Adaptive IIR Filter Based on Penalized Spline

Abstract

The purpose of this study is to develop a technique for real-time implementation of spline adaptive filters (SAFs). The P-SAF proposed in this paper is based on a recurrent penalized (P) spline and, by analogy with the classical SAF, consists of a linear dynamic component and nonlinear static component. To adapt the P-SAF, computational circuits with different topologies are developed. The proposed approach makes it possible to adapt knots and evaluate spline coefficients simultaneously. This improves the efficiency of the P‑SAF as compared to the classical SAF, as well as reduces the computational costs. The efficiency of the P‑SAF in terms of MSE[dB] on model and real-time series is equal to or higher than that of the classical SAF.

INTRODUCTION

In recent years, the interest of the scientific and engineering community in nonlinear and adaptive models, as well as in applied problems solved using them, has increased. This interest is due to the nonlinear nature of many real-world processes.

Nonlinear adaptive models and filters have good flexibility and high performance [1–3]. The nonlinearity of these models reflects the nonstationary nature of the processes, while the adaptability of the models improves their efficiency. The related computational costs are determined mainly by the principles of model adaptation and training.

One of the popular ideas in nonlinear filter development is based on updating (adapting) the coefficients of linear filters. The most popular adaptation algorithms are based on the least squares method and its modifications, as well as on the affine projection method. The first group of methods has low computational complexity, while the second one has good convergence.

Other ideas for adaptive learning are based on artificial neural networks [5], adaptive Volterra filters [4], kernels [7], functional links [8], extended Kalman filters [6], etc. However, these filters are efficient only for systems with low nonlinearity. High nonlinearity negatively affects the convergence of adaptation algorithms and complicates computations, because it implies an increase in the order of a model.

Real-time systems impose the most rigid requirements for the execution speed of information processing algorithms and, therefore, for their computational complexity. Hence, in these systems, adaptive models that have a training stage or use numerical methods are of little use.

In this situation, an approach based on adaptive splines is of interest for nonlinear modeling. The idea itself, as well as the term spline adaptive filter (SAF), was proposed by M. Scarpiniti in 2013 [9]. The concept of SAF implies the sequential combination of a linear filter and an adaptable nonlinear algorithm. The nonlinear part in the basic SAF design is represented by an interpolation spline, the structure of which remains fixed. For interpolation, basis (B) splines and Catmull–Rom splines with a fixed matrix of coefficients for a local spline link are used.

Structurally, the SAF model belongs to the block-oriented representation with linear and nonlinear components [10, 11]. The linear part of the model is time-invariant (dynamic), whereas the nonlinear model is static (see Fig. 1).

Fig. 1. [Images not available. See PDF.]

Structure of the spline adaptive filter.

The topology of SAF blocks can also be different. For instance, the SAF model [9], called the Wiener model, is a linear-nonlinear (LN) model, which includes a linear filter and a static nonlinear adaptation function. Another popular model, the Hammerstein model, is a nonlinear-linear (NL) model where the dynamic and nonlinear-static blocks are reversed [12]. There are also models that combine LN and NL components, providing flexibility and diversity in processing nonlinearities of different types.

The concept of SAF proved quite productive both for theoretical research and in practice. Various researchers developed and investigated various SAF modifications, e.g., using IIR filters [13, 14], for active control and filtering [15, 16], for non-Gaussian environments [17, 18], etc. There are many studies devoted to improving SAF stability and convergence, as well as analyzing its reliability and performance [18–20].

However, despite the large number of studies, there are certain questions regarding the practical application of SAFs.

SAFs have a good theoretical foundation and rely on a priori information about properties of input signals and noise. When adapting SAF parameters, gradient optimization methods are employed [11, 18]. However, in a real-world problem, the useful signal is often not known and the objective function is usually multimodal. Another problem with SAFs is the duration of machine learning, which makes it difficult to use them in real time.

The purpose of this study is to develop a technique for real-time implementation of SAFs. For this purpose, a modification of the penalized (P) spline, called P-SAF, is proposed. Instead of a fixed smoothing parameter used in traditional P-splines, the P-SAF implements variable smoothing within an individual spline segment. In turn, generating a group of data samples solves the problem of spline knot selection.

Another feature of the P-SAF is the cost-effective computational circuit in the form of recurrent algebraic expressions, which allows the P-SAF to be used in real time.

Finally, the P-SAF is an analytical expression, which improves the interpretability of the models based on it.

