● 2000XEEEInternat五ona1SymposⅢ、 o n 五ntemgentS五gnalProcessmgand Commumcat五onSystems ● lSPACS2000 P r o c e e d i n g s SheratonWaikikiHoteLHonolulu,Hawaii,U、S、A、 〃 November5-8.2000 S p o n s o r e d b y : ZソZeSocjetyqf〃b7、mat伽The岬α冗肌sA卯"cat伽s(SOC.q/I別ノ The加伽teqfEIe伽mcs,J7l/brmat伽α"dCbmm伽Cat伽助9mee7s(IEICEノ: Z1echmcaIGm叩onDj9伽JSj9”αIP7℃cess航9 WiththeTbchnica1CGsponsorshipof: ZソieZEEECo77ZmUmCa伽71SSOCiety lnCooperationwith: TheIEEEJ叩α冗Ch叩ter: Commwzjca伽nsSocjety S d 卯 α I P 7 ℃ c e s s m 9 S o c j e t z ノ α7℃蝋Sα”dSyStemsSOCiety The血s伽teqfEIect7ⅧCs:〃blw2at伽aMCommⅧCat伽E叩mee汚(IEICEノ: meC伽CaIC7℃叩O〃Cbmm伽Cat伽SyStemS Z1eC伽caIGm叩onα7℃“tSaMSyStemS mec伽jcaJGm叩o冗YLSIDesj9”Z1ec伽ojO9jes mec伽caIGm叩onSpeec月 mec伽jcaIGmo叩onJmQ9eE”9伽eerj叩 霧削鰯釜淵謡撫撫職;瀞 一Honolulu,Hawaii,U、SA.,November5-8,2000 B6-3-2 DESIGNOFVARIABLEDIGImkLFIIノrERSBASEDONSTⅢE・服ACEREALIZATIONS 脇MsjljMA耐UXAI脳伽αノソ叩Zy"〃KAW64MA別 D e p a m e n t o f E l e c t r o n i c E n g i n e e r i n g , G r a d u a t e S c h o o l o f E n g i n e e r i n g T b h O k u U n i v e r s i t y A o b a y a m a O 5 , S e n d a i , 9 8 0 8 5 7 9 , J a p a n P h o n e : + 8 1 2 2 2 1 7 7 0 9 5 , F a x : + 8 1 2 2 2 6 3 9 1 6 9 , E m a i l : m a t u k a w a @ m k ・ e c e i 、 t o h o k u ・ a c j p ABSTRACT F b r l o w p a s s t o l o 岬 a s s t m n s f b r m a t i o n , T ( z ) i s g i v e n b y n l i s p a P e r p r O p o s e s a d e s i g n m e t h o d o f v a r i a b l e m R d i g i t a l T ( 塵 ) = 芸 芸 伽ersbasedonbalancedrealizationsandminimumround- o E F n o i s e r E a l i z a t i o n s o f d i g i t a l m t e r s ・ H i g h l y a c c u r a t e v a r i ‐ a b I e d i g i t a l f h e r s a r e e a s i l y d e r i v e d b y t h e p r O p o s e d m e t h o d ・ 一一 ︽宮阜口、︾ mecoef過cientmaロicesofbothrealizationsofsecond-order d i g i t a l f ] 力 e r s a r e O b t a i n e d d i r c c d y f T o m p r o t o t y P e r c a l i z a ・ t i o n s ・ T h e m t e r c o e 髄 c i e n t s o f v a r i a b l e d i g i t a l m 剛 c a n b e ( 3 ) ( 4 ) 、 。 i n ( 進 ヂ 空 ) O b t a i n e d b y f i E q u e n c y t m n s f b r m a t i o n s t o t h e r e a l i z a t i o n s ・ W h e r e U c i s t h e c u t o f f f r e q u e n c y o f t h e d e s 舵 d l o w p a s s n l e m t e r c o e f f i c i e m s a r e p r c s e n t e d a s t r u n c a t e d T h y l o r s e ‐ 1 i e s f b r t h e p u r p ◎ s e o f r e d u c i n g a n u m b e r o f c a l c u l a t i o n s t o t u n e t h e c o e f 五 c i e n t s ・ H o w e v e r t h e p r o p o s e d m t e r s h a v e h i g h l y a c c u r a t e v a r i a b l e c h a r a c t e r i s t i c s a g a i n s t t h e c o e f E - d i g i t a l m 卿 A f t e r a p p 助 i n g t h e l o w p a s s t o l o w p a s s t r a n § f b r m a t i o n , t h e Ⅷ s f e r f i m c t i o n H ( z ) w i l h v a r i a b l e c h a x a c t e r i s t i c s c a n b e O b t a i n e d ・ T h e c o e f f i c i e n t s O f H ( z ) a r e c o m p l i c a t e d 伽 c ・ t i o n s o f t h e v a r i a b l e p a m m e t e r を . n e r e