A new online secondary path modeling method based on delay coefficient technique

2.1. Step size variation

Usually ${\mu }_w\left(n\right)$ is set to a small value in order to prevent the system suffering from diverging. However, in our study, we employ the FXLMS algorithm instead of the Modified FXLMS algorithm in order to realize improved performance at a reasonable computational cost. And thus, the convergence speed of the active control filter would be degraded due to slow convergence of FXLMS algorithm. In order to improve the convergence of the active control filter, we vary the value of the step size ${\mu }_w$ according to the status of the AVC system. The convergence of the AVC system is relevant to the power of original vibration and the power of the generated white noise [8]. From further analysis, we can know not only the power of the generated white noise $v\left(n\right)$ but also the estimation error of the active control filter could affect the convergence of the AVC system. From the derivation of $f\left(n\right)$, we can know the estimation error $∆\mathbf{W}\left(n\right)$ included in the error signal $f\left(n\right)$. In other words, the $f\left(n\right)$ used as the error signal of the AVC controller, reflects the convergence status of the AVC system [11]. Consequently, in order to improve the performance, the adaptive filter step size ${\mu }_w\left(n\right)$ adaptively changes following power variations of the reference signal, the generated white noise and the estimation error of the active control filter alike. By considering the above situations, we propose the following procedure to vary the step size ${\mu }_w\left(n\right)$ during system operation:

The power of these signals is estimated as:

where $\lambda$ is the forgetting factor (0.9$<\lambda <$1).

When the power of the reference signal $x\left(n\right)$ is increased, a small value of ${\mu }_w\left(n\right)$ can prevent the system from diverging. When the power decreases, a large value of ${\mu }_w\left(n\right)$ improves the vibration attenuation and convergence rate of the adaptive filter. When the power of the generated white noise increases, the convergence rate of secondary path modeling would increases. This allows using a larger ${\mu }_w\left(n\right)$ during system operation. For the similar reason, a large estimation error of the active control filter means the AVC system is far$$from steady state, this moment, a small value of ${\mu }_w\left(n\right)$ is necessary to improve the system performance.

As mentioned before, the modeling filter of the proposed system estimates the secondary path transfer function using a modified version of the VSS-LMS algorithm. Studies have shown that the value of step-size parameter ${\mu }_s\left(n\right)$ has important influence on the modeling accuracy and the system convergence, consequently, ${\mu }_s$ should be varied in accordance with the state of the SPM filter and its distance to the steady state. From the discussion about $f\left(n\right)$ above, we can know that $[v^{\text{'}}\left(n\right)-v_s^{\text{'}}\left(n\right)]$ carries error information about the iteration of ${\mathbf{S}}^{\mathbf{\text{'}}}\left(n\right)$, but $\left[d\left(n\right)-y^{\text{'}}\left(n\right)\right]$ acts as interference to the SPM filter ${\mathbf{S}}^{\mathbf{\text{'}}}\left(n\right)$. The step size should be calculated as the rule trying to reduce the influence of disturbance signal $\left[d\left(n\right)-y^{\text{'}}\left(n\right)\right]$.

If the signal $[v^{\text{'}}\left(n\right)-v_s^{\text{'}}\left(n\right)]$ could be gotten directly, the secondary path modeling filter ${\mathbf{S}}^{\mathbf{\text{'}}}\left(n\right)$ should be adapted as follows:

So we can consider:

Further derivation of ${\mu }_s$ can be gotten:

Thus, the bad influence on online secondary path modeling from the disturbance signal $\left[d\left(n\right)-y^{\text{'}}\left(n\right)\right]$ can be reduced. The signal $v\left(n\right)$ is uncorrelated with $x\left(n\right)$. Further derivation of the ratio can be expressed as:

where $f\left(n\right)={\mathbf{X}}^T∆\mathbf{W}\left(n\right)\mathrm{*}\mathbf{S}\left(n\right)+{\mathbf{V}}^T\left(n\right)∆\mathbf{S}\left(n\right)$.

