Proportionate Affine Projection Algorithm Based on
Coefficient Difference
Zongsheng Zheng
School of Electrical Engineering
Southwest Jiaotong University
Chengdu, China
bk20095185@my.swjtu.edu.cn
Haiquan Zhao, IEEE Member
School of Electrical Engineering
Southwest Jiaotong University
Chengdu, China
hqzhao@home.swjtu.edu.cn
Abstract—This paper proposes a proportionate affine
projection algorithm based on coefficient difference (DPAPA).
Its adaptation gain for each tap is proportional to the absolute
value of the difference between the current tap weight estimate
and the previous tap weight estimate. In addition, a simple
version is introduced to decrease the amount of calculation. The
simulation results show that the proposed algorithm has better
performance than the other algorithms, e.g. the proportionate
affine projection algorithm (PAPA) and improved proportionate
affine projection algorithm (IPAPA).
Index terms—Adaptive filter, echo cancellation, proportionate
affine projection algorithm (PAPA).
I. INTRODUCTION
As one of the common applications of adaptive filtering,
echo cancellation can be considered as a system identification
problem, and the unknown system to identify is the echo path.
In general, the impulse response of the echo path is sparse, i.e.,
a fraction of coefficients get large magnitudes while the rest
are close to zero. By utilizing the sparse property of the
impulse response, the proportionate normalized least mean
square (PNLMS) algorithm has been proposed by Duttweiler
[1]. It updates each coefficient of the filter respectively by
adjusting the adaptation step size in proportion to the estimated
filter coefficient. Following this method, several versions of
the PNLMS algorithm were developed [2, 3]. The improved
PNLMS (IPNLMS) algorithm was proposed by Benesty [2]
with the introduction of an adjustable parameter to combine
the normalized least mean square (NLMS) and PNLMS.
Furthermore, the μ-law PNLMS (MPNLMS) was proposed in
[3]. Its adaptation step size of each tap is proportional to the
logarithm of the coefficient magnitude. Recently, a
proportionate normalized least mean square algorithm based
on coefficient difference (DPNLMS) [4] was proposed by
using the absolute value of the difference between the current
tap weight estimate and the previous tap weight estimate as the
step size at each tap.
It is well-known that the affine projection algorithm (APA)
[5] improves convergence rate in comparison with the NLMS
algorithm for coloured input, especially for speech signal.
With the introduction of proportionate ideas from the
proportionate-type NLMS algorithms, several
Fig. 1. The model for echo cancellation
proportionate-type APAs were also developed [6, 7]. The
proportionate affine projection algorithm (PAPA) was
proposed for echo cancellation owing to its fast convergence
rate and well tracking ability [6]. Besides, the improved
proportionate APA (IPAPA) [7] was proposed with the
approach used in the IPNLMS.
In this paper, a proportionate affine projection algorithm
based on coefficient difference (DPAPA) is proposed by
combining the APA with the approach used in the DPNLMS.
Moreover, a simple version is obtained by reducing the
repetitive computation in the update process. The
improvement in convergence performance is demonstrated
through the simulations in the context of echo cancellation.
II. P
ROPORTIONATE-TYPE NLMS ALGORITHMS
Consider the model for echo cancellation shown in Fig.1,
where the impulse response of the echo path and its estimate
are defined as
01 1
[,,..., ]
T
L
hh h
−
=h ,
01
ˆˆ
ˆ
() [ (), (),...,nhnhn=h
1
ˆ
()]
T
L
hn
−
, respectively. Superscript T denotes transposition,
constant
L is the length of the adaptive filter, and variable n is
the time index. Then, the desired response
()dn is obtained
as
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