A Dual-Mode Synchronous/Asynchronous CORDIC Processor

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1 A Dual-Mode Synchronous/Asynchronous CORDIC Processor Eckhard Grass, Bodhsatya Sarker and Koushk Maharatna IHP Im Technologepark Frankfurt (Oder), Germany grass@hp-mcroelectroncscom Abstract For applcaton n a software defned rado a CORDIC processor has been developed that can operate both n synchronous and asynchronous mode Each mode of operaton has advantages and drawbacks Dependng on the actual applcaton, an optmal trade-off can be acheved by selectng the mode of operaton that fts best wth system demands We beleve that for a system developer ths addtonal degree of freedom sgnfcantly ncreases the applcaton space The desgn has been mplemented on a 05 µm SGe:C BCMOS process Smulaton results usng post-layout extracted parameters ndcate that the desgn s compettve both wth purely synchronous and purely asynchronous mplementatons 1 Introducton Fourth generaton (4G) wreless and moble systems are currently the focus of research and development They wll allow new types of servces to be unversally avalable to consumers and for ndustral applcatons Broadband wreless networks wll enable packet based hgh datarate communcaton sutable for vdeo transmsson and moble Internet applcatons Ths paper s based on a project whch ams to develop a sngle-chp wreless broadband communcaton system n the 5 GHz band, complant wth the Hperlan/ [1] and IEEE 8011a [] standards Both standards specfy broadband communcaton systems usng Orthogonal Frequency Dvson Multplexng (OFDM) wth data rates rangng from 6-54 Mbt/s The duraton of one OFDM symbol, whch can carry 4 to 16 bts of nformaton s 4 µs For such a system a large porton of the dgtal baseband processor has to operate on complex sgnals whch are sampled at a rate of 0 MS/s The most dffcult to mplement functonal blocks of the baseband processor are FFT/IFFT processor, Vterb Decoder, and Numerc Controlled Oscllator (NCO) The NCO s used to perform a frequency correcton on the ncomng sgnals A Coordnate Rotaton Dgtal Computer (CORDIC) offers an elegant way of mplementng an NCO To open a broad market for consumer products, low cost of the requred hardware s essental One way to realse low-cost systems s to reduce system complexty and mplement all functons of a wreless modem n a sngle chp Potentally, a sngle-chp soluton s also advantageous n terms of performance and power dsspaton when compared wth mult-chp mplementatons Fewer sgnals have to be routed va slow and power-hungry pad drvers In addton, short nterconnectons allow faster operaton of the system Our n-house 05 µm SGe:C BCMOS technology enables the ntegraton of complex dgtal baseband functonalty together wth the analog RF front-end (AFE) Ths process technology forms the bass for the sngle-chp mplementaton of a complete wreless modem Ths sngle-chp wreless modem wll hence comprse analog crcutry, processor cores as well as dedcated hardware For the sngle-chp modem, a number of crtcal desgn problems have to be solved One major ssue s the reducton of crosstalk from the dgtal part to the analog crcutry The hgh energy whch s delvered by clock drvers can nfluence