2013 年度 第 26 回 信號處理合同學術大會論文集 第 26 卷 1 號
Constrained Affine Projection Sign Algorithm
한윤기, 송우진
포항공과대학교 전자전기공학과
im815@postech.ac.kr, wjsong@postech.ac.kr
초록: A new constrained affine projection sign
algorithm (CAPSA) is proposed, which has
robustness
against
heavy-tailed
impulsive
interference in linearly constrained adaptive filter
problem. The conventional constrained affine
projection
algorithm
suffers
performance
degradation
in
the
presence
of
impulsive
interferences. The proposed CAPSA is based on the
L1 -norm minimization criterion. Simulations in a
system identification show that the proposed CAPSA
has robustness against impulsive interferences.
주제어: Linearly Constrained Adaptive Filter, Affine
Projection Algorithm, Impulsive Interference
Ⅰ. Introduction
Linearly constrained adaptive filters play an
important role in many signal processing applications
such as adaptive beamforming, blind interference
suppression in code-division multiple-access (CDMA)
systems, and system identification [1]. The linear
constraints are used to reflect a prior knowledge of
certain parameter or properties of problem in the
signal processing applications. For example, linear
constraints reflect knowledge of direction of arrival
of desired user signals in adaptive beamforming [2],
user spreading codes in blind multiuser detection [3],
and linear phase feature of a plant in system
identification [4]. The constrained least-meansquare (CLMS) algorithm [1] is most popular for its
simplicity; however convergence speed can be
deteriorated seriously for correlated input signals or
in the presence of heavy-tailed impulsive
interferences [5].
To address the deterioration of convergence
speed by colored input signals, the constrained affine
projection algorithm (CAPA) [4] have been
developed. The CAPA updates the adaptive filter
coefficients using the previous L input vectors. The
normalized CLMS (NCLMS) and normalized data
reusing CLMS (BNDR-CLMS) algorithm [6] are
special case for L  1 and L  2 , respectively. As the
number of using previous input vectors increases,
the convergence speed increases and so does the
computational
complexity.
Therefore
many
computational
efficient
methods
have
been
developed to reduce the computational complexity,
such as the fast affine projection algorithm (FAPA)
[7].
Several studies of unconstrained algorithm have
shown that lower-order norms lead to robustness
against heavy-tailed impulse interference. The least
mean p -norm (LMP) algorithm based on the L p
norm is developed in [8]. Among all lower-order
algorithms, the sign algorithms based on L1 -norm
have merits of low computational cost and easy
implementation. Many variant of the sign algorithm
have been developed, including the normalized sign
algorithm (NSA) [9]; however, it is hard to find
literatures for constrained algorithm with robustness
against heavy-tailed impulse interference.
In this work, we propose a constrained affine
projection sign algorithm (CAPSA) based on the L1 norm optimization criterion with the previous input
vectors in linearly constrained adaptive filter
problem. Since the proposed algorithm takes benefits
of both the constrained affine projection algorithm
and the sign algorithm, the proposed algorithm has
fast convergence speed in heavy-tailed impulse
interference environments.
This paper is organized as follows. Section II
reviews
the
conventional
constrained
affine
projection. In Section III, the derivation of CAPSA is
presented. Simulations of the algorithm are shown in
Section IV, and conclusions are summarized in
Section V.
Ⅱ. Conventional Constrained Affine Projection
Algorithm
In this section, we review the conventional
constrained affine projection algorithm [4] for
linearly constrained filtering problem.
Fig.1 shows the system identification problem.
The output signal from an unknown system with a
weight coefficients vector
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w o is d (k )  wo x(k )  v(k ) ,
2013 年度 第 26 回 信號處理合同學術大會論文集 第 26 卷 1 號
where
x( k )   x ( k )
x(k  N  1)  is
x(k  1)
T
the input signal vector of length
N , and v (k ) is the
background noise plus interference signal. The a
priori error vector and a posteriori error vector are
defined as follows:
e(k )   e(k ) e(k  1)
e(k  N  1) 
e p (k )   (k )  ( k  1)
 (k  N  1) 
T
1
e(k ) . The step-size 
controls amount of update. The matrix P is the
projection matrix for a projection onto homogeneous
hyperplane defined by CT w  0 , and vector F
moves back to the constraint hyperplane, as given
below:
P = I  C(CT C)1 CT
(1)
T
(7)
1
(8)
F = C(C C) f
Since accumulation of round-off errors may cause
projection into wrong hyperplane, the simplification
of the correction term Pw ( k )  F to w ( k ) should
e(k )  d(k )  XT (k )w(k )
e p (k )  d(k )  XT (k )w(k  1)
instant

T
These errors vector can be computed as
where

with t (k )  XT (k )PX(k )
(2)
be avoided [4].
w (k ) is the weight coefficient vector at time
k , and d(k ) , X(k ) are the desired signal
vector and the input signal matrix, respectively. The
d(k ) and X(k ) are defined by
J
d(k )   d (k ) d (k  1)
d (k  L  1) 
X(k )   x(k ) x(k 1)
x(k  L  1)
T
(3)
Fig. 1. Block Diagram of System Identification
The constraints are given by the following set of
equations in the system identification:
CT w  f
(4)
where C is an N  J constraint matrix and f is a
vector containing the J constraint values. These
constraints represent the linear feature of system in
the system identification.
The optimization problem of constrained affine
projection algorithm is derived as following:
Ⅲ. Constrained Affine Projection Sign Algorithm
To obtain robustness against heavy-tailed
impulse interference, the proposed algorithm based
on minimizing L1 -norm of a posteriori error vector:
w(k  1)  arg min d(k )  XT (k )w
subject to
where
w (k  1)  arg min w  w ( k )
subject to
where
2
coefficient
2
2
CT w  f , d(k )  XT (k )w  0
means
vector
(5)
w(k  1)  P  w(k )   X(k )t(k )  F
means L1 -norm, and
2
(9)
is parameter
that controls change of weight coefficients. Using
the
method
of
Lagrange
multipliers,
the
unconstrained cost function can be obtained,
L2 -norm. We find new
w (k  1) that minimizes the
change of the consecutive filter coefficient vectors
while it satisfies zero a posteriori error constraint
and linear feature constraint simultaneously.
The method of Lagrange multipliers is employed
to obtain solution of (5). The update equation of the
CAPA can be obtained as following:
1
1
C w  f , w  w (k ) 2   2
2
T
J (w)  e p (k )  1  w  w(k ) 2   2   ΛT2 CT w  f 
1


