Elsevier

Signal Processing

Volume 83, Issue 1, January 2003, Pages 121-134
Signal Processing

Computational load reduction of fast convergence algorithms for multichannel active noise control

https://doi.org/10.1016/S0165-1684(02)00382-1Get rights and content

Abstract

In this paper, the computational load of fast convergence recursive least-squares algorithms for multichannel active noise control (ANC) is reduced by the use of an inverse model of the acoustic plant between the actuators and the error sensors. The complexity reduction applies to both classical recursive least-squares algorithms or their fast time series or order-recursive implementations. To develop the new algorithm, a comparison of several control structures (filtered-x, adjoint, filtered-ε, inverse filtered-x, delay-compensated) available for the training of adaptive FIR filters in ANC is performed, based on three main factors that affect the convergence speed of the learning algorithms: correlation of input signals and acoustic plant, delay between the filters and the error signals, and filtering of the error signals. Stochastic gradient descent algorithms and recursive least-squares algorithms are combined with the different structures, and the resulting algorithms are compared based on the three factors. Several of the resulting algorithms have never been published, but of those new algorithms only one algorithm has the potential for optimal convergence speed, based on the three factors. Not only can this algorithm provide fast convergence, but for multichannel systems it also provides a large reduction of the computational load compared to the previously published algorithm with the fastest convergence. Therefore it is introduced in detail in the paper, and simulation results are presented to validate the convergence behavior of the new proposed algorithm.

Introduction

The concept of active sound cancellation (or active noise control, ANC) has been known for more than 50 years. Active sound cancellation works on the principle of destructive interference between an original “primary” disturbance sound field and a “secondary” sound field that is generated by some control actuators. For example, a classical application of active sound cancellation is the control of sound waves in a small duct. Fig. 1 shows such a duct with an actuator (a loudspeaker in this case), a reference sensor (possibly an accelerometer or a tachometer) and an error sensor (a microphone in this case). The reference sensor is used to measure an advanced information on the disturbance sound wave that propagates in the duct, and the error sensor is used to monitor the performance of the active sound cancellation system, thus providing feedback to a control algorithm. The control structure of Fig. 1 is called feedforward, because the controller feeds the actuator with a signal based on the advanced information obtained from the reference sensor. If the controller works properly, the signal sent to the actuator will generate a sound wave that will cancel the primary field sound wave at the location of the error sensor. Since usually only plane waves propagate in small ducts, the sound field will be uniform in any section of the duct, and the sound will be reduced, from the error sensor location to the end of the duct. Detailed presentations of active sound cancellation theory and applications can be found in [8], [9], [10], [13], [17], [18], [20], [22]. Active sound cancellation is certainly not limited to monochannel, feedforward, one-dimensional sound field systems: there are also multichannel, feedback or three-dimensional active sound cancellation systems. Also, the principle of destructive interference is not limited to the control of acoustic waves, and it has been successfully applied to the control of vibration [12], [13].

Even though the concept is simple, it is mostly with the development of fast low-cost digital signal processors during the last 15 years that the implementation of practical active sound and vibration control systems has become feasible. Digital technology is well suited for adaptive control systems where a lot of precision is required in order to achieve a good performance. Adaptive linear filtering techniques [14], [24], [25] have been extensively used for the active control of sound and vibration, and many of today's implementations of active control use those techniques. A popular adaptive filtering algorithm (to be described in more detail in Section 2) is the multichannel filtered-x least-mean-squares (LMS) algorithm (or the multi-error LMS algorithm) [6], [11], because of its simplicity and its relatively low computational load. This algorithm is a steepest descent algorithm that uses an instantaneous estimate of the gradient of the cost function (i.e. the mean squared error signals). Fig. 2 shows a block diagram of a monochannel implementation of this algorithm for ANC.

From our previous experience in the field of adaptive FIR filtering for multichannel active noise control [2], [3], [4], [5], [6], [7], three important factors that affect the convergence speed of the adaptive FIR filters learning algorithms have been identified. The first one is the well-known eigenvalue spread in the global autocorrelation matrix of the (typically filtered) input signals used to update the FIR filters [6], [14], [24]. In multichannel ANC, this eigenvalue spread can have a few causes: the spectral dynamics in the input signals of the control system (for example the signals from reference sensors in feedforward active noise control systems), the coupling between the different input signals, and also the spectral dynamics and the coupling of the multichannel plant between the acoustic transducers (typically loudspeakers) and the error sensors (typically microphones). The plant has an effect because a model of this plant is often used to filter the reference signals before they are used to adapt the FIR filters (as in Fig. 2).

A second factor that affects the convergence speed is the delay between the adaptive FIR filters and the error signals to be mean-squares or least-squares minimized by the learning algorithms. Indeed, for each update to the adaptive FIR filters, it takes some time for the effect of this update to be stabilized in the error signals, because of the propagation through the acoustic plant. This does not mean that the learning algorithm has to be stopped after every update so that the system can stabilize. However it does mean that a smaller adaptation gain (i.e. step size) has to be used in the learning algorithm, which slows down the convergence [6], [14], [24].

