† ‡ ∗ †

† ‡ ∗

{ }

nN

k kk Nk nk nk {dk(n),uk(n)}

wo(n) dk(n)uk(n) M

dk(n) uk(n)dk(n) = uT

k (n)wo(n−1)+vk(n)vk(n)

u�(n) � = 1, 2, · · ·, k, · · ·, N σ2vk

Nk k

N1

2

�

3

{dk(n),uk(n)}

{d1(n),u1(n)}

{d2(n),u2(n)}

{d�(n),u�(n)}

{dN (n),uN (n)}

{d3(n),u3(n)}

N n k{dk(n),uk(n)} k

Nk = {1, 2, �, k} nk = 4

k ψk(n)wk(n−1)

Nk

wk(n)

ψk(n)=wk(n−1)+μk(n)uk(n)[dk(n)−u

T

k (n)wk(n−1)]

wk(n)=∑

�∈Nk

c�k(n)ψ�(n),

μk(n) �μk

δ + ‖uk(n)‖2,

μk δ‖ · ‖ ψk

wk Mc�k(n) � ∈ Nk

k k

978-1-4673-1026-0/12/$31.00 ©2012 IEEE

•

k ψk(n)ψk(n− 1)

• ψk(n)

ψk(n) kwk(n)

Nk k kbk nk = nk−1

Nk b(m)k , m = 1, . . . , nk

mth bk

mth k kNk = {1, 2, �} bk = [ 1 2 � ]T b

(3)k = �

ck(n) ck(n)nk nk

k ckk(n)k

ck(n) = [c1k(n) c2k(n) c�k(n) ckk(n)]T

ck(n)=[c1k(n) c2k(n) c�k(n)]T

ck(n) 1T ck(n) = 1T ck(n) +ckk(n) = 1 1

wk(n) =wk(n) −wo(n)

k(n) = ‖wk(n)‖2

(n) =1

N

N∑k=1

k(n),

n →∞

ck(n) = arg min E‖wk(n)−wo‖2

1Tck(n) = 1

k = 1, 2, . . . N E

wo

wo

• {ψk(n), n ≥ 0}Ψk EΨk = wo

k ∈ {1, 2, . . . , N}

• �, m ∈ Nk E [ψ�(n)−wo]T [ψm(n)−wo]

≈ [ψ�(n)− ψ�(n− 1)]T [ψm(n)− ψm(n− 1)] .

k wo

ψk(n) = ψk(n− 1) + μk(n)uk(n)ek(n),

k = 1, . . . , N μk(n)ek(n)

ek(n) = dk(n)− uT

k (n)ψk(n− 1) � dk(n)− yk(n).

ψk(n)n− 1

wk(n) = ckk(n)ψk(n) +

nk∑m=1

c(m)k (n)w

b(m)k

(n−1).

wk(n)

ck(n) k = 1, 2, . . . , Nck(n)

ckk(n) = 1−

nk∑m=1

c(m)k (n).

ck(n)

Jk(n) =n∑

i=1

β(n, i)e2k(n, i),

β(n, i)

ek(n, i) = dk(i)− yk(n, i),

yk(n, i) =

nk∑m=1

c(m)k (n)y

k,b(m)k

(i)+

[1−

nk∑m=1

c(m)k (n)

]yk(i)

= yk(i) +

nk∑m=1

c(m)k (n)

[y

k,b(m)k

(i)− yk(i)]

k iNk

nyk,p(n) = u

T

k (n)wp(n− 1), p ∈ Nk.

ek(n, i) = ek(i) +

nk∑m=1

c(m)k (n)

[yk(i)− y

k,b(m)k

(i)].

c(�)k (n) � = 1, 2, . . . , nk

∂Jk(n)

∂c(�)k (n)

= 2

n∑i=1

β(n, i)ek(n, i)[yk(i)− y

k,b(�)k

(i)].

n∑i=1

nk∑m=1

β(n, i)c(m)k (n)y

k,b(m)k

(i)yk,b

(�)k

(i)

=n∑

i=1

β(n, i)ek(i)yk,b

(�)k

(i),

yk,p(n) � yk,p(n)− yk(n) p ∈ Nk

k nk

Pk(n)ck(n) = zk(n),

Pk(n) nk

[Pk(n)]m,�

=n∑

i=1

β(n, i)yk,b

(m)k

(i)yk,b

(�)k

(i),

m, � = 1, 2, . . . , nk zk(n)nk �th

z(�)k (n) =

n∑i=1

β(n, i)ek(i)yk,b

(�)k

(i),

� = 1, 2, . . . , nk

ck(n) = P−1k (n)zk(n).

