necessarily zero initial conditions),l _p stability, and three other important types of stability. In particular, conditions are given under which asymptotically periodic inputs produce asymptotically periodic outputs with the same period.     

Optimizing Spatial filters for Robust EEG Single-Trial Analysis

Due to the volume conduction multichannel electroencephalogram (EEG) recordings give a rather blurred image of brain activity. Therefore spatial filters are extremely useful in single-trial analysis in order to improve the signal-to-noise ratio. There are powerful methods from machine learning and signal processing that permit the optimization of spatio-temporal filters for each subject in a data dependent fashion beyond the fixed filters based on the sensor geometry, e.g., Laplacians. Here we elucidate the theoretical background of the common spatial pattern (CSP) algorithm, a popular method in brain-computer interface (BCD research. Apart from reviewing several variants of the basic algorithm, we reveal tricks of the trade for achieving a powerful CSP performance, briefly elaborate on theoretical aspects of CSP, and demonstrate the application of CSP-type preprocessing in our studies of the Berlin BCI (BBCI) project.     

Toward high stability in active filters

About four years ago, the consensus among network theorists was that active synthesis was headed nowhere because of high sensitivities. At that time, active filters had a well-deserved reputation among design engineers for poor stability. Practical active filters were plagued with self-oscillations, nonadjustability, nonreproducibility, and high thermal coefficients. Active network theory could accomplish nothing except explain the poor results. Since then, however, the picture has changed from total gloom to overjoyed optimism. The sensitivity problem now has several solutions, and active filters are more stable than passive filters, at least in some cases.     

Asymptotic stability of two-dimensional digital filters underquantization

The stability of 2-D digital filters implemented with two's complement quantization is investigated. The difference equation model is considered for the following cases: (i) quantization after each product and (ii) quantization after a complete sum of products. For both cases, sufficient conditions are established for asymptotic stability. The 2-D state-space Roesser's (1975) model is also considered under two's complement quantization, and a sufficient condition is derived for its stability. Some examples are given to illustrate the results     

On the stability of two-dimensional continuous filters

The Hermite stability criterion for one-dimensional filters is applied to prove the stability theorems for two-dimensional continuous filters. It is shown that the problem in the stability theorems for the two-dimensional continuous filters when the leading coefficient becomes zero is inherent in the Hermite stability criterion for the one-dimensional filters. It is in sharp contrast to the Schur-Cohn criterion which does not have such a problem.     

Exponential stability of filters and smoothers for hidden Markovmodels

We address the problem of filtering and fixed-lag smoothing for discrete-time and discrete-state hidden Markov models (HMMs), with the intention of extending some important results in Kalman filtering, notably the property of exponential stability. By appealing to a generalized Perron-Frobenius result for non-negative matrices, we are able to demonstrate exponential forgetting for both the recursive filters and smoothers; furthermore, methods for deriving overbounds on the convergence rate are indicated. Simulation studies for a two-state and two-output HMM verify qualitatively some of the theoretical predictions, and the observed convergence rate is shown to be bounded in accordance with the theoretical predictions     

Robust pole assignment and optimal stability margins

A robust pole assignment by linear state feedback is achieved in state-space representation by selecting a feedback which minimises the conditioning of the assigned eigenvalues of the closed-loop system. It is shown here that when this conditioning is minimised, a lower bound on the stability margin in the frequency domain is maximised.     

