Elsevier

Signal Processing

Volume 93, Issue 5, May 2013, Pages 1210-1220
Signal Processing

Efficient adaptive identification of linear-in-the-parameters nonlinear filters using periodic input sequences

https://doi.org/10.1016/j.sigpro.2012.12.012Get rights and content

Abstract

This paper introduces computationally efficient NLMS and RLS adaptive algorithms for identifying non-recursive, linear-in-the-parameters (LIP) nonlinear systems using periodic input sequences. The algorithms presented in the paper are exact and require a real-time computational effort of a single multiplication, an addition and a subtraction per input sample. The transient, steady state, and tracking behavior of the algorithms as well as the effect of model mismatch is studied in the paper. The low computational complexity, good performance and broad applicability make the approach of this paper a valuable alternative to the current techniques for nonlinear system identification.

Highlights

► Adaptive algorithms for linear-in-the-parameters (LIP) nonlinear filters are discussed. ► NLMS and RLS algorithms for LIP filters using periodic sequences are derived. ► Computational cost is only three arithmetic operations per sample time. ► Transient, steady state, and tracking behavior of the algorithms is analyzed. ► Effects of a model mismatch between unknown and adaptive system are also analyzed.

Introduction

This paper considers the problem of identifying or tracking non-recursive linear-in-the-parameters (LIP) nonlinear systems. This class of nonlinear systems and filters is composed of all systems whose input–output relationships depend linearly on the system parameters [1]. The class includes, in addition to linear filters, truncated Volterra filters [2], [3], extended Volterra filters [4], FLANN filters [5] based on trigonometric, polynomial, and piece-wise linear expansions [6], [7], [8], [9], radial basis function networks [10], [11], interpolated Volterra filters [12], [13], [14], generalized memory polynomial filters [15], and many other nonlinear structures [1]. Different approaches have been proposed in the literature to identify these systems. A fundamental difficulty with the identification of these nonlinear systems is the complexity of the system model and the correspondingly large computational complexity of the identification algorithm. For example, a generic pth order truncated Volterra system model with N sample memory has O(Np) coefficients. The lowest computational complexity for adaptive identification and tracking available today is O(Np) arithmetical operations per input signal sample. Another fundamental problem is the relatively slow learning speed of the identification algorithm. Even when the input signal is white Gaussian, the autocorrelation matrix of the input data is often non-diagonal and ill-conditioned [3].

Recently, the first author of this paper presented an algorithm for the identification and tracking of linear FIR systems using periodic input sequences [16]. The algorithm was derived on the basis of the early work of Antweiler on the NLMS algorithm with perfect periodic inputs1 [17], [18], [19]. In [16], efficient NLMS and RLS algorithms that have a real-time computational effort of a single multiplication, an addition and a subtraction per input sample were discussed. These algorithms do not evaluate the coefficients of the underlying system directly. Instead, they determine the coefficients of an equivalent representation, from which the impulse response can be easily computed. The paper also showed that the algorithms have convergence and tracking properties that can be better than or comparable to the NLMS algorithm for white noise input.

In this paper, we extend the approach of [16] to the identification and tracking of nonrecursive LIP nonlinear systems (some preliminary results were presented in [20]). The resulting systems preserve the low computational complexity as well as the good convergence and tracking properties exhibited by their linear counterparts. The derivation of the algorithms as well as the analysis techniques shares some similarity with those for linear systems. However, there are substantial differences in the influence of the input signal as well as its design for the nonlinear case. Similarly, there are differences in the transient, steady-state and tracking analyses as well as model mismatch analyses between the linear and nonlinear cases. We will refer to the derivations in [16] in cases they are similar to the linear case and concentrate our attention on aspects that are novel with respect to the linear case. Of particular interest is the discussion of the characteristics and design of an ideal periodic sequence suitable for nonlinear system identification.

