Research on nonlinear active noise control (NANC) revolves around the investigation of the sources of nonlinearity as well as the performance and computational load of the nonlinear algorithms. The nonlinear sources could originate from the noise process, primary and secondary propagation paths, and actuators consisting of loudspeaker, microphone or amplifier. Several NANCs including Volterra filtered-x least mean square (VFXLMS), bilinear filtered-x least mean square (BFXLMS), and filtered-s least mean square (FSLMS) have been utilized to overcome these nonlinearities effects. However, the relative performance and computational complexities of these algorithm in comparison to FXLMS algorithm have not been carefully studied. In this paper, systematic comparisons of the FXLMS against the nonlinear algorithms are evaluated in overcoming various nonlinearity sources. The evaluation of the algorithms performance is standardized in terms of the normalized mean square error while the computational complexity is calculated based on the number of multiplications and additions in a single iteration. Computer simulations show that the performance of the FXLMS is more than 80% of the most effective nonlinear algorithm for each type of nonlinearity sources at the fraction of computational load. The simulation results also suggest that it is more advantageous to use FXLMS for practical implementation of NANC.
The increase in environmental noise and the need for a low-cost noise control system promote the growth of active noise control (ANC) applications. Transportation and road noise from motor vehicles, aircraft, and rail are major contributors to environmental noise [
Linear ANC techniques are limited in that the system exhibits performance degradation when dealing with nonlinearities [
The nonlinear models used in NANC can be updated adaptively using various algorithms. In this paper, the performance of the nonlinear VFXLMS, BFXLMS and FSLMS algorithms are compared with the linear FXLMS, algorithms in terms of noise reduction and computational load. The algorithm with the best performance for each type of nonlinearity is identified. The paper is organized as follows: Section
The nonlinearity sources in active noise control can be generally classified into three types. The first type is due to system actuators such as the loudspeaker, microphones, preamplifiers, analogue to digital converters (ADC), and digital to analogue converters (DAC). The second type is due to the noise process when the dynamic behaviour of the noise generation is nonlinear. The third type is due to the acoustic propagation paths, where the acoustic signals propagate under the presence of nonlinearities, such as high pressure, temperature variations, or nonhomogeneous media [
In many practical ANC applications, the actuator that generates the antinoise signal is a loudspeaker, and the sensor that detects the error signal is a microphone. It is possible that the primary noise amplitude level is so high that the error and reference sensors are saturated due to their low-power characteristics. The saturation and clipping effects due to over driving the loudspeakers can produce extra odd harmonics, thus, affecting the convergence speed of the NANC controller [
In loudspeakers, the nonlinearity in the form of limited dynamic range causes electrical distortion when output amplitude level is high. The two most important forms of distortion in the loudspeakers are (i) harmonic distortion and (ii) intermodulation distortion [
In order to deliver sufficient power to drive the loudspeaker, power amplifier is used to amplify the control signal. The amplifier should ideally provide nominally flat response between 20 Hz and 20 kHz and generate a minimum harmonic distortion. However, nonlinear harmonic distortion which consists mainly of cubic terms having 5 to 10 percent amplitude of the total output amplitude can occur, especially when dealing with small loudspeakers that operate at high volumes [
The nonlinearities of the power amplifier, loudspeaker, microphone, and preamplifier can be represented as a lumped model system using second-order Volterra series [
However, the representation in (
A raised cosine function has also been used to model the nonlinear characteristics of the amplifier-loudspeaker system [
Recently, chaotic signal processing has received much interest. Some signals, like radar sea clutter, speech, and indoor multipath effects, are better represented by deterministic chaotic rather than stochastic process. Understanding the chaotic process is useful in applications such as radar surveillance, secure communication, and narrowband interference cancellation in chaotic spread-spectrum system [
In ANC system, the noise that is generated from a dynamic system may be nonlinear and deterministic chaotic rather than stochastic, white, or tonal noise processes [
Among these chaotic noises, the Logistic type is the simplest and the most useful test signal, and it can be described as a second-order white and predictable nonlinear process. The Logistic process can be generated using the following equation [
The white noise signal
Primary and secondary propagation paths may exhibit nonlinear impulse responses. Nonlinear distortion between the reference sensor and the error sensor occurs in ducts where the noise propagates with high sound pressure [
The nonlinear noise source at the cancelling point can be represented by a third-order polynomial equation given as [
In (
In general, FXLMS structure and algorithm are used both in linear and nonlinear feedforward ANC systems. The reference signal must be filtered for correct adaptation. In the next section, the feedforward ANC with FXLMS algorithm is presented.
The basic single channel ANC system can be represented using an adaptive filtering block diagram, as shown in Figure
ANC System using FXLMS Algorithm.
The FXLMS scheme is depicted in Figure
The performance of linear ANC algorithm (FXLMS) degrades due to the existence of nonlinearity sources. Therefore, NANC algorithms are needed. Such algorithms are utilised in adapting nonlinear models such as truncated Volterra series, functional link neural networks, bilinear filters, and the NARMAX model. The model order represented by the number of weights is an important issue in hardware implementation. It affects both the computational load and the memory requirements of the control algorithm. Typically, in polynomial models such as the Volterra, bilinear, and NARMAX models, the model order is set to a large value causing over parameterization. However, only a few parameters are dominant to model the system. Therefore, several strategies have been employed to tackle the over parameterization problem like akaike information criterion (AIC) and bayesian information criterion (BIC) [
FLNN uses the functional expansion structure to replace the task performed by the hidden layers in traditional neural networks. This replacement has the advantage of making the structure for hardware easier to implement. Moreover, FLNN involves less computational complexity compared to the multilayer artificial neural network (MLANN).