Robust design of envelope-constrained filters in the presence ofinput uncertainty

The maximum robustness design of envelope-constrained filters with uncertain input (ECUI) is formulated as a minimax optimization problem in which the set of feasible filters is characterized by nonsmooth matrix inequalities. An efficient design algorithm is developed to solve this problem by use of a newly developed transformation technique. The attractive feature of this algorithm is that the ECUI filter thus designed can allow for the greatest uncertainty in the input signal while still achieving the acceptable filter performance     

Robust least-squares estimation with a relative entropy constraint

Given a nominal statistical model, we consider the minimax estimation problem consisting of finding the best least-squares estimator for the least favorable statistical model within a neighborhood of the nominal model. The neighborhood is formed by placing a bound on the Kullback-Leibler (KL) divergence between the actual and nominal models. For a Gaussian nominal model and a finite observations interval, or for a stationary Gaussian process over an infinite interval, the usual noncausal Wiener filter remains optimal. However, the worst case performance of the filter is affected by the size of the neighborhood representing the model uncertainty. On the other hand, standard causal least-squares estimators are not optimal, and a characterization is provided for the causal estimator and the corresponding least favorable model. The causal estimator takes the form of a risk-sensitive estimator with an appropriately selected risk sensitivity coefficient.     

Exponential asymptotic stability of time-varying inverse predictionerror filters

It is a classical result of linear prediction theory that as long as the minimum prediction error variance is nonzero, the transfer function of the optimum linear prediction error filter for a stationary process is minimum phase, and therefore, its inverse is exponentially stable. Here, extensions of this result to the case of nonstationary processes are investigated. In that context, the filter becomes time-varying, and the concept of “transfer function” ceases to make sense. Nevertheless, we prove that under mild condition on the input process, the inverse system remains exponentially stable. We also consider filters obtained in a deterministic framework and show that if the time-varying coefficients of the predictor are computed by means of the recursive weighted least squares algorithm, then its inverse remains exponentially stable under a similar set of conditions     

Linear stability analysis of dispersion-managed solitons controlledby filters

We present a linear stability analysis of dispersion-managed (DM) solitons controlled by inline narrow-band filters. We show that the filters can destabilize the pulse if they are unsuitedly located in the dispersion map, which is contrary to the case of standard solitons in fibers with constant dispersion. We also show that for such an instability to take place the pulse energy and the dispersion-map strength should be significantly larger than those usually required for practical long-distance transmission. The filter-induced instability of DM solitons will be an issue in the operation of stretched-pulse mode-locked fiber lasers     

Robust stability analysis for current-programmed regulators

Uncertainty models for the three basic switch-mode converters: buck, boost, and buck-boost are given in this paper. The resulting models are represented by linear fractional transformations with structured dynamic uncertainties. Uncertainties are assumed for the load resistance R=R/sub O/(1+/spl delta//sub R/), inductance L=L/sub O/(1+/spl delta//sub L/), and capacitance C=C/sub O/(1+/spl delta//sub C/). The interest in these models is clearly motivated by the need to have models for switch-mode DC-DC converters that are compatible with robust control analysis, which require a model structure consisting of a nominal model and a norm-bounded modeling uncertainty. Therefore, robust stability analysis can be realized using standard /spl mu/-tools. At the end of the paper, an illustrative example is given which shows the simplicity of the procedure.     

Robust stability condition for continuous-time systems

New linear matrix inequalities (LMIs) are presented for robust stability of uncertain continuous-time systems. The LMIs do not involve any product of the Lyapunov matrix and the system dynamic matrix thus enables the checking of stability using parameter-dependent Lyapunov matrices. The condition is less conservative than previously presented conditions. The reduction in conservativeness is achieved by the introduction of an instrumental variable. A simple numerical example is given to illustrate the results.     

Robust schur stability with polynomial parameters

We propose a stability test for discrete-time systems whose coefficients depend polynomially on some bounded parameters. The test is a particular form of Positivstellensatz, appeals to sum-of-squares polynomials and can be implemented as a semidefinite programming problem. Although implementable only in relaxed form, due to the necessity of limiting the degrees of the polynomial variables involved, the experiments show a good accuracy with degrees smaller than for other tests.     

Finite word length effect and stability of multidimensional digital filters

In this paper, the effect of multiplier quantization on unstable transition in n-dimensional filters is studied using a qualitative model. This model is used to determine a priori, for a worst-case design, the finite word length of the multipliers sufficient to retain Stability.