iaf-mpapa para cancelamento de eco acústico

8/12/2019 IAF-MPAPA Para Cancelamento de Eco Acstico

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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: [email protected],

[email protected]) .*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:[email protected]:[email protected])mailto:[email protected])mailto:[email protected]






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






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)






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.






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.






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.






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

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.

iaf-mpapa para cancelamento de eco acústico

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