Generalized adaptive comb filters/smoothers and their application to the identification of quasi-periodically varying systems and signals☆
Introduction
We will consider the problem of identification of quasi-periodically varying complex-valued systems governed by where denotes the normalized discrete time, denotes the system output, denotes regression vector, made up of the past input samples, denotes measurement noise, and is the vector of time-varying system coefficients, modeled as weighted sums of complex exponentials The following three types of real-valued quantities are incorporated in (2): the instantaneous angular frequencies , the instantaneous amplitudes , and the time-invariant phase shifts . With a slight abuse of terminology, the complex-valued vectors will be further referred to as ‘complex amplitudes’.
Under certain circumstances (in the presence of several strong reflectors) the model (1)–(2) can be used to describe rapidly fading mobile radio channels (Bakkoury et al., 2000, Giannakis and Tepedelenlioǧlu, 1998, Tsatsanis and Giannakis, 1996). In this case denotes the sampled baseband signal received by the mobile unit, denotes the sequence of transmitted symbols, and denotes channel noise.
We will assume that the frequencies are harmonically related, namely where denotes the slowly varying fundamental frequency and are integer numbers. Such multiple frequencies, called harmonics, appear in the Fourier series expansions of periodic signals. For example, if parameter trajectory is periodic with period , it admits the following Fourier representation: .
The notion of ‘time-varying harmonics’ can be regarded as a natural extension of the Fourier analysis to quasi-periodically varying systems, such as (1)–(2). The choice of the multipliers , depends on our prior knowledge of the system time variation. When all harmonics are expected to be present, one should set . In the presence of odd harmonics only, the natural choice is , etc.
In the special case where and , Eqs. (1), (2), (3) describe a complex-valued harmonic signal buried in noise The problem of either elimination or extraction of harmonic signals buried in noise can be solved using adaptive comb filters (Nehorai and Porat, 1986, Regalia, 1995). For this reason the system identification/tracking algorithm described below can be considered a generalized comb filter.
The problem of causal identification (tracking) of single-mode quasi-periodically varying systems was studied in Niedźwiecki and Kaczmarek, 2004, Niedźwiecki and Kaczmarek, 2005a, Niedźwiecki and Kaczmarek, 2005b.
In the recent conference paper (Niedźwiecki & Meller, 2011a), the results presented earlier were extended to noncausal identification (smoothing). Additionally, a more sophisticated frequency estimation scheme was proposed, incorporating frequency rate tracking/smoothing and yielding better results in practice.
All papers published so far focus on the identification of single-mode systems, i.e., systems with parameters that can be modeled as complex sinusoids (cisoids) with slowly varying amplitudes and a slowly varying instantaneous frequency.
This paper extends results presented in Niedźwiecki and Meller (2011a) to nonstationary systems with quasi-harmonically varying parameters, i.e., to systems with several frequency modes governed by the same slowly varying fundamental frequency. In practice such harmonic modes of variation often arise in oscillatory systems with nonlinear elements and/or loads (Neimark, 2003).
In principle, quasi-harmonically varying systems can be identified using the multiple-frequency versions of the algorithms mentioned above. Such algorithms are made up of several single-frequency sub-algorithms that work in parallel and are driven by the common prediction error. Since the estimated frequencies are in this case regarded as mutually unrelated quantities, the harmonic structure of the system/signal time variation is not exploited in any way. In this paper we present algorithms that take advantage of such a prior information, i.e., the algorithms that perform a coordinated frequency search. This allows one to improve estimation results considerably.
Section snippets
Generalized adaptive notch filter — overview of known results
Suppose that the identified nonstationary system has a single frequency mode , i.e., it is governed by where and are slowly varying quantities. Furthermore, suppose that:
- (A1)
The measurement noise is a zero-mean circular white sequence with variance .
- (A2)
The sequence of regression vectors , independent of , is nondeterministic, wide-sense stationary and ergodic with known correlation matrix2
Multiple-frequency GANF
Denote by , where , and is an estimate of , the output of this subsystem of (1) which is associated with the frequency . If the signals were measurable, one could design independent GANF algorithms of the form (6), each taking care of a particular subsystem. Since it holds that , the final parameter estimate could be easily obtained by combining the partial estimates.
