Analysis of Incremental Augmented Affine Projection Algorithm for Distributed Estimation of Complex-Valued Signals

Abstract

In this paper the aim is to solve the problem of distributed estimation in an incremental network when the measurements taken by the nodes follow a widely linear model. The proposed algorithm, which we refer to as incremental augmented affine projection algorithm (incAAPA), utilizes the full second order statistical information in the complex domain. Moreover, it exploits the spatio-temporal diversity to improve the estimation performance. We derive steady-state performance metric of the incAAPA in terms of mean-square deviation. We further derive sufficient conditions to ensure mean-square convergence. Our analysis illustrates that the proposed algorithm is able to process both second-order circular (proper) and non-circular (improper) signals. The validity of the theoretical results and the good performance of the proposed algorithm are demonstrated by several computer simulations.

Appendices

Appendix A

Now, to develop a distributed solution for (5), we turn the constrained minimization into an unconstrained one as

$$\begin{aligned} J({\mathbf {h}}_i,{\mathbf {g}}_i)\triangleq f({\mathbf {h}}_i,{\mathbf {g}}_i) + \sum _{k=1}^{K}{{\mathop {\mathfrak {R}}\nolimits }} \left\{ {{{\left( {{\mathbf {d}}_{k,i}} - {\mathbf {X}}_{k,i}^{{\mathsf {T}}} {{\mathbf {h}}_{i}} - {\mathbf {X}}_{k,i}^{{\mathsf {H}}}{{\mathbf {g}}_{i}}\right) }^{{\mathsf {H}}}} {\pmb {\beta }}_{k,i}}\right\} \end{aligned}$$

(50)

where $T \times 1$ vector ${\pmb {\beta }}_{k,i}$ comprises Lagrange multipliers. We can see that (50) is real-valued function of complex variables ${\mathbf {h}}$ and ${\mathbf {g}}$. Obviously, each node can use its local data $\{d_{k}(i), {\mathbf {x}}_{k,i}\}$ to calculate an estimate of $\{{\mathbf {h}}^{\circ }, {\mathbf {g}}^{\circ }\}$. However, when a set of nodes has access to data, we can take advantage of node cooperation and space-time diversity to improve the estimation performance. To solve (50) we firstly compute the gradient of $J({\mathbf {h}}_i,{\mathbf {g}}_i)$ with respect to weight vectors ${{\mathbf {h}}^{*}}$ and ${{\mathbf {g}}^{*}}$ which are defined as

$$\begin{aligned} \nabla _{{{\mathbf {h}}^{*}}}J({\mathbf {h}}_i,{\mathbf {g}}_i)= & {} \frac{\partial J({\mathbf {h}}_i,{\mathbf {g}}_i)}{\partial {\mathbf {h}}_{i}^{*}}= \frac{1}{2}\left( {\frac{{\partial J({\mathbf {h}}_i,{\mathbf {g}}_i)}}{{\partial {\mathbf {h}}^{\mathfrak {R}}_{i}}} + j \frac{{\partial J({\mathbf {h}}_i,{\mathbf {g}}_i)}}{{\partial {\mathbf {h}}_{i}^{{{\mathop {\mathfrak {I}}\nolimits }}}}}} \right) \end{aligned}$$

(51)

$$\begin{aligned} _{{\mathbf {g}}^{*}}J({\mathbf {h}}_i,{\mathbf {g}}_i)= & {} \frac{\partial J({\mathbf {h}}_i,{\mathbf {g}}_i)}{\partial {\mathbf {g}}_{i}^{*}}= \frac{1}{2}\left( {\frac{{\partial J({\mathbf {h}}_i,{\mathbf {g}}_i)}}{{\partial {\mathbf {g}}_{i}^{\mathfrak {R}}}} + j\frac{{\partial J({\mathbf {h}}_i,{\mathbf {g}}_i)}}{{\partial {\mathbf {g}}_{i}^{\mathop {\mathfrak {I}}\nolimits }}}} \right) \end{aligned}$$

(52)

The required gradients are given by

$$\begin{aligned} \frac{\partial J({\mathbf {h}}_i,{\mathbf {g}}_i)}{\partial {\mathbf {h}}_{i}^{*}}&= {\mathbf {h}}_{i} - {\mathbf {h}}_{i-1} -\frac{1}{2}\sum _{k=1}^{K}{{\mathbf {X}}_{k,i}^{*}} {\pmb {\beta }}_{k,i} \end{aligned}$$

(53)

$$\begin{aligned} \frac{\partial J({\mathbf {h}}_i,{\mathbf {g}}_i)}{\partial {\mathbf {g}}_{i}^{*}}&= {\mathbf {g}}_{i} - {\mathbf {g}}_{i-1} -\frac{1}{2}\sum _{k=1}^{K}{{\mathbf {X}}_{k,i}} {\pmb {\beta }}_{k,i} \end{aligned}$$

