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2014 27 27 1
:
diffusion least mean
square (LMS)
.
adaptive gain adaptation stage
diffusion least mean square
. step
size weight error
mean-square gain
,
.
: Adaptive network, adaptive estimation, sparse vector, proportionate-type LMS.
.
. i k
( )kd i 1 M ,k iu
. ,( ) ( )ok k i kd i v iu w
ow
1L , ( )kv i
0 2,v k k
.
.
Adapt-then-Combine (ATC) diffusion LMS [1]
adaptation
update
adaptation
combination update .
local ,
cooperation
.
.
LMS
step size gain
proportionate-type NLMS [2]-[4] ,
sparsity constraint sparse LMS [5],
[6] . proportionate-
type NLMS step size
sparse LMS update
term steady state error
.
sparsity constraint diffusion LMS
[7], [8]
, proportionate gain
. proportionate-type
NLMS [4] weight error
mean square gain
z2-proportionate NLMS ATC diffusion LMS
adaptation stage .
.
II.
III. implementation
. IV.
V.
.
. 2z -proportionate ATC Diffusion LMS
ATC diffusion LMS
adaptation stage proportionate-gain
. ATC
diffusion LMS update adaptation
stage proportionate gain matrix ,k iG
proportionate ATC diffusion LMS update equation
.
*, 1 , , , , , ,
1
, 1 , , 1
( ( ) )
k
N
k i k i k k i l k l i l l i k il
k i l k k il N
c d i
a
w G u u w
w
y
y (1)
,k iG L L gain
k node k step size, ,l kc ,l ka
adaptation combination coefficient .
, (1) ,k iG L L
ATC diffusion LMS .
ATC diffusion LMS
.
Adaptation stage
2z
- 1 -
2014 27 27 1
, 1k iy update combination stage
combination
update
. [4] z2-proportionate gain
, ,1 ,2 ,diag{ ( ), ( ),..., ( )}k i k k k Lg i g i g iG s
, ( )k sg i s
mean-square error .
2
,, 2
,1
E ( )( ) 1 E ( )
k sk s L
k mm
z ig i
z iL (2)
, ( )k sz i , ,o
k i k iz w w weight
error s .
gain allocation
tap gain optimum weight
proportionate-type NLMS
diffusion LMS .
. Practical Implementation
update
gain weight error
mean square practical
implementation
.
1. Weight error
gain matrix
weight error mean-square .
.
, ,
, ,
,1
( ) ( )( )
( 1) ( ) ( )
k k k i k i
k i k i k
Lk k ll k
e i d iv i
u i l z i v i
u wu z
(3)
weight error
, weight
error .
, , , , ,1( ) ( ) ( ) ( ) ( ) ( ) ( )L
k s k k s k l k l k s klu i e i u i u i z i u i v i (4)
2, , ,E ( ) ( ) E ( )k s k u k k su i e i z i (5)
,
, 2,
E ( ) ( )E ( ) k s k
k su k
u i e iz i (6)
(6)
time averaging method . 2
, , , ,( ) ( 1) (1 ) ( ) ( ) /k s k s k s k u kp i p i u i e i (7)
, ( )k sp i ,E ( )k sz i .
weight error mean-square
weight error mean
. 22
, ,E ( ) E ( )k s k sz i z i (8)
adaptive filter
[4], [9].
2. Adaptive convex gain combination
time averaging method
weight error mean (2), (8) gain
. weight error
steady state
[4]. (2) , state
weight error time
averaging fluctuation
, steady state
fluctuation gain .
adaptive convex gain combination
. (2) gain L L LI convex combination
.
, ,(1 )k i k k i k LIG G (9)
mixing parameter k
.
2,
2 2 2, , ,1
min 1,( ( ))
v kk L
u k k s v ksp i
(10)
1 user parameter .
mixing parameter ,
weight error
, k 0
, steady state
k 1 .
ATC diffusion LMS
.
.
.
50 . 1.
20
,
. ,k iu 0 white Gaussian
2, ,u k u k MR I .
0 white Gaussian
2,v k .
adaptation
(C I ), combination uniform rule [10]
adaptation weight ,l kc combination
weight ,l ka . 100
.
ATC diffusion
LMS . ow10 1 0
. step size
0.1 . 2.
, ATC diffusion LMS
. gain
ATC diffusion LMS steady state error
.
- 2 -
2014 27 27 1
1. Network topology( ), 2,u k ( , ),
2,v k ( , )
[7] sparse diffusion LMS:
ZA diffusion LMS, RZA diffusion LMS
. 2. ow . ATC
diffusion LMS step size 0.035 , ZA
diffusion LMS 0.065, RZA diffusion LMS
0.1 . Sparse diffusion LMS
regularization function weight ZA
diffusion LMS 310 , RZA diffusion LMS
30.25 10 . parameter
steady state error
. 3.
.
ow 1w ,
2w
. 1w
, 0 50 1 ,
2w 50 1
. 1 .
1w
, 2w dispersive
. 4. , RZA diffusion LMS
2, 3
,
ATC diffusion LMS .
2. ATC diffusion
LMS MSD
3. ZA, RZA ATC
diffusion LMS, ATC diffusion LMS
MSD
4. ATC diffusion
LMS, RZA ATC diffusion LMS MSD
( 1w 2w )
- 3 -
2014 27 27 1
. step size
weight error mean square gain
proportionate diffusion LMS .
implementation weight error
adaptive convex gain combination
.
,
.
This research was supported in part by the MSIP
(Ministry of Science, ICT&Future Planning), Korea,
under the CITRC (Convergence Information
Technology Research Center) support program
(NIPA-2014-H0401-14-1001) supervised by the
NIPA (National IT Industry Promotion Agency), and
in part by the National Research Foundation of Korea
(NRF) grant funded by the Korea government
(MEST) (2012R1A2A2A01011112).
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strategies for distributed estimation,"
vol. 58, pp. 1035-1048, Mar.
2010.
[2] D. Duttweiler, "Proportionate normalized least-
mean-squares adaptation in echo cancelers,"
., vol. 8, no. 5,
pp. 508-518, Sep. 2000.
[3] J. Benesty and S. Gay, "An improved PNLMS
algorithm," in
, Orlando, FL,
USA, 2002, pp. 1881-1884.
[4] K. Wagner and M. Doroslovacki, "Proportionate-
type normalized least mean square algorithms
with gain allocation motivated by mean-square-
error minimization for white input,"
., vol. 59, no. 5, pp. 2410-2415,
May. 2011.
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., Taipei, Taiwan,
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., vol. 60, no. 8, pp. 4480-4485,
Aug. 2012.
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., vol. 11, no. 2, pp.
132-135, Feb. 2004.
[10] V. D. Blondel, J. M. Hendrickx, A. Olshevsky, and
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