2014 27 27 1

: diffusion LMS

combination periodic

,

oscillation data-saved periodic

combination .

.

: Distributed estimation, diffusion LMS,

periodic combination

.

distributed estimation

. N

, k i scalar desired

response ( )kd i 1 M regressor vector ,k iu . k

k kN .

( )kd i ,k iu

ow .

Diffusion least mean square (LMS) [1]-[2] .

Diffusion LMS adaptation combination

. Combination

,

.

power resource

.

[3]-[5].

periodic combination

oscillation

.

.

. Diffusion LMS

k i

desired response ( )kd i regressor vector ,k iu

.

,( ) ( )ok k i kd i u w v i . (1)

( )kv i zero-mean measurement noise

2,v k

. ( )kv i ,l ju , , ,k l i j independent .

Diffusion LMS adaptation

combination .

adaptation ( )kd i ,k iu .

combination adaptation

,

. combination

.

adapt-then-combine (ATC)

diffusion LMS adaptation combination

. *

, , 1 ,

, , ,

( ) (Adaptation)

(Combination)k

k i k i k k i k

k i l k l il

w u e i

w al N

(2)

, , 1( ) ( )k k k i k ie i d i u w . (3)

,l ka node k l combination

weight nonnegative

.

, ,1

0 if , and 1N

l k k l kl

a l aN . (4)

, (2) combination

.

diffusion LMS combination adaptation

stand-alone LMS

[2].

- 1 -

2014 27 27 1

0 200 400 600 800 1000-35

-30

-25

-20

-15

-10

-5

0

Number of iteration

(a) Conventional diffusion LMS (p=1)(b) Diffusion LMS with periodic combination (p=3)(c) Diffusion LMS with periodic combination (p=8)

(a) (b)

(c)

1. p Diffusion LMS with

periodic combination Network

MSD

. Diffusion LMS with periodic combination

(2) combination

,

.

combination

oscillation

. adaptation

combination p

diffusion LMS

.*

, , 1 ,

, ,

,

,

( )

for mod( , ) 0

otherwisek

k i k i k k i k

l k l il

k i

k i

w u e i

a i pw l N

(5)

p combination

diffusion LMS ( 1p )

1/ p .

.

1

. 1 y Network mean square

deviation (MSD) , .

2

,1

1Network MSD( ) = EN

ok i

ki w w

N(6)

steady state MSD

periodic

combination .

adaptation combination

transient state steady state

. transient state

,k iw ow

adaptation ( )kd i ,k iu

2. Periodic combination

data-saved periodic combination

( 3p )

. Combination

. , steady state

,k iw .

adaptation steady state

combination

steady-state error .

( stand-alone LMS diffusion LMS

steady-state error .)

periodic combination

, combination

adaptation ,k iw

. Network MSD

, p combination

network MSD .

steady state

.

network MSD

data-saved periodic

combination .

1. Data-saved periodic combination

diffusion LMS iadaptation combiantion .

3p (5) periodic combination

2 adaptation

2

network MSD .

( )kd i ,k iu combination

padaptation . 2

. 1i 2i adaptation , ( )kd i

,k iu combination

3i adaptation 3

combination .

- 2 -

2014 27 27 1

1i 2i

combination

3 network MSD

.

adaptation

.

,

.

. Performance analysis

periodic combination

diffusion LMS

. variance relation

steady-state Network MSD .

p ,ikw ,k i pw

.

k

(7)

weight error vector ,i ,o

k k iw w w ,

weight error vector global

weight error vector .

1, 1,

i i

, ,

,i i

N i N i

ww

w

(8)

global vector (7) matrix

equation . 1

0( , ) ( , ) g

pT T

i p i i p mm

w i p i w i p i p mA A M

(8)

(8) matrix .

1

* *1, 1 ,

1, ,

*, , ,

diag , ,

col ( ), , ( )

diag , ,

N

i i N i N

i i N i

k i k i k i

I I

g u v i u v i

R R R

R u uA I

M

A

(9)

A matrix combination weight,l ka ( , )l k

element , kronecker product

, diag{} diagonal element

diagonal matrix col{}

column vector .

( , )i j characteristic

function .

1( 1, ) ( ), ( , )( , ) ( , ) ( , ) for 0

ii i I R i i Ii j i k k j i k j

M (10)

(8) hermitian

matrix weighted norm

variance relation . 2 2

, ,

1

0

1

0

E E

,E

,

T Ti p ii p i i p i

TpT

i p mm

pT

i p mm

w w

i p i p m g

i p i p m g

A A

A M

A M

(11)

(11)

. *1

0

,Tr E

,

Tpi p m i p m

Tm

i p i p m g g

i p i p m

A M

M A

(12)

expectation expectation

approximation . 2 2

,, ,

1

0

Tr

T Tk i p ii p i i p i

pm mT

m

E w E w

I R g I R

A A

A M M M M A

(13)

(13) *{ }i ig E g g { }iR E R . (13)

vectorization operation

. 2 2

1

0

E E

vec

i p i

p Tm mT

m

w w

I R g I R

F

A M M M M A

(14)

F .

E , ,

E , ,

T T

T T

i p i i p i

i p i i p i

I p I R p R I

F A A

A A

M M A A

(15)

(15) approximation small step-size

[2]. Steady state ,k iw

, (14)

. 12

(I ) 0

E vecp Tm mT

mw I R g I R

FA M M M M A (16)

1( )I F Steady state

Network MSD .

12

(I ) 01

1 1E [vec

( ) vec( )]

p Tm mT

mw I R g I R

N NI I

FA M M M M A

F

(17)

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2014 27 27 1

3.

0 200 400 600 800 1000-35

-30

-25

-20

-15

-10

-5

0

Number of iterations

(a) Conventional diffusion LMS(b) [3](c) [4](d) [5](e) Proposed

(a)

(c)

(d)

(e)

(b)

4.

. 3 30

.

Step-size 0.05 .

4 periodic combination

.

diffusion LMS 25%

. ,

steady-state network MSD .

5

steady-state network MSD (17)

.

. diffusion LMS

combination periodic

.

.

0 200 400 600 800 1000-35

-30

-25

-20

-15

-10

-5

0

Number of iterations

(a) Practical (p=2)(b) Theoretical (17) (p=2)(c) Practical (p=5)(d) Theoretical (17) (p=5)

(d)

(b)

(c)(a)

5. Theoretical network MSD (17)

Acknowledgement

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 and in part by the National Research

Foundation of Korea (NRF) grant funded by the

Korea government (MEST)

(2012R1A2A2A01011112)

[1] C. G. Lopes and A. H. Sayed, “ Diffusion least

-mean squares over adaptive networks: Formu

lation and performance analysis,”

, vol. 56, no. 7, pp.

3122– 3136, July. 2008.

[2] F. Cattivelli and A. H. Sayed, “ Diffusion LMS

strategies for distributed estimation,”

, vol. 58, no. 3,

pp. 1035-1048, Mar. 2010

[3] C. G. Lopes and A. H. Sayed, “ Diffusion

adaptive networks with changing topologies, ”

las Vegas, USA,

Apr. 2008, pp. 3285– 3288.

[4] X. Zhao and A. H. Sayed , “ Single-link diffusion

strategies over adaptive networks, ”

, Kyoto, Japan, Mar. 2012,

pp. 3749– 3752.

[5] Qyvind Lunde Rortveit, John Hakon Husoy, and A.

H. Sayed , “ Diffusion LMS with Communication

Constraints, ”

, Pacific Grove,

USA, Nov. 2010, pp. 1645– 1649.

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