f b r e , n l e y 咽 u i r e a cienttruncationsincebalancedrealizationsandminim皿m r o m d o f f n o i s e r E a l i z a t i o n s h a y e v e r y l o w c o e f n c i e n t s e n - hugenmnberofcalculationstOmnethecoe価cients・Ijne a r a p p m x i m a t i o n i s a p p l i e d t o t h e c o e f 過 c i e n t s i n o r d e r t o s i t M t i e s , w h i c h a r e i n v a r i a n t u n d e r t h e f r B q u e n c y t r a n s f b F m a t i o n s ・ N m n e r i c a l e x a n m 1 e s s h o w t h e e 齢 c t i v e n e s s o f t h e v a r i a b l e d i g i t a l E l t e r s d e s i g n e d b y t h e p m p o s e d m e t h o d . r e d u c e 血 e n u m b e r o f t h e c a l c u l a t i o n s ・ n i s a p p r D x i m a t i o n i s b a s e d o n t h e 脚 l o r s e g i e s 露 p a n s i o n w i l h r E s p e c t t o g . T h e c o e 価 c i e n t s o f t h e t r a n s f E r h m c t i o n H ( z ) a r e e x p a n d e d 1.INTRODUCmON i n t o 耐 y l o r s e g i e s , o f w h i c h h i g h e r 迄 o r d e r t e n n s a r e m m c a t e d V h r i a b l e d i g i t a l m t e r s a r e f i c q u e n c y s e l e c t i v e a l t e r s w i t h t u n a b l e f T e q U e n c y c h a m c t e r i s t i c s ・ n e y a r e o f t e n U S e d i n m a n y s i g m a l p r o c e s s i n g a p p l i c a t i o n s , s u c h a s t e l e c o m m u n i ・ u n d e r t h e a s s u m p t i o n に │ < 1 . H o w e v 鋤 1 h e l i n e a r a p p m x i ‐ m a t i o n c a u s e s m a g n i t u d e d e g r a d a t i 。 n . m l i s d e g 届 a d a t i o n c a n b e d e c r e a s e d b y a d O p t i n g l o w c o e f 五 c i e n t s e n s i t i v i t y s 伽 c ‐ c a t i o n s , d i g i t a l a u d i o e q U i p m e n t a n d a d a p t i v e s y s t e m s ・ M a n y tureS・ s伽ctmesanddes卿meロユodsofvariabledigital鋤ershave ahBadybea1proPosed・Mostoflhemareclassifedand compa歴dbyStoyanov皿dKawamata[1]・ V h r i a b l e d i g i t a l f l t e r s a r c u s u a l l y d e s i 即 e d b y e m p l o y ・ i n g l h e f h 巳 q U e n c y 伽 n s f b r m a t i o n s o f C o n s t a n t i n i d e s 間 . T h e 館nsfmnationsarEbasedomhesubstitution: z−1→T(z)(1) h1hispap眺weemploybalancedz巴alizationsandmin- i m u m r o u n d o f f n o i s e r e a l i z a t i o n s t o d e s i g n v a r i a b l e d i g i t a l E 1 t e r s ・ B o d l r B a l i z a t i o n s a r B 1 m o w n a s v e r y l o w s e n s i l i v i t y s t m c t u r e s 、 K a w a m a t a , I w a t s u k i a n d H i g U c h i 【 3 ] p m v e d t h a t b a l a n c e d 記 a l i z a t i o n s h a y e m i n i m u m s t a t i s t i c a l s e n s i t M t i e s ・ Onlheotherhand,minimumrDund-offnoiser巳aliza且ons i n v e s t i g a t e d b y M u l l i s 麺 d R O b e r t s 【 4 , 5 ] , H w a n g 【 6 ] a n d Bames[7]haveminimumstatisticalsensitivitiesunderdle w h e m c T ( z ) i s a 5 r s t o r s e c o n d o r d e r a n p a s s 剛 s f E r 伽 c ‐ t i o n ・ A P p l y i n g t h e 廿 a n s f b n n a t i o n s t o a p r o t o t y p e l o w p a s s n h e r w i t h t h e 伽 n s f b r h m c t