iaf-mpapa para cancelamento de eco acústico
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8/12/2019 IAF-MPAPA Para Cancelamento de Eco Acstico
1/92329-9290 (c) 2013 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See
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his article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation informatio
10.1109/TASLP.2014.2318519, IEEE/ACM Transactions on Audio, Speech, and Language Processing
1
AbstractAn individual-activation-factor memory
proportionate affine projection algorithm (IAF-MPAPA) is
proposed for sparse system identification in acoustic echo
cancellation (AEC) scenarios. By utilizing an individual activation
factor for each adaptive filter coefficient instead of a global
activation factor, as in the standard proportionate affine
projection algorithm (PAPA), the adaptation energy over the
coefficients of the proposed IAF-MPAPA can achieve a betterdistribution, which leads to an improvement of the convergence
performance. Moreover, benefiting from the memory
characteristics of the proportionate coefficients, its computational
complexity is less than the PAPA and improved PAPA (IPAPA).
In the context of AEC and stereophonic AEC (SAEC) for highly
sparse impulse responses, simulation results indicate that the
proposed IAF-MPAPA outperforms the PAPA, IPAPA, and
memory IPAPA (MIPAPA) in terms of the convergence rate and
tracking capability when the unknown impulse response suddenly
changes.
Index TermsAdaptive filtering, proportionate affine
projection algorithm, sparse impulse response, sparse system
identification, individual activation factor.
I. INTRODUCTION
or acoustic echo cancellation (AEC), a key attitude is that
the impulse response of the echo path is identified by an
adaptive filter. However, the echo paths in AEC scenarios
are typically sparse in nature, i.e., most of the coefficients of the
echo path are close to zero (inactive coefficients) with a few of
large value (active coefficients), which causes the well-known
adaptive filtering algorithms, such as least mean square (LMS)
and normalized LMS (NLMS), to converge slowly. To address
this problem, some proportionate adaptive filter algorithms
[1]-[11] were developed by assigning a different step size inproportion to the estimated magnitude of each adaptive filter
Manuscript received Jul. 24, 2013. This work was partially supported by
National Science Foundation of P.R. China (Grant: 61271340, U1134205,
U1234203, U1134104 and 61071183), the Sichuan Provincial Youth Science
and Technology Fund (Grant: 2012JQ0046), and the Fundamental Research
Funds for the Central Universities (Grant: SWJTU12CX026).
Haiquan Zhao, Yi Yu, Shibin Gao, Xiangping Zeng, and Zhengyou He arewith the School of Electrical Engineering at Southwest Jiaotong University,
Chengdu, 610031, China. (e-mail: hqzhao@home.swjtu.edu.cn,
yuyi_xyuan@163.com) .*Corresponding author.
coefficient.
In comparison with the standard NLMS algorithm, the
proportionate NLMS (PNLMS) algorithm has fast initial
convergence for sparse impulse responses [1]. Regrettably, the
PNLMS algorithm suffers from slow convergence after the
initial fast convergence, and its performance is degraded as the
sparseness decreases. Although the improved PNLMS
(IPNLMS) algorithm [3] is suitable for moderate sparseness, itdoes not provide the same fast initial convergence as the
PNLMS algorithm for highly sparse impulse responses. To
overcome the uneven convergence rate of PNLMS during the
estimation process, the -law PNLMS (MPNLMS) algorithm
[5] was proposed, which can achieve faster convergence over
the whole adaptation process, at the cost of increasing
computational complexity.
It is widely accepted that the affine projection algorithm
(APA) provides better convergence performance than the
NLMS algorithm for colored input signals, especially for
speech input signals. To meet the requirement of sparse
impulse responses, the proportionate APA (PAPA) [2],
improved PAPA (IPAPA) [4], and -law PAPA (MPAPA) [6]
were developed by incorporating the proportionate ideas of the
PNLMS, IPNLMS, and MPNLMS algorithms into the APA,
respectively. Recently, Paleologu et al.[8] proposed a memory
IPAPA (MIPAPA) via taking into account the memory of the
proportionate coefficients. Compared to the IPAPA, the
MIPAPA not only speeds up the convergence rate, but also
reduces the computational complexity. In [11], a -law
MIPAPA (MMIPAPA) was proposed to achieve a significant
improvement of the convergence speed at the cost of higher
computational complexity as compared with MIPAPA.
Similar to the PNLMS algorithm, the performance of PAPA
in terms of convergence rate and steady-state error depends onsome predefined parameters that control proportionality and
initialization. Therefore, a key problem is how to select optimal
values for these parameters. Moreover, the parameters are
associated with an algorithm!s variable called the activation
factor, having the task to prevent the updated filter coefficients
from stalling when their magnitudes are zero or significantly
smaller than the largest coefficient. In the PNLMS and PAPA
algorithms, the activation factors are common to all adaptive
filter coefficients, computed sample-by-sample, and depend on
the instantaneous l"-norm of the estimated coefficient vector.
