On the convergence of real-time active noise control systems
Introduction
Active noise control (ANC) systems aim at producing an anti-noise in such a way that the residual noise at the desired silence zone, which is the net combination of the anti-noise and environmental noise, is minimized. In order to cope with any change in the physical plant, these systems utilize an adaptation algorithm to update parameters used in the estimation of the anti-noise. It has been shown that a traditional adaptation algorithm such as least mean square (LMS) algorithm does not have acceptable convergence behavior in ANC systems [1], [2]. This is mainly because of the existence of a signal channel called the secondary channel in the path of the anti-noise signal to the desired silence zone. In order to solve this problem, the Filtered-x LMS (FxLMS) algorithm was proposed by Widrow in 1981 [1]. In this adaptation algorithm, the reference signal is filtered using an estimate of the secondary path before being used by the LMS algorithm. Although this filtering task improves the stability of the adaptation process, it may reduce the convergence rate of the adaptation process when the secondary path has long impulse response. However, the convergence rate can be adjusted by changing a scalar parameter called the adaptation step-size (). Increasing improves the convergence rate, but this parameter should be increased carefully because there is an upper-bound for the step-size () beyond which the ANC process becomes unstable.
As can be seen in Table 1, there have been several contributions made in studying the effect of the adaptation step-size on the stability or convergence behavior of the adaptation process in ANC systems. However, most of these studies were based on a number of simplifying assumptions regarding the secondary path and the unwanted noise. These assumptions are not usually applicable in practice. A major group of these analyses have been conducted for the case that noise is a deterministic and narrow-band process so that the reference signal can be modeled as the sum of sinusoids [3], [4], [5]. In [3], Bjarnason conducted an analysis for this case and derived an expression for under the simplifying assumption that the secondary path is a pure delay system. Later, Vicente conducted another analysis for this simplified case and showed that is inversely proportional to the product of the filter length, L, and , where is the delay in the secondary path [4]. Recently, Xiao tried to study this case for a general secondary path but, as he reported, the theoretical results were not in good agreement with the simulation results [5].
Another major group of available analyses assumes that the unwanted noise is a broad-band white signal. Elliott showed that for a pure delay secondary path and broad-band white noise, is inversely proportional to the sum of the filter length, L, and [6], [7]. This result was derived based on simulation experiments and was not supported by a comprehensive theoretical analysis. Long conducted a theoretical analysis for this simplified case and showed that is inversely proportional to the sum of L and [8], [9]. Bjarnason conducted a comprehensive convergence analysis for the stochastic FxLMS algorithm and derived a similar to that obtained by Long [3]. It has to be noted that values of which have been proposed by Elliott, Long and Bjarnason, were derived for the simplified case with the pure delay secondary path. However, such a simple secondary path does not exist in practice. As a step forward to the estimation of a more reliable , the authors studied convergence of stochastic ANC systems with a moving average secondary path [10]. Results, which were proposed in [10], have been found to be promising in simulation experiments, but practical experiments have shown that a realistic is smaller than that suggested in. This is because in practice, the noise is a band-limited flat spectrum (white) signal, rather than a broad-band white signal. Even if the noise process is white over a wide frequency range, the reference signal is required to be sampled or filtered with a sampling frequency higher than the maximum frequency of the noise. Therefore, in practice, the reference signal has not a perfect flat spectrum over its entire frequency range. It seems that in order to match theoretical and practical results, another step in the generalization of the analysis given in [10] should be taken.
The aim of this paper is to conduct a comprehensive theoretical convergence analysis for ANC systems in practical conditions and verify the obtained theoretical results by using a fully implemented real-time ANC system. The main contributions of this research can be summarized as:
- 1.
Conducting a comprehensive theoretical convergence analysis for ANC systems with a general secondary system and band-limited flat spectrum reference signal.
- 2.
Derivation of a reliable for the general case described in 1.
- 3.
Derivation of the step-size leading to the fastest convergence rate.