A third factor that can affect the convergence speed in some algorithms is the minimization of filtered error signals instead of the original error signals. Indeed, some learning algorithms for control structures such as the adjoint-LMS [6], [23] and the filtered-ε LMS [25] algorithms (both to be described in Section 2) rely on filtering the error signals by adjoint (i.e. time reversed) models of the plant or by inverse models of the plant. In these algorithms, the original error signals are indirectly minimized through the minimization of the filtered error signals, but the convergence speed may be slowed down, because the statistics of the filtered signals can be different from the statistics of the original error signals (i.e. different power spectra, coupling between the signals, etc.). Obviously, filtering and reducing the level of some very energetic frequency bands from the original error signals would lead to a slower convergence for those frequency components. Note that some filtering of the error signals can sometimes be performed to add a frequency-dependent weighting to the error signals (such as when a dBA weighting is to be applied) [19]. In this case the filtered error signals are the ones which actually need to be minimized, not the original ones, and this type of desirable filtering should not be confused with the filtering of the adjoint or filtered-ε structures.

To overcome these three factors slowing down the convergence speed, different approaches have been used. To reduce the eigenvalue spread in the global autocorrelation matrix of the (typically filtered) input signals used to update the adaptive FIR filters, recursive-least-squares algorithms [4], [6], and transform or frequency domain algorithms [5] have been proposed. Another approach that can reduce the eigenvalue spread is to compensate for the spectral dynamics and the coupling of the multichannel plant by using a multichannel inverse model of the plant [7]. Similarly, using algorithms that filter the error signals (as in the adjoint-LMS and the filtered-ε LMS algorithms) instead of filtering the signals from the reference sensors is another way to typically reduce the eigenvalue spread found in the global autocorrelation matrix, because in this case the dynamics and coupling of the multichannel plant are not found in the matrix.

To overcome the problem of the delay between the adaptive FIR filters and the error signals to be minimized, a delay-compensated filtered-x (or “modified filtered-x”) structure has been proposed by different authors [1], [6], [15], [21]. In this structure, alternative error signals with the same steady-state statistics as the original error signals (assuming accurate plant models) are minimized, and an update to the adaptive FIR filters has an instantaneous effect on those alternative error signals (no delay, therefore an increased step size can be used and faster convergence speed is obtained). On the opposite end, note that algorithms using filtering of the error signals (adjoint LMS or filtered-ε LMS algorithms) can increase the overall delay between the adaptive FIR filters and the filtered error signals, in particular when filtering with the adjoint plant models [6]. For the error signals filtering problem (i.e. the third factor that slows down the convergence speed of adaptive FIR filters learning algorithms for multichannel ANC), obviously all that is needed to overcome this problem is to use adaptive algorithms that do not require the filtering of the error signals.

In Section 2, different control structures that can be used for adaptive FIR filters in ANC such as filtered-x, adjoint, filtered-ε, inverse filtered-x and delay-compensated structures are combined with stochastic gradient descent algorithms, as an introduction to the use and meaning of the different structures. Then, based on the three previously mentioned factors affecting the convergence speed of the learning algorithms, and realizing that recursive least-squares algorithms (or their fast implementations) can also be combined with the different control structures, a comparison of several adaptive FIR filters algorithms for multichannel ANC (either previously published or unpublished) is done. Based on the three factors affecting the convergence speed, only one algorithm from the unpublished algorithms has the potential for optimal convergence speed. Since this algorithm uses an inverse structure and has the potential to provide a large reduction of the computational load compared the previously published algorithm with the fastest convergence speed, the algorithm will therefore be introduced in detail in Section 3. Section 4 will then show simulation results comparing the convergence behavior of the new proposed algorithm with some previously published algorithms, to validate the convergence behavior of the proposed algorithm.

Section snippets

Multichannel filtered-x LMS algorithm for ANC

In this paper, bold variables are used to represent vectors, and upper case bold variables are used to represent matrices. The multichannel filtered-x LMS algorithm [6], [11] will first be described. The instantaneous or stochastic gradient computed by the multichannel filtered-x LMS algorithm can be defined by [2]J(n)=12k=1Kek(n)2gradientΔ(n)=m=0M−1∂J(n)w(n−m)with ek(n) being the signal at time n from the kth error sensor in the active control system (see Fig. 1, Fig. 2), K the total number

The multichannel delay-compensated inverse filtered-x RLS algorithm for ANC

It was mentioned in the previous section that based on the three factors which slow down the convergence of adaptive FIR filter learning algorithms for multichannel ANC, the multichannel delay-compensated inverse filtered-x RLS algorithm for ANC has the potential to achieve the fastest convergence, while having a lower computational load than the multichannel delay-compensated filtered-x RLS algorithm [6]. Using the developments of Section 2 for the delay-compensated inverse filtered-x

Simulation results

Simulations were performed to verify the convergence behavior of the new delay-compensated inverse filtered-x RLS algorithm. The performance of the new algorithm was compared with the performance of the delay-compensated filtered-x RLS algorithm [6]. Two standard steepest descent algorithms for ANC (filtered-x LMS and delay-compensated filtered-x LMS algorithms [6], [11]) were also tested, as a basis of comparison. Fig. 9 shows the structure that was used for the simulations, with additional

Conclusion

In this paper, a new algorithm was proposed for multichannel ANC. Based on three factors that were mentioned to affect the convergence speed of adaptive FIR filter learning algorithms for ANC, the new proposed algorithm was chosen from several unpublished algorithms because it has the potential for the fastest convergence speed, yet with a significantly reduced computational load compared to the previously published fastest convergence speed algorithm. The new proposed algorithm uses a

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