Memory Proportionate APA with Individual
Activation Factors for Acoustic Echo
Cancellation
Haiquan Zhao*,Member, IEEE, Yi Yu, Shibin Gao, Xiangping Zeng, and Zhengyou He, SeniorMember, IEEE
F
mailto:hqzhao@home.swjtu.edu.cnmailto:yuyi_xyuan@163.com)mailto:yuyi_xyuan@163.com)mailto:hqzhao@home.swjtu.edu.cn -
8/12/2019 IAF-MPAPA Para Cancelamento de Eco Acstico
2/92329-9290 (c) 2013 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See
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his article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation informatio
10.1109/TASLP.2014.2318519, IEEE/ACM Transactions on Audio, Speech, and Language Processing
2
This way of computing the activation factor leads to a gain
distribution over the adaptive filter coefficients that is not
entirely in line with the concept of proportionality, which is the
desired attribute of the PNLMS algorithm. Recently, to address
this issue, an individual activation factor PNLMS
(IAF-PNLMS) algorithm [7], [9], [10] was proposed by
assigning an individual activation factor to each adaptive filter
coefficient, wherein each individual activation factor is
computed in terms of past and current values of thecorresponding estimated coefficient magnitude.
In this paper, motivated by the recently proposed
IAF-PNLMS algorithm, we propose an
individual-activation-factor memory PAPA (IAF-MPAPA) by
extending the individual-activation-factor and memory ideas to
PAPA. Subsequently, the IAF-MPAPA is extended to
stereophonic AEC (SAEC) [12], which can be modeled as a
single-input/single-output system with complex random
variables by using the widely linear (WL) model [13]-[15].
In contrast to the standard PAPA, the proposed IAF-MPAPA
algorithm has the following features:
a) An individual activation factor is employed for each filter
coefficient.
b) Each individual activation factor, calculated by past and
current values of the corresponding coefficient,
incorporates inherent memory characteristics associated
with the corresponding coefficient magnitude.
c) The computational complexity is reduced in comparison
with the PAPA and IPAPA, due to the memory of the
proportionate coefficients is taken into account.
Consequently, the convergence performance of the proposed
IAF-MPAPA is enhanced. For highly-sparse impulse response
and colored input, results obtained from computer simulations
have shown that the proposed algorithm not only has faster
convergence than the existing PAPA, IPAPA, and MIPAPAalgorithms, but also obtains better tracking capability when the
unknown impulse response suddenly changes.
This paper is organized as follows. Section II briefly reviews
the standard PAPA. In Section III, we firstly analyze the impact
of the activation factor on the PAPA performance, and then the
IAF-MPAPA is derived. In Section IV, the IAF-MPAPA is
extended to the SAEC. In Section V, numerical simulations in
the context of AEC and SAEC confirm the improved
performance of the proposed IAF-MPAPA. Finally, Section VI
presents concluding remarks.
II. BRIEF REVIEW OF PAPA
For sparse system identification in an AEC, as shown in Fig.
1, the adaptive filter that estimates the unknown sparse impulse
response o o, 0 o, 1 o, 1( ), ( ), ..., ( )T
Mw k w k w k ! "#$ %w
is defined
by & '0 1 1( ) ( ), ( ), ..., ( )T
Mk w k w k w k #w at time k, where
superscript Tdenotes transposition and Mis filter length. The
input matrix ( )kX is defined as thePmost recent input vectors
( )kx , i.e., & '( ) ( ), ( 1), ..., ( 1)k k k k P # (X x x x , where
& '( ) ( ), ( 1), ..., ( 1) Tk x k x k x k M # (x , andPis the affine
projection order. The desired response ( )d k of the adaptive
filter is given by
o( ) ( ) ( )Td k k v k x w# ( (1)
where v(k) is the measurement noise. Then, the error vector
( ) [ ( ), ( 1), ..., ( 1)]T
k e k e k e k P e # ( can be obtained as
( ) ( ) ( ) ( )T
k k k k # e d X w (2)
where & '( ) ( ), ( 1), ..., ( 1) Tk d k d k d k P # (d is thedesired response vector of the adaptive filter, containing the P
successive past desired responses.
o
( )v k
( )k
( )x k
( )d k +
( )e k
( )k+
+
Fig. 1. Structure of an acoustic echo canceller.
As in [2], the weight coefficient update of the standard PAPA
is expressed by the following set of equations
( ) ( ) ( )k k k#P G X , (3)
1
( 1) ( ) ( ) ( ) ( ) ( )Tk k k k k k !
! "( # ( () *$ %w w P X P I e (4)
where and I are the step-size and identity matrix,
respectively, ! is a small regularization parameter to avoid an
ill-conditioned matrix ( ) ( )T k k! ") *$ %X P , and the proportionate M#
M diagonal matrix + ,0 1 1( ) diag ( ), ( ), ..., ( )Mk g k g k g k #G assigns an individual step size to update each filter coefficient
( ), 0, 1, ..., 1mw k m M # .
