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

Abstract

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.

Introduction

We will consider the problem of identification of quasi-periodically varying complex-valued systems governed by

$$y\left(t\right)=\sum _{i=1}^n{\theta }_i\left(t\right)u(t-i+1)+v\left(t\right)={\phi }^T\left(t\right)\theta \left(t\right)+v\left(t\right)$$

where $t=1,2,\dots$ denotes the normalized discrete time, $y\left(t\right)$ denotes the system output, $\phi \left(t\right)={[u\left(t\right),\dots ,u(t-n+1)]}^T$ denotes regression vector, made up of the past input samples, $v\left(t\right)$ denotes measurement noise, and $\theta \left(t\right)={[{\theta }_1\left(t\right),\dots ,{\theta }_n\left(t\right)]}^T$ is the vector of time-varying system coefficients, modeled as weighted sums of complex exponentials

$$\theta \left(t\right)=\sum _{k=1}^K{\beta }_k\left(t\right)e^{j{\sum }_{i=1}^t{\omega }_k\left(i\right)}$$$${\beta }_k\left(t\right)={[b_{k1}\left(t\right),\dots ,b_{kn}\left(t\right)]}^T$$$$b_{ki}\left(t\right)=a_{ki}\left(t\right)e^{j{\nu }_{ki}}\text{,}i=1,\dots ,n\text{.}$$

The following three types of real-valued quantities are incorporated in (2): the instantaneous angular frequencies ${\omega }_k\left(t\right)$, the instantaneous amplitudes $a_{ki}\left(t\right)$, and the time-invariant phase shifts ${\nu }_{ki}$. With a slight abuse of terminology, the complex-valued vectors ${\beta }_k\left(t\right)$ 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 $y\left(t\right)$ denotes the sampled baseband signal received by the mobile unit, $\left\{u\left(t\right)\right\}$ denotes the sequence of transmitted symbols, and $v\left(t\right)$ denotes channel noise.

We will assume that the frequencies ${\omega }_k\left(t\right)$ are harmonically related, namely

$${\omega }_k\left(t\right)=m_k{\omega }_0\left(t\right)\text{,}k=1,\dots ,K$$

where ${\omega }_0\left(t\right)$ denotes the slowly varying fundamental frequency and $m_k$ are integer numbers. Such multiple frequencies, called harmonics, appear in the Fourier series expansions of periodic signals. For example, if parameter trajectory $\theta \left(t\right)$ is periodic with period $L$, it admits the following Fourier representation: $\theta \left(t\right)={\sum }_{k=0}^{L-1}{\beta }_k{e}^{jk{\omega }_{0}t},{\omega }_0=\left(2\pi \right)/L$.

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 $m_k,k=1,\dots ,K$, depends on our prior knowledge of the system time variation. When all harmonics are expected to be present, one should set $m_k=k$. In the presence of odd harmonics only, the natural choice is $m_k=2k-1$, etc.

In the special case where $n=1$ and $\phi \left(t\right)\equiv 1$, Eqs. (1), (2), (3) describe a complex-valued harmonic signal $s\left(t\right)=\theta \left(t\right)$ buried in noise

$$y\left(t\right)=s\left(t\right)+v\left(t\right)\text{,}s\left(t\right)=\sum _{k=1}^K{b}_k\left(t\right)e^{j{\sum }_{i=1}^t{\omega }_k\left(i\right)}\text{.}$$

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.