the AFE and deterorate ts sgnal to nose (S/N) rato The most predomnant crosstalk mechansms are substrate nose, supply voltage varatons as well as electro-magnetc effects (EMR) [3] Secondly, system power dsspaton s a crtcal ssue Snce many functonal blocks are actve for a lmted perod only, there s potental to reduce power dsspaton drastcally A token-flow approach [4] s adopted by addng sgnals whch ndcate the valdty of data at nputs and outputs of functonal blocks Dstrbutng the control of data flow, ths token-flow approach allows easy mplementaton of clock gatng and lends tself to asynchronous mplementatons of blocks Currently a number of extensons to the standards for 5 GHz broadband communcaton systems are beng dscussed by the standardsaton commttee Therefore, t

2 w ou ld b e b e n e fc a l fo r c u s to m e rs a n d s upp le rs o f tho s e s y s te m s f h a rd w a re w ou ld b e a b le to acc o m m od a te fu tu re c o mm un ca to n s ta nd a rd s A s o ftw a re d e fn e d ra d o, n p rn c p a l, o ff e rs th s n ee d e d fle x b lty H e re w e e x te nd th e d ea o f s o ftw a re d e fn e d ra d o s u c h th a t th e s o ftw a re d e v e lop e r ca n c ho s e ce rta n h a rd w a re acce le ra to rs to op e ra te e th e r s yn c h ro n ou s ly o r a s yn c h ro n ou s ly T h s a dd ton a l d e g ree o f fr e e do m n c rea s e s th e a pp lca ton s p ace o f th e c o m p le te s y s te m F u rth e rm o re, a ls o w th a g v e n s ta nd a rd, d ff e re n t a pp lca ton s ce n a ro s re q u re d ffe re n t p ro p e rte s o f th e h a rd w a re W e a re c onv n c e d th a t h a v ng th e fr ee d o m fo r ce rta n k e y b lo c k s o f c h o o s ng b e tw ee n s yn c h ro nou s a n d a s yn c h ro n ou s o p e ra ton a llo w s fn d ng a s u ta b le tra d e -o ff b e tw ee n po w e r d ss p a ton, c ro ss ta lk, d a ta th rou g hpu t, la te n c y a nd (s e lf-) te s ta b lty A s a n e x a m p le, t m a y b e d e s ra b le w th v e ry s m a ll s gn a l a m p ltud e s on th e rece v e r to re d u ce th e m p ac t o f th e C L K s gn a l by op e ra tng th e C O R D IC n a s yn c h ro nou s ( w h s p e r ) m od e O n th e o th e r h a nd, f th e rece v e s g n a l s v e ry s trong a nd fo r c on tnu o u s rece p to n o f d a ta, th e m o re p o w e r e ff c e n t s y n c h ro n ou s op e ra to n n c o n jun c to n w th c lo c k -g a tn g ca n b e u s e d T h e p a p e r d s c u ss e s a du a l m od e m p le m e n ta ton o f a C O R D IC p ro ce ss o r a nd c o m p a re s th e m a n p a ra m e te rs o f th s d e s gn w th a pu re ly s yn c h ro n ou s a s w e ll a s a pu re ly a s yn c h ro n ou s m p le m e n ta ton It s d e m on s tra te d th a t a s yn c h ro n ou s c rc u ts ca n b e m p le m e n te d n VHD L u s ng s yn c h ro n ou s too ls VHD L s yn th e s s a nd la y ou t a re s uppo rte d by a nu m b e r o f a ttrbu te s a n d c on s tra n ts w h c h a re s p ec fc to th e to o ls u s e d T h e h a rd w a re ov e rh ea d o f th e du a l m o d e C O R D IC p ro ce ss o r c o m p a re d to th e s yn c h ro n ou s v e rs o n s a bou t 1 % It s ho w s a s m la r p e rf o rm a n ce w h e n c o m p a re d to ts s yn c h ro n ou s