- 4.
FPGA design and full implementation of a real-time adaptive ANC system in an acoustic duct using National Instrument CompactRIOTM Control and Acquisition System [11].
- 5.
Empirical verification of the theoretical results using the implemented system.
The rest of this paper is organized as follows. Section 2 describes the mathematical model of the adaptation process performed by the FxLMS algorithm in ANC systems. Section 3 develops a linear model for the convergence of this adaptation process. Section 4 applies the obtained model to derive a number of mathematical expressions for some important parameters of the adaptation process. Section 5 shows the validity of the theoretical results using computer simulation. Section 6 describes the FPGA design and implementation of a real-time ANC system using CRIOTM embedded system and shows the successful application of the obtained theoretical results in practice. Finally, Section 7 states concluding remarks.
Section snippets
Mathematical model
Block diagram of an ANC system with the FxLMS adaptation algorithm is shown in Fig. 1. In this figure, x(n) is the reference signal, y(n) is the anti-noise signal generated by the ANC controller, d(n) is the acoustic noise in the silence zone, and e(n) is the residual noise signal. In practice, x(n) is picked up by a microphone called the reference microphone, y(n) is played by a loudspeaker called the canceling loudspeaker and e(n) is measured by another microphone called the error microphone.
Modeling excess MSE function
For the adaptation process performed by an FxLMS algorithm, the MSE function is expressed aswhere Jmin is the optimum MSE and is the excess MSE (EMSE) function. For a stationary noise, Jmin is constant and can be calculated from the statistics of the primary noise [1]. In [3], it was shown that for a general stationary reference signal, the EMSE function isThe difference of the MSE function can be defined asIn [10],
Steady state MSE
The steady state MSE, Jss is defined asSubstituting Eq. (30), (60) results inThe second term in this equation can be obtained by solvingSubstituting Eqs. (57), (60) into (62), and solving the obtained equation results inwhereNow, let us define Equivalent Delay of secondary path asUsing this definition, Eq. (63) can be expressed as
Simulation results
In order to verify the validity of the analytical results obtained in the previous section, several computer simulation experiments have been performed. This section shows the agreement between the simulation results and analytical results, particularly for the MSE (prediction) model given in Eq. (58), the value of the steady state MSE function given in Eq. (60), the upper-bound of the step-size given in Eq. (72), and the step-size leading to the fastest convergence given in Eq. (82). In these
Real time implementation
Fig. 8 shows the schematic diagram of the implemented real-time adaptive ANC system in an acoustic duct and Fig. 9 shows a photo of the actual system. The acoustic duct with dimensions of was constructed from 18 mm thick medium density fiber-board, with carpeted interiors. This duct was equipped with the following electro-acoustical components.
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NI 9104: Reconfigurable FPGA Chassis which utilizes an embedded Xilinx FPGA chip (clocked at 40 MHz) to synthesize custom control and
Conclusion
The convergence behavior of practical ANC systems does not match to that obtained from available theoretical convergence analyses. This is mainly because available analyses have been conducted based on simplifying assumptions regarding the secondary path or the reference signal. This paper applied Lyapunov stability theory in order to conduct a more general analysis for ANC systems. This analysis led to derive an upper-bound for the adaptation step-size beyond which the system becomes unstable
Acknowledgments
The authors would like to thank Mr. Matthew McCallum and Rhys Farrand for their contribution regarding the implementation of the real-time ANC system applied in this research.
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2018, Applied Soft Computing JournalCitation Excerpt :ANC systems are effectively used for reducing low-frequency noise [5], while passive noise control systems are used for attenuating high frequency noise interferences through sound-absorbing materials [6]. Mostly the main power of industrial noise is based on the low frequencies for which a filtered extended least mean square (FXLMS) algorithm is implemented for viable ANC system [7–11] but with limitations of more prone to get stuck in local minimum. Additionally, these adaptive strategies require identification of the secondary path and normally applicable to linear ANC system.