The individual gain ( )mg k of the proportionate matrix
( )kG is calculated as
1
0
( )( ) , 0, 1, ..., 1
( )
mm M
mm
kg k m M
k
"
"
#
# #
-. (5)
And the proportionality function ( )m k" is defined by
+ ,( ) max ( ), ( )m mk q k w k " # (6)with the activation factor
+ ,( ) max , ( )q k k# $.
# w (7)
where./ is the infinity norm, and the initialization parameter
$ and proportionality parameter # prevent the coefficients
( )mw k from stalling when their magnitudes are initialized to
-
8/12/2019 IAF-MPAPA Para Cancelamento de Eco Acstico
3/92329-9290 (c) 2013 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See
http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
his article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation informatio
10.1109/TASLP.2014.2318519, IEEE/ACM Transactions on Audio, Speech, and Language Processing
3
zero, i.e., (0)#w 0 , and are significantly smaller than the
largest coefficient.
It can be seen from (7) that the activation factor ( )q k not
only depends on the adaptive filter coefficient vector ( )kw ,
but also is conditioned by parameters $and # . From (5) and
(6), the individual gain ( )mg k can be rewritten as
+ ,1( ) max ( ), | ( ) |( )m mg k q k w k
c k# (8)
with1
0
( ) ( )M
m
m
c k k"
#
#- . (9)
Here, we highlight some problems of the PAPA by analyzing
(8) [7], [9], [10]. First, if ( ) | ( ) |mq k w k 0 , the mth coefficient
( )mw k is inactive and its associated gain can be expressed as
inactive ( )( )
( )
q kg k
c k# . (10)
There is a common gain
inactive
( )g k is assigned to allinactive coefficients since the activation factor ( )q k is
common to all adaptive filter coefficients. However, this
characteristic is undesirable for sparse system identification
and needs to be eliminated.
Second, if ( ) | ( ) |mq k w k 1 , the mth coefficient ( )mw k is
active and its associated gain can be expressed as
active
active
| ( ) |( )
( ) ( ) ( )m
m
mm C
w kg k
M M q k w k2
# (-
(11)
where activeM is the number of active coefficients and Cis the
set of indices associated with their positions. In (11), the gain of
each active coefficient depends on its magnitude, as well as the
activation factor q(k). Therefore, we conclude that the
activation factor given by (7) affects the gains assigned to both
active and inactive coefficients, which does not entirely satisfy
the concept of proportionality.
III. PROPOSED IAF-MPAPA
A. Algorithm DesignTo overcome the above-mentioned problems of the PAPA,
an individual-activation-factor memory PAPA (IAF-MPAPA)
utilizing the approach of [7], [9], [10], is proposed by assigning
an individual activation factor ( )mq k to each inactivecoefficient instead of a common value as in the PAPA. Thus,
the proportionality function and activation factor given by (6)
and (7) are modified to, respectively,
+ ,( ) max ( ), | ( ) |m m mk q k w k " # (12)and
1 1( ) ( 1), , =1, 2,...
( ) 2 2
( 1), otherwise
m mm
m
w k k k = nM nq k
q k
"344 ( 4#544 46
(13)
where each individual activation factor qm(k) is initialized by a
small positive constant such that (0) 0mq 0 (typically,
2(0) 10 /mq M
# ) to avoid the freezing of coefficients wm(k).
In (13), the activation factor ( )mq k is periodically updated only
by the interval of M iterations which is equal to the adaptive
filter length. Therefore, the instantaneous magnitude of each
estimated coefficient ( )mw k is proportional to the magnitude
of the corresponding unknown coefficient o, mw .
By analyzing (12) and (13) [7], [9], [10], the proposed
IAF-MPAPA has the following properties.
P1): Each coefficient in either active or inactive, and has an
associated individual activation factor,
( ) 0, 0, 1, , 1mq k m M 0 # 7 , with inherent memory
associated with the mth coefficient.
P2): Each individual activation factor ( )mq k converges to
the corresponding coefficient magnitude ( )mw k as adaptive
processing goes on, i.e.,
lim ( ) lim ( ) , 0, 1, , 1m mk kq k w k m M 8. 8.# # 7 . (14)
Proof of P2):Firstly, we consider (0) (0)m mq" # when the
adaptive filter coefficient vector is initialized to the zero vector.
Then, if the mth coefficient is active at time k = n M , the
activation factor ( )mq k and the proportionality function
( )m k" are, respectively, calculated as [7], [9], [10]
1 1( ) ( 1) ( )
2 2m m mq nM w nM w nM # ( (15)
and
1( ) max ( 1)
21
( ) , ( )2
m m
m m
nM w nM
w nM w nM
"344# 5
446944( :44;
. (16)
If the mth coefficient is inactive at time k = n M , the activity
factor ( )mq k is given by
1
2
1 1( ) (0) ( )
2 2
1(2 ) ...
