filtr adaptacyjny do rekonstrukcji sygnałów stereofonicznych

filtr adaptacyjny do rekonstrukcji sygnałów stereofonicznych

 ! ! " ##$#% &'()*% (+ ,!-.!.!-
2003
Poznañskie Warsztaty Telekomunikacyjne
Poznañ 11-12 grudnia 2003
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; 8 (
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--
0 0 .(
/
A -
5 -
5 0 ! " ! ! "
# C [N × N ] k ! k ∈< 1, N >
$ C<k| % C|k> C<k|k> = (C<k| )|k> =
(C|k> )<k| [N − 1 × N ]% [N ×
N − 1] [N − 1 × N − 1] C k % k k k &
" & $' % & " (y(t))
(s(t)) *
(z(t))
% T
% + y(t) = yL (t) yR (t) % s(t) =
T
T
sL (t) sR (t) % z(t)= zL (t) zR (t) % L R ,&$' & (y(t))
(s(t)) & ' +
y(t) = s(t)
+ z(t)
sL (t)
zL (t)
yL (t)
=
+
,
yR (t)
sR (t)
zR (t)
-./
zL (t) ∼ N (0, σz2 ), zR (t) ∼ N (0, σz2 ) 0
L
R
R(t) z(t)
R(t) =
σz2
σzLR
L
E[z(t)z(t)T ] =
. 0 σzRL σz2
R
(s(t)) & &$'+
s(t)
= s(t) + n(t) p
nL (t)
sL (t)
aLi sL (t − i)
i=1
=
+
,
p
sR (t)
nR (t)
i=1 aRi sR (t − i)
-1/
aL1 , . . . , aLp % aR1 , . . . , aRp 2 p% n(t) =
T
nL (t) nR (t)
!
* -nL (t) ∼ N (0, σn2 L )% nR (t) ∼ N (0, σn2 R )/
0 0
T
& z(t) -E[n(t)z(t) ] =
= 0/
0 0
0 Q(t)
n(t) σn2
σnLR
L
Q(t) = E[n(t)n(t)T ] =
.
σnRL σn2
R
- L R /
⎡
⎡
⎤
⎤
sL (t)
sR (t)
⎢ sL (t−1) ⎥
⎢ sR (t−1) ⎥
⎢
⎢
⎥
⎥
ϕL (t) = ⎢
⎥, ϕR (t) = ⎢
⎥,
⎣
⎣
⎦
⎦
sL (t−q+1)
⎡
aL 1
⎢ aL 2
⎢
⎢ ⎢ ⎢
θL = ⎢
⎢ aL p
⎢ 0
⎢
⎢ ⎣ 0
⎡
aR 1
⎢ aR 2
⎢
⎢ ⎢ ⎢
θR = ⎢
⎢ aR p
⎢ 0
⎢
⎢ ⎣ 0
⎤
sR (t−q+1)
q×1
⎤
⎡
⎥
⎥
⎥
⎥
⎥
⎥,
⎥
⎥
⎥
⎥
⎦
q×1
θLT
⎢ 1 0 ... ...
⎢
⎢
AL = ⎢ 0 1 0 . . .
⎢ ⎣ 0 0 ... 1
0
0
0
⎥
⎥
⎥
⎥,
⎥
⎦
q×q
q×1
⎤
⎥
⎥
⎥
⎥
⎥
⎥,
⎥
⎥
⎥
⎥
⎦
⎡
⎢
⎢
BL = ⎢
⎣
θRT
⎢ 1 0 ... ...
⎢
⎢
AR = ⎢ 0 1 0 . . .
⎢ ⎣ 0 0 ... 1
q×1
1
0
0
⎤
⎡
0
0
0
⎤
⎡
⎢
⎥
⎢
⎥
⎥, BR = ⎢
⎣
⎦
q×2
0
0
0
1
0
0
0
0
⎥
⎥
⎥
⎥,
⎥
⎦
q×q
⎤
⎥
⎥
⎥,
⎦
0
q×2
&$ -./ -1/ & ' +
AL 0
ϕL (t)
ϕL (t − 1)
=
+
ϕR (t)
0
AR
ϕR (t − 1)
BL
nL (t)
-3/
,
nR (t)
BR
ϕL (t)
T
z
yL (t)
BL BTR
=
+ L . -4/
zR
yR (t)
ϕR (t)
5 +
ϕL (t)
AL 0
BL
ϕ(t) =
, A=
, B=
,
ϕR (t)
0
AR
BR
2q×1
2q×2q
2q×2
'+
ϕ(t)
y(t)
= A ϕ(t − 1) + B n(t),
= BT ϕ(t) + z(t).
