Electrocardiogram ECG signal is the electrical recording of heart activity. The Electrocardiogram ECG reflects the activities and the attributes of the human heart and reveals very important hidden information. The information is extracted by means o
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In the design of water treatment plant a secondary treatment (called Biological Treatment) has been given to the raw water next to primary water treatment.The treatment units installed to remove th...Full description
PSPICE FILTERS
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By: Douglas L. Jones
Adaptive Filters
By: Douglas L. Jones
Online:
CONNEXIONS Rice University, Houston, Texas
L2
L2 xk
dk
⇔ E [yk ] = E [yk+d ] ryz (l) = E [yk zk+l ] ∀k, l : ( ryy (0) < ∞)
E 2 dk
xk
M −1
yk =
(wl xk l ) −
l=0
2
argminE [2 ] = E (dk − yk ) wl
2
M −1 l=0
M −1 l=0
(wl E [dk xk l ]) + −
= E M −1 m=0
E 2 = rdd (0) − 2
2
(wl xk l ) −
((wl wmE [xk l xk
M −1
M −1 l=0
dk −
(wl rdx (l)) +
l=0
−
m
−
M −1
M −1
l=0
m=0
= E dk 2 −
]))
(wl wm rxx (l − m))
rdd (0) = E dk 2
rdx (l) = E [dk X k l ] −
rxx (l − m) = E [xk xk+l
m]
−
E 2 = rdd (0) − 2PWT + WT RW
P
R=
=
rdx (0) rdx (1)
rdx (M − 1)
rxx (0)
rxx (1)
rxx (1)
rxx (0)
rxx (M − 1)
...
.
∇=
...
rxx (M − 1)
rxx (0)
rxx (1)
rxx (1)
rxx (0)
...
...
W
∂ ∂w 0 ∂ ∂w 1
∂ ∂w M −1
2 2
2
∇ = − (2P) + 2 RW d dW
R
T
W = AT
A
d dW
(WM W) = 2M W
W opt R = P ⇒ W opt = R
⇒
M 1
−
P
R
W opt = R
1
−
P
xk
dk
R
P
xk dk
k ˆ(l ) rxx
N →
→
ˆ opt W
x d N
N > M
O M 2
R M O (M ) Rk+1
Rk
O (M )
O (M ) O M 2
N
E 2 = rdd (0) − 2PT W + WT RW R
E 2 (w0 , w1 , . . . , wM
1)
−
N
R
R P
L
1
−
P
ε
E ∇ = ∂ ∂w i
2
E
M
R
W
i
∇ W
µ
i+1
= Wi − µ∇i
µ
µ µ W 0
i = 1, ∞
∇i = − (2P ) + 2 RW i W i+1 = W i − µ∇i
W opt = W
∞
O M 2
O (M )
N −1
1 rxxˆ(l) ≈
N
rxdˆ(l) ≈
(xk xk+l )
k=0 N −1
1 N
ˆ opt = Rˆ 1 = P ˆ W
(dk xk l ) −
k=0
−
k
rxxˆ(l)
rdxˆ(l)
ˆk W opt k ≈ R
1
−
k
=
=
1 N
1 N
N −1
(xk
m xk −m−l )
−
m=0 N −1
(xk
m−l dk−m )
−
m=0
ˆk P
k k 1 rxx (l) = rxx (l) + xk xk −
l
−
− xk
N xk−N −l
−
k 1 (1 − α) rxx k (l) = αrxx (l) + xk xk −
l
−
{xk } {dk } W
E k 2
M −1
k = dk − yk = dk −
T
(wi xk i ) = dk − X k W k −
i=0
X k =
xk xk
xk
1
−
M +1
−
k
W =
w0k w1k
k wM
1
−
=0
∇k
∂ ∂W
E k 2
=
= E 2k −X k
T
= E −2 dk − X k W k X k = − 2E dk X k
+ E X k
T
W
= −2P + 2RW
⇒ W opt = R W opt
1
−
P
W k+1 = W k − µ∇k R
P
P ⇒
R
∇k = k 2 k ∂ ∇ˆk =
∂W
T
E k 2
2
k 2 = 2 k
∂ E k 2 ∂W
∂ T dk − W k X k ∂W
M −1
= 2 k −X k = − 2k X k
yk = W k X k = i=0 wik xk i k = dk − yk W k+1 = W k − µ∇ˆk = W k − µ −2k X k = W k + 2µk X k wik+1 = wik + 2µk xk
V k+1 = V k − 2µRV k + (− (2µRW opt )) + 2 µP W opt = R
1
−
V k+1 = V k − 2µRV k + − 2µRR V k
