AN OPERATOR-BASED AND SPARSITY-BASED APPROACH TO ADAPTIVE SIGNAL
SEPARATION
Xiaolei Yi, Xiyuan Hu, Silong Peng
∗
Institute of Automation, Chinese Academy of Sciences, Beijing, 100190, P.R. China
ABSTRACT
An operator-based and sparsity-based approach is proposed
to adaptively separate a signal into additive subcomponents.
The proposed approach can be formulated as an optimization
problem. Since the design of the operator can be adaptively
customized to the target signal, we can propose different types
of operators for different types of signals. The subcompo-
nents are a kind of local narrow band signals in the null space
of an adaptive operator and a residual signal which is a sparse
signal in some sense. Our experiments, including simulated
signals and a real-life signal, demonstrate the efficacy and ac-
curacy of the proposed approach.
Index Terms— Signal separation, adaptive operator, the
null space, ℓ
1
constraint, sparse signal
1. INTRODUCTION
Recently, single-channel signal separation has attracted a lot
of interests since it has affected many applications. Many ap-
proaches have been proposed to decompose a single-channel
signal into a mixture of several additive coherent subcompo-
nents. The different definitions of subcomponents lead to dif-
ferent kinds of decomposition methods. For example, Em-
pirical Mode Decomposition (EMD) [1, 2] decomposes an
oscillatory signal into a summation of intrinsic mode func-
tions (IMFs); Matching Pursuit (MP) [3] decomposes a sig-
nal into a summation of time-frequency atoms; Null Space
Pursuit (NSP) [4], which is an operator-based approach, de-
composes a signal into some local narrow band signals which
are defined in the null space of adaptive operators. Among
those approaches, we have a particular interest in NSP.
The NSP approach [4], uses an adaptive operator Γ
S
to separate a signal S into additive subcomponents: R and
U(U = S − R). It can be formulated as an optimization
problem:
min
R
{∥Γ
S
(S−R)∥
2
+λ
1
(∥R∥
2
+γ∥S−R∥
2
)+F(Γ
S
)}, (1)
where the operator Γ
S
can be adaptively estimated from the
signal S, and the last term is the Lagrange term for the param-
∗
This work was supported by the Natural Science Foundation of China
(61201375, 61201050) and the State Key Program of Natural Science Foun-
dation of China (Grant No.61032007).
eters of the operator Γ
S
. Minimizing the term ∥Γ
S
(S − R)∥
2
can ensure that S − R is in the null space of the operator
Γ
S
. NSP can be used to decompose the residual signal R re-
peatedly. Hence, S can be represented as the summation of
subcomponents in the null spaces of a sequence of operators
derived from the corresponding sequence of residual signals.
Peng et al. proposed two types of singular operator: an in-
tegral operator and a differential operator [5]. The charming
features of NSP are that the design of the operator Γ
S
can
be customized to the target signal S, and that the operator’s
parameters and the Lagrangian multipliers can be adaptively
estimated [4]. Although NSP has a solid mathematical foun-
dation, it cannot decompose some signals successfully either.
For example, if one subcomponent of a signal S is a piecewise
smooth signal, S is difficult to be separated by NSP effec-
tively. This is because the ℓ
2
− norm, which is very sensitive
to singular points of a piecewise smooth signal, has been used
in NSP.
Here, we are interested in the type of signals:s(t) =
s
1
(t) + s
2
(t), where s
1
(t) is a narrow band signal in the
null space of an operator, and s
2
(t) is a sparse signal in some
sense. In order to decompose this type of signals, we consider
measures of sparsity rather than measures of energy used in
NSP. In fact, a very simple and intuitive measure of sparsity
of a vector x ∈ R
m
is the ℓ
0
norm: ∥x∥
0
= {i : x
i
= 0}, if
∥x∥
0
≪ m, x is sparse. But the minimization problem:
(P
0
) : min
x
∥x∥
0
subject to b = Ax. (2)
is a non-convex combinatorial optimization problem, and in-
deed, it has been proved that (P
0
) is, in general, NP-Hard [6].
According to [7], in most cases, (P
0
) can be equivalent to the
(P
1
) problem,
(P
1
) : min
x
∥x∥
1
subject to b = Ax. (3)
where ∥x∥
1
=
∑
i
|x
i
|. The problem (P
1
) can be cast as a
standard linear programming (LP) problem, and solved us-
ing simplex methods, modern interor-point methods, or other
techniques, such as homotopy methods [8].
In this paper, we adopt ℓ
1
− constraint instead of ℓ
2
− con-
straint in NSP, and propose an operator-based and sparsity-
based approach to adaptive signal separation. We demonstrate
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