Steady-state analysis of distributed incremental variable fractional tap-length LMS adaptive networks

Abstract

Recently, cooperative and distributed processing has been attracted a lot of attention, especially in wireless sensor networks, to prolong the network’s lifetime. So, distributed adaptive filtering, which operates in a distributed and adaptive manner, has been established. In the distributed adaptive networks, in addition to filter coefficients, the length of the adaptive filter is also unknown in general, and it should be estimated. The distributed incremental fractional tap-length (FT) algorithm is an approach to determine the adaptive filter length in a distributed scheme. In the current study, we analyze the steady-state behavior of the distributed incremental variable FT LMS algorithm. According to the analysis, we derive the mathematical expression for the steady-state tap-length at each particular sensor. The obtained results indicate that this algorithm overestimates optimal tap-length. Numerical simulations are provided to confirm the theoretical analyses.

Appendix A.

In [26], we have presented the concept of a short-length DILMS algorithm, and we have provided an expression for its steady-state MSD. This steady-state MSD is needed for the evaluation of the term $E\{{\Vert {\ddot{\mathbf{P}}}_{k-1}\Vert }^2\}$ in (30), so the main results are included here. For this aim, we assume a set of N sensors in order to find an $L_{opt}\times 1$ unknown vector $\mathbf{w}^o_{L_{opt}}$ with unknown length $L_{opt}$ from multiple measurements collected at N sensor nodes in the network. As previously mentioned, to find the length of the unknown parameter, a variable tap-length algorithm is needed. But, we assume that such an algorithm is not applicable for reasons such as energy storage (since energy consumption is an essential issue in WSNs). So, at each sensor, a conjectural length M for the unknown parameter is considered. More clearly, each sensor is equipped with an adaptive filter with M coefficients ($M<L_{op}$). Now at each sensor node, only the algorithm (3) is applicable in which all vectors have length M.

Now, we present a steady-state analysis for this deficient length case. To perform this analysis, first, we partition the unknown parameter $\mathbf{w}^o_{L_{opt}}$ as follow:

$$\begin{aligned} \left[ \begin{array}{c} {\dot{\mathbf{w}}}_M \\ {\ddot{\mathbf{w}}}_{L_{opt}-M} \end{array} \right] \end{aligned}$$

(59)

where ${\dot{\mathbf{w}}}_M$ is the partition of $\mathbf{w}^o_{L_{opt}}$ that is modeled by ${\varvec{\psi }}^{(i)}_k$ in each sensor, and ${\ddot{\mathbf{w}}}_{L_{opt}-M}$ is the partition of $\mathbf{w}^o_{L_{opt}}$ that excluded in the estimation of $\mathbf{w}^o_{L_{opt}}$. The partitioning simplifies the work with the variable-length vectors. Regarding the partitioning and using the data model (11), the update equation (3) is expressed as:

$$\begin{aligned}{\varvec{\psi }}^{(i)}_k&={\varvec{\psi }}^{(i)}_{k-1}-{\mu }_k\mathbf{u}^*_{k,i}{} \mathbf{u}_{L_{opt}k,i}{\overline{\varvec{\psi }}}^{(i)}_{L_{opt}, k-1}\nonumber \\&\quad +{\mu }_kv_k(i)\mathbf{u}^*_{k,i}\ \end{aligned}$$

(60)

where the vector ${\overline{\varvec{\psi }}}^{(i)}_{L_{opt},k-1}$ with length $L_{opt}$, computes the difference among the weight at sensor $k-1$ and $\mathbf{w}^o_{L_{opt}}$ as:

$$\begin{aligned} {\overline{\varvec{\psi }}}^{(i)}_{L_{opt},k-1}=\left[ \begin{array}{c} {\varvec{\psi }}^{(i)}_{k-1} \\ \mathbf{O}_{(L_{opt}-M)\times 1} \end{array} \right] -\left[ \begin{array}{c} {\dot{\mathbf{w}}}_M \\ {\ddot{\mathbf{w}}}_{L_{opt}-M} \end{array} \right] \end{aligned}$$

(61)

By padding the $L_{opt}-M$ zeros in vectors with length M in (60), and subtracting the unknown parameter $\mathbf{w}^o_{L_{opt}}$ from both of its sides results in:

$$\begin{aligned} {\overline{\varvec{\psi }}}^{(i)}_{L_{opt},k}={{\Lambda }}_k(i) {\overline{\varvec{\psi }}}^{(i)}_{L_{opt},k-1}+{\mu }_kv_k(i) \left[ \begin{array}{c} \mathbf{u}^*_{k,i} \\ \mathbf{O}_{(L_{opt}-M)\times 1} \end{array} \right] \ \end{aligned}$$

(62)

where

$$\begin{aligned} {{\Lambda }}_k(i)=I_{L_{opt}}-{\mu }_k\left[ \begin{array}{c} \mathbf{u}^*_{k,i} \\ \mathbf{O}_{(L_{opt}-M)\times 1} \end{array} \right] \mathbf{u}_{L_{opt}k,i} \end{aligned}$$