i o n H b ( z ) a n d t h e c u t o f f f i e ‐ I 2 n o r m s c a l i n g c o n s l r a i n t [ 8 ] ・ H i g h l y a c c u r a t e v a r i a b l e d i g ‐ qUency‘Jcp,anewtmn伽伽ctionH(z)canbeObtained 2.REVIEWOFBALANCEDREAI五ZAmONSOF italmterscanbederivedfTombothr巳alizations. DIGImALmLIERS aS H ( z ) = 島 ( z ル ー ' = T ( z ) . ( 2 ) ● F b r t h e p u r p o s e o f t h e f b l l ⑥ w i n g d i s c u s s ユ o n s , w e e x − 153 w h e r e 8 n ( n = 1 , 2 , … , Ⅳ ) a r e t h e s e c o n d o r d e r m o d e s , W h i C h a r e t h e s q u a r e r o o t s o f e i g e n v a l u e s o f t h e m a 粒 i x p r o d ‐ uctKIWノ.nesecond-ordermodesarcinvariantunderthe 舵 q u e n c y t r a n s f b r m a t i o n s ・ n 1 e r e f b I a K I a n d V V J a r e a l s o i n v a Z i a n t u n d e r t h e f t e q u e n c y t r a n s f b r m a t i o n s . ) 冗 ( u 3.DESIGNOFBALANCEDREA皿ZAmONSOF SECOND・ORDERDIGImkLFIIjIERS 3 . 1 . E i g e n v e c t o r e x p a n s i o n s o f c o e 鯛 c i e n t m a t r i c e s B a l a n c e d r e a l i z a t i o n s o f s e c o n d o I d e r d i g i t a l f l t e r s c a n b e O b t a i n e d d i I c c n y f r o m l h e t r a n S f b m m c t i o n b y t h e d e s i g n m e d l o d p m p o s e d i n t h i s s e c t i o n ・ W e c o n s i d e r a s e c o n d ‐ o r d e r d i g i t a l f l t e r w i t h t h e t r a n s f b r f i m c t i o n F i g m J r e l : m e s i 甑 a l n o w g r a p h r e p r B s e n t a t i o n O f d l e 恥=弐十三二釜累十吟('1) s e c o n d o r d e r s t a t e S p a c e r e a l i z a t i o n . WherB p l a i n s t a t e s p a c e r B a l i z a t i o n s o f d i g i t a M i l t e r s ・ F b r a g i v e n 入p=。p+”P(12) αp=αpァ+japj.(13) Mh-ordert『an蜘伽ctionH(z),thedi夢taMiltercanbe e x p r 巳 s s e d b y t h e f b n o w i n g s t 誠 e e q U a t i o n s : 記(叶1)=A垂(泥)+伽(")(5) ")=caB(”)+血(72)(6) Thecoe伍Cientmatricesoflhemteraregivenby A , │ : : 蓋 鍾 : ! ( ' 4 ) ‘,=慨1(is) w h … 諺 ( ” ) i s l h e M h o r d e r s t a t e v e c t o r 5 秘 ( " ) i s t h e s c a l a r i 叩 u t s e q u e n c e , " ) i s 地 e s c a l a r o u t p u t s e q u e n 鴎 a n M , b,canddarB血eⅣ×Ⅳ,Ⅳ×1,1×jVandl×1rBal c o e 缶 c i 圏 n t m a 面 i c e s , r e S p e c t i v e l y ・ 厨 9 , 】 r e l s h o w s l h e s i g n a l a o w g ” p h 麺 p 唾 s e n t a t i o n o f t h e s e c o n d o r d e r s 伽 e S p a c e r e ‐ a l i z a t i o n ・ m e t r a n s f 色 r f m c t i o n H ( z ) c a n b e d e s c g i b e d a s H(z)=c(zI−A)b+d、 ( 7 ) , cp=[cp1cp2].('の Inadditionthesecond-ordermodesaエe錘prBssedbyBames mas 9,,82=、/戸冨 コア±R('7) ThecovanancematrixKandthenoisemaなixW′arede− 伽 e d a s 血 e s O 1 u t i o n s o f 曲 e f b l l o w i n g l y a p u n c v e q u a t i o n s : K=AKA#+bbt W=AtWA+Ctc. wherB ( 8 ) ”TL笹戸('8) ( 9 ) …=壷。側 nema廿icesKandVVdependon曲eE1terstruch】rc・ n e t r a n s f b r h m c t i o n H ( z ) i s i n v a r i a n t u n d e r n o n s i n ・ g U l a r 廿 a n s f b n n a t i o n m a 園 C e s T o f t h e s t a t e , i 、 e 、 , i f 記 ノ ー T − 1 z , t h e n t h e n e w r B a l i z a t i o n ( T − ユ A T , r − 1 b , c T , 。 ) F r o m E q S . ( 8 ) − ( 1 0 ) , t h e c o n s h a i n t s o n t h e c o e 鑓 c i e n t m a ‐ 1ricesA,bandccanbeObtainedas i s a n e q U i y a l e n t r c a l i z a t i o n t o ( A , 6 , c , 。 ) o f t h e 征 a n s f b r -a12=α21 悪麗I融避;I裂職著鰯患謡測誤 a n d T t W T , r 巴 s p e c t i v e 1 y ・ B a l a n c e d r 巴 a l i z a t i o n s c a n a l s o b e d e r i v e d b y t h e s u i t a b l e ( 2 0 ) 6,=c1 C 1 ) b2=−c2. ( 2 2 ) e q u i v a l e n t t r z m s f b n n a t i o n s ・ B a l a n c e d r e a 1 i z a t i o n s a r e d i g ‐ Underthecons鯛intgivenbyEq.(20),thema面xAcanbe italmterswhichhavethefbI1owingcovarianceandnoise assumedas K'=W'=diag(81,82,…,9Jv) 作I ( 1 0 ) 154 ︽ ma画iCes: 編3;'1側 w h e r e 〃 i s a p a r a m e t e r d e t e r m i n e d b y t h e c o n s t r a i n t s o n t h e ( 3 3 ) . T h e e l e m e n t s o f t h e c o v a r i a n c e m a t r i x K a r e g i v e n b y cOvanancematrixandthenoisemalIix・ k,,=2{Ki+Re(恥)} m l e n w e e m p l o y t h e e i g e n v e c t o r e x p a n s i o n s o f c o e f 五 一 c i e n t m a t I i c e s I 7 ] 、 A n e i g e n v e c t o r o f t h e c o e f E c i e n t m a t r i x Aisgivenby 作│為l 恥 雲 膨 迦 = 声 価 十 R 陰 仰 十 j ) ) ' ( 3 8 ) 伽 = 2 1 K m + 古 馳 { 恥 十 が } 1 ( 3 , ) ( 2 4 ) where 昨畿“。) a n d l h e a s s o c i a t e d r o w e i g e n v e c t o r i s g i v e n b y 脇=島㈹ ● 妙 = ; [ ‘ ′ j − 1 / Tエ ー ア ] . ( 2 5 ) n e e l e m e n t s o f t h e n o i s e m a t r i x W a r e g i v e n b y ● n l e r o w e l g e n v e c t o r リ I i s n o n n a l i z e d s o t h a t “=妙寧ず=1. ( 3 7 ) l+シ2 ”,,=TWi+Re{W2(1+j")2}(42) ( 2 6 ) ⑩12=⑩2ユ n e c o e f 五 c i e n t m a t r i c e s A , b a n d c c a n b e e x p a n d e d o n t h e e i g e n v e c t o r a n d t h e r o w e i g e n v e c t o r a s f b I 1 o w s : j j9 j 7 8 22 く く2 く A=)W'+入・‘*妙* b=β‘+β掌ゆゅ c=7秒+'γ寧秒率 ー 毎 垂 1 " W i + R ・ 側 , − , ) } ' ㈹ 1+〃2 ⑩22=T{Wi-Re(W§)}(44) where wi=鍋㈱ where β=β,.+jβi ( 3 0 ) γ='γ『+j髄. ( 3 1 ) wh=鈴㈹ U n d e r n 1 e c o n s t r a i n t s g i v e n b y E q S . ( 1 0 ) , ( 3 5 ) a n d ( 3 6 ) , t h e f b I 1 o w i n g e q u a t i o n s c a n b e o b t a i n e d : mecoefEcientma廿icesbandccanbedenotedas ‘ − . │ 窯 I c=い−”′Wr干預]. ( 3 3 ) 一偶 脇=竿、/両§㈹ ( 3 2 ) wi=21/戸F三が. FromEqs.(34),(47)and(48),wenndthat 一一 ソ 3.2.Designprocedure m e p 画 圏 m e 8 e r s β a n d ' y m u s t s a t i s f y t h e c o n s 面 a i n t s g i v e n b y E q s . ( 1 1 ) , ( 2 1 ) a n d ( 2 2 ) . I n o r d e r t o s a t i s f y E q . ( 1 1 ) , β wheIB ( 3 4 ) ( 4 9 ) ( 5 0 ) and'γmustbechosenasfbllows: βγ=α、 ( 4 8 ) n u s , t h e p a r a m e t e r 〃 i s d e t e r m i n e d ・ ThegBfbr巳,thecoef五cientmaなicesoflhesecond-oIder F r O m E q s . ( 2 1 ) a n d ( 2 2 ) , γ m u s t b e e x p r e s s e d a s balancedr巳alizationarederivedas 』 炉 陣 三 : わ 的 津 吻 l 側 炉命(船"則(3s) 炉古(-州〃(36) 心,={鐙士躍’側 n e c o v a n a n c e a n d n o i s e m a 画 c e s a 】 ℃ r e p r e s e n t e d b y t h e a s 副 1 m e d c o e 缶 c i e n t m