DESCRIPTION AND TOPOLOGY OF THE P-SAF

For real-time SAF implementation, a mathematical model in the form of a recurrent spline function is proposed. It is the recurrent mathematical description that makes it possible to use the P-SAF in real-time.

Most of the well-known SAFs are based on smoothing splines; however, penalized splines and basis splines are also used [21, 22]. Smoothing splines have high computational complexity, which makes their real-time implementation difficult [23]. For instance, the SAF model first proposed in [9] uses basis splines. For basis and penalized splines, the optimal choice of knots significantly affects the complexity of their implementation [24]. Even though real-time adaptation is a time-consuming process, the SAF efficiency is significantly improved.

In the digital filter (DF) theory, two approaches can be distinguished: the DF with finite impulse response (FIR) and the DF with infinite impulse response (IIR). SAFs mainly use FIR filters. Their advantages are linear phase response and stability. These and other advantages of FIR are due to the absence of feedback on filter output parameters.

The structure of IIR filters includes feedback, which leads to low stability and, often, low convergence. However, unlike FIR filters, IIR filters can make the transient zone of signal passage and suppression sharper with the DF order similar to FIR filters [25].

A DF can be described using a difference equation. In this study, we use the recurrent form of the penalized spline, derived using the variational approach [26]. In the classical approach, the smoothing cubic spline S(t) can be obtained by solving the minimization problem over the entire observation interval [a, b] by samples y_i, i = :

1

where λ is the smoothing factor associated with the Tikhonov regularization parameter. The terms in (1) determine, respectively, the minimum curvature of the spline and the minimum residuals [27]. Cubic spline S(t) is a function from the Sobolev space s ∈ . The degree of smoothness, or the penalty for smoothness of S(t), is determined by parameter λ; hence the term: penalized spline. The range of parameter λ is not known and is generally large [10^–9, 10⁹]; however, for λ → 0, the spline tends to an interpolation spline.

For the real-time implementation of the P-spline, criterion (1) is modified and adapted individually for each ith segment of the spline:

2

The input data in (2) are collected into groups of h values between the extreme samples , of a group for each ith segment.

The proposed P-SAF allows one to vary the smoothness within a spline segment by using parameter ρ, thereby improving the flexibility of nonlinear models of individual segments, in contrast to traditional P-splines (1) with a single smoothing parameter.

The form of criterion (2) is known as the Tikhonov block regularization [28]; in this case, it defines a block as a group of h samples. The minimum group size h = 3 (i.e., four data samples) depends on the order of the cubic spline. However, generally h > 3, meaning that the sample is redundant and there is a positive effect on data balance for nonlinearity description [10].

Another feature of criterion (2) is the adaptation of the penalty parameter on interval ρ ∈ [0, 1]. The normalization of the smoothing factor ρ reduces the complexity of its selection in accordance with its physical meaning: from the maximum smoothness at ρ = 0 to the interpolating spline at ρ = 1.

The transition from criterion (2) to functional J(S) makes it easier to find unknown spline coefficients:

3

Sampling step Δt balances the dimensions of the terms, while functional (3) itself becomes dimensionless. Then, Δt = 1.

To obtain the coefficients , , , of recurrent spline for the ith segment,

4

we use conditions of two types:

(a) The conditions of equality of continuous derivatives for adjacent spline segments = is used to derive recurrence relations for continuous coefficients , (k = 0.1) of adjacent (i – 1)th and ith segments.

(b) The condition , = 0 is used to find discontinuous coefficients , .

Mathematical relations for P-spline coefficients , , , are algebraic expressions [26] and do not require additional resolution methods (analytical or numerical ones).

A distinctive feature of the proposed P-spline (4) is the possibility of coupling adjacent segments at any point q = within the ith segment (k = ) (see Fig. 2a). This feature is unique to both the spline theory with sequential segment coupling and the real-time implementation with a sliding window.

Fig. 2. [Images not available. See PDF.]

Topology of P-SAF computational circuits.

The instances of coupling spline q and evaluating τ are parameters of the P-SAF computational circuits. Based on their mutual ordering, it is possible to develop several topologies of spline computational circuits [29], including a sequential circuit (Fig. 2b) and multiple filtering (Fig. 2c). All three circuits combine the recurrence of spline coefficients, which corresponds to the adaptation of the linear FIR filter in the traditional SAF, and the locality to a group of samples within a segment.

The universal computational circuit shown in Fig. 2a is of primary interest for real-time implementation. For arbitrary q and τ, the difference equation of the DF corresponding to this circuit is an equation with variable parameters, with the order of the equation corresponding to the value of τ [30]. It is parameters q and τ that determine (as topology parameters) the structural adaptation of the P-SAF.

