P-Norm Based Subband Adaptive Filtering Algorithm: Performance Analysis and Improvements

Abstract

The normalized subband adaptive filtering algorithm provides fast convergence rate for colored input signals as compared to the normalized least mean square algorithm, but it suffers from a poor convergence issue in the $\alpha$-stable noise. In light of this, the normalized subband p-norm (NSPN) algorithm, which is based on the least mean p-power error (MPE) criterion, is proposed in this study. This technique is not only robust against impulsive noise samples, but it also maintains a fast convergence rate when colored input signals are used. In addition to this, we develop both the steady-state and the transient models of the NSPN algorithm and provide some insights. Then, in order to solve the problem of making a choice regarding the order p in the NSPN algorithm, we design an autonomous system and come up with the NSPN algorithm with a variable p-norm (VP-NSPN). In addition, we offer the TFC-based VP-NSPN algorithm with a fast convergence rate and low steady-state misadjustment simultaneously by making use of the tap-weights feedback-based convex combination (TFC) scheme. In conclusion, simulation results on system identification and acoustic echo cancellation are undertaken in order to validate the superiority of the proposed algorithms and validate the usefulness of the theoretical analysis.

Appendices

Derivation of Term (a) in (14)

To compute the conditional expectation in (14), by applying the total probability rule over the CG noise model shown in the assumption 2, we obtain the following relation:

$$\begin{aligned} \begin{aligned} \ \text {E}&\Big \{q_{i}\big (e_{i,D}(k)\big )\frac{\varvec{u}_{i}(k)}{||\varvec{u}_i(k)||_p^p}|\widetilde{\varvec{w}}(k)\Big \}\\&=P\{z_i(k)=1\}\text {E} \Big \{q_{i}\big (e_{i,D,1}(k)\big )\frac{\varvec{u}_{i}(k)}{||\varvec{u}_i(k)||_p^p} |\widetilde{\varvec{w}}(k)\Big \}\\&\quad +P\{z_i(k)=0\}\text {E}\Big \{q_{i}\big (e_{i,D,2}(k)\big ) \frac{\varvec{u}_{i}(k)}{||\varvec{u}_i(k)||_p^p}| \widetilde{\varvec{w}}(k)\Big \}\\&=p_v\text {E}\Big \{q_{i}\big (e_{i,D,1}(k)\big ) \frac{\varvec{u}_{i}(k)}{||\varvec{u}_i(k)||_p^p}|\widetilde{\varvec{w}}(k)\Big \}\\&\quad +(1-p_v)\text {E}\Big \{q_{i}\big (e_{i,D,2}(k)\big )\frac{\varvec{u}_{i}(k)}{||\varvec{u}_i(k)||_p^p}|\widetilde{\varvec{w}}(k)\Big \}, \end{aligned} \end{aligned}$$

(63)

where $e_{i,D,1}(k)=e_{i,a}(k)+v_{g,i,D}(k)+\eta _{i,D}(k)$ and $e_{i,D,2}(k)=e_{i,a}(k)+v_{g,i,D}(k)$. For a larger length L of adaptive filter, $e_{i,a}(k)$ at the iteration k can be approximately regarded as a Gaussian random variable [8]. Thus, based on the assumption 2, $e_{i,D,1}(k)$ and $e_{i,D,2}(k)$ can be assumed to be Gaussian [2].