a t r i c e s g i v e n b y E q s . ( 2 3 ) , ( 3 2 ) a n d c , = [ b p 1 6 p 2 ] 155 ( 5 3 ) w h e r e j j p 1 a n d 脚 2 a r e d e s c r i b e d a s f b n o w s : Thblel:ThenUmbersofcalculationstotunelhecoefEcients 凡 ( │ α p l a p § ) 仰1= 勘2= 2 , / 画 票 ・Sign(αpr). o f s e c o n d o r d e r v a r i a b I e l o w ・ p a s s f i l t e 応 ( 5 4 ) ( 5 5 ) F r o m t h e s e c o n d o r d e r b a l a n c e d r e a l i z a t i o n g i v e n b y E q s . ( 5 1 ) − ( 5 5 ) , t h e s e c o n d o r d e r m i n i m u m r o l m d o f f n o i s e r e ‐ a l i z a t i o n c a n b e O b t a i n e d b y t h e e q U i v a l e n t t m n s f b r m a t i o n s ・ O n e ◎ f t h e e q u i v a l e n t t r a n s f b r m a t i o n m a 廿 i c e s T i s d e s c r i b e d ‘‘ハ狂lNrealization,,denotestheminiml]mround-offnOise as r e a l i z a t i o n . T = , / 亙 寧 1 1 別 ( 5 6 ) ㈹㈹㈹㈹ r i v e d b y l h e s a m e e q u i v a l e n t t r a n s f b r m a t i o n o f w h i c h t h e matrixTisgiyenbyEq.(5句・necoe紐cientmal五cesof l h e v a x i a b l e d i g i t a l m t e r a r e e 髄 p I B s s e d a s ( T − 1 A T , T' b , cT,。),whereA,b,canddaregiVenbyEqS.(61)−(6の. nehigh録ordervaziabIelow・pass伽egsc麺1alSobe ObtainedbyemPloyingthepmposedm曲od・WMes埴n anMhoordervariablelow-pass魁terwhegBjV>2aslhe where Wg 一一十J一 ( 5 8 ) p a r a l l e l s 画 u c 伽 c o f f r s t a n d s e c o n d o r d e r s u b E 1 t e r S w h i c h a r c d e s i g n e d b y t h e p m p o s e d m e d l o d . ( 5 9 ) 1+2°p〈+M2学 食、 onminimumround-offnoiSe密1伽tiorlscanalsobede‐ H(z)=急十弐十.(57) αprf+(αpro+αp鋤)と2 ( 6 9 ) T h e s e c o n d o r d e r v a r i a b l e l o w p a s s d i g i t a M i l t e r b a s e d t o 1 o w ・ p a s s 伽 n s f b r m a t i o n t o H ] , ( z ) y i e I d s a n e w t r a n s f b r h m c t i o n H ( z ) : 。=dp−2 2 orderprcto卯e鰯msfErhmction・App】yingthelow,pass 食、1J W e e m P l O y H b ( z ) g i v e n b y E q S . ( 1 1 ) ( 1 3 ) a s a s e c O n d ‐ 唯バリ唯娼卿 山油︾︾︾伽 4.ANEWDESIGNMETHODOFVAmARIE LOWbrASSFYmERS ︾一いい恥州 wilhthesecond-orderminimumround-offnoiserealization pmposedbyBamesm. 一一一一一一一一一一十 伽”仙伽刀。 n 1 i s r e a l i z a t i o n ( T − ' A P T , T − ユ b p , C p T , 。 p ) c o i n c i d e s 22一一十f桃 − 1︲2ァ吟 内内均牌吟r’ where 5.NUMBERSOFCALCUIAITONSTOTUNE FⅢⅢERCOEFmCIENTS ( 6 0 ) I n t h i s s e c t i o n , w e c o m p a r B t h e n u m b e r s o f c a l c u l a t i o n s mlebalancedrcalizationofthelransfErfmctioncanbeOb‐ totunetheCoef五cientsoflhesecond-ordervarigHelow- t a i n e d b y t h e m e d m o d e X p l a i n e d i n l h e p r E v i o u s S e c d o n ・ T h e n t h e l i n e a r a p p r o x i m a l i o n i s a P p I i e d t o a n e l e m e n t s o f d 1 e c o ‐ p a s s E 1 t e r s d e s i g n e d f m m t h e P r o p o s e d r E a l i z a l i o N 1 s , d i r B c t f m m l I , p a r a n e l a l l や a s s s m 』 c 伽 e 回 a n d l a t t i c e f b r m I 1 0 1 . efEciemtmalriceS・Thercfbrelhecoe伍cientmaなicesofthe s e c o n d c r d e r v a r i a b l e l o w ・ p a s s d i g i t a l m t e r a r e 2 ‘ v e n b y T息blelshowsthexmmbersoflhecal“lations・nenmnbers A = I _ 輸 繊 皇 ‘ ) W 呈 工 麓 ③I ‘ = { 塁 1 − │ 膳 │ ± 催 │ : 劉 I 1 o f l h e c a l c u l a t i o n s o f t h e p r o p o s e d E 1 t e r S d o n o t c o n t a i n d l e numberoflhecalculationof〃givenbyEq.