## Year

2004

## Degree Name

Doctor of Philosophy

## Department

Department of Electrical, Computer and Telecommunications Engineering - Faculty of Engineering

## Recommended Citation

Ma, Liang Suo, Multichannel blind deconvolution, PhD thesis, Department of Electrical, Computer and Telecommunications Engineering, University of Wollongong, 2004. http://ro.uow.edu.au/theses/398

## Abstract

This thesis studies some issues in the multichannel blind deconvolution (MBD) problem. MBD studies the problem of recovering the original latent source signals from a set of observation data, which is the convolutive mixture of the latent sources and an unknown dynamical system. Assumptions are usually adopted in deriving MBD algorithms to simplify the problem. Common assumptions include: the dynamical system is assumed to be linear and time invariant; the latent sources are assumed to contain at most one Gaussian distributed signal; the latent sources are statistically independent. There are, however, a number of additional assumptions introduced because a particular approach is followed. In our case, we follow the state space approach in representing the unknown dynamical system. This introduces a number of additional assumptions: (I) the mixing environment is assumed to be noise free; (II) the number of sources is assumed to be known; (III) the number of sources is assumed to be equal to the number of sensor measurements; (IV) the number of the states of the mixer is assumed to be known; (V) the latent sources are assumed to be super-Gaussian distributed. Assumption (IV) is specific to the state space approach, while the other assumptions also occur in other approaches. Obviously, the above assumptions are not necessarily true in practice. Our main aim in this thesis is to relax these assumptions so that the unknown dynamical system will be more accurately modelled and the MBD algorithms will be more suitable for practical applications. We propose to relax these five assumptions, one by one, through a number of novel algorithms. Balanced parametrization of linear time invariant systems originates in the field of system identification, system reduction and H1 control. Using a balanced canonical realization of the linear time invariant system, we will derive three versions of a balanced MBD algorithm in discrete time domain, continuous time domain, and unified discrete time and continuous time domains respectively. All these three versions of balanced algorithms can estimate the number of states in the mixer by considering the identified singular values in the balanced parametrization, thus relaxing assumption (IV). It is relatively easy to extend this formulation to include situations when the number of sensor measurements is greater than the number of latent sources, thus relaxing assumption (III) partially. The more difficult situation when the number of sensor measurements is less than the number of latent sources is not considered in this thesis. Most parameter estimation algorithms for the MBD problem include a nonlinear activation function in the algorithm. Dependent on the approach used in the derivation of the parameter estimation algorithm, the nonlinearity can take various forms, e.g., hyperbolic tangent function. However, normally in the derivation, it is implicitly assumed that the latent sources are super- Gaussian distributed, thus the hyperbolic tangent function is implicitly used as the nonlinearity. Unfortunately, in practice, it is seldom known in advance that the latent sources are super-Gaussian distributed. We will investigate a number of flexible source models, which will allow to separate both super-Gaussian and sub-Gaussian distributed sources. Through our empirical studies, we conjecture that the recovery of the latent sources is relatively insensitive to the probability distribution of the source signals, as long as some common nonlinearity is used in the parameter estimation algorithm in the MBD problem. We have empirically verified this conjecture for a set of commonly used nonlinear functions. Hence, assumption (V) is relaxed to an extent that the nonlinearity can be designed to be adaptive, according to the mixture of probability distribution of the latent sources, provided that the latent sources stay either super-Gaussian or sub-Gaussian for sufficiently long for the parameter estimation algorithm to converge sufficiently. The number of sources estimation problem can be formulated as a model comparison problem, which may be solved by evaluating marginal likelihood. However, it usually involves the evaluation of multiple variable integral expressions, which is well known to be difficult to evaluate computationally. Following a variational Bayesian (VB) approach, we overcome this difficulty in MBD problem by deriving a VB MBD algorithm, which has the following features: first, it allows to enclose noises in the system model; secondly, it allows to employ model comparison and automatic relevance determination to estimate the number of sources. Hence assumptions (I) and (II) are relaxed using this approach. This approach is applied to the estimation of the number of sources in artificially mixed speech signals, and then to electroencephalograph signals, the number of sources of which is not known a priori.

**Unless otherwise indicated, the views expressed in this thesis are those of the author and do not necessarily represent the views of the University of Wollongong.**