c oun te rp a rt T h e re m a n n g p a p e r s s tru c tu re d a s fo llo w s : In S ec ton th e a rc h tec tu re o f th e C O R D IC p ro ce ss o r s d e rv e d fro m th e a lgo rth m c s p ec fca ton S ec ton 3 d s c u ss e s s tra te g e s fo r m p le m e n tn g th e d e s gn In S ec ton 4, a n u m b e r o f ss u e s re la te d to th e d e s gn flo w fo r a s yn c h ro n ou s m p le m e n ta ton s a nd th e u s e d too ls a re h gh lgh te d F n a lly n S ec ton 5 th e re s u lts o f o u r m p le m e n ta ton a re re p o rte d a n d c on c lu s on s a re d ra w n n S ec ton 6 C O RD I C P r o c e ss o r A rc h te c tu re T h e v e rs a tlt y o f C O R D IC s upp le m e n te d w th ts m u ltp le rle ss fea tu re h a s p ro p e lle d d g ta l d e s gn e rs to n c o rp o ra te t n s e v e ra l a rth m e tc p ro ce ss o rs a s w e ll a s to fo rm u la te a n d m p le m e n t m a n y m od e rn D SP a lgo rth m s [5-9 ] th a t re qu re c o m p le x n u m b e r m u ltp lca ton, e le m e n ta ry p la n e ro ta ton o r la ttce op e ra ton s In e ss e n ce, th e C O R D IC acce p ts th ree npu t v a ra b le s x0, y 0, a n d z 0 a nd g e n e ra te s th e o u tpu t x 1, y 1 a n d z 1 It ca n b e o p e ra te d n tw o d ff e re n t m o d e s v z, th e ro ta to n a nd th e v ec to rn g m od e w h e re, v a ra b le s z a n d y a re fo rc e d to ze ro re s p ec tv e ly E ac h o f th e s e m o d e s ca n b e u tls e d n c rc u la r, ln ea r a nd h yp e rbo lc c o o rd n a te s y s te m s T h s llu s tra te s th e v e rs a tlt y o f th e C O R D IC a nd e xp la n s w hy t s re g a rd e d a s on e o f th e m a jo r too ls n d g ta l c o m p u te r a rth m e tc H o w e v e r, fo r o u r p u rp o s e t s s u ff c e n t to u s e th e ro ta ton a n d v ec to rng o p e ra ton s n th e c rc u la r c oo rd n a te s y s te m on ly H e n ce w e c o n ce n tra te on th e rea lza ton o f a c rc u la r C O R D IC p ro ce ss o r It s op e ra ton re le s on th e p rn c p le o f v ec to r ro ta ton T h e b a s c op e ra ton s d e s c rb e d b e lo w : T h e ro ta ton o f a v ec to r [ x0 y 0 ] T n th e C a rte s a n c o - o rd n a te s y s te m ca n b e d e s c rb e d a s, Ø x1 ø Œ œ º y 1 ß Ø cos q Œ º - sn q sn q ø Ø x 0 ø cos q œ Œ œ ß º y 0 ß w h e re, [x 1 y 1 ] T s th e fn a l v ec to r a n d q s th e a ng le o f ro ta ton ( q ) In th e a bov e e qu a to n t w a s a ss u m e d th a t th e ro ta to n ta k e s p lace n c lo c k w s e d rec ton N o w, th e ta rg e t a n g le q ca n b e e xp re ss e d a s th e s u mm a ton o f a nu m b e r o f e le m e n ta ry a ng le s a s o th a t b - 1 q s a ( ) 0 w h e re, b s th e b t p rec s on o f th e m a c h n e n w h c h th e op e ra to n s to b e m p le m e n te d a n d s { 1, - 1 }, s kno w n a s th e d rec ton o f ro ta ton In th e c o nv e n to n a l c rc u la r C O R D IC op e ra ton a s e xp re ss e d a s, a tan ( ) (3 ) S ub s ttu tng ( ) n (1 ) a nd u b - Ø x 1 1 ø Œ œ º y 1 ß a nd z 0 cos Ø 1 Œ Œº - s s ng (3 ) o n e m a y w rte - s 1 - ø Ø x œ Œ œß º y ø œ ß (1 ) a (4 ) - 1 s (5 ) Sgn [ a - a r r z + s (6 ) T h e p h y s ca l m ea n ng o f th e s e e qu a to n s s th a t th e ta rg e t a ng le g v e n a