-6/
-7/
% & +
T
ϕ(0)= ϕTL (0) ϕTR (0)
& n(t) z(t) t
+
T
T
E[ϕ(0)] = E ϕTL (0) ϕTR (0) = ϕ0= ϕTL0 ϕTR0 ,
ΣL0 ΣLR0
.
E[(ϕ(0)−ϕ0 )(ϕ(0)−ϕ0 )T ] = Σ0 =
ΣRL0 ΣR0
8 !
ϕ(t) -
ϕ(t)
s(t) s(t) . . . , y(1)] =
Y(t)
= [y(t),
yL (t)
yL (1)
YL (t)
,...,
=
yR (t)
yR (1)
YR (t)
4
5 "5 1"
" -5 , 1, - - | 0) = ϕ(0) = [y(0), y(−1), . . . ,y(−p +1)]T
ϕ(0
| 0) = 0 6
3 Σ(0
| 0) , Σ(0
3
T
! " " ""
# $ %
| t) = ϕ(t
| t − 1) + L(t)(t),
ϕ(t
&
| t−1) = Aϕ(t
− 1 | t − 1),
ϕ(t
'
T
| t − 1),
(t) = y(t) − B ϕ(t
(
T ¹½
L(t) = Σ(t | t−1)B[B Σ(t | t−1)B+ R(t)] , )*
T,
| t−1) = AΣ(t−1
Σ(t
| t−1)AT + BQB
| t) = Σ(t
| t − 1) − LBT Σ(t
| t − 1),
Σ(t
))
)+
L(t)
(t)=
,
L (t | t−1)
L (t)
yL (t) T T ϕ
=
− BL BR
R (t | t−1)
ϕ
R (t)
yR (t)
"
-" " .
| t)
ϕ(t
| t − 1)
ϕ(t
/
- "
" %
ϕ(t)%
L (t | t)
ϕ
E[ϕL (t) | Y(t)]
| t) =
ϕ(t
=
,
R (t | t)
ϕ
E[ϕR (t) | Y(t)]
- "- )
(t | t−1)
ϕ
E[ϕL (t) | Y(t−1)]
| t − 1)= L
ϕ(t
=
.
R (t | t−1)
ϕ
E[ϕR (t) | Y(t−1)]
| t)
Σ(t
0
| t−1)
Σ(t
"
"% " 1" "
" ϕ(t)
2- 1"
3
.
%
| t) =
Σ(t
1
σn2
Z(0) = [z(0),z(−1), . . . ,z(−p +1)] ,
-" ϕ(0) sL (0)
s (−1)
ϕ(0)
=
s(t) , L
,...,
sR (0) sR (−1)
T
sL (−p+1)
. ! sR (−p+1)
ϕ(t) " | t), Σ(t | t))
Y(t)% p (ϕ(t) | Y(t)) = N (ϕ(t
83 3
"
ϕ(t) " 5 1" R(t)
Q(t) 9 1"
7 3
1
5 3 -5 3 3
"- 4-, 3
.
%
σn2
R
= σn2
L
, 3
γC
γLR
γRL
- "
κRL
κLR
κC , Q(t)
R(t)
- %
κL κC
1 γC
, R(t)
=
Q(t)
=
.
γC
1
κC κR
/ 3 5 % γ , κ , κ , κ ,
C
L
R
C
- - 3 .
/
.
γC - γC = 0 8
3
13" " 5 3
: "
1 : 6 "
1
γC = 1,
0 " . " 5 .
-
Σ(t | t),
;
" - L
. "- .-
1
Σ(t | t − 1),
σn2
L
"
Σ(t | t) Σ(t | t − 1) "
"
R(t)
" - 4"
Q(t)
. " - 1"
5 1 3 " | t − 1) =
Σ(t
3
.
Q(t)
R(t)%
,
1
1
γLR
Q(t) =
γRL γR
σn2
L
= σnLR /σn2 , γRL = σnRL /σn2
Q(t)
=
γLR
σn2 /σn2 ,
.- 1 .- < "
- .1 .
1- " -
" "5 .
5 5 " , " 1- 5 5
γR =
"5 " 0
" κL κLR
1
R(t) =
κRL κR
σn2
L
"
κ
= σz2 /σn2 , κLR = σzLR /σn2 , κRL =
L
L
L
L
σzRL /σn2 , κR = σz2 /σn2 + / "
R
L
L
L
R(t)
=
L
R
L
,
,
5 "
"
,
"
7 5 -5 4 ! " #
! " ! ! $ %
" !
