Maciej Niedwiecki

Estimation and tracking of complex-valued quasi-periodically varying systems

The problem of identification/tracking of quasi-periodically varying complex systems is considered. This problem is a generalization, to the system case, of a classical signal processing task of either elimination or extraction of nonstationary sinusoidal signals buried in noise. The proposed solution is based on the exponentially weighted basis function (EWBF) approach. First, the basic EWBF algorithm is derived. Then its frequency-decoupled, parallel-form and cascade-form variants, with highly modular structure and reduced computational requirements, are described. Finally, the frequency-adaptive versions of all schemes are obtained using the recursive prediction error method.

Optimal and suboptimal smoothing algorithms for identification of time-varying systems with randomly drifting parameters

Noncausal estimation algorithms, which involve smoothing, can be used for off-line identification of nonstationary systems. Since smoothing is based on both past and future data, it offers increased accuracy compared to causal (tracking) estimation schemes, incorporating past data only. It is shown that efficient smoothing variants of the popular exponentially weighted least squares and Kalman filter-based parameter trackers can be obtained by means of backward-time filtering of the estimates yielded by both algorithms. When system parameters drift according to the random walk model and the adaptation gain is sufficiently small, the properly tuned two-stage Kalman filtering/smoothing algorithm, derived in the paper, achieves the Cramér–Rao type lower smoothing bound, i.e. it is the optimal noncausal estimation scheme. Under the same circumstances performance of the modified exponentially weighted least-squares algorithm is often only slightly inferior to that of the Kalman filter-based smoother. c

Brief paper: On the lower smoothing bound in identification of time-varying systems

In certain applications of nonstationary system identification the model-based decisions can be postponed, i.e. executed with a delay. This allows one to incorporate in the identification process not only the currently available information, but also a number of “future” data points. The resulting estimation schemes, which involve smoothing, are not causal. Assuming that the infinite observation history is available, the paper establishes the lower steady-state estimation bound for any noncausal estimator applied to a linear system with randomly drifting coefficients (under Gaussian assumptions). This lower bound complements the currently available one, which is restricted to causal estimators. 2007 Elsevier Ltd. All rights reserved.

On noncausal weighted least squares identification of nonstationary stochastic systems

In this paper, we consider the problem of noncausal identification of nonstationary, linear stochastic systems, i.e., identification based on prerecorded input/output data. We show how several competing weighted (windowed) least squares parameter smoothers, differing in memory settings, can be combined together to yield a better and more reliable smoothing algorithm. The resulting parallel estimation scheme automatically adjusts its smoothing bandwidth to the unknown, and possibly time-varying, rate of nonstationarity of the identified system. We optimize the window shape for a certain class of parameter variations and we derive computationally attractive recursive smoothing algorithms for such an optimized case.

Parallel frequency tracking with built-in performance evaluation

The problem of estimation of instantaneous frequency of a nonstationary complex sinusoid (cisoid) buried in wideband noise is considered. The proposed approach employs a bank of adaptive notch filters, extended with a nontrivial performance assessment mechanism which automatically chooses the best performing filter in the bank. Additionally, a computationally attractive method of implementing the bank is proposed. The new structure allows one to improve tracking results considerably, especially in nonstationary conditions. In terms of accuracy of frequency estimates, the proposed scheme outperforms existing ones considerably.

On cheap smoothing opportunities in identification of time-varying systems

In certain applications of nonstationary system identification the model-based decisions can be postponed, i.e. executed with a delay. This allows one to incorporate into the identification process not only the currently available information, but also a number of “future” data points. The resulting estimation schemes, which involve smoothing, are not causal. Despite the possible performance improvements, the existing smoothing algorithms are seldom used in practice, mainly because of their high computational requirements. We show that the computationally attractive smoothing procedures can be obtained by means of compensating estimation delays that arise in the standard exponentially weighted least squares, least mean squares and Kalman filter-based parameter trackers. c

Generalized adaptive comb filters/smoothers and their application to the identification of quasi-periodically varying systems and signals

The problem of both causal and noncausal identification of linear stochastic systems with quasi-harmonically varying parameters is considered. The quasi-harmonic description allows one to model nonsinusoidal quasi-periodic parameter changes. The proposed identification algorithms are called generalized adaptive comb filters/smoothers because in the special signal case they reduce down to adaptive comb algorithms used to enhance or suppress nonstationary harmonic signals embedded in noise. The paper presents a thorough statistical analysis of generalized adaptive comb algorithms, and demonstrates their statistical efficiency in the case where the fundamental frequency of parameter changes varies slowly with time according to the integrated random-walk model.

Brief paper: Generalized adaptive notch filters with frequency debiasing for tracking of polynomial phase systems

Generalized adaptive notch filters are used for identification/tracking of quasi-periodically varying dynamic systems and can be considered an extension, to the system case, of classical adaptive notch filters. For general patterns of frequency variation the generalized adaptive notch filtering algorithms yield biased frequency estimates. We show that when system frequencies change slowly in a smooth way, the estimation bias can be substantially reduced by means of post-filtering of the frequency estimates. The modified (debiased) algorithm has better tracking capabilities than the original algorithm.

Multichannel self-optimizing narrowband interference canceller

The problem of cancellation of a nonstationary sinusoidal interference, acting at the output of an unknown multivariable linear stable plant, is considered. No reference signal is assumed to be available. The proposed feedback controller is a nontrivial extension of the SONIC (self-optimizing narrowband interference canceller) algorithm, developed earlier for single-input, single-output plants. The algorithm consists of two loops: the inner, control loop, which predicts and cancels disturbance, and the outer, self-optimization loop, which automatically adjusts the gain matrix so as to optimize the overall system performance. The proposed scheme is capable of adapting to slow changes in disturbance characteristics, measurement noise characteristics, and plant characteristics. It is shown that in the important benchmark case - for disturbances with random-walk-type amplitude changes - the designed closed-loop control system converges locally in mean to the optimal one. The algorithm, derived and analyzed assuming a single-tone, complex-valued disturbance with known frequency, can be extended to cope with a range of realistic applications, such as real-valued disturbances, multitone signals, and unknown frequency.