Memory Proportionate APSA with Individual
Activation Factors for Highly Sparse System
Identification in Impulsive Noise Environment
Yi Yu
School of Electrical Engineering
Southwest Jiaotong University
Chengdu, China
yuyi_xyuan@163.com
Haiquan Zhao, Member, IEEE
School of Electrical Engineering
Southwest Jiaotong University
Chengdu, China
hqzhao@home.swjtu.edu.cn
Abstract—This paper proposes a memory proportionate
affine projection sign algorithm (IAF-MP-APSA) by assigning an
individual activation factor to each filter coefficient. In this
algorithm, each individual activation factor is calculated by past
and current values of the corresponding coefficient magnitude.
Moreover, taking into account the memory property of the
proportionate factors leads to a decrease in computational
complexity. Simulation results to estimate impulse responses
having high sparseness have demonstrated that the IAF-MP-
APSA not only inherits good robustness of the affine projection
sign algorithm (APSA) against impulsive noise, but also
outperforms other existing proportionate APSAs in terms of
convergence rate and tracking capability.
Keywords-Proportionate affine projection sign algorithm;
sparse impulse response; impulsive noise; individual activation
factor
I. INTRODUCTION
In several practical applications such as wireless
communications, acoustic echo cancellation (AEC), network
echo cancellation (NEC), and underwater acoustic channel
estimation, etc, sparse impulse responses that we wish to
estimate by using an adaptive FIR filter are often encountered
[1]. An impulse response with this property is categorized as
sparse, that is, only a small portion of coefficients (active
coefficients) have large magnitude while the remaining
coefficients (inactive coefficients) are very close or even equal
to zero. Aiming to such impulse responses, traditional
adaptive filtering algorithms that assign the common step-size
to all filter coefficients to update the filter coefficient vector
suffer from slow convergence, typically, the normalized least
mean square algorithm (NLMS). To solve this problem,
Duttweiler firstly proposed proportionate NLMS algorithm
(PNLMS) [2], in which each filter coefficient obtains an
independent step-size that is proportionally to its estimated
magnitude. Subsequently, to further improve convergence
performance, many modified versions of the PNLMS
algorithm were proposed, such as improved PNLMS
(IPNLMS) [3], μ-law PNLMS (MPNLMS) [4] and PNLMS
with individual activation factors (IAF-PNLMS) [5]-[6]
algorithms.
It is well-known that the affine projection algorithm (APA)
proposed by Ozeki and Umeda [7] provides a faster
convergence rate than the NLMS algorithm for correlated
input signals (also termed colored inputs), especially for
speech signals in AEC application. Similarly, by exploiting
the sparsity of the impulse responses, some proportionate-type
APAs were successively developed, including proportionate
APA (PAPA) [8], improved PAPA (IPAPA) [9], memory
IPAPA (MIPAPA) [10], improved MIPAPA (IMIPAPA)
based on the l
0
-norm [11] and memory PAPA with individual
activation factors (IAF-MPAPA) [12].
Regrettably, the above-mentioned algorithms suffer from
performance degradation in the presence of impulsive noise
and even divergence, due to essentially they came from the l
2
-
norm optimisation. Aiming to this limitation, Shao et al.
proposed an affine projection sign algorithm (APSA) [13] by
combining the benefit of the APA and the l
1
-norm
optimization criterion. Moreover, the APSA has much lower
computational burden than the conventional APA with the
same affine projection, since its adaptive processing does not
involve matrix inversion operation. In [14], the proportionate
ideas of the PAPA and IPAPA (or the PNLMS and IPNLMS)
are directly extended to the APSA, respectively, and the
resulting proportionate APSA (RP-APSA) and improved RP-
APSA (RIP-APSA) achieve faster convergence rate than the
APSA in sparse system. Afterwards, a memory RIP-APSA
(MIP-APSA) [15] was proposed which obtains a lower steady-
state misalignment than the RIP-APSA.
However, the performance of the RP-APSA depends on
two predefined parameters. To this end, a critical issue is how
to set proper values for these parameters. Besides, since the
activation factor is common to all filter coefficients, all
inactive coefficients receive the same step-size. The property of
the RP-APSA slows convergence over the whole adaptation
This work was partially supported by National Science Foundation o
f
P.R. China (Grant: 61271340, U1134205, U1234203, U1134104 an
d
61071183), the Sichuan Provincial Youth Science and Technology Fun
d
(Grant: 2012JQ0046), the Fundamental Research Funds for the Central
Universities (Grant: SWJTU12CX026), and the Postgraduate Innovative
Experimental and Practice Project of Southwest Jiaotong University (Grant:
YC201403211)
978-1-4799-7338-5/14/$31.00 ©2014 IEEE 916
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