A Modified RLS Algorithm for ICA with Weighted Orthogonal Constraint

Abstract

Independent component analysis (ICA), as an important data processing technique, is widely employed in many areas. The objective of the ICA is to recover independent components from observed signals. Several algorithms, such as equivariant adaptive separation via independence algorithm, least-mean-square (LMS)-type algorithms and recursive least-squares (RLS)-type learning rules, are proposed to solve the ICA problem. In the present paper, a modified RLS algorithm for ICA with weighted orthogonal constraint is developed to implement source separation based on the local convergence analysis of the available algorithm. Comparative experiment results demonstrate that the proposed algorithm is better than existing learning rules in the aspect of the accuracy of separation and stability.

Appendix A: Proof of Theorem 1

Considering formulation (11), Tang [18] pointed out that the ${\mathbf {B}}_{\mathrm{opt},t}$ at time t is the stable point with the weighted orthogonal constraint (3). When iteration (11) converges, considering the following expression in (10)

$$\begin{aligned} {\mathbf {P}}_{t}=(\lambda {\mathbf {P}}^{-1}_{t-1}+{\mathbf {z}}_{t}{\mathbf {z}}^\mathrm{T}_{t})^{-1} \end{aligned}$$

(23)

applying $E\{\cdot \}$ operator to (23), we can obtain that

$$\begin{aligned} {\mathbf {P}}_{\mathrm{opt},t}=(1-\lambda )E\{{\mathbf {z}}_{t}{\mathbf {z}}^\mathrm{T}_{t}\}^{-1} \end{aligned}$$

(24)

Utilizing the (24) to the ${\mathbf {P}}_{t}$ in (11) yields

$$\begin{aligned} \lim _{t\rightarrow \infty }E\{{\mathbf {Q}}_{t}{\mathbf {z}}_{t}{\mathbf {z}}^\mathrm{T}_{t}\} =E\{{\mathbf {I}}-\lambda {\mathbf {I}}\} \end{aligned}$$

(25)

Combining result (25) with mathematical expectation on two sides of the final time of (11), we have the following as:

$$\begin{aligned} \lim _{t\rightarrow \infty }\frac{E\{{\mathbf {B}}_{t}-{\mathbf {B}}_{\mathrm{opt},t}\}}{E\{{\mathbf {B}}_{t-1}-{\mathbf {B}}_{\mathrm{opt},t}\}}=\lambda \end{aligned}$$

(26)

here $\lambda$ is the forgetting factor $(0\ll \lambda <1)$. It represents that our algorithm is linear convergence. Further, (9) demonstrates that $E\{{\mathbf {z}}_{t}{\mathbf {z}}^\mathrm{T}_{t}\}=E\{{\mathbf {y}}_{t}{\mathbf {z}}^\mathrm{T}_{t}\}$.