(54)

After setting the gradient of $J({\mathbf {h}}_i,{\mathbf {g}}_i)$ with respect to weight vectors ${{\mathbf {h}}^{*}}$ and ${{\mathbf {g}}^{*}}$ equal to zero we get

$$\begin{aligned} {{\mathbf {h}}_{i}}&= {{\mathbf {h}}_{i-1}} + \frac{1}{2}\sum _{k=1}^{K}{{\mathbf {X}}_{k,i}^{*}} {\pmb {\beta }}_{k,i} \end{aligned}$$

(55)

$$\begin{aligned} {{\mathbf {g}}_{i}}&= {{\mathbf {g}}_{i-1}} + \frac{1}{2}\sum _{k=1}^{K}{{\mathbf {X}}_{k,i}} {\pmb {\beta }}_{k,i} \end{aligned}$$

(56)

Clearly, the solution given by (55) and (56) is not a distributed solution for (50). To have a distributed and adaptive solution for (50), we need to apply the following modifications

• 1. Split the update Eqs. (55) and (56) into N separate steps whereby each step adds one term to summation. Then for $\forall k \in {\mathcal {K}}$ we have

  $$\begin{aligned} {{\mathbf {h}}_{1,i}}&= {{\mathbf {h}}_{i-1}},\ \ \ {{\mathbf {g}}_{1,i}}= {{\mathbf {g}}_{i-1}} \end{aligned}$$

  (57a)
  $$\begin{aligned} {{\mathbf {h}}_{k,i}}&= {{\mathbf {h}}_{k-1,i}} + \frac{1}{2}{{\mathbf {X}}_{k,i}^{*}} {\pmb {\beta }}_{k,i} \end{aligned}$$

  (57b)
  $$\begin{aligned} {{\mathbf {g}}_{k,i}}&= {{\mathbf {g}}_{k-1,i}} + \frac{1}{2}{{\mathbf {X}}_{k,i}} {\pmb {\beta }}_{k,i} \end{aligned}$$

  (57c)
  $$\begin{aligned} {{\mathbf {h}}_{i}}&= {{\mathbf {h}}_{N,i}} \ \ {{\mathbf {g}}_{i}}= {{\mathbf {g}}_{N,i}} \end{aligned}$$

  (57d)

  where ${\mathbf {h}}_{k,i}$ and ${\mathbf {g}}_{k,i}$ denote local estimates of ${\mathbf {h}}^{\circ }$ and ${\mathbf {g}}^{\circ }$ at node k at time i respectively.

• 2. Eliminate the Lagrange multiplier vectors ${\pmb {\beta }}_{k,i}$. To this end we substitute (57b) and (57c) in the constraint relation of Eq. (5) to obtain

  $$\begin{aligned}&{{\mathbf {d}}_{k,i}} - {\mathbf {X}}_{k,i}^{{\mathsf {T}}} \Big ({{\mathbf {h}}_{k-1,i}} + \frac{1}{2}{{\mathbf {X}}_{k,i}^{*}} {\pmb {\beta }}_{k,i} \Big ) \nonumber \\&\quad -{\mathbf {X}}_{k,i}^{{\mathsf {H}}} \Big ({{\mathbf {g}}_{k-1,i}} + \frac{1}{2}{{\mathbf {X}}_{k,i}} {\pmb {\beta }}_{k,i} \Big ) =0 \end{aligned}$$

  (58)

  Solving (58) in terms of ${\pmb {\beta }}_{k,i}$ yields

  $$\begin{aligned} {\pmb {\beta }}_{k,i}=2\Big ({\mathbf {X}}_{k,i}^{{\mathsf {H}}}{\mathbf {X}}_{k,i}+{\mathbf {X}}_{k,i}^{{\mathsf {T}}}{\mathbf {X}}_{k,i}^{*} \Big )^{-1}{\mathbf {e}}_{k,i} \end{aligned}$$

  (59)

  where ${\mathbf {e}}_{k,i}$ is defined in (10). Finally, we replace (59) in (57b) and(57c) and also introduce a convergence factor $\mu >0$ in order to trade-off final misadjustment and convergence speed to obtain the following update equation

  $$\begin{aligned} {{\mathbf {h}}_{k,i}}&= {{\mathbf {h}}_{k-1,i}} + \mu _k {{\mathbf {X}}^{*}_{k,i}} \Big ({\mathbf {X}}_{k,i}^{{\mathsf {H}}}{\mathbf {X}}_{k,i}+{\mathbf {X}}_{k,i}^{{\mathsf {T}}}{\mathbf {X}}_{k,i}^{*} \Big )^{-1}{\mathbf {e}}_{k,i} \end{aligned}$$

