Krzysztof Cisowski, Filtr adaptacyjny do rekonstrukcji sygnalow stereofonicznychPolitechnika Gdańska, Wydział ETI, Katedra Systemów Automatyki

Pełen tekst

(1)

(2) ! ! " ##$#% &'()*% ( + , !-.!.!-. 2003. Poznañskie Warsztaty Telekomunikacyjne Poznañ 11-12 grudnia 2003.

(3)

(4)

(5)

(6)

(7)

(8) !

(9)

(10) "

(11) # # $ ! # #%. .

(12) / 0 .. . -0 1. -( . - ( . ! 2 0 .( . . 0 -/ /- - - ( 0 + ( ! 3 $ / . 0 0 4 / /1 .05 ( . 0 ! 6 .78 . 1 -0( -/ 9 : -/ 9 ; (. - . 0 /1 - - 1 - . 0:! ( - 0 -07 0 ( . - ( - ! < 1 7 0( - ( 0 / -/ . ( ! 2 0

(13) 9="

(14) : . . / 05 ( 0 5 -/( 05 0 7( . /1 0 0 / ! > 0?! @( - 5 -0 01 -/ -/ ! - 0 - ;7 5 . 0 17 . 9 0 -/ : - 0 ; 07 ="

(15) 0 -/ 05 ( 01 >- 1? 0. 7 ! -/ . .5 -0 8 1 -;5 A !

(16) ;( 0 / ; . . 0 / 1 -!+ /( 4 1! 05 -- . - BCD BED -/ . 9": 178 ( 5 . ; > 8? - .( 05. 9 92 ::! " ; 5 ; 9-. - 0 ( : . 7 0 5 / ! 2 ; 8 -- . . / ( . . 7 0 ( 0 - 7 - ="

(17) ! 05 - - - . 0/ 0 -/. . . /-7 --( 05. / 1- 05. - ="

(18) ! "1- ( 05 92: 9 1 ; . 0 0 -( 2: . 05. -/ . 1 . 9:! @ - ( -/ / -/ - ( -/ - 05. ="

(19) 9 - - -/ 0 0 0 0 - .:!

(20) - ( - ; ( 8 2 1 0 1 ( ="

(21) - 05 "!

(22) ( 0 ="

(23) ; 8 05 ( -0 / -0! . -- ; 8 ( .05 F A 7 -/ 5 -4 ! -- 0 0 .( / A - 5 - 5 0 .

(24)

(25)

(26)

(27)

(28)

(29)

(30)

(31)

(32)

(33)

(34)

(35)

(36)

(37)

(38)

(39) !

(40) "

(41)

(42)

(43) !

(44) !

(45)

(46) " # C

(47) [N × N ] k ! k ∈< 1, N > $ C<k| % C|k> C<k|k> = (C<k| )|k> = (C|k> )<k|

(48) [N − 1 × N ]% [N × N − 1] [N − 1 × N − 1]

(49)

(50)

(51) C

(52)

(53) k

(54) %

(55) k

(56) k

(57) k . .

(58)

(59)

(60)

(61) .

(62) & " & $'

(63) % & " (y(t)) (s(t)) *

(64)

(65) (z(t))

(66)

(67)

(68)

(69)

(70)

(71) %

(72)

(73) T

(74) % + y(t) = yL (t) yR (t) % s(t) = T T sL (t) sR (t) % z(t)= zL (t) zR (t) % L R

(75)

(76)

(77)

(78) , &$'

(79) & (y(t)) (s(t)) & '

(80) + . 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

(81) R(t)

(82) z(t)

(83) R(t) = σz2 σzLR L E[z(t)z(t)T ] = . 0 σzRL σz2 R (s(t))

(84) & &$'+ 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

(85)

(86)

(87)

(88) 2 p% n(t) = T nL (t) nR (t)

(89)

(90)

(91) ! *

(92) -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

(93) Q(t)

(94) n(t)

(95) σn2 σnLR L Q(t) = E[n(t)n(t)T ] = . σnRL σn2 . . R. - L R

(96)

(97)

(98)

(99) / ⎡ ⎡ ⎤ ⎤ 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/ & '

(100)

(101) + 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

(102) + ϕL (t) AL 0 BL ϕ(t) = , A= , B= , ϕR (t) 0 AR BR 2q×1 2q×2q 2q×2.

(103) '+ ϕ(t) y(t). = A ϕ(t − 1) + B n(t), = BT ϕ(t) + z(t).. -6/ -7/. % &

(104)

(105)

(106)

(107) + T ϕ(0)= ϕTL (0) ϕTR (0)

(108) & n(t) z(t)

(109) 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

(110) !