2
1 1[( 1) ] ( )
22
m m mn n
mn
m m
q nM q w M
w M
w n M w nM
# (
( (
( (
(17)
and the proportionality function ( )m k" is
& '1 2
1 1( ) max (0) ( )
2 2
1 1(2 ) ... ( 1)
2 2
1( ) , ( )
2
m m mn n
m mn
m m
nM q w M
w M w n M
w nM w nM
"
344# (5446
( ( (
944( :44;
. (18)
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4/92329-9290 (c) 2013 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See
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his article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation informatio
10.1109/TASLP.2014.2318519, IEEE/ACM Transactions on Audio, Speech, and Language Processing
4
Assuming that the proposed algorithm has converged to
steady state, then ( 1) ( )m mw nM w nM < . Thus, by
observing (15) and (17), we can straightforwardly obtain
lim ( ) ( ) 0m mn
q nM w nM 8.
! " #$ % (19)
for both active and inactive coefficients. Thereby, equation (14)
is verified, which is our desired goal. Furthermore, the gain
( )m
g nM from (5), (16), and (18) tends to be proportional to
( )mw nM for both active an inactive coefficients, which is in
line with the proportionate requirement.
To reduce computational complexity, the matrix ( )kP given
by (3) can be approximated as '( )kP by considering the
memory of proportionate coefficients [8], i.e.,
1'( ) [ ( ) ( ), ' ( )]k k k k #P g x Pe (20)
where the operation e denotes the Hadamard product; ( )kg is
a column vector containing the diagonal elements of ( )kG , i.e.,
T0 1 1( ) [ ( ), ( ), ..., ( )]Mk g k g k g k #g , and the matrix 1' ( )kP
contains the firstP
1 columns of '( 1)k
P .As a result, the proposed IAF-MPAPA algorithm is
summarized as in Table I.TABLE I.
SUMMARY OF IAF-MPAPA.
Initialization 21 1(0) , (0) 10 /M M mq M
= =# #w 0
Parameters ,!
Proportionateprocessing
1 1( ) ( 1), , = 1, 2, ...
( ) 2 2
( 1), otherwise
m mm
m
w k k k = nM nq k
q k
"344 ( 44#544 446
+ ,( ) max ( ), | ( ) |m m mk q k w k " #
1
0
( )( ) , 0,1,..., 1
( )
mm M
m
m
kg k m M
k
"
"
#
# #
-
0 1 1( ) [ ( ), ( ), ..., ( )]T
Mn g k g k g k g #
1'( ) [ ( ) ( ), ' ( )]k k k k P g x Pe =
Adaptation
processing
( ) ( ) ( ) ( )Tk k k k e d X w#
1( ) ( 1) '( )[ ( ) '( ) ] ( )Tk k k k k k w w P X P I e ! # ( (
TABLE II
COMPUTATIONAL COMPLEXITY OF COEFFICIENT UPDATE FOR VARIOUS
ALGORITHM COMPARED TO THE PROPOSED IAF-MPAPA
Algorithms Additions Multiplications Comparisons Memory
PAPA (P2+P)M -1(P2+P+2)M +
P2 + 22M M
IPAPA (P2+P+1)M
(P 2+P+2)M
+P 20 M
MIPAPA (P2+P+1)M (P 2+3)M +P 2 0 M
IAF-
MPAPA(P2+P)M
(P2+3)M +
P 2 + 2M 3M
B. Computational complexityIn Table II, the computational complexity of the proposed
IAF-MPAPA is compared with that of other existing
algorithms in terms of the total number of additions,
multiplications, comparisons, and memory spaces. To update
the weight coefficient vector at every iteration, the proposed
IAF-MPAPA with filter length M and projection order P
requires (P2+P)M additions, (P2+3)M+P2 +2 multiplications,
Mcomparisons, and 3Mmemory spaces. In addition, all of the
algorithms require P#P direct matrix inversion (DMI). As
shown in Table II, the IAF-MPAPA requires an additional
memory of size 2M for storing both the activity factors andproportionality function values in comparison with PAPA and
IPAPA, but it eliminates (P 1)M and (P 1)M 2
multiplications, respectively. Moreover, this advantage would
become more apparent as the projection order Pincreases.
