Two Novel Arctangent Normalized Subband Adaptive Filter Algorithms Against Impulsive Interferences

Abstract

In this paper, two arctangent-based normalized subband adaptive filter (Arc-NSAF) algorithms are proposed, which are derived from two different forms of arctangent cost function and their difference is the order of the arctangent computation and the summation operation of the normalized subband error signal. Benefiting from the excellent property of the arctangent function for suppressing impulsive interferences, the proposed Arc-NSAF algorithms not only possess good robustness against impulsive interferences, but also obtain faster convergence rate for the colored input signals. To further improve the convergence performance of the Arc-NSAFs when identifying sparse system, the proportionate versions of the Arc-NSAF algorithms are also proposed. Simulation results have demonstrated the superiority of the proposed algorithms in impulsive interferences environments as compared to their respective competitors.

Appendix

In this Section, the detailed derivation of (14) is presented. Using Price’s theorem in [15] and the assumption (1), we have

$$\begin{aligned}&E\left( {\frac{e_{a,i} (k)e_{i,\mathrm{D}} (k)}{\frac{\alpha ^{2}e_{i,\mathrm{D}}^4 (k)}{\left\| {u_i (k)} \right\| ^{2}}+\left\| {u_i (k)} \right\| ^{2}}} \right) \approx P_\mathrm{r} \frac{E\left( {e_{a,i} (k)e_{i_1 ,\mathrm{D}} (k)} \right) }{E\left( {e_{i_1 ,\mathrm{D}}^2 (k)} \right) }E\left( {\frac{e_{i_1 ,\mathrm{D}}^2 (k)}{\frac{\alpha ^{2}e_{i_1 ,\mathrm{D}}^4 (k)}{\left\| {u_i (k)} \right\| ^{2}}+\left\| {u_i (k)} \right\| ^{2}}} \right) \nonumber \\&\quad +\,(1-P_\mathrm{r} )\frac{E\left( {e_{a,i} (k)e_{i_2 ,\mathrm{D}} (k)} \right) }{E\left( {e_{i_2 ,\mathrm{D}}^2 (k)} \right) }E\left( {\frac{e_{i_2 ,\mathrm{D}}^2 (k)}{\frac{\alpha ^{2}e_{i_2 ,\mathrm{D}}^4 (k)}{\left\| {u_i (k)} \right\| ^{2}}+\left\| {u_i (k)} \right\| ^{2}}} \right) \end{aligned}$$

(A1)

with

$$\begin{aligned} e_{i_1 ,\mathrm{D}} (k)=e_{a,i} (k)+\eta _i (k), \quad e_{i_2 ,\mathrm{D}} (k)=e_{a,i} (k)+v_i (k) \end{aligned}$$

(A2)

where $\eta _i (k)$ and $v_i (k)$ are zero-mean Gaussian sequences with variances $\sigma _{\eta _i }^2 =\frac{1}{N}(\kappa +1)\sigma _v^2$ and $\sigma _{v_i }^2 =\frac{1}{N}\sigma _v^2$.

According to the assumption (3), we have

$$\begin{aligned} E\left[ {e_{i_1{,\mathrm{D}}}^2 (k)} \right] =E\left[ {e_{a,i}^2 (k)} \right] +\frac{1}{N}(\kappa +1)\sigma _v^2 , \quad E\left[ {e_{i_2{,\mathrm{D}}}^2 (k)} \right] =E\left[ {e_{a,i}^2 (k)} \right] +\frac{1}{N}\sigma _v^2\nonumber \\ \end{aligned}$$

(A3)

Then, we get

$$\begin{aligned} E\left[ {e_{i{,\mathrm{D}}}^2 (k)} \right]= & {} P_\mathrm{r} E\left[ {e_{i_1{,\mathrm{D}}}^2 (k)} \right] +(1-P_\mathrm{r} )E\left[ {e_{i_2{,\mathrm{D}}}^2 (k)} \right] \nonumber \\= & {} E\left[ {e_{a,i}^2 (k)} \right] +\frac{1}{N}(P_\mathrm{r} \kappa +1)\sigma _v^2 \end{aligned}$$

(A4)

Based on the definitions of the c(k) and $e_{a,i} (k)$, we have the following equation

$$\begin{aligned} E\left( {e_{a,i}^2 (k)} \right) =\sigma _{u_i }^2 (k)c(k) \end{aligned}$$

(A5)

Therefore, combining (A1)–(A5), (14) can be derived.