s th e z v a ra b le s d rv e n to ze ro by to -a nd -fr o m o ton o f th e v ec to r A t ea c h te ra ton s te p th e s gn o f th e re s du a l a ng le (s ) s ca lc u la te d a n d acc o rd ng ly th e v ec to r s ro ta te d n c lo c k w s e o r c o u n te r c lo c k w s e d rec ton F ro m th e h a rd w a re m p le m e n ta ton p o n t o f v e w, th s op e ra ton c a n b e c a rr e d ou t u s ng on ly s h fta nd -a d d ton o p e ra ton s s n ce m u ltp lca ton o f a n y - qu a n tty b y s a s h ft o f th e b n a ry re p re s e n ta ton o f th e qu a n tty b y -b t to th e rgh t H o w e v e r, th e c onv e n ton a l C O R D IC a lgo rth m s u ff e rs fr o m tw o p rn c p a l d ra w b ac k s, v z, th e re qu re m e n t o f a s ca le fac to r a nd th e s lo w ra te o f c onv e rg e n ce In o rd e r to ov e rc o m e th e s e d ff c u lte s w e d e v e lo p e d a m od fe d C O R D IC a lgo rth m w h c h s d e s c rb e d b e lo w ]

3 In o u r s c h e m e, th e ta rg e t a n g le s ac h e v e d b y ro ta tng th e v ec to r on ly n on e d rec ton (e th e r c lo c k w s e o r c oun te rc lo c k w s e ) th ro u gh s u ff c e n tly s m a ll e le m e n ta ry a ng le s a T h e y c a n b e e xp re ss e d b y th e fo llo w ng e q u a ton s n a - (7 ) T h s a ss u m p ton m po s e s a lo w e r lm t o n th e v a lu e o f te ra ton nd e x T h e lo w e r lm t o f ca n b e ca lc u la te d fr o m th e fac t th a t th e la rg e s t v a lu e n e g lec te d n th e e xp a n s on o f th e s n e s e re s s ( 3 + log 6 ) - ( 3 + / T hu s, th e c o nd ton fo r p re s e rv n g th e acc u rac y d u rn g s u c h a pp rox m a ton b ec o m e s b o r Ø (b ) / 3 ø p (13 ) (s n ce ca n a d o p t on ly n te g e r v a lu e s ) T h e u p p e r lm t o f s b - 1 s n ce a rgh t s h ft o f a ny b -b t nu m b e r b y b -b t w ll re s u lt n m ac h n e ze ro U n d e r th e a bov e c rc u m s ta n ce s, th e v a lu e s o f s n a a n d c o s a ca n b e e xp re ss e d a s, s n a - (14 ) c o s a ( + 1 ) (15 ) C on s d e rng th e a p p ro x m a ton, s + 1 (s n ce th e s gn o f th e re s du a l a ng le s a lw a y s th e s a m e fo r a ll te ra ton s ) a nd c lo c k w s e ro ta to n o f th e v ec to r, e qu a to n (1 ) m a y b e re w rtte n a s b - 1 Ø x 1 ø Œ œ º y 1 ß p Ø1 - Œ - º - - ( + 1 ) 1 - ) - - ( + 1 ) ø Ø x œ Œ ß º y ø œ ß (16 ) E qu a ton (16 ) ca n b e rea lze d n p rac tce u s ng on ly s h ft-a nd -a d d op e ra ton s F u rth e rm o re, c on tra ry to e qu a ton (4 ), n o s ca lng te rm a p p ea rs n e qu a to n (1 6 ), w h c h m p le s a s ca lng fr ee C O R D IC op e ra ton T h e e lm n a ton o f th e s ca lng te rm s a s gn fca n t s te p, s n ce t a vo d s th e re q u re m e n t o f e x tra po s t-p ro ce ss ng c rc u try a nd s a v e s p ro ce ss ng tm e T h e d a ta p a th c o m pon e n ts fo r m p le m e n tng th e op e ra to n o f e qu a to n (1 6 ) a re s h o w n n F gu re 1 T h s ca n b e v e w e d a s a n e le m e n ta ry ro ta to n a l s ta g e re nd e rn g a ro ta to n o f a to ts npu t v ec to r T h s re qu re s fo u r s h fte rs, tw o s ub trac to rs a n d tw o a dd e r / s ub trac to rs a s d a ta p a th c o m po n e n ts T hu s, c o m p a re d to