R
R Q
1
−
ΛQ
→∞
1
−
P
→0
+ 2µP = ( I − 2µR) V k
Λ
λi
R
Q
R V k+1 = I − 2µ Q
1
−
Q
ΛQ
V k
QV k+1 = ( Q − 2µΛQ) V k = (1 − 2µΛ) QV k V = QV V
V k+1 = (1 − 2µΛ) V k m
V
V
R
1 − 2µΛ
V
→0
M
∀i, i = [1 , 2, . . . , M ] : V i k+1 = (1 − 2µλi ) V i k |1 − 2µλi | < 1
∀i : (|µλi | < 1) µ
∀i : µ <
V
1
λi
µ<
1 λmax λmax
M
tr (R) =
M
rii =
i=1
i=1
λi ≥ λmax
λi
∀i, i ∈ {1, M } : (rii = r (0)) r (0)
tr (R) = M r (0) = M E [xk xk ]
O (1)
1
µ<
ˆ M r (0)
µ
(1 − 2µλi )k 1 − 2µλmax 1 − 2µλmin λmin R
2
E k
= E
k T
dk − W
2
k
X
T
T
T
= E dk 2 − 2dk X k W k − W k X k X k W k T
T
= rdd (0) − 2W k P + W k RW k
W opt = R min 2
1
−
P
= E 2 =
rdd (0) − 2P T R
= rdd (0) − P T R
1 P
−
+ P T R
1 RR−1 P
−
1 P
−
V = Q [W − W opt ] k T
2
k T
E [k ] = rdd (0) − 2W P + W
k
k T
RW + − W
T
RW opt
Q
1
−
ΛQ = R
− W opt T RW k + T
W opt T RW opt + W k RW opt + W opt T RW k − W opt T RW opt = rdd (0) + V k RV k − T T T P T R 1 P = min 2 + V k RV k = min 2 + V k Q 1 QRQ 1 QV k = min 2 + V k ΛV k −
−
−
N −1
E k
E vjk
2
2
2
= min +
2
λj E vjk
j =0
V k
E V k V k
W k+1 = W k + 2µk X k V k+1 = W k + 2µk QX k
k V +1
= E V k+1 V k+1
T
T
= E 4µ2 k 2 QX k X k QT =
k
V
k
+ 2µ k QX V
k T
k T
k
+ 2µ k V X
T
Q
2
2
k
k T
+ 4µ E k QX X
T
T
Q
T
k = dk − W k X k = dk − W opt T − V k QX k
k
E k QX V
k T
k
= E dk QX V
k T
T
= 0 + 0 − QX k X k T
= − QE X X = − ΛV k
T
E k 2 QX k X k QT
k
− W opt X QX V k T
k
k
QT V k V
T
k T
k
Q E V V
k T
k T
− V
k T
k
QX V
k T
k 2 T
k = dk − W k X k
X W k ≈ W opt W k
2
k
k T
E k QX X
T
Q
= E k
2
X k X k k T
k
E QX X
Q
T
T
= E k 2 Λ
k 2 = min 2 + V k ΛV k 2
E k
k+1
V
2
= min + E = min 2 +
= ( I − 4µΛ) V k + 4µ2
λj V j k
λj V jjk
λj V jjk Λ + 4µ2 min 2 Λ V ∞
⇒ 4µΛV ⇒
∞
∞
V
= 4 µ2
=µ
2
= V
+1
∞
(λj V jj ) + min 2 Λ
(λj V jj ) + min 2 I
X k
T
∞
V ii
=µ
∞
V ii
V ii
1−µ ∞
V ii
2
∞
=
µmin 2 1 − µ λj
= min
2
=
2
2
∞
∞
∞
µmin 2P (λj ) λj 1−µ
= min 2 +
1 − µN σx 2
= µmin 2
λj
= min 2 1
E
2
= min 2 + E V λV
E
λj + min
∞
µ
−
1P
P
λj
1
1−µtr(R) 1 min 1−µrxx (0)N
= min 2
1 1 − µN σx 2 µ
µ
µ
k+1
V
= ( I − 4µΛ) V k + 4µ2
∀i : (4 µλi < 2) k+1 V ii
µ<
= (1 −
k+1
V ii
λj V jjk Λ + 4µ2 min 2 Λ
k+1
k +1
V ii
4µλi ) V iik
2
+ 4µ λi
λj
= (1 − 4µλi ) V iik + 4µ2 λi
k ≤ (1 − 4µλi ) V iimax + 4µ2 λi k
V jjmax
= 1 − 4µλi V ijk
1 2λmax
1 − 4µλi V ii
k+1
V ij
λj
k
V jjmax
1 − 4µλi + 4µ2 λi 4µ2 λi
k
V jj
+ 4µ2 min 2 λi
λj
k
V jj
= 1 − 4µλi + 4µ2 λi
λj < 1
λj < 4µλi
λj
k
V jjmax
µ<
P1
λj
1
=
tr(R)
=
Nr xx (0)
=
Nσ x 2
1 1
µ µ= xk dk
µ 3Nσ x 2
R
1 P
−
= W ∞
dk = xk ∗ hk =
(xk i hi ) −
i=0
=
P
∞
i=0
E [dk xk
(hi E [xk i xk j ]) −
−
j
−
]
=
E [(
=
∞
i=0
∞
i=0
(xk i hi )) xk j ] −
−
(rxx ( j − i))
r (M − 1) | r (M ) r (M + 1) . . .
