Mirror Fourier Nonlinear Filters Research Articles

Study of non-linear filter-based algorithms for active noise control of machinary noises

Sometimes, machinary noise comprises of non-linear deterministic noise. Therefore, it is expected that for such cases of noises, non-linear filter based active noise control algorithms can provide improved performance in terms of reduction of noise. Various non-linear filter based algorithms explored in this paper are: Volterra filtered-x least mean square algorithm, Hammerstein algorithm, Fourier algorithm utilising Fourier non-linear filter, even mirror Fourier non-linear filter and CN filter, bilinear algorithm, Legendre algorithm, filtered-s least mean square using functional link artifical neural network (FLANN) filter, generalised FLANN, exponential FLANN and recursive FLANN. Noise from various workshop machines like band saw, compressor and welding process were recorded using a microphone and utilised to study the algorithms. Two cases of primary paths, (a) linear primary path and (b) non-linear primary path with cubic non-linearity, are considered for simulation study. Performance of various algorithms is evaluated based on steady-state residual noise.

Improved adaptive recursive even mirror Fourier nonlinear filter for nonlinear active noise control

In this paper, we analyze the theoretical mechanism for the effectiveness of recursive filters including both linear and nonlinear feedforward and feedback subsections to accurately approximate nonlinear distributions in the active noise control (ANC) system. As a result, we improve the recursive even mirror Fourier nonlinear (REMFN) filter by adding an additional linear section termed REMFNL filter together with its channel reduced form (CRREMFNL) to explore better control performance. As for the linear feedback section introduced, we have studied the bound-input bound-output (BIBO) stability condition for the REMFNL filter and designed a stable control scheme to guarantee the filter satisfying the stability criteria. The performance of the proposed filters equipped with a filtered-x least mean square (FXLMS) algorithm is validated through analyses of computational complexity and simulations of various nonlinearities for nonlinear ANC (NANC) systems. Simulation results demonstrate that the proposed REMFNL and CRREMFNL filters can achieve better performance over the bilinear, recursive second-order Volterra (RSOV), recursive functional link artificial neural network (RFLANN) and standard REMFN filters based on the FXLMS algorithm, which often outperform the conventional Volterra FXLMS (VFXLMS) and filtered-s least mean square (FSLMS) algorithms.

Sparse Modeling of Nonlinear Secondary Path for Nonlinear Active Noise Control

In many practical active noise control systems, the nonlinear secondary path (NSP) model can offer broader applications than the linear secondary path (LSP) model for the widespread existence of nonlinearities in the secondary path. However, the computational burden often limits the application of the NSP algorithms. In this paper, we apply a sparse Volterra model for the LSP and NSP instead of the complete Volterra model, which may be over parameterized for the nonlinear active noise control (NANC) system. The developed new control algorithms greatly result in reduction of computational complexity. We also provide an analysis of the NANC system to show that the noise at the canceling point could be approximated by the function expansion filters when the secondary path is modeled as the second-order Volterra series. In addition, two new function expansion forms, the even mirror Fourier nonlinear filter with a linear finite-impulse response section and the Chebyshev filter, are explored to process the nonlinearities in the NANC system using the filtered-x least mean square and filtered-error least mean square algorithm structures. The computational complexity analysis and the computer simulations validate that the new control algorithms with a sparse Volterra model for the secondary path can significantly reduce computational load without sacrificing the control performance.

A study about Chebyshev nonlinear filters

The paper studies a novel family of nonlinear filters based on Chebyshev polynomials of the first kind, the Chebyshev nonlinear filters. This family shares many of the characteristics of the recently introduced Legendre and even mirror Fourier nonlinear filters, but has also peculiar properties. Chebyshev nonlinear filters belong to the class of linear-in-the-parameters nonlinear filters. Their basis functions are polynomials, specifically, products of Chebyshev polynomial expansions of the input signal samples. According to the Stone–Weierstrass theorem, they are universal approximators for causal, time-invariant, finite-memory, continuous, nonlinear systems. Their basis functions are mutually orthogonal for white input signals with a particular nonuniform distribution. They admit perfect periodic sequences, i.e., periodic input sequences that guarantee the mutual orthogonality of the basis functions on a finite period. Using perfect periodic input signals, an unknown nonlinear system and its most relevant basis functions can be identified with the cross-correlation method. It is shown in the paper that the perfect periodic sequences of Chebyshev nonlinear filters are simply related to those of even mirror Fourier nonlinear systems. Experimental results involving a real nonlinear system illustrate the potentialities of these filters.