IV. EXTENSION TO SAEC
In this section, the proposed IAF-MPAPA is extended to the
SAEC. In the classical SAEC setup [12], there are two input or
loudspeaker signals denoted by x1(k) and x2(k) ($left% and
$right%), and two output or microphone signals denoted by d1(k)
and d2(k), which can be expressed as
1 1 1( ) ( ) ( )d k y k v k # ( (21)
2 2 2( ) ( ) ( )d k y k v k # ( (22)
wherey1(k) andy2(k) are the stereo echo signals, and v1(k) and
v2(k) are the noise or near-end signals. The echo signals are
given by
1 11 1 21 2( ) ( ) ( )T T
y k k kw x w x# ( (23)
2 12 1 22 2( ) ( ) ( )T T
y k k kw x w x# ( (24)
where w11, w12, w21, w22 are M-dimensional vectors of the
loudspeaker-to-microphone ($true%) acoustic impulse
responses that need to be estimated to cancel the echo,
and & '1 1 1 1( ) ( ), ( 1), ..., ( 1)T
k x k x k x k M # (x and
& '2 2 2 2( ) ( ), ( 1), ..., ( 1)T
k x k x k x k M # (x are input
signal vectors. Recently, a classical two-input/two-output
system with real random variables was converted to a
single-input/single-output system with complex random
variables by using the widely linear (WL) model [13]-[15], as
shown in Fig. 2.
Fig. 2. The WL model for SAEC.
-
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5/92329-9290 (c) 2013 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See
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his article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation informatio
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5
The complex output signal is defined as,
1 2( ) ( ) ( )
( ) ( )
d k d k jd k
y k v k
# (
# ( (25)
where 1j # ,y(k) =y1(k) +jy2(k), and v(k) = v1(k) +jv2(k).
Also, let us define the complex input vector
& '
1 2( ) ( ) ( )
( ), ( 1), ..., ( 1)
T
k k j k
x k x k x k M
# (
# (
x x x (26)
wherex(k) = x1(k) + jx2(k). Thus, the complex echo signal can
be expressed as
t t( ) ( ) ( )H H *y k k k># (w x w x (27)
where the superscripts Hand * denote transpose-conjugate and
conjugate, respectively, and
t1 t2jw w w# ( (28)
t1 t2j> > ># (w w w (29)
with
11 22 21 12t1 t2, ,
2 2
w w w ww w
( # # (30)
11 22 21 12t1 t2,
2 2 (> ># #w w w ww w . (31)
Thereby, the (25) can be rewritten as [13]-[15]
( ) ( ) ( )H
d k k v k w x%# ( (32)
where ( ) ( ), ( )T
T Hk k k! "# ) *$ %x x x% , and
t t,T
T T'! "# ) *$ %
w w w is a
complex acoustic impulse response, which is estimated by a
2M-length adaptive filter ( )kw . In light of the above
introduced WL model, some proportionate APAs such as
PAPA, IPAPA, MIPAPA, IAF-MPAPA, etc., can be easily
extended to SAEC by using their complex variants [13]-[14].
It is well-known that there is a nonunique solution problem
for SAEC, owing to the two input signals, i.e., x1(k) andx2(k),
which are obtained by filtering a common source [12]-[13].
Therefore, to achieve a unique solution, it may be necessary to
preprocess the input signals to weaken the coherence between
these two signals in order to obtain estimation of the true
acoustic impulse responses. A positive and negative half-wave
rectifier can be met on each input signal respectively as follows
[13]-[15]:
1 1
1 1
( ) ( )( ) ( )
2
x k x kx k x k b
(> # ( (33)
2 2
2 2
( ) ( )( ) ( )
2
x k x kx k x k b
> # ( (34)
where b is a parameter used to control the amount of
nonlinearity. Also, experiments have shown that the stereo
perception is not affected by the above methods even with bas
large as 0.5 [13]-[15].
Thus, along with (33) and (34), the proposed IAF-MPAPA is
generalized for SAEC as shown in Table III, where
( ) ( ), ( 1), ..., ( 1)k k k k P ! "# () *$ %X x x x% % % .
V. SIMULATION RESULTS
To verify the effectiveness of the proposed IAF-MPAPA,
numerical simulations in the context of AEC and SAEC are
carried out in the following.
0 50 100 150 200 250 300 350 400 450 500-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Sample number
Magnitude
Fig. 3. Impulse response with sparsity 0.9357$# .
TABLE III.SUMMARY OF THE PROPOSED IAF-MPAPAFOR SAEC.
Initialization 22 1 2 1(0) , (0) 10 / (2 )M M mq M
= =# #w 0
Parameters ,!
Preprocessing of
input signals
1 11 1
( ) ( )( ) ( )
2
x k x kx k x k b
(> # (
2 22 2
( ) ( )( ) ( )
2
x k x kx k x k b
> # (
1 2( ) ( ) ( )x k x k jx k> ># (
( ) [ ( ), ( 1), ..., ( 1)]Tk x k x k x k M # (x
( ) [ ( ), ( )]T H Tk k k#x x x%
Proportionate
processing
1 1( ) ( 1), , = 1, 2, ...