(69).msis b e c a u s e 山 e c a l c u l a t i o n o f 7 7 i s n o t n e c e s s a r y m m a n y c a s e S sincethesignof刃doesnotchangeundertheassumption l割く1. T耐ecalculationsoftheproposedm脈麺erBducedby t h e l i n e a r a p p m x i m a t i o n ・ H o w e v 甑 t h e v a r i a b l e E 1 t e g s d e - ( 6 1 ) d=dp−2apアビ 鋤鋤聞 くくく c=[b1-b2] si即edbylheproposedmeUlodneedmorecalculationsto 156 −−−IdOaI --Balancedmal値ation −−ldeal -DiredformIl 0 四℃星@つ且置口画室 画ロ星①℃。壱胃乏 −0.2 -60 -0.2 0 −0.50.2 0.50.2 5 Requencyのinmms Requencyのin面units ( a ) § ( b ) −−−ldeg 一一ldBal −l盆断E⑧form -PBmllOlBIl=pBBs 0 0 画℃星@℃星匡卑画室 、つ星の画星匡画因至 − 印0 -0 、 2 −60 -02 0 0.2 0.50.2 0.5 0.2 5 FmquencyQoin派uni鱈Frequencyのin症units ( c ) § ( d ) T h e m a g n 伽 d e r e s p o n s e s o f t h e s i x t h o r d e r v a r i a b l e d i g i t a l m t e r s . ( a ) n l e p a 別 F i g U r 巳 2 : T h e m a 厚 1 i m d e r e s p o n s e s o f t h e s i x t h o r d e r v a r i a b l e d i g i t a l m t e r s . ( a ) n l e p a r a n e l s 剛 c t u r B o f s e c o n d o r d e r b a l a n c e d r e a l i z a t i o n s . ( b ) n l e p a r a n e l s t r u c t u 記 o f s e c o n d ・ o r d e r d i r e c t f b r m n . ( c ) m l e p a m l l e l a l l p a s s s t r u c t u r e . ( d ) n e cascadesncn1reofsecond-orderlatticefbnn. t u n e l h e c o e 鐙 c i e n t s t h a n t h e o 1 h e r m t e r s s i n c e t h e p r o p o s e d mtershayemorecoef過cientsthantheothers. 6.NUMEmCALE叉AMPLES W b d e s i 印 t h e v a r i a b l e d i g i t a l E 1 t e r s b a s e d o n b a l a n c e d r e 副 i z a t i o n s a n d m i n i m u m m u n d o f f n o i s e r e a l i z a t i o n s , a n d comparetheproposedmterswiththevanablemterswhich a r E d e s i g n e d f m m t h e p a m n e l s t r u c t u r e o f d i r c c t f m n l I , t h e p a m n e l a l l p a s s s t r u c t u r e [ 9 ] a n d t h e c a s c a d e s t r u c t u r e o f u r B s 2 ( b ) , 2 ( c ) a n d 2 ( d ) s h o w t h e m a g n i t u d e r e S p o n s e s o f t h e o t h e r t h r e e v a r i a b l e m t e r s , r e s p e c t i v e l y ・ I t i s c o n E r m e d t h a t t h e v a r i a b l e l O w p a s s E 1 t e r b a s e d onbalancedrBalizationshasmoreaccumtemagnimde心 s p o n s e s t h a n t h e o d l e r m t e I s . 、 l e m a g n i m d e d e g r a d a t i o n o f t h e p a r a l l e l s t r u c m r e o f d i r e c t f b m n l l i s c a u s e d c o n s i d . e顕blyinthecaseof《=士0.2.memagnitudereSponses o f t h e p a r a n e l a l l p a s s s t r u c t u r e d e v i a t e i n t h e s t o p b a n d d u e t o t h e h i g h s t o p ・ b a n d s e n s i t M t y o f t h e s 面 u c 伽 e ・ T h e m a g ‐ f l 1 e r も n i t u d e r B s p o n s e s O f t h e c a s c a d e s l r u c t u r e o f l a t t i c e f b n n a r e c o m p l e t e l y d e s t r o y e d i n t h e a n c a s e o f を . I n a d d i t i o n t h e v a r i a b l e l o w p a s s m t e r b a s e d O n m i n i m u m r o u n d o f f n o i s e r e a l i z a t i o n s h a s t h e m a g n i m d e r e s p O n s e s w h i c h a r e a s a c c u ‐ F i g u r e 2 ( a ) S h ◎ W s t h e m a g l i t u d e r c s p o n s e s o f t h e p r 。 ‐ p o s e d 団 t e r b a s e d o n b a l a n c e d r E a l i z a t i o n s w h e n t h e p a r a m ・ e0erfvaIiesfTom-062to0.2.Ontheotherhand,Fig. r a t e a S t h e r B S p o n s e s s h o w n i n F i g u r e 2 ( a ) . F i g m 巳 3 p l o t s l h e s t a t i s t i c a l s e n s i t M t i e s o f t h e t h r e e d i f f E r 巳 n t v a r i a b l e l o w p a s s 伽 e r s a s f i m c t i o n s o f t h e v a r i a b l e l a t t i c e f m m [ 1 0 ] ・ W e a d o P t a s i x t h o r d e r B Ⅲ e r w o r t h l o w ・ p a s s E l 8 e r w i d M h e c u t P o f f f t B q u e n c y 0 . 1 T a s t h e p m t o t y p e 157 [ 2 ] A ・ G ・ C o n s t a n t i n i d e s , “ S p e c t r a l t r a n s f b r m a 箇 o n s f b r 1 0 4 − 一 一 − 4 “I 3 ,︾n︾ ●・●●Q●● 1 5 9 0 , A u g u s t l g 7 0 . [ 3 ] M ・ K a w a m a t a , M ・ I w a t s u k i a n d 工 H i g 』 c h i , " B a l a n c e d r e a l i z a t i o n s a s x m n i m u m s e n s i t i v i l y s t r u c m l E s i n I 血 e a r s y s 0 e m s ; , T h a n s a c t i o n s o f t h e S o c i e 句 o f l n s t r u m e n t andContml画哩neers,vol、21,no、9,pp、900-906, e . 、 9 . . 0 . G . 0 . 0 . . . 6 . 0 . G . ・ ・ 0 . , . Q ○ . 0 Q ・ o e D o ・ ■ D o g ・ 0 ・ o Q O 0 Q c ● 。 G8●●●●OO0e0 S e p t 副 n b e r l 9 8 5 、 2 10 昏一皇一の匡①の一⑮。一重誌あ ●■O●000● d i g i t a l f i l 血 e r s , f , P r o c ・ 肥 E , v o l 、 1 1 7 , n o 、 8 , p p 、 1 5 8 5 - ParalIelbaIancedrealizaMon ParalIeIMRNre劃izaUon Par a l Il eI D i re c t fo r m lf IormlIl l r a I e I D i r e c t 14]CmMullisandR.A、ROberts,“Symhesisofm血i‐ m u m r o u n d o f f n o i s e E x e d p o i n t d i g i t a l m t e r s ; , 凪 E E T r a n s ・ C i r c u i t s S y s t 。 , v o l ・ C A S 2 3 , n o 、 9 , p p , 5 5 1 5 6 2 , D q Septemb画1976. ; 日 1 三0.2−0.100.10.2 ■ ● [5]CmMullisandR.A・ROberts,‘Roundoffnoisein 。 d i g i t a l n l t e r s : I 蛇 q U e n c y t r a n s f b r m a t i o n s a n d i n v a r i P a n t s , ? , 肥 E E T t a n s ・ A c o u s L , S p e e c I L S i g l a l P r o c e s s ‐ 層 F i g m e 3 : n l e s t a t i s t i c a l s e n s i t i v i t i e s o f t h e s i x o r d e r v a r i a b l e d i g i t a l m t e r s . " B 凪 N r e a l i z a t i o n , ' d e n o t e s t h e m i n i m u m ing,vol・ASSP-24,no、6,卯.538-550,December l 9 7 6 . round-offnoiser理1i”tirm= [61s・YHwang,‘‘Minimumuncogrelatedunitnoisein p a m m e b e r g . T h e s t a t i s t i c a l s e n s i t i v i t y 6 p r o p o s e d b y K a w a ‐ s t a t e s p a c e d i g i t a l E 1 t e r i n g , ? , 肥 E E T r a n s 、 A c o u s t . , s p e e c h , S i g n a l P m c e s s i n g , v o l 、 A S S P 2 5 , n o 、 4 , 卯 . mataetaL[3],isdefnedas 2 7 3 2 8 1 , A u g U s t l 9 7 7 . S={tr(K)+1}{tr(W)+1}.