Let us consider a special case of this computational circuit for τ = q + 1, q = . To derive the difference equation, it is required to specify a unified time sample for each ith segment of the spline. Basically, it can be any sample of the ith segment , j = . Here, instant is selected, which is denoted by i = .

Let us analyze the components of functional (3) with respect to the selected instant. By denoting y_i = , S_i = , sequences {S_i}, {y_i}, i = 1, 2, 3, … can be regarded as lattice functions with quantization interval Δt. With the introduced notation, the difference equation of the P-SAF is written as follows:

5

where

Equation (5) is the simplest case of difference equation for the P-SAF and corresponds to τ = q + 1, q = 0. It is a first-order equation with constant coefficients γ_j, j = . At τ > q + 1 and τ = const for all spline segments, the difference equation preserves constant parameters, but its order is (τ – h). In this case, the coefficients γ_i, j = , of difference equation (5) do not depend on the parameters q and τ of the computational circuit; they depend only on the parameters h and τ of the spline itself. These parameters determine the adaptive properties of the P-SAF, i.e., they are used in the process of parametric adaptation.

STABILITY AND TRANSIENT PROCESSES OF THE P-SAF

The concept of SAF proved to be quite useful, which is due to the ability of the DF to model nonlinear systems, taking into account the low complexity of SAFs themselves. However, the SAF convergence is still not sufficiently fast [31].

The recurrent P-SAF, as a mathematical tool for real-time information processing, must meet the convergence, stability, and accuracy requirements [32]. To assess the efficiency of the P-SAF in steady-state and transient modes, it is reasonable to use methods of linear dynamical systems. As in the case of any DF, the P-SAF description uses a mathematical apparatus that includes instrumental and system filtering functions.

In contrast to FIR filters, characterized by linear phase and stability, IIR filters can be unstable. Hence, it is imperative to analyze the stability of the recurrent spline filter in the region of variation of its parameters.

Based on the z-transform of the right-hand and left-hand sides of difference equation (5), the system function of the spline filter W(z) = is derived analytically [31]:

6

which is an analogue of the frequency response function (FRF) of continuous systems.

Despite the simple form of the system function, it is difficult to synthesize the P-SAF directly on its basis. An alternative way to synthesize the P-SAF is to represent the FRF as links of an automatic control system with direct, parallel, or cascade connections [33]. For this purpose, system function (6) is represented in the traditional form of polynomial relation

7

with coefficients ; ; β_κ = γ₃(k – 1)² + γ₄(k – 1)³.

Then, the block diagram for the direct implementation of the P-SAF is represented as in Fig. 3.

Fig. 3. [Images not available. See PDF.]

Block diagram of the recurrent P-SAF.

Feedback in the P-SAF structure, being an advantage of recursive DFs, can lead to its instability, i.e., to the presence of roots of the characteristic equation outside the unit circle < 1. To estimate the stability regions, we rewrite the characteristic polynomial of FRF (6) taking into account the bilinear w-transform. By substituting z = , we obtain

8

where

For second-order equation (8), the Hurwitz–Mises criterion determines the stability conditions for discrete systems.

Taking into account the notation used in (5) for the coefficients γ_j,, of the difference equation, the stability of the P-SAF is completely determined by spline parameters h and τ. For the selected topology of the computational circuit q = 0, τ = 1, the P-SAF is stable at any values of smoothing parameter ρ ∈ [0, 1] for any group size h.

In Fig. 4, the regions of instability are not shaded, the regions of absolute stability (i.e., at all values of ρ ∈ [0, 1]) are completely shaded, and the regions of partial stability indicate the lower range value of ρ for certain h and q. The variation in topology parameter q > 0 noticeably narrows the stability regions of the R‑SAF. The stability is observed only for ρ → 1, with the ranges of ρ being quite small: [0.99–1], [0.84–1]. Finally, at q → h, the P-SAF is always unstable.

Fig. 4. [Images not available. See PDF.]

P-SAF stability regions.

When adjacent segments are coupled at the beginning of the current segment (q = 0), the P-SAF always remains stable even for τ > 1. In practice, the coupling at the beginning of the segment is most often used.

It is also reasonable to estimate the convergence of the P-SAF by using indirect criteria, i.e., investigate instrumental function g(t). It is well-known that the instrumental function of the DF is a weight function of filter components. As the length of the DF weight increases, the computational complexity of the filtering and adaptation processes increases sharply.

