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,"

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2010.

[2] D. Duttweiler, "Proportionate normalized least-

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[3] J. Benesty and S. Gay, "An improved PNLMS

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, Orlando, FL,

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type normalized least mean square algorithms

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May. 2011.

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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

J. N. Tsitsiklis, "Convergence in multiagent

coordination, consensus, and flocking," in

Seville, Spain, Dec.

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