Accordingly, we can take advantage of the Price’s theorem [27] shown in (64) to calculate the expectations in (63)

$$\begin{aligned} \begin{aligned} \text {E}\big \{xq(y)\big \}=\frac{\text {E}\{xy\}}{\{\text {y}^2\}}\text {E}\big \{yq(y)\big \}. \end{aligned} \end{aligned}$$

(64)

Specifically, by applying (64) and assumptions 2 and 3, i.e., letting $x=\frac{\varvec{u}_{i}(k)}{||\varvec{u}_i(k)||_p^p}$ and $y=e_{i,D,1}(k)$, we obtain

$$\begin{aligned} \begin{aligned} \ \text {E}&\Big \{q_{i}\big (e_{i,D,1}(k)\big )\frac{\varvec{u}_{i}(k)}{||\varvec{u}_i(k)||_p^p}|\widetilde{\varvec{w}}(k)\Big \}\\&=\frac{\text {E}\Big \{\frac{e_{i,D,1}(k)\varvec{u}_i(k)}{||\varvec{u}_i(k)||_p^p}\Big \}}{\text {E}\big \{e_{i,D,1}^2(k)\big \}}\text {E}\big \{e_{i,D,1}(k)q_i\big (e_{i,D,1}(k)\big )\big \}, \end{aligned} \end{aligned}$$

(65)

where

$$\begin{aligned} \begin{aligned} \ {\text {E}\Big \{\frac{e_{i,D,1}(k)\varvec{u}_i(k)}{||\varvec{u}_i(k)||_p^p}\Big \}}&=\text {E}\Big \{\frac{\varvec{u}_i(k)\big (\varvec{u}_i^{\text {T}}(k)\widetilde{\varvec{w}}(k)+v_{i,D}(k)\big )}{||\varvec{u}_i(k)||_p^p}\Big \}\\ {}&=\text {E}\Big \{\frac{\varvec{u}_i(k)\varvec{u}_i^{\text {T}}(k)}{||\varvec{u}_i(k)||_p^p}\Big \}\widetilde{\varvec{w}}(k), \end{aligned} \end{aligned}$$

(66)

and

$$\begin{aligned} \begin{aligned} \ \frac{\text {E}\big \{e_{i,D,1}(k)q_i\big (e_{i,D,1}(k)\big )\big \}}{\text {E}\{e_{i,D,1}^2(k)\}}&=\frac{1}{\sqrt{2\pi }\sigma _{e_{i,D,1}}^3(k)}\int _{-\infty }^{+\infty }|e_{i,D,1}(k)|^p \\ {}&\quad \times \text {exp}\Big (\frac{e_{i,D,1}^2(k)}{-2\sigma _{e_{i,D,1}}^2(k)}\Big )d e_{i,D,1}(k)\\&=\frac{(\sqrt{2})^p(\sigma _{e_{i,D,1}}^2(k))^{\frac{p-2}{2}}}{\sqrt{\pi }}\varGamma \big (\frac{p+1}{2}\big ). \end{aligned} \end{aligned}$$

(67)

Then, by substituting (66) and (67) into (65), we compute the first expectation at the right side of (63) as

$$\begin{aligned} \begin{aligned} \ \text {E}&\Big \{q_{i}(e_{i,D,1})\frac{\varvec{u}_{i}(k)}{||\varvec{u}_i(k)||_p^p}|\widetilde{\varvec{w}}(k)\Big \}=\frac{(\sqrt{2})^p(\sigma _{e_{i,D,1}}^2(k))^{\frac{p-2}{2}}}{\sqrt{\pi }}\varGamma (\frac{p+1}{2})\text {E}\big \{\varvec{\varXi }_i(k)\big \}\widetilde{\varvec{w}}(k). \end{aligned}\nonumber \\ \end{aligned}$$

(68)

In the same way, the second expectation at the right side of (63) is calculated as

$$\begin{aligned} \begin{aligned} \ \text {E}&\Big \{q_{i}\big (e_{i,D,2}(k)\big )\frac{\varvec{u}_{i}(k)}{||\varvec{u}_i(k)||_p^p}|\widetilde{\varvec{w}}(k)\Big \}=\frac{(\sqrt{2})^p(\sigma _{e_{i,D,2}}^2(k))^{\frac{p-2}{2}}}{\sqrt{\pi }}\varGamma (\frac{p+1}{2})\text {E}\big \{\varvec{\varXi }_i(k)\big \}\widetilde{\varvec{w}}(k). \end{aligned}\nonumber \\ \end{aligned}$$