(111)

(112) ϕ(t) -

(113) ϕ(t).

(114)

(115)

(116) . s(t) s(t)

(117) . . . . , y(1)] = Y(t) = [y(t), yL (t) yL (1) YL (t)

(118)

(119) ,..., = yR (t) yR (1) YR (t) .

(120)

(121) . 4 5 "5 1"

(122)

(123) " -5 ,

(124) . 1,

(125) - - . | 0) = ϕ(0) = [y(0), y(−1), . . . ,y(−p +1)]T ϕ(0 | 0) = 0 6 3

(126)

(127) Σ(0 | 0)

(128)

(129) ,

(130) Σ(0 3. T. .

(131) ! " .

(132)

(133)

(134) " "". # $

(135) %. | 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)

(136) (t)= , L (t | t−1) L (t) yL (t) T T ϕ

(137)

(138) = − BL BR R (t | t−1) ϕ R (t) yR (t). "

(139) . . -" " .

(140) . | t) ϕ(t.

(141). | 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

(142).

(143). | t−1) Σ(t. " " % . "

(144) . 1" " " . ϕ(t). 2-

(145) 1"

(146) . 3 . .

(147)

(148) %. | t) = Σ(t. 1 σn2. Z(0) = [z(0),z(−1), . . . ,z(−p +1)]

(149) , -" ϕ(0)

(150)

(151) 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 "

(152)

(153) ϕ(t)

(154) "

(155) 5 1"

(156) .

(157) R(t) Q(t) 9

(158)

(159)

(160) 1" . 7 .

(161) 3 .

(162)

(163) 1 5

(164) 3

(165)

(166) -5

(167) 3 3 "

(168) - 4

(169) -,

(170) 3 . .

(171) %. σn2. R. = σn2. L. ,

(172) .

(173) 3

(174)

(175) . γC. γLR. . γRL.

(176)

(177) . - " . κRL. . κLR. κC ,

(178) Q(t) R(t)

(179) - % κL κC 1 γC , R(t) = Q(t) = . γC 1 κC κR / 3

(180) 5

(181) % γ , κ , κ , κ , C L R C - . .

(182)

(183)

(184) - 3

(185) .

(186) /

(187) . . . γC

(188) -

(189) γC = 0 . 8 3 .

(190) 13"

(191)

(192). " 5

(193) 3 :

(194) "

(195)

(196) 1

(197) : . . 6 ". 1. γC = 1,. . 0

(198) "

(199)

(200) .

(201)

(202) " 5 .

(203)

(204)

(205) -. Σ(t | t),. ;

(206) "

(207) - . L. .

(208)

(209) "- . -. 1 Σ(t | t − 1), σn2 L "

(210) Σ(t | t) Σ(t | t − 1) " "

(211) R(t)

(212)

(213) "

(214) - 4" Q(t).

(215) .

(216)

(217) " . -

(218)

(219)

(220) 1"

(221) . 5

(222) 1 3

(223) " . | t − 1) = Σ(t. 3 . .

(224) . Q(t).

(225) .

(226). R(t)%. ,. 1 1 γLR Q(t) = γRL γR σn2 L = σnLR /σn2 , γRL = σnRL /σn2. Q(t) = γLR σn2 /σn2 ,. . .

(227) - 1 . - <

(228) "

(229) - .1 .

(230) 1-

(231) "

(232)

(233) - "

(234)

(235) " 5 . 5 5 "

(236)

(237) , " 1-

(238)

(239)

(240) 5 5. γR =. "5 " 0 " . κL κLR 1 R(t) = κRL κR σn2 L "

(241) κ = σz2 /σn2 , κLR = σzLR /σn2 , κRL = L L L L σzRL /σn2 , κR = σz2 /σn2 .

(242) + / . "

(243) . R. L. L. . L. R(t) =. L. R. L. ,. ,.

(244) 5

(245) . . "

(246) " , ".

(247) 7 5

(248) -5

(249)

(250)

(251)

(252)

(253)

(254) 4

(255) .

(256)

(257)

(258)

(259)

(260)

(261) !

(262) " #

(263)

(264) !

(265) " ! !

(266)

(267)

(268) $

(269) %. " !

(270) !

(271) !

(272)

(273)

(274) ! !

(275) & #

(276) γC !# ! #

(277)

(278) '

(279) 3 ÷ 20 ( 0,7 ÷ 0,1 !

