Un algorithme rapide à identi cation polynomiale multivariable pour

x(n) = As(n)
AN×Ns(n)
x(n)N
A
z(n) = R−1
2
xx(n)
Rx
z(n)
Rz=IN
wtz(n)kurt(wtz(n))
wkwk= 1
wi
N
yi(n) = wt
iz(n)wi
W=W(WtW)−1
2
W=[w1,··· ,wN]N
•
•WN N
W=IN
•
1) ∀k= 1 . . . N, wk←E©z(n)(wt
kz(n))3ª−3wk
∝∂kurt(wt
kz(n))
∂wk
2) W←W¡WtW¢−1
2
Colloque GRETSI, 11-14 septembre 2007, Troyes
997

T
∧
N×T
N×N
∧
2T N
T(4N2+N) + 2N2
T
z(n)
y(n) = wtz(n)
z(n)
z(n)
y(n)
kurt(y(n)) = kurt(wtz(n))
=E©(wtz(n))4ª−3E©(wtz(n))2ª2.
E©(wtz(n))2ª=E©wtz(n)zt(n)wª
=wtE©z(n)zt(n)ªw
=wtIw
=kwk2,
kurt(y(n))
=E½PN
i1,..,i4=1 Q4 k=1 w(ik)zik(n)¾−3µPN i=1 w(i)2¶2
=
N
X
i1,..,i4=1
E½4
Y
k=1
zik(n)¾4
Y
k=1
w(ik)−3
N
X
i1,i2=1
w(i1)2w(i2)2.
w(1),· · · , w(N)
(αd)d∈D
D=nd∈ {0,· · · ,4}N\PN
k=1 d(k) = 4o
kurt (y(n)) = X
d∈D
αd
N
Y
k=1
wd(k)
k
R D d1,· · · ,dR
α1,· · · , αR(αd)d∈D
kurt(y(n)) =
R
X
r=1
αr
N
Y
k=1
wdr(k)
k.
αr
z1(n),· · · , zN(n)
(vi)i=1..R
R N
∀i= 1 . . . R, kurt(vt
iz(n)) =
R
X
j=1
αj
N
Y
k=1
vi(k)dj(k)
Mα=k
α(αr)r=1..R
M= [mij ]i,j=1..R k= (ki)i=1..R
∀i, j = 1 . . . R, mij =
N
Y
k=1
vi(k)dj(k)
∀i= 1 . . . R, ki=kurt(vt
iz(n)).
RviN
M
Rvt
iz(n)
R(αr)r=1..R
α=M−1k
z(n)E©(vt
iz(n))2ª=kvik2
vt
iz(n)kurt(vt iz(n)) =
E{(vt
iz(n))4} − 3kvik4
R
vt
iz(n) = vt iR−1/2 xx(n) = ut ix(n)
ut
i=vt iR−1/2
x
z(n)V U
(vi)i=1..R (ui)i=1..R
M
N
(vi)i=1..R
M
(αr)r=1..R
kurt(wtz(n)) = PR
r=1 αrQN k=1 wdr(k)
kw
kwk= 1
card(D) = R=N+3N(N−1)
2+
N(N−1)(N−2)
2+N(N−1)(N−2)(N−3) 24 .
998

kurt(wtz(n))
w
∂kurt(wtz(n))
∂w=
R
X
r=1
αr
∂QN
k=1 w(k)dr(k)
∂w
∂QN
k=1 w(k)dr(k)
∂w(j)=
dr(j)w(j)dr(j)−1Y
k6=j
w(k)dr(k),si dr(j)≥1
0,si dr(j) = 0
•Rx
x(n)
•U=R−1/2
xV V R
vi
•(ki)i=1..R Rut
ix(n),
i= 1..R
α= (αr)r=1..R α=M−1k
M
•W= [wij ]i,j=1..N Nwi
N
∀i, j = 1 . . . N,
wji ←PR
r=1 αrdr(j)wmax(dr(j)−1,0)
ji Qk6=jwdr(k)
ki
∀i, wi←∂kurt(wt
iz(n))
∂wi
W←W(WtW)−1
2
αrT
T
N≤7
∧
2T N
R+
RN +R2+N2
T
W= [wij ]i,j=1..N
w= (wj)j=1..N
103104105106
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103104105106
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103104105106
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1
N=2
N=3
N=4
t’/t
T
T N
10−4
t0/t T
N t0t
t0/t
N7t0/t
T
1/15 T
Colloque GRETSI, 11-14 septembre 2007, Troyes
999

103104105106
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103104105106
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1
N=5
N=6
N=7
T
t’/t
T N
E1={w∈ {0,1}N\ ∃i1t.q.
w(i1) = 1,∀i6=i1w(i) = 0}
E2={w∈ {0,1}N\ ∃i1, i2t.q.
w(i1) = w(i2) = 1,∀i6=i1, i2w(i) = 0}
E3={w∈ {0,1,2}N\ ∃i1, i2t.q.
w(i1) = 2w(i2) = 2,∀i6=i1, i2w(i) = 0}
E4={w∈ {0,1,2}N\ ∃i1, i2, i3t.q.
w(i1) = 2w(i2) = 2w(i3) = 2,∀i6=i1, i2, i3w(i) = 0}
E5={w∈ {0,1}N\ ∃i1,· · · , i4t.q.
w(i1) = . . . =w(i4) = 1,∀i6=i1,· · · , i4w(i) = 0}
E1, . . . , E5N ,
N(N−1)
2,N(N−1) ,N(N−1)(N−2) 2,N(N−1)(N−2)(N−3) 24
E=∪5
i=1Eicard(E) = P5
i=1 card(Ei) = N+
3N(N−1)
2+N(N−1)(N−2) 2+N(N−1)(N−2)(N−3) 24 =card(D)
M
N≤7
M
M
N= 2,· · · ,7 182,414,844,1605,2758,4344
R
5,15,35,70,126,210
210
0 1
50000
(αr)r=1..R
1000