Even though the outputs are
Generalized adaptive comb filter
In order to arrive at the algorithm which performs a coordinated search of the instantaneous fundamental frequency , one should minimize for where . Note that This leads to the following recursive estimation scheme which will be further referred to as a generalized adaptive comb filter (GACF)
Generalized adaptive comb smoother
The important consequence of the fact that the approximate error equations (13)–(14) are identical with those derived in Niedźwiecki and Meller (2011a) for systems with a single frequency mode of parameter variation, is that the smoothing technique proposed there is directly applicable to the multiple-frequency case. Following Niedźwiecki and Meller (2011a), suppose that a pre-recorded data block of length is available, which is typical of off-line applications,
Optimization and Cramér–Rao bounds
Consider a system (1), (2), (3) with pseudo-linear frequency changes. In order to achieve the best tracking/smoothing results, the adaptation gains of the GACF/GACS algorithms should be chosen so as to trade-off the bias and variance components in (19)–(20) and (21)–(22). Such optimal settings depend exclusively on the balance between the bias and variance error components, determined by the scalar coefficient further referred to as the rate of
ALF-based analysis
To check the validity of the analytical expressions (19)–(20), based on the approximating linear filter equations (13)–(14), the following two-tap FIR system (inspired by channel equalization applications) was simulated where denotes a white 4-QAM [quadrature amplitude modulation — see e.g. Giannakis and Tepedelenlioǧlu (1998)] input sequence and denotes a complex-valued Gaussian measurement noise.
Each of impulse response
Conclusion
The problem of identification of linear stochastic systems with quasi-harmonically varying parameters was considered. Both causal and noncausal identification algorithms were derived, referred to as generalized adaptive comb filters (GACFs) and generalized adaptive comb smoothers (GACSs), respectively. In both cases the frequency and frequency rate estimation properties of the proposed algorithms were analyzed using the method of approximating linear filter. It was shown, and later confirmed by
Maciej Niedźwiecki was born in Poznań, Poland in 1953. He received the M.Sc. and Ph.D. degrees from the Gdańsk University of Technology, Gdańsk, Poland, and the Dr. Hab. (D.Sc.) degree from the Technical University of Warsaw, Warsaw, Poland, in 1977, 1981 and 1991, respectively.
He spent three years as a Research Fellow with the Department of Systems Engineering, Australian National University from 1986 to 1989. From 1990 to 1993 he served as a Vice-Chairman of the Technical Committee on Theory
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Maciej Niedźwiecki was born in Poznań, Poland in 1953. He received the M.Sc. and Ph.D. degrees from the Gdańsk University of Technology, Gdańsk, Poland, and the Dr. Hab. (D.Sc.) degree from the Technical University of Warsaw, Warsaw, Poland, in 1977, 1981 and 1991, respectively.
He spent three years as a Research Fellow with the Department of Systems Engineering, Australian National University from 1986 to 1989. From 1990 to 1993 he served as a Vice-Chairman of the Technical Committee on Theory of the International Federation of Automatic Control (IFAC). He is currently a Professor and Head of the Department of Automatic Control, Faculty of Electronics, Telecommunications and Computer Science, Gdańsk University of Technology. His main areas of research interest include system identification, statistical signal processing and adaptive systems. He is the author of the book Identification of Time-varying Processes (Wiley, 2000).
Dr. Niedźwiecki is currently Associate Editor for IEEE Transactions on Signal Processing, a member of the IFAC committees on Modeling, Identification and Signal Processing and on Large Scale Complex Systems, and a member of the Automatic Control and Robotics Committee of the Polish Academy of Sciences (PAN).
Michał Meller received the M.Sc. and Ph.D. degrees in Automatic Control from the Gdańsk University of Technology, Gdańsk, Poland, in 2007 and 2010, respectively. Since 2007 he has been working in the Department of Signal and Information Processing, Bumar Elektronika, Gdańsk Division. In 2010 he also joined the Department of Automatic Control at the Gdańsk University of Technology, Faculty of Electronics, Telecommunications and Computer Science. His professional interests include signal processing and adaptive systems.
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This work was supported by the National Science Center. The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Wolfgang Scherrer under the direction of Editor Torsten Söderström.
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