Instrumental function g(t) is derived analytically based on system function (6) by using the inverse Fourier transform.

Being an analogue of the pulse weight function of a continuous system, the instrumental function expresses the analytical dependence of signals between the input and output signals of the DF, based on a discrete convolution equation. Visually, the instrumental function of the P-SAF is asymmetric, which is typical of IIR filters (Fig. 5a). The damping character confirms the convergence of the P-SAF when varying the spline parameters ρ and h.

Fig. 5. [Images not available. See PDF.]

P-SAF instrumental function.

However, the causality condition (g(t) = 0, t < 0) holds only if the P-SAF operates in no-delay mode, i.e., there is no delay with respect to the topology parameters of the computational circuit. In other cases, the instrumental function is not zero, which is typical of systems with delay, e.g., when processing grouped data.

The system error of the DF depends on the width Δ of the instrumental function (Fig. 5c). This width can be estimated by the following relation [34]:

9

It can be seen from the geometric illustration that the width of the instrumental function depends on spline parameters ρ and h. The width decreases both with increasing smoothing parameter ρ and with increasing spline length h. As the number of segment samples h increases, the bandwidth of the filter decreases. Hence, the width of the instrumental function decreases, which can be seen from the figure. In these cases, the smoothness of the spline at the filter output increases. However, it becomes significantly distant from the regression line, which leads to an increase in the systematic error. Thus, the width of the amplitude function is physically interpreted as a factor that affects the accuracy of measurements.

The influence of smoothing parameter ρ is much lower than that of h. For h > 15, the smoothing parameter has almost no influence.

RESULTS OF NUMERICAL EXPERIMENTS

To evaluate the performance of the proposed P-SAF, a series of computational experiments with its real-time implementation was carried out. The numerical experiments were designed to demonstrate the smoothing and filtering capabilities of the P-SAF, as well as to compare its results with other SAFs.

The proposed P-SAF based on the real-time penalized spline is compared with the classical versions of SAF-LMS [9].

To conduct the study, as the model and real-world data, it is reasonable to select well-known functions with significant nonlinearities.

For SAF performance evaluation [18], the accuracy indicator based on the mean square error expressed in decibels is used:

10

where E[*] is the average value and e(n) is the difference between the useful and reconstructed signals.

Model Input Signals

All results were obtained by averaging over 20 Monte Carlo trials. The maximum sample size for all model signals was 30 000 samples.

Figure 6 shows the performance results of the proposed P-SAF for two of the most popular examples from the SAF theory. In Fig. 6a, the input useful signal x_n is a Gaussian random process generated by the relation [9]where is white Gaussian noise with zero mean and unit variance, while r ∈ [0, 1) is the coefficient that determines the correlation between adjacent input samples x_n.

Fig. 6. [Images not available. See PDF.]

Efficiency of the P-SAF algorithm in environments with Gaussian processes.

In addition, independent white Gaussian noise with different signal-to-noise ratios (SNR = 10, 20, 30, and 40 dB) was added to the input data.

Figure 6b illustrates the filtering efficiency for the process generated by the alpha-stable distribution; for α ≠ 1, it has the following form [17, 18]:where α ∈ (0.2] is the stability index, which determines pulse intensity; –1 ≤ β ≤ 1 is the symmetry index; ρ is the position parameter; and γ > 0 is the variance parameter. Obviously, at α = 2, there is a Gaussian signal. By analogy, for other values of α ∈ (0, 2], the signal is called non-Gaussian. In the experiment, this signal is useful with parameters α = 1.6, β = 0, ρ = 0, and γ = 0.05. Additive Gaussian noise with different signal-to-noise ratios (SNR = 10, 20, 30, and 40 dB) is used.

The MSE[dB] curves allow us to estimate the overall efficiency (the approach of MSE[dB] to the steady-state MSE value) and convergence rate of the algorithms.

Figure 6b demonstrates the good performance of the P-SAF for the non-Gaussian input signal. The MSE[dB] curves confirm the convergence of the algorithm to the steady-state error value for different signal-to-noise ratios. In the steady state, the MSE[dB] curves asymptotically tend to the noise power value [9]. For SNR = 10, 20, 30, and 40 dB, they are theoretically equal: –10, –20, –30, and –40 dB, respectively.

However, in the case of the Gaussian useful signal (Fig. 6a), the MSE [dB] curves are quite far from the theoretical values; hence, the P-SAF algorithm has a significant steady-state error. For comparison, Fig. 6a shows the dashed MSE [dB] curve for the classical SAF [13] at SNR = 30 dB. The steady-state error is consistent with the signal power for a fairly large number of samples n > 2000 (see Fig. 5 in [13]).