! ! ! ! &#
γC !# ! # '
3 ÷ 20 ( 0,7 ÷ 0,1 ! ! &
! ! # &
κL " κR " κC ! ! " $ &
κC ! $ ! ! " # " !# # # (
! ! $
κL " κR ) κL *κR + ! $ *+ ! ! ! " , - ! ! !
" " ! $ ! & ! !" κL
*κR + ! " # ! " , ! !
" # ! . κL *κR + ! $ & # κL *κR + !
! ! ! & " ! ! !" , ! ! ,! ! & ! ! !# ! $" κL " κR κC /
• ! $ κL " κR κC ! "
• 0 *+1"
0*+1 ! " κL (t) =
0
0
0" κR (t) = 0" κC (t) = 0 R(t) =
"
0 0
• ! ! " * ! $ ! ! +" " ! # 2
! # ! ! " # !!" ! $ ! $
3 κL (t) 4 κR (t) $
*κL (t) = ∞ 4 κR (t) = ∞+ 2
"
! t1 ! t2 ! "
R(t1 ) R(t2 ) !
#/
1) =
R(t
∞ 0
2) =
, R(t
0 0
∞ 0
.
0 ∞
2 ! # *
*56++/
⎡
⎤
E[sL (t) | Y(t)]
⎢
⎢
⎢
⎢ E[sL (t − q + 1) | Y(t)]
| t) = ⎢
ϕ(t
⎢
E[sR (t) | Y(t)]
⎢
⎢
⎣
E[sR (t − q + 1) | Y(t)]
⎥
⎥
⎥
⎥
⎥,
⎥
⎥
⎥
⎦
! ! /
s (t−q+1 | Y(t))
s(t−q+1 | Y(t)) = L
| Y(t))
sR (t−q+1
E[sL (t−q+1) | Y(t)]
| t)
=
= CT ϕ(t
E[sR (t−q+1) | Y(t)]
ϕ
L (t | t)
= CTL CTR
,
R (t | t)
ϕ
⎡
C =
⎢
CL
⎢
" CL = ⎢
CR
⎣
0 0
⎤
⎡
0 0
⎤
⎢ ⎥
⎥
⎢
⎥
⎥
⎥, CR = ⎢ ⎥.
0 0⎦
1 0
q×2
⎣0 0⎦
0 1
q×2
7 ! τ = t−q+1 ' sL (τ )"
sR (τ ) ' Y(t) *
" Y(t)+
859 : # ! t ' ! ! ! ! 2 τ !
q −1 !
t" $ sL (τ )" sR (τ )
! q−1 . !
" # ! ;
q−1 ) sL (τ ) 4 sR (τ ) " # ! ! !
q−1 *
! ! +
& ! ! ! %< p q > p + 1
! " #
! " $ q − p − 1
p
L
R
R
C
| t) =
Σ(t
| t − 1) =
Σ(t
C
(t | t)
Σ
L
ΣC (t | t)
(t | t)
Σ
C
(t | t) ,
Σ
R
2q×2q (t | t − 1) Σ
(t | t − 1)
Σ
L
C
(t | t − 1) Σ
(t | t − 1) ,
Σ
C
R
# $# ! αL
αR
αC
=
=
=
βL
=
βR
=
βCR
=
βCL
=
(t−1 | t−1)θ +1
θLT Σ
L
L
(t−1 | t−1)θ +1
θRT Σ
R
R
T Σ
θ
(t−1
|
t−1)θ
C
R +γC
L
ΣL (t−1 | t−1)θL
<q|
(t−1 | t−1)θ
Σ
R
R
<q|
(t−1 | t−1)θ
Σ
C
R
<q|
(t−1 | t−1)θ
Σ
C
L
L
R
C
R
L
2q×2q
L∞LR (t) =
lim
κL (t) → ∞
R
L(t) = 0.
κR (t) → ∞
*-+
% &" 0 ' (
! ' $!# &#
| t) = ϕ(t
⎡ | t−1) = Aϕ(t−1
ϕ(t
| t−1)
⎤
L (t − 1 | t − 1)
θLT ϕ
⎢ϕ
L (t − 1 | t − 1)<q| ⎥
⎥.
= ⎢
⎣ θT ϕ
R (t − 1 | t − 1) ⎦
R
R (t − 1 | t − 1)<q|
ϕ
<q|
*+ ",
T
T
⎡ Σ(t | t − 1) = ATΣ(t − 1 | t − 1)A + BTQ B =⎤
βL
αC
βCL
αL
⎥
⎢β
⎢ L ΣL (t−1| t−1)<q|q> βCR ΣC (t−1| t−1)<q|q>⎥
⎥.