  (60a)
  $$\begin{aligned} {{\mathbf {g}}_{k,i}}&= {{\mathbf {g}}_{k-1,i}} + \mu _k {{\mathbf {X}}_{k,i}} \Big ({\mathbf {X}}_{k,i}^{{\mathsf {H}}}{\mathbf {X}}_{k,i}+{\mathbf {X}}_{k,i}^{{\mathsf {T}}}{\mathbf {X}}_{k,i}^{*} \Big )^{-1}{\mathbf {e}}_{k,i} \end{aligned}$$

  (60b)

To avoid singularities due to the inversion of a rank deficient matrix, a positive constant $\delta$, called the regularisation parameter, is added to the above updates, giving

$$\begin{aligned} {{\mathbf {g}}_{k,i}}&= {{\mathbf {g}}_{k - 1,i}} \nonumber \\&\qquad + {\mu _k}{{\mathbf {X}}_{k,i}}{\left[ {{\mathbf {X}}_{k,i}^H{{\mathbf {X}}_{k,i}} + {\mathbf {X}}_{k,i}^T {\mathbf {X}}_{k,i}^ * + \delta {\mathbf {I}}} \right] ^{-1}}{{\mathbf {e}}_{k,i}}\end{aligned}$$

(61)

$$\begin{aligned} {{\mathbf {h}}_{k,i}}&= {{\mathbf {h}}_{k - 1,i}} \nonumber \\&\qquad + {\mu _k}{\mathbf {X}}_{k,i}^ * {\left[ {{\mathbf {X}}_{k,i}^H{{\mathbf {X}}_{k,i}} + {\mathbf {X}}_{k,i}^T {\mathbf {X}}_{k,i}^ * + \delta {\mathbf {I}}} \right] ^{-1}}{{\mathbf {e}}_{k,i}} \end{aligned}$$

(62)

and the proof is complete.

Appendix B

If we equate the weighted energies of both sides of (25), we arrive the following space-time version of the weighted energy conservation relation for incAAPA as:

$$\begin{aligned} \Vert {\tilde{\mathbf {w}}}_{k,i}\Vert _{{\varvec{\Sigma }}_{k}}^2+{\mathbf {e}}^{{\varvec{\Sigma }}_{k}{\mathsf {H}}}_{a,k}{\mathbf {F}}_{k,i}^{-1}{\mathbf {e}}^{{\varvec{\Sigma }}_{k}}_{a,k}= \Vert {\tilde{\mathbf {w}}}_{k-1,i}\Vert _{{\varvec{\Sigma }}_{k}}^2+{\mathbf {e}}^{{\varvec{\Sigma }}_{k}{\mathsf {H}}}_{p,k}{\mathbf {F}}_{k,i}^{-1} {\mathbf {e}}^{{\varvec{\Sigma }}_{k}}_{p,k} \end{aligned}$$

(63)

Substituting (20) into (63) and rearranging the result, we obtain

$$\begin{aligned} \Vert {\tilde{\mathbf {w}}}_{k,i}\Vert _{{\varvec{\Sigma }}_{k}}^2= \Vert {\tilde{\mathbf {w}}}_{k-1,i}\Vert _{{\varvec{\Sigma }}_{k}}^2 -\mu _k {\mathbf {e}}^{{\varvec{\Sigma }}_{k}{\mathsf {H}}}_{a,k}{\mathbf {B}}_{k,i} {\mathbf {e}}_{k,i} -\mu _k {\mathbf {e}}_{k,i}^{{\mathsf {H}}} {\mathbf {B}}_{k,i} {\mathbf {e}}^{{\varvec{\Sigma }}_{k}}_{a,k}+\mu _k^2 {\mathbf {e}}_{k,i}^{{\mathsf {H}}} {\mathbf {B}}_{k,i} {\mathbf {F}}_{k,i} {\mathbf {B}}_{k,i} {\mathbf {e}}_{k,i} \end{aligned}$$

(64)