This paper compares the performance and computational complexity of three NANC algorithms, which are commonly applied to update the nonlinear controllers. The algorithms considered are Volterra filtered-x least mean square (VFXLMS), bilinear filtered-x least mean square (BFXLMS), and filtered-s least mean square (FSLMS) which is based on the functional link neural network (FLNN) model. The description of the three NANC algorithms is given in the following subsections.
The general
VFXLMS for the feedforward ANC system.
The Volterra model typically requires a large number of parameters to estimate. In practice, the model order is initially chosen to be large which could cause overparameterization. Unfortunately, overparameterization will degrades parameter estimation accuracy and the robustness of the model. Moreover, overparameterization can be responsible for several unwanted dynamic effects [
In comparison to FIR filter, IIR filter can model a linear system with fewer coefficients. These coefficients are associated with delayed input and output samples. The Volterra filters are viewed as the nonlinear generalisation of a linear FIR filter while bilinear filters are an extension of the linear IIR filter with additional coefficients associated with the input-output cross multiplied samples. Thus, bilinear filters can model nonlinear systems with fewer coefficients compared to Volterra filters. The general equation of the input-output relationship of bilinear filters is given as [
BFXLMS for the feedforward ANC system.
From Figure
In BFXLMS, steepest descent algorithm is utilized in minimizing the squared error function. The weights are updated according to [
The disadvantages of modelling using the adaptive bilinear filters are the following. The adaptive bilinear filter may not converge to the global minimum because the error function has local minima. The adaptation process may be unstable.
These problems can be avoided if the algorithm is designed by choosing the step size carefully. In terms of computational complexity, BFXLMS requires less computation compared to VFXLMS for the same amount of noise reduction and same system conditions.
In conventional MLANN, the activation functions are used to introduce nonlinearity into the network. In FLNN, the functional expansion carries out the task performed by the activation functions. As a result of this fact, FLNN has the key advantages of involving less computational complexity and a simpler structure for hardware implementation [
The block diagram of the FSLMS based on FLNN is shown in Figure
FSXLMS for the feedforward ANC system.
From Figure
The weight updates equations can be derived using the steepest descent algorithm to minimise the squared error function
The disadvantage of the FSLMS algorithm is the extra computation required by the functional expansion block. The choice of the expansion function depends on the strength of nonlinearity and sets the computational complexity of the model. Thus, numerous expansion functions have been used to achieve compromised design between computational complexity and performance [
In NANC system, appropriate nonlinear algorithm is required in compensating the nonlinearity effects. Improved noise reduction can sometimes be achieved at the expense of high computation load. It is important to understand the trade off between performance and computational load in the design process. In the next section, a performance comparison between the nonlinear algorithms is investigated by means of simulation. In achieving fair comparison, the nonlinear algorithms are applied under the same system conditions and nonlinearity sources (type of reference noise, primary and secondary path transfer functions).
The performances of the three nonlinear algorithms with the associated models are compared. The normalized mean square error (
The saturation characteristic of the loudspeaker is modelled by saturating the controller output to a certain level. This level of saturation is obtained by setting a clipping threshold to the controller output signal at 85% of the maximum signal value.
Two reference noises are used in this simulation to investigate the performances of the algorithms. The first noise process is represented by a multiharmonic signal [
Three experiments were simulated to compare the performance of the NANC algorithms with the FXLMS algorithm. In each experiment, only a single nonlinearity source is introduced in the ANC system at any time. This was done to evaluate the performance of the NANC and the FXLMS algorithms in the presence of the corresponding nonlinearity source individually. In all experiments, the memory tap of the Volterra filter
NMSE simulation results summary.
NANC algorithm | Chaotic reference noise | Nonlinear primary path | Loudspeaker saturation 85% | |||
NMSE (dB) | Step size | NMSE (dB) | Step size | NMSE (dB) | Step size | |
FXLMS | −39 | −22 | −27 | |||
VFXLMS | −39 | −24 | −26 | |||
BFXLMS | −48 | −22 | −29 | |||
FSLMS | −31 | −18 | −29 |
NANC and FXLMS algorithms performance in the presence of nonlinear chaotic noise.
NANC and FXLMS algorithms performance in the presence of nonlinear primary path.
NANC and FXLMS algorithms performance in the presence of loudspeaker saturation.
Figure
Computational complexity is an important issue when implementing the algorithm in real-time application. In feed forward ANC structure, three stages of signal processing computation are performed. These stages of computation are generating the control signal, filtering the reference signal through the estimated secondary path, updating the controller weights.
Table
Complexity comparison of NANC algorithms.
NANC algorithm | Op. | Controller output (stage 1) | Filtered x | Weights update | Total | ||
---|---|---|---|---|---|---|---|
FXLMS | × | — | 25 | ||||
+ | — | 22 | |||||
VFXLMS | × | — | 200 | ||||
+ | — | 121 | |||||
BFXLMS | × | — | 393 | ||||
+ | — | 265 | |||||
FSLMS | × | 490 | |||||
+ | — | 205 |
In this paper, the performance and computational load of three nonlinear active noise control algorithms and the conventional FXLMS algorithm are compared. The nonlinearity sources that affect the system are categorised into noise process, primary and secondary acoustic paths, and the actuators which consist of the loudspeaker, microphone, and amplifier. Three NANC algorithms, namely, VFXLMS, BFSLMS, and FSLMS were simulated on a feedforward ANC system. The results from this study suggest that, in the presence of nonlinearity sources, the FXLMS algorithm still gives an acceptable performance compared to nonlinear control algorithms at the fraction of computational load.