( ) 2 2
( 1), otherwise
m mm
m
w k k k = nM nq k
q k
"344 ( 44#544 446
+ ,( ) max ( ), | ( ) |m m mk q k w k " #
1
0
( )( ) , 0,1,..., 2 1
( )
mm M
m
m
kg k m M
k
"
"
#
# #
-
0 1 2 1( ) [ ( ), ( ), ..., ( )]T
Mn g k g k g k g #
1'( ) [ ( ) ( ), ' ( )]k k k k P g x Pe =
Adaptation
processing
( ) ( ) ( )H
d k k v k # (w x%
( ) ( ) ( ) ( )
T
k k k k ?
# e d X w 1( ) ( 1) '( )[ ( ) '( ) ] ( )Hk k k k k kw w P X P I e ! ?# ( (
A. AECFor the AEC simulations, a highly sparse impulse response
with 512 coefficients is used as shown in Fig. 3, and its
sparseness level is = 0.9357 according to the definition in [16],
[17]. The input signal is either an AR(1) process generated by
filtering a zero-mean white Gaussian signal through a
first-order system1( ) 1 / (1 0.9 )H z z# or a speech signal.
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6
The affine projection orders are selected as P= 2 andP= 8, and
the initial weight coefficient vectors are set to 0. The same step
size 0.5# is used to compare PAPA (with 0.01## and
0.01$# ), IPAPA (with 0%# and 0.01 ), MIPAPA
(with 0%# and 0.01 ), and the proposed IAF-MPAPA
(with3
(0) 10mq
# ) under the same steady-state performance.
The misalignment, 10 o o2 2
20log ( ( ) / )kw w w , is used to
measure the performance of these algorithms. The results are
averaged over 10 independent trials.
1) AR(1) correlated inputIn this subsection, the input signal is an AR(1) process with a
pole at 0.9. Fig. 4 shows the convergence performance of the
algorithms with 30-dB signal-to-noise ratio (SNR) for
projection orders P= 2 and 8. As expected, these algorithms
have the same steady-state misalignment for the same order P,
and both the convergence rate and steady-state misalignment of
each algorithm increases as the order Pincreases. Although the
MIPAPA and IPAPA outperform PAPA in terms of
convergence performance for a modest sparse and dispersive
impulse response [8], the superiority is only slight for a highly
sparse impulse response, which can be seen from Fig. 4. For an
impulse response having high sparseness, it is clearly observed
that the IAF-MPAPA not only has faster convergence thanPAPA, IPAPA, and MIPAPA, but also obtains better tracking
capability when the impulse response is multiplied by 1 at
iteration 20000. This is owing to the fact that the proposed
IAF-MPAPA assigns an individual activation factor for each
adaptive filter coefficient.
0.5 1 1.5 2 2.5 3 3.5 4
x 104
-35
-30
-25
-20
-15
-10
-5
0
5
10
Iteration number
Misalignment/dB
0.5 1 1.5 2 2.5 3 3.5 4
x 104
-30
-25
-20
-15
-10
-5
0
5
10
Iteration number
Misalignment/dB
PAPA
IPAPA
MIPAPA
IAF-MPAPA
PAPA
IPAPA
MIPAPA
IAF-MPAPA
(a) (b)Fig. 4. Misalignment performance of various APAs with SNR = 30 dB for AR(1) input.
(a)P= 2. (b)P= 8.
0.5 1 1.5 2 2.5 3 3.5 4
x 104
-30
-25
-20
-15
-10
-5
0
5
10
Iteration number
Misa
lignment/dB
0.5 1 1.5 2 2.5 3 3.5 4
x 104
-25
-20
-15
-10
-5
0
5
10
Iteration number
Misa
lignment/dB
PAPA
IPAPA
MIPAPA
IAF-MPAPA
PAPA
IPAPA
MIPAPA
IAF-MPAPA
(a) (b)
Fig. 5. Misalignment performance of various APAs with SNR = 20 dB for AR(1) input.
(a)P= 2. (b)P= 8.
-
8/12/2019 IAF-MPAPA Para Cancelamento de Eco Acstico
7/92329-9290 (c) 2013 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See
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his article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation informatio
10.1109/TASLP.2014.2318519, IEEE/ACM Transactions on Audio, Speech, and Language Processing
7
Fig. 5 describes the misalignment curve of these APAs with
SNR = 20 dB for projection orders P = 2 and 8. It is also
concluded that the proposed IAF-MPAPA is still superior to
PAPA, IPAPA, and MIPAPA in terms of the convergence rate
and tracking capability. Moreover, it can be also observed from
Figs. 4 and 5 that the steady-state misalignments of these APAs
with the same projection order increase as the SNR decreases.
