Improved functional link artificial neural network via convex combination for nonlinear active noise control☆
Introduction
Over the past decades, active noise control (ANC), which is based on the superposition principle that a noise can be canceled by another noise with the same amplitude but opposite in phase, has attracted increasing attention because of its potential use in low frequency noise control applications [1]. One of the most popular adaptive filtering algorithms is the filtered-x LMS (FXLMS) algorithm due to its simple structure and ease of implementation in ANC systems. However, in actual ANC systems, the performance of the FXLMS algorithm may be degraded, or even failed. Major reason is the fact that the FXLMS algorithm is linear in nature, and not capable of compensating nonlinear distortions under the following situations [2], [3], [4], [5]: (1) the reference noise received by a reference microphone may be a nonlinear and deterministic noise process, such as a chaotic noise rather than a stochastic, white, or tonal noise; (2) the primary path noise at the canceling point may exhibit nonlinear distortions when the primary path is nonlinear; (3) the secondary path transfer function of an ANC system has a nonminimum-phase. Consequently, in the design of these ANC systems, the nonlinear adaptive filters should be employed because it can provide satisfactory performance. But, there is no unique theory for modeling and characterizing nonlinear phenomenon.
As an effective way of compensating nonlinear distortions in nonlinear ANC systems, the filtered-s LMS algorithm-based adaptive functional link artificial neural network (FLANN) filter (FSLMS) has recently received increasing interest. By utilizing its advantages of low computational complexity and linear outputs-coefficients relationship, the FLANN filters-based trigonometric functional expansions [6], [7], polynomial expansions [8] and piece-wise linear expansions [9] have been developed as adaptive controllers in nonlinear ANC systems. To improve the performance of nonlinear ANC systems, various types of FLANN filters have been proposed in recent years, such as reduced feedback FLANN [10], recursive FLANN [11], generalized FLANN [12], nonlinear neuro-controller-based FLANN [13] and hybrid active noise control system-based FLANN [14]. It is worth noting that the FLANN filters are members of the class of causal, shift-invariant, finite memory, nonlinear filters whose output depends linearly on the filter coefficients. Therefore, as in the linear FXLMS algorithm, the FSLMS algorithm using trigonometric functional expansions is most popular in nonlinear cases [15]. However, an important limitation regarding the FSLMS algorithm is that the selection of a certain value for the step size implies a compromise between speed of convergence and steady state performance. That is, a large step size yields a fast convergence rate but a large steady state mean square error (MSE), and a small step size leads to a slow speed with a low steady state MSE. To deal with this disadvantage, some approaches including normalized FSLMS [11], [12] and fast FSLMS algorithms [16], [17] have been proposed, but it is not easy to establish the same speed vs precision compromise.
Inspired by the convex combination approaches in [18], [19], [20], [21], a combination scheme based on the normalized FSLMS algorithm (CNFSLMS) is proposed to circumvent the compromise between speed of convergence and precision for nonlinear ANC systems with the linear secondary path (LSP) and nonlinear secondary path (NSP) because of the use of a fixed step size in this paper. The proposed algorithm is constructed by the convex combination of two FLANN filters with different step sizes. Partial work of the algorithm was presented in [22] which only simply discussed the convex combination of two FSLMS algorithms for nonlinear ANC system with the nonminimum-phase LSP. In this paper, the CNFSLMS algorithm is derived in detail. Moreover, the modified CNFSLMS is presented to reduce computational complexity. Furthermore, the performance analysis (including computational complexity and steady state error) for nonlinear ANC system with the LSP and NSP is provided. Simulation results are given for nonlinear active noise control (NANC) with a LSP and NSP. Specifically, we address main contributions in this paper as the following points:
- (1)
The combination scheme based on adaptive FLANN filter is designed for nonlinear ANC systems with the LSP and NSP.
- (2)
The CNFSLMS algorithm in the filter bank form is derived to obtain an improved convergence behavior.
- (3)
To reduce the computational complexity, the modified CNFSLMS (MCNFSLMS) algorithm is proposed by replacing the sigmoid function with the modified Versorial function for nonlinear ANC systems.