So we can get approximately:

If the power of the estimation error $P_{∆S}\left(n\right)$ could be gotten, we can adapt the secondary path modeling filter ${\mathbf{S}}^{\mathbf{\text{'}}}\left(n\right)$ using the evaluation of Eq. (19). In order to get the accurate estimate value of $P_{∆S}\left(n\right)$, the delay coefficient technique is adopted in our study, which has been largely applied in the field of acoustic echo cancellation [14]. The delay coefficient technique derives from the observation that the system error tends to have a uniform distribution on its coefficients. By delaying by $D$ samples the internally generated signal $v\left(n\right)$ sent to the secondary path, the steady-state value of the first $D$ coefficients of the secondary path modeling filter becomes 0, and the instantaneous value of the first $D$ delay coefficients can be used for estimating the system distance:

where ${\mathbf{S}}_0^{\text{'}}\left(n\right)$ the first $D$ coefficients of the secondary path modeling filter ${\mathbf{S}}^{\text{'}}\left(n\right)$.

So the power of the estimation error $P_{∆S}\left(n\right)$ can be estimated as:

The last $M$ coefficients of ${\mathbf{S}}^{\mathbf{\text{'}}}\left(n\right)$, ${\mathbf{S}}_1^{\mathbf{\text{'}}}\left(n\right)$ is the estimation vector for the impulse response of the secondary path. ${\mathbf{S}}^{\mathbf{\text{'}}}\left(n\right)$ and ${\mathbf{S}}_1^{\mathbf{\text{'}}}\left(n\right)$ are all adapted with LMS algorithm, as follows:

where:

2.2. Auxiliary noise power scheduling

In general, an unavoidable problem of the online secondary path modeling methods using the auxiliary noise technique is that a large auxiliary noise could lead to a quick acquisition of an accurate secondary path modeling but will greatly increase the residual vibration [11]. In order to obtain an accurate secondary path model and reduce the influence of auxiliary noise to the residual vibration, one of the solutions is auxiliary noise power scheduling technique. Large auxiliary noise is injected to obtain an accurate model of secondary path quickly at the start-up of AVC system, or when a sudden and large change occurs in the secondary path. But the power of the auxiliary noise should be kept as low as possible when the AVC system is close to steady-state.

The training signal for the modeling filter of the secondary path is $v\left(n\right)$, which is injected at output $y\left(n\right)$ and is defined as:

where $v_m\left(n\right)$ is an internally generated zero-mean white Gaussian noise signal. $G\left(n\right)$ is the parameter that controls the injection of the white noise.

The value of the gain $G\left(n\right)$ should be varied in accordance with the power of original vibration and the modeling error of the secondary path. In other words, the strategy schedule $\mathrm{}v\left(n\right)$ based on both the convergence status of the SPM system and the power of the reference signal $x\left(n\right)$.

Here, let us first discuss the ratio $\rho \left(n\right)=P_f\left(n\right)/P_e\left(n\right)$. Since $v^{\text{'}}\left(n\right)$ is uncorrelated with $d\left(n\right)$ and $y\text{'}\left(n\right)$, further derivation of the error signal $\rho \left(n\right)$ as follows:

As stated in [11], the injected random noise $v\left(n\right)$ is low level excitation signal as compared with the reference signal $x\left(n\right)\text{.}$ Initially at $n=0\text{,}$ the canceling signal $y^{\text{'}}\left(n\right)=\text{0}\text{,}$ so $P_{\left[d\left(n\right)-y^{\text{'}}\left(n\right)\right]}\left(n\right)\gg P_{v^{\text{'}}\left(n\right)}$ and $P_{\left[d\left(n\right)-y^{\text{'}}\left(n\right)\right]}\left(n\right)\gg P_{[v^{\text{'}}\left(n\right)-v_s^{\text{'}}\left(n\right)]}$, and hence $\rho \left(n\right)\approx P_x\left(n\right)$. In steady state, when the AVC system comes to converge properly, $y^{\text{'}}\left(n\right)$ trends to $d\left(n\right)$, $v_s^{\text{'}}\left(n\right)$ trends to $v^{\text{'}}\left(n\right)$, the value of $\rho \left(n\right)$ is only correlated with the distance between ${\mathbf{S}}^{\mathbf{\text{'}}}\left(n\right)$ and the real value$\mathrm{}\mathbf{S}\left(n\right)$.

The proposed strategy can be formulated as follows:

From Eq. (14), it can be seen that the power of $v\left(n\right)$ is scheduled by taking into account the convergence status of the AVC system and the power of the primary vibration. In fact, after the active control filter comes to converge, the value of $G\left(n\right)$ becomes very small. By using Eq. (14), the power of $v\left(n\right)$ is, therefore, scheduled to a very low level. As a result, the contribution of auxiliary noise to the residual vibration is greatly alleviated. And thus improve the vibration attenuation of AVC system.