(69)

Now, by inserting (68) and (69) into (63), we complete the calculation of the conditional expectation in (14).

$$\begin{aligned} \begin{aligned}&\text {E}\Big \{q_{i}\big (e_{i,D}(k)\big )\frac{\varvec{u}_{i}(k)}{||\varvec{u}_i(k)||_p^p}|\widetilde{\varvec{w}}(k)\Big \}=\\ {}&\frac{(\sqrt{2})^p\varGamma (\frac{p+1}{2})}{\sqrt{\pi }}\Big [p_v(\sigma _{e_{i,D,1}}^2(k))^{\frac{p-2}{2}}+(1-p_v)(\sigma _{e_{i,D,2}}^2(k))^{\frac{p-2}{2}}\Big ]\text {E}\big \{\varvec{\varXi }_i(k)\big \}\widetilde{\varvec{w}}(k). \end{aligned} \end{aligned}$$

(70)

Derivation of (27)

In order to compute $\text {E}\left\{ q_{i}^2(e_{i,D}(k))\right\}$, we recall the conditional expectation to yield:

$$\begin{aligned} \begin{aligned} \ \text {E}\left\{ q_{i}^2(e_{i,D}(k))\right\} =\text {E}\Big \{\text {E}\left\{ q_{i}^2(e_{i,D}(k))\right\} |\widetilde{\varvec{w}}(k)\Big \}. \end{aligned} \end{aligned}$$

(71)

Then, by applying the law of total probability over the CG model, we obtain

$$\begin{aligned}{} & {} \text {E}\left\{ q_{i}^2(e_{i,D}(k))|\widetilde{\varvec{w}}(k)\right\} \nonumber \\{} & {} \quad =P\big \{z(n)=1\big \}\text {E}\left\{ q_{i}^2\big (e_{i,D,1}(k)\big )|\widetilde{\varvec{w}}(k)\right\} +P\big \{z(n)=0\big \}\text {E}\left\{ q_{i}^2\big (e_{i,D,2}(k)\big )|\widetilde{\varvec{w}}(k)\right\} \nonumber \\{} & {} \quad =p_v\text {E}\left\{ q_{i}^2\big (e_{i,D,1}(k)\big )|\widetilde{\varvec{w}}(k)\right\} +(1-p_v)\text {E}\left\{ q_{i}^2\big (e_{i,D,2}(k)\big )|\widetilde{\varvec{w}}(k)\right\} . \end{aligned}$$

(72)

By exploiting similar steps in Appendix A under assumptions 4, the following relations are established:

$$\begin{aligned} \ \text {E}\left\{ q_{i}^2(e_{i,D,1}(k))|\widetilde{\varvec{w}}(k)\right\}= & {} \frac{(\sqrt{2})^{2p-2}}{\sqrt{\pi }}\Big [\text {Tr}\big \{\widetilde{\varvec{W}}(k)\{\varvec{R}_u\}_i\big \}+(1+\kappa )\sigma _g^2||\varvec{h}_i||_2^2\Big ]^{p-1}\nonumber \\ {}{} & {} \quad \varGamma (p-\frac{1}{2}), \end{aligned}$$

(73)

and

$$\begin{aligned} \begin{aligned} \ \text {E}\left\{ q_{i}^2(e_{i,D,2}(k))|\widetilde{\varvec{w}}(k)\right\} =\frac{(\sqrt{2})^{2p-2}}{\sqrt{\pi }}\Big [\text {Tr}\big \{\widetilde{\varvec{W}}(k)\{\varvec{R}_u\}_i\big \}+\sigma _g^2||\varvec{h}_i||_2^2\Big ]^{p-1}\varGamma (p-\frac{1}{2}). \end{aligned}\nonumber \\ \end{aligned}$$

(74)

By substituting (72)–(74) into (71), the derivation of (27) is completed.