Pk(n)

yk(n) = [yk,b

(1)k

(n) yk,b

(2)k

(n) · · · yk,b

(nk)

k

(n)]T

zk(n) yk(n)ek(n)

β(n, i)

β(n, i) =

{1, n−i < L0, n−i > L,

L

β(n, i) P(n)z(n)

P−1k (n)

β(n, i)

L = 500

wk(−1) = ψk(−1) = 0 k = 1, 2, . . . , N.

n = 1, 2, . . .

k = 1, 2, . . . , N

yk(n) = uT

k (n)ψk(n− 1)ek(n) = dk(n)− yk(n)μk(n) = μk/(δ + ‖uk(n)‖2)ψk(n) = ψk(n− 1) + μk(n)uk(n)ek(n)

k = 1, 2, . . . , N

�, m = 1, 2, · · · , nk

yk,b

(m)k

(n) = uT

k (n)wb(m)k

(n− 1)

yk,�(n) = yk,�(n)− yk(n)

[Pk(n)]m� =

n∑i=1

β(n, i)yk,b

(m)k

(i)yk,b

(�)k

(i)

z(�)k (n) =

n∑i=1

β(n, i)ek(i)yk,b

(�)k

(i)

ck(n) = P−1k (n)zk(n)

k = 1, 2, . . . , N

wk(n) = [1− 1T ck(n)]ψk(n)

+

nk∑m=1

c(m)k (n)w

b(m)k

(n− 1)

wo

M = 32−1 1

n = 5000

uk(n), k = 1, . . . , 4,

I vk(n)uk(n)

μ1 = μ2 = μ4 = 1μ3 = 0.1

μk = 1 μk = 0.1

4(n) = E‖wo − w4(n)‖2

wo(n)

{wo(n) = wo + θ(n)θ(n) = γθ(n− 1) + q(n),

0 < γ < 1 q(n)Q

wo(n)γ = 0.99 Q = σ2

qI σq (Q) =0.01 Tr(·)

−4 −6

−8

μ3 = 0.1

E‖w

o−ψ

k(n

)‖2

(dB

)

10

0

0

−10

−20

−30

−40

2000 4000 6000 8000 10000

E‖w

o−

w4(n

)‖2

(dB

)

10

0

0

−10

−20

−30

−40

2000 4000 6000 8000 10000

0

0 2000 4000 6000 8000 10000

0.20.40.60.8

0.20.4

1.01.2 c14 c24 c34 c44

00 2000 4000 6000 8000 10000

0.2

0.4

0.6

0.8

1.0

1.2

c14 c24 c34 c44

M = 50

μk = 0.1μk = 1

wk(n) = ckk(n)ψk(n) +

nk∑m=1

c(m)k (n)ψ

b(m)k

(n).

yk,p(n)

0

0 500 1000 1500 2000 2500 3000

−8

−6

−4

−2

2

4

6

E‖w

o(n

)−w

4(n

)‖2

(dB

)

μk = 1 μk = 0.1

yk,p(n)Pk(n)

wo(n) (Q) = 10−1

(Q) = 10−3 (Q) = 10−5

μk = 0.1

−50

−40

−30

−20

−10

0

20

10

500 1000 1500 2000 2500 3000 3500 4000 4500 5000

(Q)

500 1000 1500 2000 2500 3000 3500 40005

6

7

8

9

11

12

13

10

500 1000 1500 2000 2500 3000 3500 4000−20

−15

−5

−10

0

5

15

10

500 1000 1500 2000 2500 3000 3500 4000−40

−30

−20

−10

0

20

10

(Q) = 10−1 (Q) = 10−3 (Q) = 10−5