2.3. Computational complexity analysis

Table 1 compares the computation complexity of the proposed method with that of Eriksson’s, Zhang’s, Akhtar’s, and Carini’s method.

Here $N$ and $M$ are tap-weight lengths of the control filter $\mathbf{W}\left(n\right)$, and SPM filter ${\mathbf{S}}^{\text{'}}\left(n\right)$, respectively. $H$ is tap-weight length of the third filter in the Zhang’s method. The tap-weight length of the part of the SPM filter ${\mathbf{S}}_0^{\text{'}}\left(n\right)$ is $D$ in Carini’s method, and the SPM filter ${\mathbf{S}}_0^{\text{'}}\left(n\right)$ is used for estimating the modeling error of the secondary path. The Eriksson’s method has the lowest computational cost among the other existing online secondary path modeling methods, because it’s the basic online secondary path modeling method which fixed step-sizes are employed for both the adaptive filters, and no auxiliary noise scheduling strategy is employed. The Carini’s method (the control filter $\mathbf{W}\left(n\right)$ is adapted with a fixed step-size) has the highest computational cost comparing with the other methods discussed in this paper. The main reason of the high computational cost is the using the delay coefficient technique for online estimation error of the secondary path, which equivalent to introducing the third adaptive filter. The computation complexity of Zhang’s method is also very high due to introducing the third adaptive filter. Akhtar’s method avoiding using the third adaptive filter, has lower computational cost than Carini’s method, however, it adds two extra (fixed FIR) filters for $\mathbf{W}\left(n\right)$ and $\mathbf{S}\mathbf{\text{'}}\left(n\right)$. The proposed method has lower computational cost as compared with Akhtar’s method, because no experimentally determined parameter is employed for the iteration of the adaptive filters and auxiliary noise scheduling strategy, and the variance rules for step size parameter and auxiliary noise scheduling strategy in the proposed method are simpler than that in Akhtar’s method.

3.1. Case 1

In Case 1, the reference signal $x\left(n\right)$ is a tonal signal with 50 Hz frequency. The variance of this signal is adjusted to 2.0 and the tonal signal is corrupted with a zero-mean white Gaussian noise until a 30 dB signal-to-noise ratio (SNR). In order to ensure low residual vibration in the steady state, a zero-mean white Gaussian noise of variance 0.05 is used in the secondary path modeling process. The step size of the active control filter $\mathbf{W}\left(n\right)$ is ${\mu }_w=1\times {10}^{-2}$ in all of these methods. The parameters of Eriksson’s method are:${\mu }_s=5\times {10}^{-2}$. The parameters of Akhtar’s method are: ${\mu }_{smin}=\text{3×}{\text{10}}^{\text{-2}}$, ${\mu }_{smax}=\text{0.8}$, ${\sigma }_{max}^2=\text{1}$, ${\sigma }_{min}^2=\text{1×}{\text{10}}^{\text{-2}}$. The parameters of Carini’s method are: ${\mu }_{smin}=\text{2×}{\text{10}}^{\text{-3}}$, $R\left(n\right)=\text{1}$, $N_s^{\text{'}}\left(0\right)=N_w^{\text{'}}\left(0\right)=\text{0}$.

The plots of relative modeling error $∆\mathbf{S}\left(n\right)$ are shown in Fig. 4(a). It can be seen that the proposed method can decrease the modeling error to a lower level (about –38 dB) at reasonably faster convergence rate comparing with Carini’s method and Akhtar’s method. The results about vibration reduction $R\left(n\right)$ are shown in Fig. 4(b). It can be shown that the proposed method offers a much better vibration reduction. The variation charts of the step size ${\mu }_s$ for secondary path modeling filter are shown in Fig. 4(c). In Akhtar’s method, the step size is set to a fixed value ${\mu }_{smin}$ at the start-up and later increased to a fixed value ${\mu }_{smax}$. In proposed method, the step size varied adaptively in accordance with the ratio of the power of the auxiliary noise and that of residual signal $e\left(n\right)$. After the start-up of the AVC system, the step size in proposed method increases to a relatively large value resulting in a rapid convergence of the secondary path modeling filter. In the proposed method and Carini’s method, the variation of step size has a decreasing behavior with the converging of the AVC system. That because the variation of step size ${\mu }_s$ depends upon the relative modeling error of the secondary path which starts iteration from a unit vector, and then decrease to zero with the converging of the AVC system. The corresponding curve for the step size ${\mu }_w$ is shown in Fig. 4(d).