2) Speech inputIn this experiment, a speech signal is used as the input of the
AEC. In Fig. 6, we compare the performance of the
IAF-MPAPA with that of the PAPA, IPAPA, and MIPAPA for
projection orders P= 2 and 8. Simulation results with speech
input in Fig. 6 are similar to those results with AR(1) input in
Figs. 4 and 5, which demonstrates that the proposed
IAF-MPAPA also outperforms the other existing APAs for a
speech input signal in terms of the convergence rate and
tracking ability when the unknown impulse response suddenly
changes. This result is reasonable since an individual activation
factor for each adaptive filter coefficient is utilized instead of a
global activation factor as in the standard PAPA, so that the
adaptation energy over the proposed IAF-MPAPA coefficientsis capable of achieving a better distribution. This results in a
significant improvement of the performance in terms of the
convergence rate, steady-state error, and tracking capability.
0.5 1 1.5 2 2.5 3 3.5 4
x 104
-25
-20
-15
-10
-5
0
5
Iteration number
Misalignment/dB
PAPA
IPAPA
MIPAPA
IAF-MPAPA
0.5 1 1.5 2 2.5 3 3.5 4
x 104
-30
-25
-20
-15
-10
-5
0
5
Iteration number
Misalignment/dB
PAPA
IPAPA
MIPAPA
IAF-MPAPA
(a) (b)
Fig. 6. Misalignment performance of various APAs with SNR = 30 dB for speech input.
(a)P= 2. (b)P= 8.
0.5 1 1.5 2 2.5 3 3.5 4
x 104
-35
-30
-25
-20
-15
-10
-5
0
5
10
Iteration number
Misalignment/dB
= 0.3663
= 0.8083
= 0.8713
= 0.9357
Fig. 7. Misalignment performance of the proposed IAF-MPAPA with P= 2 for
different levels of sparsity. !=0.5, 3(0) 10mq
# .
3) Different sparseness levelsHere, to evaluate the performance of the proposed
IAF-MPAPA with respect to different sparseness levels, four
impulse responses (with sparseness = 0.3663,0.8083, 0.8713,
and0.9357, respectively) are used. The input signal is an AR(1)
process with a pole at 0.9, and SNR = 30dB. It is clearly seen
from Fig. 7 that the IAF-MPAPA has good convergence rateand tracking ability when the impulse response is multiplied by
1 at iteration 20000 for a highly sparse impulse response, for
example = 0.9357. Moreover, when the sparseness of the
impulse response is higher the convergence rate of the
IAF-MPAPA is faster. However, the fluctuation range of the
steady-state misalignment of the IAF-MPAPA is larger for a
more sparse impulse response, which is a subset for future
work.
B. SAECIn SAEC experiments, we use four impulse responses of
lengthM= 512 with high sparseness as shown in Fig. 8. The
length of the adaptive filter w(k) is 2M= 1024, and its initialvalue is set to w(0) = 0. The input signals are obtained by finite
impulse response (FIR) filtering a common speech signal in the
far-end location, and SNR = 30 dB. The parameters are chosen
to get approximately the same steady-state misalignment after
convergence for all algorithms (i.e., PAPA, IPAPA, MIPAPA,
and the proposed IAF-MPAPA) with projection order P= 2.
The misalignment, 102 2
20log ( ( ) / )kw w w , is used to
measure the performance of these algorithms (one trial).
-
8/12/2019 IAF-MPAPA Para Cancelamento de Eco Acstico
8/92329-9290 (c) 2013 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See
http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
his article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation informatio
10.1109/TASLP.2014.2318519, IEEE/ACM Transactions on Audio, Speech, and Language Processing
8
0 200 400-0.5
0
0.5
1
Sample number
(a)
Magnitude
0 200 400-0.5
0
0.5
1
Sample number
(b)
Magnitude
0 200 400-0.5
0
0.5
1
Sample number(c)
Magnitude
0 200 400-0.5
0
0.5
1
Sample number(d)
Magnitude
Fig. 8. Impulse responses used in SAEC simulation.
(a) w11with sparsity 0.9452$# . (b) w12with sparsity 0.9262$# .
(c) w21with sparsity 0.9201$# . (d) w22with sparsity 0.9357$# .
Fig. 9 depicts the misalignment performance of the proposedIAF-MPAPA for SAEC (Table III) for different parameter b
values (i.e., b = 0, 0.1, 0.2, and 0.5). Obviously, the
IAF-MPAPA for SAEC with nonlinear preprocessing ( b'0)
has better filtering performance in terms of convergence rate
and steady-state misalignment than the IAF-MPAPA without
nonlinearity (b = 0). The reason for this result is that the
positive and negative half-wave rectifiers have a good
capability for reducing the coherence between the two far-end
input signals. Moreover, the capability is strengthened as the
nonlinearity parameter b increases. Thereby, the nonunique
solution problem of SAEC can be mitigated by using (33) and
(34).
0 0.5 1 1.5 2 2.5 3 3.5 4
x 105
-30
-25
-20
-15
-10
-5
0
Iteration number
Misalignment/dB
b = 0
b = 0.1
b = 0.2
b = 0.4
b = 0.5
Fig. 9. Misalignment performance of the proposed IAF-MPAPA for SAEC for
different bvalues. != 0.5, 3(0) 10mq# .