The next model signal often used in data analysis is the Doppler function defined on interval [0, 1]:

In this case, additive noise is also represented by white Gaussian noise with different signal-to-noise ratios (SNR = 10, 20, 30, and 40 dB). Figure 7a shows the useful signal (gray curve), the signal–noise mixture (black dots), and the P-SAF result (blue curve) on interval [0, 1] at SNR = 10 dB for 1000 data samples. The parameters of the P-SAF algorithm are indicated in the figure. The zoomed region shows the result of the algorithm (red curve) on interval [0, 0.3] for h = 3 and ρ = 0.5. For these signals with significant variation in both frequency and amplitude, the parameters of the P-SAF have a substantial effect on its performance. Figure 7b shows the MSE[dB] curves for different signal-to-noise ratios at h = 3 and ρ = 0.5. The convergence of the algorithm is confirmed by the approach of the MSE[dB] curves to the steady-state value. Moreover, the convergence rate for low noise with SNR =40 dB is an order of magnitude lower than for SNR = 10 dB.

Fig. 7. [Images not available. See PDF.]

Efficiency of the P-SAF algorithm for the Doppler function.

Real Time Series

As input data, two real datasets from the DaISy database [11] were used. This database contains a large volume of real statistical data from different domains: mechanical, biomedical, industrial, environmental, etc.

Dataset 96-008 (wing flutter data) contains 1024 values. Figure 8a shows (in gray) the informative part of the time series: 512 values. Wing flutter is the vibration of an aircraft’s wing during flight. Flutter is characterized by high-frequency vibrations, which affects flight safety. The red curve in Fig. 8a represents the result of the P-SAF algorithm for h = 3 and ρ = 0.8. When processing these high-frequency signals, the P‑SAF settings play an important role. The influence of the spline parameters is mutual and combined. In the proposed P-SAF algorithm, parameter ρ largely determines the amplitude of the reconstructed signal, while parameter h affects the frequency. For h = 3, the frequency of the signals completely coincidences over the entire time interval: the blue curve in the sampling range [60÷160] and the green curve in [250÷350]. In addition, Fig. 8b shows the influence of the sample group size h on the accuracy of signal reconstruction. It is obvious that the accuracy improves with decreasing h.

Fig. 8. [Images not available. See PDF.]

Efficiency of the P-SAF algorithm on 96-008 DaISy.

Another DaISy dataset is 96-004 (data of the ball-and-beam setup in SISTA) with 1000 values. In Fig. 9a, the original dataset is represented by the gray curve, while the result of the algorithm is represented by the green curve. To expand the research [11], the time series is additionally investigated under conditions of additive white Gaussian noise with different signal-to-noise ratios (SNR = 10, 20, 30, and 40 dB). Figure 9b demonstrates the good convergence of the P-SAF algorithm for all signal-to-noise ratios.

Fig. 9. [Images not available. See PDF.]

Efficiency of the P-SAF algorithm on 96-004 DaISy.

In the absence of noise, MSE[dB] varies in the range of (–55, –60) dB for different h and ρ. These values can be regarded as a systematic error of the proposed P-SAF algorithm for this time series.

CONCLUSIONS

IIR filters are popular among researchers due to their wide applicability to real-time data processing. IIR filters based on splines attract particular attention as a nonlinear processing tool, known as SAF.

The P-SAF proposed in the paper is based on the recurrent P-spline by analogy with the classical SAF proposed by Scarpiniti [9]. The P-SAF consists of the linear dynamic and nonlinear static components. For P-SAF adaptation, computational circuits with different topologies, which specify methods for knot adaptation and spline coefficient evaluation, have been developed. This improves the efficiency of the P-SAF as compared to the classical SAF, as well as reduces the computational costs.

The frequency and time characteristics of the recursive P-SAF have been analyzed and the conditions for its convergence have been investigated. It has been found that, when varying its parameters, the P‑SAF remains low-frequency.

The comparative analysis of the proposed P-SAF with other SAFs has been carried out on some model and real-world data from the DaISy repository. The values of MSE[dB] for the P-SAF are at the level of or higher than those of the classical SAF for deterministic or real-world high-frequency signals. This is an advantage of the P-SAF: the short recursive part and the analytical model.

FUNDING

This work was supported by the Russian Science Foundation, project no. 23-21-00259.

CONFLICT OF INTEREST

The authors of this work declare that they have no conflicts of interest.

Translated by Yu Kornienko

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