By using the error signal ${\mathbf {e}}_{k,i}={\mathbf {U}}_{k,i}^{{\mathsf {H}}} {\tilde{\mathbf {w}}}_{k-1,i}+{\mathbf {v}}_{k,i}$ we have

$$\begin{aligned} \Vert {\tilde{\mathbf {w}}}_{k,i}\Vert _{{\varvec{\Sigma }}_{k}}^2= & {} \Vert {\tilde{\mathbf {w}}}_{k-1,i}\Vert _{{\varvec{\Sigma }}_{k}}^2-\mu _k {\tilde{\mathbf {w}}}_{k-1,i}^{{\mathsf {H}}}{\varvec{\Sigma }}_{k}{\mathbf {U}}_{k,i}{\mathbf {B}}_{k,i}\left( {\mathbf {U}}_{k,i}^{{\mathsf {H}}}{\tilde{\mathbf {w}}}_{k-1,i}+{\mathbf {v}}_{k,i}\right) \nonumber \\&-\, \mu _k \left( {\tilde{\mathbf {w}}}_{k-1,i}^{{\mathsf {H}}}{\mathbf {U}}_{k,i}+{\mathbf {v}}_{k,i}^{{\mathsf {H}}}\right) {\mathbf {B}}_{k,i}{\mathbf {U}}_{k,i}^{{\mathsf {H}}}{\varvec{\Sigma }}_{k}{\tilde{\mathbf {w}}}_{k-1,i} \nonumber \\&+\,\mu _k^2\left( {\tilde{\mathbf {w}}}_{k-1,i}^{{\mathsf {H}}}{\mathbf {U}}_{k,i}+{\mathbf {v}}_{k,i}^{{\mathsf {H}}}\right) {\mathbf {B}}_{k,i}{\mathbf {F}}_{k,i}{\mathbf {B}}_{k,i}\left( {\mathbf {U}}_{k,i}^{{\mathsf {H}}}{\tilde{\mathbf {w}}}_{k-1,i}+{\mathbf {v}}_{k,i}\right) \end{aligned}$$

(65)

Taking expectations of both sides of (65) and applying the Assumptions. 1 and 2 we obtain

$$\begin{aligned} {\mathbb {E}} \Big [\Vert {\tilde{\mathbf {w}}}_{k,i}\Vert _{{\varvec{\Sigma }}_{k}}^2\Big ]&={\mathbb {E}} \Big [\Vert {\tilde{\mathbf {w}}}_{k-1,i}\Vert _{{\varvec{\Sigma }}'_{k}}^2\Big ]+\mu _k^2 {\mathbb {E}}\Big [{\mathbf {v}}_{k,i}^{{\mathsf {H}}}{\mathbf {B}}_{k,i}{\mathbf {F}}_{k,i}{\mathbf {B}}_{k,i}{\mathbf {v}}_{k,i}\Big ] \end{aligned}$$

(66)

where

$$\begin{aligned} {\varvec{\Sigma }}'_k&={\varvec{\Sigma }}_k-\mu _k{\mathbb {E}} \Big [{\varvec{\Sigma }}_k {\mathbf {U}}_{k,i} {\mathbf {B}}_{k,i} {\mathbf {U}}_{k,i}^{{\mathsf {H}}}\Big ] -\mu _k {\mathbb {E}} \Big [{\mathbf {U}}_{k,i} {\mathbf {B}}_{k,i} {\mathbf {U}}_{k,i}^{{\mathsf {H}}} {\varvec{\Sigma }}_k \Big ]+\mu _k^2 {\mathbb {E}} \Big [{\mathbf {U}}_{k,i} {\mathbf {B}}_{k,i} {\mathbf {U}}_{k,i}^{{\mathsf {H}}} {\varvec{\Sigma }}_k {\mathbf {U}}_{k,i}{\mathbf {B}}_{k,i}{\mathbf {U}}_{k,i}^{{\mathsf {H}}}\Big ] \end{aligned}$$

(67)

Then, (66) and (67) can be rewritten as

$$\begin{aligned} {\mathbb {E}} \Big [\Vert {\tilde{\mathbf {w}}}_{k,i}\Vert _{{\varvec{\Sigma }}_{k}}^2\Big ]&={\mathbb {E}} \Big [\Vert {\tilde{\mathbf {w}}}_{k-1,i}\Vert _{{\varvec{\Sigma }}'_{k}}^2\Big ] +\mu _k^2 {\mathbb {E}}\Big [{\mathbf {v}}_{k,i}^{{\mathsf {H}}}{\mathbf {C}}_{k,i}{\varvec{\Sigma }}_{k}{\mathbf {C}}_{k,i}^{{\mathsf {H}}}{\mathbf {v}}_{k,i}\Big ]\end{aligned}$$

(68)

$$\begin{aligned} {\varvec{\Sigma }}'_k&={\varvec{\Sigma }}_k-\mu _k {\varvec{\Sigma }}_k {\mathbb {E}} [{\mathbf {D}}_{k,i}] \mu _k {\mathbb {E}} [{\mathbf {D}}_{k,i} {\varvec{\Sigma }}_k]+\mu _k^2 {\mathbb {E}} [{\mathbf {D}}_{k,i} {\varvec{\Sigma }}_k {\mathbf {D}}_{k,i}] \end{aligned}$$

(69)

and the proof is complete.