Modified Incremental LMS with Improved Stability via Convex Combination of Two Adaptive Filters

Abstract

In distributed networks, the conventional incremental mode of cooperation between the nodes may suffer instability due to two major reasons: (1) large local errors due to accidental problems, and (2) instability due to link failure or noisy link. This causes error propagation through the entire network resulting in divergence. In this research, we propose a novel incremental least mean square algorithm with improved stability by employing convex combination of two filters. Adaptation of one filter is based on the estimate of the adjacent node (incremental type), while that of the other is based on the estimate of the current local node at previous time instant. These two filters are then fused together by using a suitable mixing parameter. An adaptive mixing parameter is further proposed for this convex combination, ensuing dynamic assignment of the weights for the two combining filters. Steady state excess mean square error is derived for the proposed convex combination, and simulations are presented to validate the proposed claims.

A Derivation of First-Order and Second-Order Moments

For the sake of simplicity, let $\varDelta \psi _{k}(n)=x$. Hence, x is central Chi-squared centered random variable with M degrees of freedom and its probability density function (pdf) is [21]:

$$\begin{aligned} f_{x}(x)=\frac{1}{\sigma ^{M} 2^{M/2}\varGamma (M/2)}x^{M/2-1}e^\frac{-x}{2\sigma ^{2}} \end{aligned}$$

(54)

Consequently, the first-order moment of $h_{k}(n)$ can be evaluated as:

$$\begin{aligned} E[h_{k}(n)]=\frac{1}{\sigma ^{M} 2^{M/2}\varGamma (M/2)}\int ^{\infty }_{0}\frac{1}{(1+e^{-ax})}x^{M/2-1}e^\frac{-x}{2\sigma ^{2}}\mathrm{d}x \end{aligned}$$

(55)

after some straight forward calculations, (55) can be set up as

$$\begin{aligned} E[h_{k}(n)]=\frac{1}{\sigma ^{M} 2^{M/2}\varGamma (M/2)a^{M/2}}\int ^{\infty }_{0}\frac{x'^{M/2-1}e^{-px'}}{(1+e^{-ax'})}\mathrm{d}x',\quad \quad p>-1 \end{aligned}$$

(56)

where $x'=ax$ and $p=\frac{1}{2\sigma ^{2}a}-1$. The closed for solution of (56) can be evaluated as [12]:

$$\begin{aligned} E[h_{k}(n)]=\frac{(M/2-1)!}{\sigma ^{M}2^{M/2}\varGamma (M/2)a^{M/2}}\sum ^{\infty }_{n=1}\frac{(-1)^{n-1}}{(p+n)^{M/2}} \end{aligned}$$

(57)

In the same manner, the second-order moment of $h_{k}(n)$ can be set up as:

$$\begin{aligned} E[(h_{k}(n))^{2}]=\frac{1}{\sigma ^{M} 2^{M/2}\varGamma (M/2)}\int ^{\infty }_{0}\frac{1}{(1+e^{-ax})^{2}}x^{M/2-1}e^\frac{-x}{2\sigma ^{2}}\mathrm{d}x \end{aligned}$$

(58)

after some straight forward calculations (58) are found to be:

$$\begin{aligned} E[h_{k}(n)]=\frac{1}{\sigma ^{M} 2^{M/2}\varGamma (M/2)a^{M/2}}\int ^{\infty }_{0}\frac{x'^{M/2-1}e^{-\frac{x'}{2a\sigma ^{2}}}}{(1+ e^{-x'})^{2}}\mathrm{d}x' \end{aligned}$$

(59)

The closed-form solution of (59) can be written as [12]:

$$\begin{aligned} E[(h_{k}(n))^{2}]= & {} \frac{1}{\sigma ^{M}2^{M/2}\varGamma (M/2)a^{M/2}}\nonumber \\&\left[ \varGamma \left( M/2( \varPhi (-1;M/2-1,1/2\sigma ^{2}a)-p\varPhi (-1;M/2,1/2\sigma ^{2}a))\right) \right] \nonumber \\ \end{aligned}$$

(60)