2
(10)
where 1 , Λ 2 are Lagrange multipliers. The derivative
of the cost function (10) with respect to weight
coefficient vector is
(6)
J (w )
  X(k ) sgn  e p (k )   21  w  w (k )   CΛ 2
w
(11)
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2013 年度 第 26 回 信號處理合同學術大會論文集 第 26 卷 1 號
where sgn 

denotes the sign function. Setting the
constraint as
derivative equal to zero, we get
w (k  1)  w (k ) 
 I ( N 1)/ 2 


C   0T 
  J ( N 1)/ 2 


1 
CΛ 2  X(k ) sgn  e p ( k )  
21 
(12)
f  0 0
Substituting (12) into constraint (9), the Lagrange
multipliers can be obtain by
1 


T
T
1 sgn e p (k ) X (k ) PX( k ) sgn  e p ( k ) 
2
2

Λ 2  CT C

1
CT X(k ) sgn  e p ( k ) 
with initial weight coefficient vector,
(18)
0
T
(19)
where J is and identity matrix with all rows in
reversed order. The input is generated by filtering a
white, zero-mean Gaussian random sequence
through a first-order system G( z )  1/ (1  0.99 z 1 ) .
(13)
(14)
w (0)  F .
Substituting (13) and (14) into (12), the update
equation is then:
w (k  1)
An independent white Gaussian noise is added to the
system background with a 30 dB signal-to-noise
ratio (SNR). A strong interference signal is also
added to the system output with -30 dB signal-tointerference ratio (SIR). The interference signal is
modeled by the Bernoulli-Gaussian (BG) distribution
[10]. BG distribution is generated as product of a
Bernoulli process with given probability, Pr  0.001
and a zero mean Gaussian process. The mean-
2


square-deviation (MSD) define by E w o  w ( k ) is

X
(
k
)
sgn
e
(
k
)


p
F
 P  w (k ) 
evaluated by ensemble averaging over 100

T
T
sgn
e
(
k
)
X
(
k
)
PX
(
k
)
sgn
e
(
k
)
 p  
p

independent trials.


(15)
The a posterior error vector e p ( k ) is not available
due to dependency with w ( k  1) . It is reasonable to
approximate e p ( k ) with a prior error vector e( k ) .
The update equation of proposed constrained affine
projection sign algorithm becomes
w (0)  F
w (k  1)
(16)


 X(k ) sgn  e(k ) 
F
 P  w (k ) 

T
T
sgn e (k ) X (k ) PX( k ) sgn  e( k )  




In Fig. 2 shows the MSD learning curves of
proposed algorithm with various order (the number
of the previous input vectors used). The step-size is
set to   0.05 . From the Fig. 2, it is noted that
larger L leads to faster convergence speed, and
lower steady-state error.
The comparisons of the proposed algorithm with
the conventional CAPA are shown in Fig. 3. In the
case of L  1 , the proposed algorithm converges
faster and achieves smaller steady-state error than
conventional algorithm. The conventional algorithm
with L  16 is likely to diverge while the proposed
algorithm
is
robust
against
the
impulsive
interference.
(17)
Since no matrix inversion is needed for proposed
algorithm, the update equation of proposed algorithm
is much simpler in implementation than conventional
CAPA.
IV. Simulation Results
Consider the system identification problem as
shown in Fig. 1. The unknown system to be
identified with linear phase feature is modeled with
randomly generated weight coefficients. The
adaptive filter has a length N  33 taps. In order to
fulfill the linear phase requirement, we use
- 3 -
Fig. 2. MSD curves of the proposed algorithm
with varying orders.
2013 年度 第 26 回 信號處理合同學術大會論文集 第 26 卷 1 號
(NIPA-2013-H0401-13-1008) supervised by the
NIPA (National IT Industry Promotion Agency).
Reference
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Fig. 3. Comparison of conventional CAPA and
proposed Algorithm
We also exams the effect of the step size on the
MSD of the proposed algorithm, as shown in Fig. 4,
where   1.0, 0.05, 0.0025, 0.001 , are used. All
parameter expect step-size is same as simulation for
Fig. 1. A small step-size leads to slow convergence
speed with low steady-state error. In contrast, a
large step-size speeds up convergence rate with
higher steady-state error.
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Ⅴ. Conclusion
This paper has proposed a constrained affine
projection sign algorithm based on the minimizing L1
-norm criterion. The proposed CAPSA need no
matrix inversion, so it has benefit to implementation.
The simulation results show that the proposed
CAPSA
exhibits
greater
performance
than
conventional CAPA in impulsive interference
environments.
Acknowledgement
This work was supported by the National
Research Foundation of Korea (NRF)grant funded by
the
Korea
government
(MEST)
(2012R1A2A2A01011112), and
by the MSIP
(Ministry of Science, ICT&Future Planning), Korea,
under the C-ITRC (Convergence Information
Technology Research Center) support program
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algorithm,” IEEE Trans. Signal Process., vol. 56,
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