Mean-Square Performance of the Modified Filtered-x Affine Projection Algorithm

Abstract

The modified filtered-x affine projection (MFxAP) algorithm is effective for active noise control owing to its good convergence behavior and medium computational burden. The transient and steady-state performances of the MFxAP algorithm have been analyzed in previous studies, which presented a relatively good agreement between the theory and measured results. However, the correlation between the weight-error vector and the past noise vectors is disregarded in the existing methods. Hence, a more accurate theoretical analysis for the MFxAP algorithm is presented herein, in which the effect of the past noise vector on the weight-error vector is considered comprehensively. Simulation results indicate that the proposed theoretical results match the experimental results more precisely than the previous studies, in particular, at the steady state.

Appendix A

From (17), we expand the term $\prod \nolimits _{l = 0}^{k - 1} {\left( {{\mathbf{{I}}_L} - \mu \mathbf{{P}}(n - l)} \right) }$ as follows:

$$\begin{aligned} \begin{aligned}&\prod \limits _{l = 0}^{k - 1} {\left( {{\mathbf{{I}}_L} - \mu \mathbf{{P}}(n - l)} \right) } \\&\quad = \left( {{\mathbf{{I}}_L} - \mu \mathbf{{P}}(n)} \right) \left( {{\mathbf{{I}}_L} - \mu \mathbf{{P}}(n - 1)} \right) \cdots \left( {{\mathbf{{I}}_L} - \mu \mathbf{{P}}(n - k + 1)} \right) \\&\quad = {\mathbf{{I}}_L} - \mu \left( {\mathbf{{P}}(n) + \mathbf{{P}}(n - 1) + \cdots \mathbf{{P}}(n - k + 1)} \right) \\&\qquad + {\mu ^2}\left( {\mathbf{{P}}(n)\mathbf{{P}}(n - 1) + \mathbf{{P}}(n)\mathbf{{P}}(n - 2) + \cdots \mathbf{{P}}(n - k)\mathbf{{P}}(n - k + 1)} \right) \\&\qquad + \cdots + {( - \mu )^k}\left( {\mathbf{{P}}(n)\mathbf{{P}}(n - 1) \cdots \mathbf{{P}}(n - k + 1)} \right) \\&\quad = {\mathbf{{I}}_L} - \mu \sum \limits _{{n_0} = 0}^{k - 1} {\mathbf{{P}}(n - {n_0})} + {\mu ^2}\sum \limits _{{n_0} = 0}^{k - 2} {\sum \limits _{{n_1} = {n_0} + 1}^{k - 1} {\mathbf{{P}}(n - {n_0})\mathbf{{P}}(n - {n_1})} } + \cdots \\&\qquad + {( - \mu )^k}\sum \limits _{{n_0} = 0}^0 {\sum \limits _{{n_1} = {n_0} + 1}^1 { \cdots \sum \limits _{{n_{l - 1}} = {n_{l - 2}} + 1}^{k - 1} {\mathbf{{P}}(n - {n_0})\mathbf{{P}}(n - {n_1}) \cdots \mathbf{{P}}(n - {n_{l - 1}})} } }\\&\quad = \sum \limits _{l = 0}^k {{{( - \mu )}^l}{} \mathbf{{G}}_k^l\left( {\mathbf{{P}}(n)} \right) }, \end{aligned} \end{aligned}$$

(35)

where $\mathbf{{G}}_k^l\left( {\mathbf{{P}}(n)} \right)$ is the coefficient related to ${( - \mu )^l}$ which can be written as

$$\begin{aligned} \mathbf{{G}}_k^l\left( {\mathbf{{P}}(n)} \right) = \sum \limits _{{n_0} = 0}^{k - l} {\sum \limits _{{n_1} = {n_0} + 1}^{k - l + 1} { \cdots \sum \limits _{{n_{l - 1}} = {n_{l - 2}} + 1}^{k - 1} {\mathbf{{P}}(n - {n_0})\mathbf{{P}}(n - {n_1}) \cdots \mathbf{{P}}(n - {n_{l - 1}})} } }.\nonumber \\ \end{aligned}$$

(36)

In particular, $\mathbf{{G}}_k^0\left( {\mathbf{{P}}(n)} \right) = {\mathbf{{I}}_L}$.