- (4)
The analysis of the computational complexity is discussed.
- (5)
The steady state performance is analyzed.
The rest of the paper is organized as follows. In Section 2, the brief FLANN filter is presented. In Section 3, we briefly review the normalized FSLMS algorithm. The combination scheme based on adaptive FLANN filter is presented for nonlinear ANC systems in Section 4, where the CNFSLMS and MCNFSLMS algorithms are derived, respectively. In Section 5, the steady state performance of the proposed algorithms is presented. The computational complexity is analyzed in Section 6. The simulation results are provided for nonlinear active noise control (NANC) with a LSP and NSP in Section 6. Finally, we present a brief summary and discussion in Section 7.
Section snippets
Brief FLANN filter
As a computationally efficient single layer network, the FLANN filter has been developed as an alternative architecture to the nonlinear adaptive filter. Fig. 1 depicts the schematic diagram of the FLANN filter, which is a fat net without any need for a hidden layer. Where n is the time index.
In a FLANN, each input x(n) to network is expanded by a suitable set of linearly independent functions in the functional expansion block (including Chebyshev, Legendre, and trigonometric), and then the
Normalized FSLMS algorithm
Fig. 2 describes the block diagram of the nonlinear ANC system-based adaptive FLANN filter using the normalized FSLMS algorithm with the filter bank implementation that can reduce the computational complexity.
As illustrated in Fig. 2, the output signal y(n) of the secondary path in a filter bank form is generated by adaptive FLANN filter expressed aswhere yi(n) is the output of the ith sub-filter, and given byand wi(n) represents the corresponding weight
Novel nonlinear ANC system based on convex combination of two adaptive FLANN filters
To cope with the compromise between the speed of convergence and the steady state error of the NFSLMS algorithm in nonlinear ANC systems with the LSP and NSP, the novel nonlinear ANC system based on the adaptive FLANN filter using the convex combination of two normalized filtered-s LMS algorithms (CNFSLMS) is designed, as shown in Fig. 3. To achieve a good performance from the convex combination scheme, two adaptive FLANN filters with different step sizes are adapted individually. Moreover, the
Analysis of the convergence of the proposed algorithm
In this section, the MSEs of the overall filter and its component filters are analyzed and discussed.
To begin with, substituting (32) into the cost function J(n) of the overall filter, we have
Next, assuming that λ(n) is independent of the component filter errors in steady state. When using a reduced adaptation speed for a(n), this assumption is reasonable, and therefore
Analysis of computational complexity
As in [6], [7], it is a well-known fact that the FSLMS algorithm requires N(2P + 1)(L + 3) − L multiplications to calculate the output of the filter and to update its weights, and 2NP to calculate the functional expansion by sin(·)/cos(·) functions. Since the proposed combination algorithm combines two FLANN filters, it needs 2N(2P + 1)(L + 3) − L multiplications for the adaptation of the component FLANN filters, 2NP to compute the sin(·)/cos(·) functions, and more multiply products to calculate the output
Simulation tests
To illustrate the effectiveness of the proposed combination scheme in applications of nonlinear ANC systems with the LSP and NSP, three experiments (including two examples of NANC systems with a LSP and one example of NANC systems with a NSP) are carried out in this section.
For performance comparisons, we use the NMSE achieved by each adaptive controller versus the number of iterations defined aswhere is the power of the primary noise at the canceling point.
All the
Conclusion
In this paper, a convex combination scheme-based adaptive FLANN filter was developed for nonlinear ANC systems with the LSP and NSP to alleviate the compromise between fast convergence rate and small steady state error. Two new adaptive algorithms with different step sizes, CNFSLMS and MCNFSLMS, were derived, respectively. Simulation results demonstrated that the proposed algorithms outperform the conventional NFSLMS and VFXLMS algorithms in controlling nonlinear noise processes with the LSP
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This work was partially supported by National Science Foundation of P.R. China (No. 61271340, 61571374, and 61433011), the Sichuan Provincial Youth Science and Technology Fund (No. 2012JQ0046), and the Fundamental Research Funds for the Central Universities (No. SWJTU12CX026).