0 1 2 3 4 5 6 7 8
x 105
-25
-20
-15
-10
-5
0
5
10
Iteration number
Misalignment/dB
PAPA
IPAPA
MIPAPA
IAF-MPAPA
Fig. 10 Misalignment performance of various APAs using nonlinear
preprocessing for SAEC. PAPA:!= 0.5,"= 0.01, = 0.01. IPAPA:!= 0.5, #=
0, $= 0.01. MIPAPA: != 0.5, #= 0, $= 0.01. IAF-MPAPA: != 0.5,
4(0) 5 10mq# = .
In Fig. 10, the misalignment results of various APAs arepresented, where the input signals are preprocessed using (33)
and (34) with b= 0.2. The algorithm parameters are chosen to
get approximately the same steady-state misalignment for all
algorithms after convergence. As can be clearly seen from Fig.
10, the IAF-MIPAPA still has the fastest convergence rate
among these algorithms for SAEC and the best tracking
capability when four impulse responses are multiplied by 1 at
iteration 400000. Although the IAF-MIPAPA is only slightly
superior to the PAPA on the convergence performance, it
reduces the computational complexity. Moreover, this
advantage is very commendable for real-time echo cancellation
applications, and would become more apparent as the
projection orderPincreases.
VI. CONCLUSION
We have proposed an individual-activation-factor memory
PAPA (IAF-MPAPA) to enhance AEC convergence
performance for highly sparse impulse responses by
incorporating the individual activation factor idea proposed in
[9] into the PAPA. Moreover, the algorithm has a lower
computational burden than the PAPA and IPAPA by
employing the memory of the proportionate coefficients.
Simulation results in AEC and SAEC applications have shown
that the proposed algorithm not only has faster convergence
than existing algorithms such as PAPA, IPAPA and MIPAPAfor impulse responses with high sparseness and colored input,
but also obtains better tracking capability when the unknown
impulse response suddenly changes.
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Haiquan Zhao (M!11) was born in Henan
Province, China, in 1974. He received the B.S.
degree in applied mathematics in 1998, the M.S.degree and the Ph.D degree in signal and
information processing all at Southwest Jiaotong
University, Chengdu, China, in 2005 and 2011,respectively. Since August 2012, he was a
Professor with the School of Electrical Engineering,
Southwest Jiaotong University, Chengdu, China.His current research interests include adaptive
filtering algorithm, adaptive Volterra filter,
nonlinear active noise control, nonlinear systemidentification and chaotic signal processing. At present, he is the author or
coauthor of more than 50 journal papers and the owner of six invention patents.
Prof. Zhao is a member of the IEEE Computational Intelligence Society. Hehas served as an active Reviewer for several IEEE RANSACTIONS, The
Institution of Engineering and Technology, and other international journals.
Yi Yu received the B.E. degrees in School of
Electrical & Information at XiHua University,
Chengdu, China, in 2011. Now he is pursuiting the
master!s degree in the field of signal and
information processing at the School of Electrical
Engineering, Southwest Jiaotong University,
Chengdu, China. His current research interest is
adaptive signal processing.
Shibin Gaowas born in Hubei Province, China, in
1963. He received the B.S. degree, M. S. degree and
the Ph.D. degree from Southwest Jiaotong
University, Chengdu, China, in 1985, 1988 and2004, respectively. And he is a Professor, the
Director of the Key Lab of Maglev Technology andMaglev Train of Ministry of Education, and the
President with the school of Electrical Engineering
in Southwest Jiaotong University. His research
interests are in the area of signal process and
information theory and its application in electrical
power system, power system protection andsubstation automation, and system identification. At present, he is the author or
coauthor of more than 60 journal papers.
Xiangping Zengwas born in Sichuan Province,China, in 1974. She received the B.E. degrees in
School of Electrical Engineering at Southwest
Jiaotong University in 1998, the M.S. degrees inCollege of Information Engineering at Graduate
School of Chinese Academy of Sciences in 2006,
and the Ph.D degree in School of InformationScience & Technology at Southwest Jiaotong
University in 2013. Her current research interests
are video image processing and intelligent
information processing.
Zhengyou He (M!10-SM!13) was born in Sichuan
Province, China, in 1970. He received the B.S.degree in 1992, the M. S. degree in 1995 fromChongqing University, Chongqing, China, and
received the Ph.D. degree in 2001 from Southwest
Jiaotong University, Chengdu, China.Since February 2002, he has been a Professor at
the department of Electrical Engineering in
Southwest Jiaotong University. His research
interests are in the area of signal process and
information theory and its application in electrical
power system, and application of wavelet transforms in power system. Atpresent, he is the author or coauthor of more than 100 journal papers and the
owner of two invention patents.
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