⎢
T
T
βCR
αR
βR
⎦
⎣ αC
(t−1| t−1)
(t−1| t−1)
Σ
βCL Σ
β
C
R
R
<q|q>
<q|q>
*.+
/ " | t − 1) # 0
Σ(t
*-+ ", q + 1 # q + 1 $ 1 $ #!
q − 1 '
, Σ (t − 1 | t − 1)
Σ (t − 1 | t − 1)
(t−1 | t−1) ( # Σ(t
| t − 1) Σ
&" 0 '
( ! ,
L
C
L
% ! ! ! # " &" # ' ( )# $& ! !# *Σ (t | t)
Σ (t | t)
Σ (t | t − 1)
Σ (t | t − 1)+ $!
*Σ (t | t) Σ (t | t − 1)+,
L
% ! & 3 κ (t)
κ (t)
κ (t) ! ! ' ( * ! $
κ (t) = 0 # # +
4 "
t #
# ! 3
κ (t) = 0
κ (t) = 0
q 5 | t) # $
' Σ(t
6! $# ! $! $ q
& *.+
| t) $ # #
Σ(t
0 ' ( $!
,
T
1 0 ··· 0 0 0 ··· 0
L(t) =
7
0 0 ··· 0 1 0 ··· 0
" $!
' 8 & ' # $ ! & % #
!&
κ (t) = ∞
κ (t) =
∞
L(t) *2+ &"
R
C
⎤ αR+κR(t) −(αC+κC(t))
αL αC
⎢ βL βCR⎥
−(αC+κC(t)) αL+κL(t)
⎥
L(t) =⎢
⎣ αC αR ⎦ (α +κ (t))(α +κ (t))−(α +κ (t))2
L
L
R
R
C
C
βCL βR
⎡
*2+
/ " (t | t) ϕ
(t | t) ! & ϕ
! ! ! 1
$& # q − 1 0 ! /
$!# ! $ ϕ (t | t) (t | t) $!# $ ϕ
! ! # ##
$ ! 0 % $ * + $! " 1 ! 9:1 $! !
q − p / &
# p ! # ! 8 # !# $ 9:1 & ! &! 0 ; $!# ! !3
L
R
L
R
! κR (t) = ∞
L(t) "#$% L∞R(t)= limκ (t)→∞ L(t)
⎡ R
⎤
αL αC ⎡ 1
⎢ βL βCR ⎥ αL
⎥⎣
=⎢
⎣ αC αR ⎦
0
βCL βR
⎤
⎡
1
⎢ βL /αL
⎦=⎢
⎣ αC /αL
0
βCL /αL
0
0
0
0
0
⎤
⎥
⎥.
⎦
"#&%
' ( ! )
* " % + *
( ,- L (t).
! !
0) ! * 6 * ! * 0) *
0 !
"0)% 5000
0
−5000
−10000
0
50
100
150
200
250
300
350
0
50
100
150
200
250
300
350
0
50
100
150
200
250
300
350
5000
0
−5000
5000
0
−5000
| t)=Aϕ(t−1
| t−1)+L
∞R(t)(t)
⎡ T ϕ(t
⎤
⎡
⎤
L (t − 1 | t − 1)
θL ϕ
1
0 ⎢ϕ
⎢ βL /αL 0 ⎥ L (t)
L (t − 1 | t − 1)<q| ⎥
⎥
⎢
⎥
.
=⎢
+
⎣ θT ϕ
R (t − 1 | t − 1) ⎦ ⎣ αC /αL 0 ⎦ R (t)
R
R (t − 1 | t − 1)<q|
βCL /αL 0
ϕ
"#/%
) ! αC ! γC ! 0* )! 0! , #. * * * % *%
! % ! ! ) * * ! !! " *0* % ) 7 . * 1 "300 * % " % 0 ! , # " 0! * 44100 2 )! 16 *%
3, 0 12
γC
0 0,5 1 ) * 4 ,- * * *
* ) * 8#9 : ; - < =3> ** > * 3, 3,:3 ?
>
@@
A
: #BB$
CD/CA&
! 0 * * * *
0 0 * 5 * 0! ! 8D9 : ; =E * > = ?
1
F
#G#A H*
#BB$
I FFF
#&@B#&CD
8A9 < -. , J( J( DGGG
8@9 : ; - < =H ?
> @B
#G
7* DGG#
DD&DDD/D