State Space Blind Source Recovery of Non-minimum Phase Environments

The paper describes the use of the state space and the natural gradient for the demixing of sources mixed in a non-minimum phase convolutive environment. Non-minimum phase implies that some or all of the zeros of the mixing environment lie outside the unit circle, and as such the theoretical inverse or the requisite demixing system becomes unstable due to the presence of poles outside the unit circle. These unstable poles are required to cancel out the non-minimum phase transmission zeros of the environment. In order to avoid instability due to the existence of these poles outside the unit circle, the natural gradient algorithm may be derived with the constraint that the demixing system is a double sided FIR filter, i.e., instead of trying to determine the IIR inverse of the environment, we will approximate the inverse using an all zero non-causal filter. The use of the state space warrants that the derived framework is rich in structure while at the same time compact in representation. This results in derivation of update laws that can invariably handle most mixing scenarios. Some simulations illustrating the performance of the algorithm are also provided.