^{1}

^{1}

^{1}

Active noise controls are used in a wide field of applications to cancel out unwanted surrounding noise. Control systems based on the feedback structure however have the disadvantage that they may become unstable during run-time due to changes in the control path—in this context including the listener’s ear. Especially when applied to active noise cancellation (ANC) headphones, the risk of instability is associated with the risk of harmful influence on the listener’s ear, which is exposed to the speaker in striking distance. This paper discusses several methods to enable the analysis of a feedback ANC system during run-time to immediately detect instability. Finally, a solution is proposed, which identifies the open loop behavior parametrically by means of an adaptive filter to subsequently evaluate the coefficients regarding stability.

Noise can interfere, adversely affect or even be harmful. In this context, the use of hearing protection or headphones with active noise cancellation (ANC) derives a substantial gain in comfort. Besides pilot headsets, this technology is also increasingly being offered in commercially available headphones to reduce noise especially when traveling or at work. While the current selection of ANC headphones is mainly limited to circumaural models, the concept has been extended to the more compact in-ear headphones in recent years. For the technical implementation of ANC systems based on the feedback structure, the interference signal of noise and speaker sound is measured by an internal microphone and fed back to a controller. The problem with the feedback structure is the risk of instability, which is expressed in an overdrive and eventual destruction of the speakers or even damage of the user’s inner ear. In [

In the following, a feedback control system for active noise cancellation in in-ear headphones is investigated. A schematic setup of the system is shown in

The secondary path, however, continuously changes even by slight variations of the headphones' position in the ears, possibly causing the control loop to become unstable [

The analysis of the system in the frequency domain initially requires a Fourier transform of the signals. To perform this during run time, the short-time FFT is used [

This transfer function can, for example, be displayed in a Nyquist plot and subsequently be evaluated geometrically. The stability of the system is then estimated by applying the simplified Nyquist criterion, according to which the locus of a stable system must intersect the real axis to the right of the critical point

Applied to the investigated system, first of all the signals of the speaker

When using the Nyquist stability criterion, the system can alternatively be analyzed in a Bode diagram. The estimation of its behavior is then based on the amplitude and phase component of the transfer function. Still, the critical point is crucial, which results in the constraint for stability

At the gain crossover frequency

Since the proposed methods in the frequency domain inherently cause a high computational effort due to the FFT, alternative methods operating in the time domain (which hence do not require the transfer function of the open loop) are investigated in the following. Now, the power

can be derived, according to which the control loop is presumed unstable if the power changes by more than a positive value δ.

Since this threshold is difficult to quantify, in the first approach an adaptive tolerance band is defined instead, which adapts to the current power level with a selectable fixed delay k. When the control loop becomes unstable, the power is expected to increase rapidly enough to cross the upper boundary of the tolerance band. The boundaries are estimated from prior samples of the power

with the tolerable difference―the width of the band―

Instead of evaluating the degree of the increase―determined by δ―, an alternative approach is to quantify the total amount of increases in a fixed time interval. This is actualized by counting the positive and negative changes in slope per interval and checking, whether within the respective interval a certain amount of positive changes is exceeded. For ease of application, the estimated power is first discretized logarithmically; that is, the range of discrete power values is equally spaced per decade. Then two counters detect the rising and falling edges _{C}) has passed, both edge counters are reset to zero. If, on the other hand, a certain amount of rising edges is reached within the interval, the control loop is presumed unstable and the estimated state is displayed respectively.

As mentioned in the previous discussions, a real-time implementation of the system analysis in the frequency domain is problematic because of the required FFT. On the other hand, the methods evaluating the signal power in the time domain substantially depend on the parameter sets and are not necessarily correlated with the control loop behavior. Consequently, a method which permits detection of unstable behavior in the time domain based on the transfer function of the open loop is desirable.

In the following approach, the open loop is identified parametrically using an adaptive FIR filter

which, in the case of an FIR filter, merely consists of the identified filter's coefficients

In this section, the proposed methods are tested in simulations. To induce the unstable state in the simulation model, a delay time is added to the secondary path after half the simulation time (t = 0.5 s). The methods are now supposed to detect the state change at this time.

First, the “Nyquist method” is tested. For this, the required signals

Now, the power estimation based methods are investigated.

enlarged detail reveals how the adaption of the band ensures that irrelevant increases of the power (which are not correlated with the system behavior) do not cross the boundaries. The increase caused by the instability (t ≥ 0.5 s) on the other hand, results in a peak of the estimated state (

_{C} and threshold values for

The previous simulations have shown that the approaches in the frequency and time domain both generally enable the estimation of the system state and thus the detection of instability. Their inherent disadvantages (required FFT and dependency on parameters), which make an implementation difficult, are supposed to be omitted by the combined method using the parametrical identification of the open loop. However, simulations have shown that the frequency response of the model filter does not approximate the one of the real open loop sufficiently. This problem has been attributed to the properties of the disturbance

Because of the previously mentioned issues with the open loop identification, the Jury stability test also yields no feasible results at first glance. On closer inspection, however, there is a change in the run-time of the algorithm depending on the system state. More precisely, the index―meaning the value of the loop variable within the algorithm―, at which the iterative estimation of the determinants (see [

This paper proposes methods to evaluate the signals of a control loop for ANC in in-ear headphones to detect an occurring instability of the system at run-time. It was shown that the analysis in the frequency domain―on the basis of the open loop behavior―generally allows the appliance of conventional methods such as the Nyquist criterion. However, regarding the sampling frequency of the DSP (f_{S} = 40 kHz), the high computational effort of the necessary FFT makes a real-time implementation not feasible.

On the other hand, the methods in the time domain―on the basis of signal power estimation―require significantly lower effort in computation and implementation. Yet, their functionality and hence their robustness substantially depend on the parameter sets. Furthermore, the effect of the rapid increase in signal power used for instability detection is not necessarily correlated with the control loop behavior.

Finally, the combination of the frequency- and time domain-based methods led to a solution, which evaluates the coefficients of an adaptive filter approximating the open loop behavior parametrically. To subsequently check the stability of the model filter, the determinant criterion of Jury was used. Here, however, there were difficulties because the convergence of the adaptive filter and thus the identification of the open loop were impaired by the highly transient properties of the disturbance. Nevertheless, the detection of instability by means of the Jury test was generally possible. Furthermore, it was shown that the state of the control loop can alternatively be estimated by simply observing the distribution of the coefficients of the model filter.

Firstly, the two combined approaches are to be implemented and evaluated experimentally with the real ANC system to substantiate the presented simulation results. However, since the principle of the filter based identification and evaluation of the open loop is suitable for all linear systems, the proposed solution can potentially be utilized for other control applications also.

Furthermore, if the “unstable” state is detected, the system has to be stabilized subsequently. This problem could be solved by observing and controlling the gain of the actuating variable―yet resulting in a decrease of the performance of the control system. An alternative approach would be to compensate for the change in the control path―meaning the distortion of the frequency response―which has caused the closed loop to become unstable originally. This issue is to be researched in detail in future work, too.

SvenHöber,ChristianPape,EduardReithmeier, (2015) Real-Time Detection of Unstable Control Loop Behavior in a Feedback Active Noise Cancellation System for In-Ear Headphones. Engineering,07,796-802. doi: 10.4236/eng.2015.712069