(280)

(281)

(282)

(283) ! &

(284) ! ! #

(285)

(286)

(287) &

(288) κL " κR " κC ! ! "

(289)

(290)

(291)

(292) $ &

(293) κC

(294)

(295) ! $ !

(296) ! " . #

(297) " !# # # ( !

(298) !

(299)

(300) $

(301)

(302) κL " κR ) κL *κR +

(303)

(304) ! $ * + !

(305) ! !

(306) " , - !

(307) !

(308)

(309)

(310)

(311) !

(312)

(313) " " !

(314) $ !

(315) &

(316) ! !" κL *κR + !

(317) " # !

(318) " , !

(319)

(320) !

(321)

(322) " # !

(323) .

(324)

(325)

(326) κL *κR + !

(327)

(328)

(329) $ & # κL *κR + ! ! !

(330)

(331) !

(332) & " ! !. !" ,

(333)

(334) !

(335)

(336) ! , !

(337) !

(338) &

(339)

(340) ! !

(341) !# ! $"

(342)

(343) κL " κR κC / • !

(344) $

(345)

(346) κL " κR κC

(347) !

(348) " •

(349)

(350)

(351) 0 *+1"

(352) 0*+1 !

(353) "

(354) κL (t) = 0 0 0" κR (t) = 0" κC (t) = 0 R(t) = " 0 0 • ! ! "

(355) .

(356) * ! $ !

(357)

(358) !

(359) +"

(360)

(361) "

(362)

(363) ! . #. 2 ! #

(364)

(365) ! ! .

(366) . " # !!" ! $ !

(367)

(368) $ 3

(369) κL (t) 4 κR (t)

(370)

(371)

(372) $ *κL (t) = ∞ 4 κR (t) = ∞+ 2 " ! t1

(373) !

(374) t2

(375) !

(376) " R(t1 ) R(t2 ) !

(377) #/ 1) = R(t. ∞ 0 2) = , R(t 0 0. ∞ 0 . 0 ∞.

(378)

(379) . 2 !

(380) # * *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)]. ⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎥ ⎦.

(381) !

(382)

(383)

(384) ! / 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

(385) ! τ = t−q+1

(386) ' sL (τ )" sR (τ ) ' Y(t) * " Y(t)+ 859 : #

(387)

(388) ! . t '

(389) !

(390) ! !

(391) !

(392)

(393) 2

(394) τ !

(395) q −1

(396) !

(397)

(398)

(399) t"

(400) $ sL (τ )" sR (τ )

(401) !

(402)

(403) . q−1

(404) . !

(405) " #

(406) !

(407) ; q−1

(408) )

(409) sL (τ ) 4 sR (τ )

(410) "

(411) # !

(412)

(413)

(414) ! !

(415)

(416)

(417) q−1

(418)

(419)

(420) * ! !

(421) + & ! !

(422) !

(423)

(424) %< .

(425)

(426) p q > p + 1 ! " # ! " $ q − p − 1 p.

(427) . 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

(428) ! 9:1 $! ! q − p / & # p ! # ! 8 # !# $ 9:1 & ! &! 0 ; $!# ! !3 L. R. L. R.

(429)

(430)

(431)

(432) ! κR (t) = ∞. L(t)

(433) "#$% 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. ⎤ ⎥ ⎥. ⎦. "#&% ' ( ! )

(434) * " % + * ( ,- L (t).. ! ! 0) !

(435) * 6

(436) * ! * 0) *

(437) 0 ! " 0) %

(438) 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. "#/% ) !

(439) αC

(440) ! γC ! 0* )! 0 ! . , #. *

(441) *

(442) * % *% !

(443) % ! ! )

(444) * * ! !! " *0* % ) 7

(445) .

(446) * .

(447) . . 1 "300

(448) * % " % 0 ! , # " 0 !

(449) * 44100 2 )! 16 *

(450) % 3, 0 12

(451) γC 0

(452) 0,5 1

(453) ) *

(454) 4 ,- * * * * ) * . 8#9 : ; - < =3 > ** > * 3, 3,:3 ?.

(455)

(456) > @@ A : #BB$ CD/CA&.

(457) ! 0 *

(458)

(459) *

(460)

(461) *

(462) * 0 0 * 5

(463) * . 0 ! ! . 8D9 : ; =E * > = ?.

(464)

(465)

(466) 1 F #G#A H * #BB$ I FFF #&@B#&CD 8A9 < -.

(467)

(468)

(469)

(470)

(471) , . J( J( DGGG 8@9 : ; - < =H ?

(472)

(473) . > @B #G 7 * DGG# DD&DDD/D.

(474)