Recursive Even Mirror Fourier Nonlinear Filters and Simplified Structures

In this paper, a novel class of recursive nonlinear filters is presented as an extension of finite-memory even mirror Fourier nonlinear filters. These filters are characterized by two relevant properties: i) they are able to arbitrarily well approximate any discrete-time, time-invariant, causal, infinite-memory, continuous, nonlinear system and ii) they are always stable according to the bounded-input-bounded-output criterion. Even though recursive models can represent many systems with fewer coefficients than their finite-memory counterparts, it is still possible to further reduce their computational complexity. In fact, while in general simplified structures lead to a loss of performance, it is pointed out in the paper that in various common real-world situations, they are able of giving remarkable complexity reductions without negatively affecting the modeling capabilities.

Legendre nonlinear filters

The paper discusses a novel sub-class of linear-in-the-parameters nonlinear filters, the Legendre nonlinear filters. The novel sub-class combines the best characteristics of truncated Volterra filters and of the recently introduced even mirror Fourier nonlinear filters, in particular: (i) Legendre nonlinear filters can arbitrarily well approximate any causal, time-invariant, finite-memory, continuous, nonlinear system; (ii) their basis functions are polynomials, specifically, products of Legendre polynomial expansions of the input signal samples; (iii) the basis functions are also mutually orthogonal for white uniform input signals and thus, in adaptive applications, gradient descent algorithms with fast convergence speed can be devised; (iv) perfect periodic sequences can be developed for the identification of Legendre nonlinear filters. A periodic sequence is perfect for a certain nonlinear filter if all cross-correlations between two different basis functions, estimated over a period, are zero. Using perfect periodic sequences as input signals permits the identification of the most relevant basis functions of an unknown nonlinear system by means of the cross-correlation method. Experimental results involving identification of real nonlinear systems illustrate the effectiveness and efficiency of this approach and the potentialities of Legendre nonlinear filters.

Unconstrained linear combination of even mirror Fourier non‐linear filters

In this study, the unconstrained linear combination of the outputs of even mirror Fourier non-linear filters is considered. These filters are new members of the class of causal, shift-invariant, finite-memory and linear-in-the parameters non-linear filters. Their name derives from the even symmetry of their trigonometric basis functions. Even mirror Fourier non-linear filters are universal approximators for causal, time invariant, finite-memory and continuous non-linear systems. Moreover, their basis functions are mutually orthogonal for white uniform input signals in the interval [−1, +1]. The authors show in this study how to exploit these characteristics, in the framework of the unconstrained linear combination of non-linear filters, for modelling unknown non-linear systems. In particular, they show that the filters whose outputs are combined can be adapted avoiding the choice of the step sizes, by using a simple algorithm presented in this study. The analysis of the proposed structures is accompanied by a set of simulation results that confirm the good performance obtained in different situations.

Perfect periodic sequences for even mirror Fourier nonlinear filters

The paper deals with the identification of a class of nonlinear filters recently introduced in the literature, the even mirror Fourier nonlinear filters, and shows that perfect periodic sequences can be developed for these filters. A periodic sequence is perfect for a nonlinear filter if all cross-correlations between two different basis functions, estimated over a period, are zero. By applying perfect periodic sequences as input signals to even mirror Fourier nonlinear filters, it is possible to model unknown nonlinear systems using the cross-correlation method. Moreover, the most relevant basis functions, i.e., those that guarantee the most compact representation of the nonlinear system according to some information criterion, can be easily estimated. Experimental results on the identification of real nonlinear systems illustrate the main advantage of the proposed method which is the remarkable reduction in the computational complexity.

Fourier nonlinear filters

In this paper, two new sub-classes of linear-in-the-parameters nonlinear discrete-time filters, derived from the truncation of multidimensional generalized Fourier series, are presented. The filters, called Fourier nonlinear filters and even mirror Fourier nonlinear filters, are universal approximators for causal, time-invariant, finite-memory, continuous nonlinear systems, according to the Stone–Weierstrass approximation theorem. Their properties and limitations are discussed in detail. In particular, we show, by means of appropriate simulation examples, that an orthogonality property they satisfy for white uniform input signals is useful for improving the identification of nonlinear systems.