(70) [7]C・WEames,“Onthedesignofop伽alstate-Space AsshowninFigure3,thevariablem0ersdesig画edby曲e r e a l i z a t i O n s o f s e c o n d 戸 o I d e r d i g i t a l m t e r s d , ’ 正 E E T r a n s ・ C i r c u i t s S y s t . , v o l ・ C A S 3 1 , n o 、 7 , 卯 . 6 0 2 6 0 8 , p r o p o s e d m e d m d h a v e v e r y l o w s t a t i s t i c a l s e n s i t i v i t i e s ・ T h e p a m u e l s t r u c t u r B o f d i x B c t f b r m Ⅱ h a s h i g h s e n s i t i v i t y w h i c h m c r e a s e s w i 曲 と . T h e s e n s i t M t i e s o f t h e p a m l l e l s t r u c n 】 x e o f b a l a n c e d r c a l i z a t i o n s a n d t h e p a r a l l e I s l r u c n 』 r e o f m i n i m u m r o u n d o f f n o i s e r e a l i z a t i o n s a l s o d e p e n d o n 噂 , b u t t h e J u l y l 9 8 4 . [8]M・KawamataandmHiguchi,‘§Auninedapproachto U l e o P t i m a l s y m h e s i s o f f x e d p o i n t s t a t e S p a c e d i g i t a l m t e r s , ? , 正 E E T r a n s ・ A c o u s t . , s p e e c h , S i g l a l P r o c e s s ‐ i n g , v o l ・ A S S P 3 3 , n o , 4 , p p 、 9 1 1 9 2 0 , A u g u s t l 9 8 5 . variationsofthesensitMtiesaresmallerthanlhatofthe paraⅡelslructureofdirectfmnⅡ、ItisconEnnedthatthe v a r i a b l e d i g i t a l m t e r s d e s i g n e d b y t h e p r o p o s e d m e t h o d p r e ‐ servetheinvarianceofthestatisticalsensitMtiesunderthe [9]S、K・Mitm,YNeuvoandH・Roiyainen,"Designof f i e q U e n c y t r a n S f b r m a t i O n o f b a l a n c e d r e a l i z a t i O n s a n d m i n ‐ r e c u r s i v e d i g i t a l n l t e r s w i l h v a r i a b l e c h a r a c t e r i s t i c s , ? , imumround-offnoiser唱塑、l伽tmn里 7.CONCLUSIONS I n t e m a t i o n a l J O u m a l o f C i r c u i t T h e O I y a n d A P p l i c a ‐ t i o n s , v o l , 1 8 , p p , 1 0 7 1 1 9 , 1 9 9 0 . [ 1 0 ] N ・ M u r a k o s h i , E ・ W a t a n a b e a n d A 、 N i s h i h a m , ‘ § A s y n ‐ thesisofvariable皿digital、胸sj,,IEICETrans・Fm damentals,vol・E75-A,no、3,卯.362-368,March W e h a y e p r o p o s e d t h e d e s i g n m e d l o d o f v a r i a b l e d i g i t a l mtersbased皿balancejTE塑Qlizatio醒麺dminimumround- l 9 9 2 . o f f n o i s e r e a l i z a t i o n s ・ T h e v a r i a b l e d i g i t a l m 0 e r s d e 罰 g n e d bythepmpoSedmed1odhayeverylowcoefEcientsensi‐ t i v i t i e s , a n d p r e s e r v e t h e i n v a r i a n c e o f t h e s e n s i t i v i t y u n d e r t h e f i E q U e n c y t r a n s f b n n a t i o n s o f t h e p r o t o 可 p e r e a l i z a t i o n s ・ N U m e g i c a l e m 【 a m p l e s s h o w t h e e f f も c ロ v e n e s s o f t h e p m p o s e d medlod. RErERENCES [ 1 ] G ・ S t o y a n o v a n d M ・ K a w a m a t a , “ V a r i a b l e d i g i t a l n l ‐ t e r s ; , J O u m a l o f S i g n a l P r o c e s s m g , v o l 、 1 , n o 、 4 , p p 、 2 7 5 2 9 0 , J u l y l 9 g 7 . 158
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