Fig. 4Simulation results in Case 1: a) Modeling error ∆S(n) (dB), b) Vibration reduction R(n) (dB), c) The time-varying step size μs, d) The time-varying step size μw

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Fig. 5Simulation results in Case 2: a) Modeling error ∆S(n) (dB), b) Vibration reduction R(n) (dB), c) The time-varying step size μs, d) The time-varying step size μw

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Fig. 6Simulation results in Case 3: a) Modeling error ∆S(n) (dB), b) Vibration reduction R(n) (dB), c) The time-varying step size μs, d) The time-varying step size μw

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3.2. Case 2

In Case 2, the reference signal $x\left(n\right)$ is a broadband vibration signal obtained by filtering a zero-mean white Gaussian noise with band pass filter having pass-band 100-400 Hz. The variance of the reference signal is adjusted to 2. A zero-mean white Gaussian noise of variance 0.1 is used for the secondary path modeling. The parameters have been set as follows: The step size of the active control filter $\mathbf{W}\left(n\right)$ is ${\mu }_w=\text{5×}{\text{10}}^{\text{-4}}$ in all of these methods. The parameters of Eriksson’s method are: ${\mu }_s=\text{1×}{\text{10}}^{\text{-3}}$. The parameters of Akhtar’s method are: ${\mu }_{smin}=\text{2×}{\text{10}}^{\text{-3}}$, ${\mu }_{smax}=\text{0.8}$, ${\sigma }_{max}^2=\text{4}$, ${\sigma }_{min}^2=\text{0.001}$. The parameters of Carini’s method are: ${\mu }_{smin}=\text{2×}{\text{10}}^{\text{-3}}$, $R\left(n\right)=\text{1}$, $N_s^{\text{'}}\left(0\right)=N_w^{\text{'}}\left(0\right)=\text{0}$.

Comparison results of the simulation experiment are shown in Fig. 8. As can be seen, we notice a strong performance improvement with the proposed method in the convergence speed of the algorithm, in the estimation accuracy of the secondary path modeling filter, and in the power of error microphone signal. The simulation results for broadband input show similar superiority as for fixed frequency input in Case 1. The reason is also similar with Case 1.

3.3. Case 3

Studies show the norms of the adaptive filters $\mathbf{W}\left(n\right)$ and $\mathbf{S}\mathbf{\text{'}}\left(n\right)$ are almost not affected by changes in the primary vibration after system convergence. However, they are sensitive to changes in the secondary path or the primary path [8].

In Case 3, we consider both the primary and the secondary paths transfer functions are suddenly changed during operation. The reference signal is same as described in Case 1. The system is started with the same conditions as described for Case 1. In the first 40000 simulation samples we assume to have impulse responses of the primary path and of the secondary path which are the same of the first two set of simulations. A sudden and large variation is introduced to the secondary path at the 40000th sample. The FIR filter coefficients of the primary path and secondary path are shown in Fig. 3.

The results about modeling error $∆S\left(n\right)$ for the four methods are shown in Fig. 6(a). As can be seen, the proposed method decreases the modeling error to about –28 dB after the vibration path change, while Akhtar’s method can reduce the modeling error to about –34 dB. So the system equipped with Akhtar’s method gets a smaller modeling error. But the system equipped with proposed method provides a much faster convergence of the SPM filter after the vibration path change. Carini’s method has low tracking capability for the vibration path perturbation, and the convergence of the SPM filter is slow. The results regarding vibration reduction $R\left(n\right)$ are shown in Fig. 6(b). It can be seen that the proposed method achieves higher vibration reduction $R\left(n\right)$ as compared with the other three methods. In proposed method, due to the fast convergence of SPM filter, the interference effect of $[v^{\text{'}}\left(n\right)-v_s^{\text{'}}\left(n\right)]$ is quickly neutralized from the error signal $f\left(n\right)$ for the active vibration control filter $\mathbf{W}\left(n\right)$. The variation charts of the step size ${\mu }_s$ for secondary path modeling filter are shown in Fig. 6(c). The corresponding curve for the step size ${\mu }_w$ is shown in Fig. 6(d).