Blind Source Separation with Variable Step-Size
Method Based on a Reference Separation System
Pengcheng Xu, Yuehong Shen, Qiao Su
College of Communications Engineering
PLA University of Science & Technology
Nanjing, China
b71087@126.com
Abstract
—Variable step-size methods are effective methods
to solve the problem of choosing step size in adaptive blind
source separation process. This paper proposes a novel
variable step-size method based on a reference separation
system for blind source separation. In view of the correlation
between the estimated source signals and original source
signals increases along with iteration, the method introduces a
reference separation system to approximately estimate the
correlation which is utilized to update the step-size. The
performance in terms of cross-talking error of the proposed
algorithm is analyzed. Simulation results show that the
proposed method exhibits superior convergence and better
steady-state performance compared with the fixed step-size
method in the noise free case, and converges faster than
classical variable step-size methods in both stationary and
non-stationary environments.
Index Terms
—Blind source separation, Variable step-
size method, Reference separation system, Correlation.
I.
INTRODUCTION
Blind Source Separation (BSS) aims to extract the latent
unknown source signals from their observed mixtures by an
array of sensors without priori knowledge of the original
source signals and the mixing coefficients, which makes the
BSS becoming a versatile tool used in many multi-sensor
systems such as antenna arrays in acoustics or
electromagnetism, chemical sensor arrays, electrode arrays
in electroencephalography, etc [1].
BSS methods generally require some contrasts (or
contrast functions) [2]-[4] and optimization algorithms. A
variety of optimization algorithms can be categorized into
batch-based and adaptive (sequential) algorithms [5, 6]. In
this paper, we consider the latter, which have practical
advantage compared with batch-based algorithms due to
their computational simplicity and latent ability in tracing a
non-stationary environment [7].
However, the traditional adaptive BSS algorithms such
as Equivariant Adaptive Separation via Independence
algorithm (EASI) [8] and Natural Gradient Algorithm
(NGA) [9] usually assume that the step size is a small
positive constant, which leads to an inevitable conflict
between the learning rate and stability performance, i.e. the
problem of slow convergence speed or large steady state
error. A simple way to solve the conflict is reducing the
learning rate as the iteration goes on [10]-[11], but it brings
about another new problem: if the learning rate decreases
too fast before source components are extracted, the
separation system will probably fail to separate sources. To
improve the learning rate and stability performance, variable
step-size algorithms have been proposed. In contrast, the
variable step-size algorithms can exploit the online
measurements of the state of the separation system from the
outputs and the parameter updates. In [12]-[14], traditional
NGA, EASI and S-NGA algorithms have been modified to
derive the variable step-size algorithms. Zhang et al. put
forward a grading learning algorithm based on the
measurements of correlation of the separating signals, the
learning rate of which is determined by the state of
separating [15]. Hsieh et al. proposed an effective learning
rate adjustment method based on an improved particle
swarm optimizer [16]. These algorithms resolve the conflict
to some extent and are possessed of the ability of tracking
the time-varying mixing system. But in these variable step
size algorithms, the initial convergence is still slow and
improper initial values of learning rate usually leads to large
steady state error or even divergence.
In order to improve the initial convergence and stability
performance, we consider adding a reference separation
system to approximate the correlation between original
sources and estimated sources, which is used as the
measurement to update the step-size adaptively. The
proposed technique is shown to increase the convergence
speed and the steady-state performance, and the use of
“mini-batches” can reduce the computational load in the
whole adaptive process. The remainder of this paper is
organized as follows. In Section 2, the adaptive BSS
problem and traditional adaptive algorithms are briefly
summarized. Our algorithm is proposed in Section 3.
Numerical stimulation results and discussions are provided
in Section 4. At the end of the paper, a concise conclusion is
given.
II.
ADAPTIVE ALGORITHMS FOR BSS
In the noise-free instantaneous case, we assume that
n
unknown statistically independent zero mean source signals,
with at most one having a Gaussian distribution, contained
within
1
( ) [ ( ), , ( )]
n
tst st=
T
s …
pass through an unknown
mixing system
mn×
∈ℜA
(
m
≥
n
), therefore
m
mixed signals
1
() [ (), , ()]
m
txt xt=
T
x …
can be modeled as
() ()tt=xAs
(1)
where
t
is the time index, and T is the vector transpose
operator. To simplify the problem, we further assume that
the number of sources matches the number of mixtures i.e.
m
=
n
, an exactly determined problem.
978-1-4799-5274-8/14/$31.00 ©2014 IEEE
110
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