(63)

To demonstrate the steady-state MSD, first we write the ${\Vert {\overline{\varvec{\psi }}}^{(i)}_{L_{opt},k}\Vert }^2$ as:

$$\begin{aligned} \begin{array}{l} {\Vert {\overline{\varvec{\psi }}}^{(i)}_{L_{opt},k}\Vert }^2={{\overline{\varvec{\psi }}}^{*}}^{(i)}_{L_{opt},k-1}{{\mathrm {\Lambda }}^*}_k(i) {\mathrm {\Lambda }}_k(i){\overline{\varvec{\psi }}}^{(i)}_{L_{opt},k-1} \\ +{\mu }_kv_k(i) {{\overline{\varvec{\psi }}}^{*}}^{(i)}_{L_{opt},k-1} {{\mathrm {\Lambda }}^*}_k(i)\left[ \begin{array}{c} \mathbf{u}^*_{k,i} \\ \mathbf{O}_{(L_{opt}-M)\times 1} \end{array} \right] \\ +{\mu }_kv_k(i)\left[ \begin{array}{cc} \mathbf{u}_{k,i} &{} \mathbf{O}_{1\times (L_{opt}-M)} \end{array} \right] {\mathrm {\Lambda }}_k(i){\overline{\varvec{\psi }}}^{(i)}_{L_{opt},k-1} \\ +{\mu }^2_kv^2_k(i){\Vert \mathbf{u}_{k,i}\Vert }^2 \end{array} \end{aligned}$$

(64)

Considering the assumptions 1, 2, and employing the mathematical expectations from both sides of (64), after some tedious algebra leads to:

$$\begin{aligned}&E\{{\Vert {\overline{\varvec{\psi }}}^{(i)}_{L_{opt},k}\Vert }^2\} ={\beta }_kE\{{\Vert {\overline{\varvec{\psi }}}^{(i)}_{L_{opt},k-1}\Vert }^2\}\nonumber \\&\quad +({\eta }_k-{\beta }_k){\Vert {\ddot{\mathbf{w}}}_{L_{opt}-M}\Vert }^2+{\tau }_k \end{aligned}$$

(65)

where

$$\begin{aligned} \begin{array}{c} {\beta }_k=1-2{\mu }_k{\sigma }^2_{u,k}+{\mu }^2_k{\sigma }^4_{u,k}(M+2) \\ {\eta }_k=1+{\mu }^2_k{\sigma }^4_{u,k}M \end{array} \end{aligned}$$

(66)

and

$$\begin{aligned} {\tau }_k={\mu }^2_k{\sigma }^2_{v,k}{\sigma }^2_{u,k}M \end{aligned}$$

(67)

Now we write (65) in the brief form of below:

$$\begin{aligned} E\{{\Vert {\overline{\varvec{\psi }}}^{(i)}_{L_{opt},k}\Vert }^2\} ={\beta }_kE\{{\Vert {\overline{\varvec{\psi }}}^{(i)}_{L_{opt},k-1}\Vert }^2\}+f_k \end{aligned}$$

(68)

where

$$\begin{aligned} f_k=({\eta }_k-{\beta }_k){\Vert {\ddot{\mathbf{w}}}_{L_{opt}-M}\Vert }^2+{\tau }_k \end{aligned}$$

(69)

According to the steady-state analysis, as $(i \rightarrow \infty )$ in (68), and assuming ${\mathbf{P}_{L_{opt},k}=\overline{\varvec{\psi }}}^{(\infty )}_{L_{opt},k}$, (68) is rewritten as:

$$\begin{aligned} E\{{\Vert \mathbf{P}_{L_{opt},k}\Vert }^2\}={\beta }_kE\{{\Vert \mathbf{P}_{L_{opt},k-1}\Vert }^2\}+f_k \end{aligned}$$

(70)

This equation has a similar structure with (47), therefore in the same manner that was used for the solving of the recursive equation (47) we have:

$$\begin{aligned} E\{{\Vert \mathbf{P}_{L_{opt},k-1}\Vert }^2\}={(1-\Pi _{k,1})}^{-1}s_k \end{aligned}$$

(71)

where

$$\begin{aligned}&\Pi _{k,\ell }{\triangleq \beta }_{k-1}{\beta }_{k-2}\dots {\beta }_1{\beta }_N{\beta }_{N-1}\dots \nonumber \\&\quad {\beta }_{k+\ell }{\beta }_{k+\ell -1},\ \ \ \ell =1,\dots ,N \end{aligned}$$

(72)

and

$$\begin{aligned} \begin{array}{l} \begin{array}{c} s_k\triangleq \Pi _{k,2}\end{array} f_k+ \begin{array}{c} \Pi _{k,3} \end{array} f_{k+1}+\dots + \begin{array}{c} \Pi _{k,N-1} \end{array} f_{k-3}+ \begin{array}{c} \Pi _{k,N} \end{array} f_{k-2} \\ + \begin{array}{c} f_{k-1} \end{array} \end{array} \end{aligned}$$

(73)