The properties of pseudorandom multilevel sequences, i.e., of periodic sequences, used to identify Volterra and extended Volterra filters were studied in [4]. It was shown that a pseudorandom multilevel sequence of degree D [4] can persistently excite an extended Volterra filter of order P and memory length N if and only if it has at least P+1 distinct levels and DN. An efficient algorithm for the least-square parameter estimation was also proposed in [4]. In [21], a Wiener model was estimated using a multilevel sequence. The approach of this paper differs from those of [4], [21] in two ways: (1) our derivations are applicable to the broader class of LIP nonlinear filters and (2) this paper deals with the adaptive NLMS and RLS algorithms.

The paper is organized as follows. In Section 2 the class of LIP nonlinear filters is reviewed and a representation using periodic inputs is introduced. In Section 3 the efficient NLMS and RLS algorithms are derived. The transient, steady-state and tracking behaviors of the algorithms are analyzed in Section 4. The effect of a model mismatch between the unknown system and the adaptive filter is discussed in Section 5. Simulation results are presented in Section 6. Concluding remarks are given in Section 7.

Throughout the paper, lowercase boldface letters denote vectors, uppercase boldface letters denote matrices, the symbol indicates the Kronecker product, E[·] denotes the statistical expectation, · denotes the Euclidean norm, ·F represents the Frobenius norm, · is the largest integer smaller than or equal to the argument, amodb is the remainder of the division of a by b, Cond2(X) is the condition number in the 2-norm of matrix X, CondF(X) is the condition number in the Frobenius norm of X, XT is the transposed inverse of X, and I is an identity matrix.

Section snippets

A review of LIP nonlinear filters

The algorithms described in the next section apply to non-recursive LIP models with a finite memory of N samples. The input–output relationship of such models can be expressed in the vector form asy(n)=hT(n)xF(n),where h(n) is a length M coefficient vector and xF(n) is an input data vector of the same length. Let us assume that the vectorxF(n)=[xF1(n),xF2(n),,xFM(n)]is composed of M terms formed with any linear or nonlinear combination and/or nonlinear expansion of the N most recent samples of

Efficient NLMS and RLS algorithms

To derive the NLMS algorithm, we apply the standard procedure used to develop the LMS algorithm and we show that it results in a normalized LMS algorithm for the filter structure of this paper.

We want to find the coefficients ci(n) that minimize the following minimum-mean-square cost function:J(n)=E[(d(n)y(n))2],where d(n) is the desired signal and y(n) is as given in (11).

The coefficients are adapted with the gradient methodci(n+1)=ci(n)μ2J(n)ci(n).By approximating J(n) with (d(n)y(n))2

Transient, steady-state, and tracking analyses

In this section, we first study the transient and steady-state behavior of the algorithms in the identification of time-invariant LIP systems. Later we study the algorithms' tracking properties in the case of time-varying LIP systems. Finally, we discuss the design of periodic sequences suitable for identification and tracking of LIP nonlinear systems. We consider algorithms of the form (20), noting that replacing ρ(m) by μ in (20) leads to the NLMS algorithm, and when ρ(m) varies as defined

Model mismatch analysis

Here we analyze the behavior of the algorithms when there is a model mismatch between the adaptive filter and the unknown nonlinear system. This analysis is performed under the simplifying assumption that there is no measurement noise, i.e., ν(n)=0. Under this condition, the NLMS algorithm with μ=1 and the RLS algorithm converge in just M samples. Sincec(M)=[d(0),d(1),,d(M1)]T=d,we have from (10) that the adaptive filter converges toh(M)=Wd.We proceed by considering four different cases.

Case 1

The

Identification of nonlinear systems

In this subsection we provide some simulation results for the identification of a second order FLANN filter of memory length 50 samples, a second order truncated Volterra filter of memory length 20 samples, and a second order extended Volterra filter of memory length 5 samples without the constant term.3 All filters have approximately the same number of coefficients, i.e. 250 for the FLANN

Concluding remarks

This paper discussed computationally efficient, exact NLMS and RLS algorithms for identifying and tracking a class of nonlinear systems that are linear in their parameters. The algorithms require a computational effort of just a multiplication, an addition and a subtraction per input sample. We analyzed the transient, steady-state, and tracking behavior of the algorithms and derived exact expressions for the steady-state values of the MSE(n), MSDc(n), and MSDh(n). We also studied the behavior

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