The state space framework for blind dynamic signal extraction and recovery

Abstract

The paper describes a framework in the form of an optimization of a performance index subject to the constraints of a dynamic network, represented in the state space. The performance index is a measure of statistical dependence among the outputs of the network, namely, the relative entropy also known as the Kullback-Leibler divergence. The network is represented as (either discrete or continuous time) state space dynamics. Update laws are derived in the general cases. Moreover, in the discrete-time case, they are shown to specialize to the FIR and IIR network representations.

Blind Source Recovery in a State-Space Famework: Algorithms for Static and Dynamic Environments

“…In this case, even if the environment is unstable (due to the existence of non-minimum phase zeros), the overall augmented network may represent overall a non-linear adaptive dynamic system, which may converge to the true parameters as a stable equilibrium point. Thus achieving the global task of blind source extraction [13,18,19].…”

“…In order to derive the update law, we formulate the following optimization problem [19,20], note the notation has been altered for convenience. Minimize…”

“…For the linear time-invariant case the ordinary stochastic gradient (SG) update laws for the states, co-states and the parametric matrices A, B, C, and D are given by [19,20].…”

Blind source recovery for non-minimum phase surroundings

“…For convenience, we will assume that the demixing network is represented in Canonical Form I (or the controller canonical form) [2,4].…”

“…The choice of the state space representation has several advantages, which include a compact representation capable of handling both time delayed and filtered versions of signals [2,7,6], possibility of multiple efficient and generalized internal structural representations which allows for proficient identifiability [2]. Further, The inverse for a state space representation is well defined and ensures the existence of a solution for the BSR structure or recoverability [6].…”