rxx (0)
r (1)
r (1)
r (0)
|
...
r (2)
r (1)
|
...
...
...
. . . r (0)
r (1)
|
r (2)
r (3)
...
r (M − 1) r (M − 2) . . . r (1)
r (0)
|
r (1)
r (2)
...
H
h (m) = h (m + 1) = · · · = 0
M
P = Rh
W opt = R
1
−
1
−
P = R
1
−
(Rh) =
h
M
= =
h (0) h (1) h (2)
W H ≈ z δ (n − ∆)
∆
−
W ≈
z −∆ H
W
sk
sk sˆk sk
∆0
−
∆0
−
sˆk
< 75%
dk
nk nk
nk
E k 2 = E (sk + nk − yk ) sk nk
nk
2
2 = E sk 2 + 2E [sk (nk − yk )] + E (nk − yk )
sk
E [sk (nk − yk )] = E [sk ] E [nk − yk ] = 0
nk
nk
E k 2 = E sk 2 + E (nk − yk ) sk
W = Rn n
1
−
P nn
2
E k 2
∆
sk
∆ sk
sk
∆
−
• • •
W opt = R
1
−
P
R
1
P
−
O N 2
O (N )
R R
QRQ
1
−
W opt
Q
R R R O (NlogN )
Q N O (logN )
N
L
yk =
L
vnk yk−n
n=1
W k =
U k =
+
wnk xk
n=0
v1k v2k
k vL
w0k w1k
k wL
yk
−
yk
−
yk
1 2
L
−
xk
xk
xk
n
−
1
−
L
−
k = dk − yk = dk − W k T U k
E k 2 = k 2
ˆk = ∇
∂ ∂ k 2 = 2 k ( k ) = 2 k ∂W k ∂W k
∂ ∂ (yk ) = k ∂v i ∂v ik
L
ˆk = ∇
β 1k
β 2k
k . . . βL
∂ ∂v 1k
∂ ∂ k
= −2k
w1k
L
vnk yk−n
L
wnk xk−n
+
n=1
L
n
−
+
...
L
wnk xk−n
+
k αL
n=1
∂ ∂w ik
( yk )
= xk
−i
+
k−j
vjk αi
L
β ik = yk
i +
−
k−j
vjk β i
j =1
( yk )
∂ k ∂v L
(yk )
∂ k ∂w 0
(yk )
∂ k ∂w L
( yk )
∂ (yk ∂v ik
∂ ∂w ik
n)
−
n)
−
+ xk
+0
n
−
(yk ) β ik =
∂ ∂v ik
yk = W k T U k
j =1
ˆ k = −2k ∇
∂ ∂v 1k
∂ ( yk ∂w ik
αik =
( yk )
L
αki
vnk
=
n=0
∂ ∂v ik T
vnk
n=1
L
n=1
αk0
= yk
n=0
vnk yk−n
(k )
∂ ∂ (yk ) = k ∂w i ∂w ik
k
β 1
k
β 2
...
k
α0
k
α1
...
αkL
T
ˆk W k+1 = W k − U ∇ µ
dk
(yk )
2
k = ( |yk |) − A2
yk = W kH X k ek = dk − yk W k+1 = W k + 2µek X k ∗