th e d a ta p a th o f th e e le m e n ta ry ro ta to n a l s ta g e o f a n un s ca le d C O R D IC, t re qu re s tw o s h fte rs a nd tw o s ub trac to rs m o re H o w e v e r, fo r th e e le m e n ta ry ro ta ton a l s ta g e s c o rr e s po n d ng to b /, th e s e e x tra s h fte rs a nd s ub trac to rs ca n b e o m tte d a s th e rg h t s h ft o f th e n p u t qu a n tty b y ( + 1 ) b e c o m e s m ac h n e z e ro a n d thu s d o e s no t a ff ec t th e ac c u rac y a ny m o re H o w e v e r, e q u a ton (7 ) m p le s th a t th e c onv e rg e n ce ra ng e o f th e C O R D IC w ll b e s m a ll T h e c onv e rg e n ce ra ng e ca n b e e x te nd e d ov e r th e e n tre c o o rd n a te s p ace by re p ea tng ce rta n te ra to n s te p s a nd by e x p lo tng th e s y mm e try o f th e c o o rd n a te a x e s In p rn c p le, w e fo u nd th a t to c ov e r th e w ho le c oo rd n a te s p ace, t s s u ff c e n t to c o m pu te th e a ng le s n th e n te rv a l [0, 5 ] T h e re s u lts o f C O R D IC ro ta ton s fo r a n y a ng le b e tw ee n 5 a n d ca n b e e x tra po la te d fro m th e re s u lt o f a ro ta ton c o rr e s po n d ng to [0, 5 ] x (- 1 ) y (- 1 ) x (- 1 ) y (- 1 ) b t s h fte r + + x () -b t s h fte r -b t s h fte r + 1 -b t s h fte r y () F gu r e 1 B lo c k d a g r a m o f a n e le m e n ta r y d a ta p a th c o m pon e n t ( r o to r s ta g e ) T h e c o m p le te C O R D IC p ro ce ss o r ca p a b le to op e ra te n th e ro ta ton a nd v ec to rng m od e ov e r th e e n tre c oo rd n a te s p ace h a s th ree p a rts, v z, s gn a nd d o m a n d e tec to n c rc u try, b a s c C O R D IC s ec ton w th c onv e rg e n ce ra ng e 5 a n d th e o u tpu t c rc u try T h s s s ho w n n F gu re T h e s gn a n d do m a n d e tec ton c rc u try (d e s gn a te d a s s gn_d e tec t) d e tec ts th e s gn o f th e npu t a n g le, q u a d ra n t a nd th e d o m a n (po s ton o f th e v ec to r n a qu a d ra n t) o f th e npu t v ec to r A cc o rd ng ly, th e n p u t v a ra b le s a re p ro ce ss e d a nd th e y m a y b e re g a rd e d a s th e m o d fe d n pu t to th e b a s c C O R D IC p ro ce ss o r T h e d e tec ton o f s gn a n d do m a n, p re -p ro ce ss ng a n d a p p ro p ra te npu t o p e ra ton s re qu re tw o c lo c k c y c le s T h e b a s c C O R D IC p ro ce ss o r (d e s gn a te d a s c o rd_p p e ) s a b t p a ra lle l p p e ln e d m p le m e n ta ton w th n te rn a l w o rd le n g th o f 1 6 b ts F ro m e qu a ton (13 ), th e v a lu e o f p fo r a 16 -b t m p le m e n ta to n s a pp rox m a te ly 4 T h e p p e ln e s 1 7 s ta g e s long w h e re th e e le m e n ta ry ro ta ton a l s ta g e c o rre s po n d ng to - 4 s re p ea te d 6 tm e s O u r a m s to a p p rox m a te z a nd y fo r th e ro ta ton a n d th e v ec to rng m od e re s p ec tv e ly w th n a n acc u rac y o f O ( - 16 ) In th e c on s tru c to n o f th e e le m e n ta ry ro ta ton a l s ec ton s, th e c o nv e n to n a l tw o s c o m p le m e n t a dd ton s u s e d T h e n fo rm a ton a bou t th e qu a d ra n t a nd do m a n o f th e n ta l v ec to r d e tec te d a t th e s gn_d e tec t m odu le s tra n s fe rr e d b e tw ee n tw o c on s ec u tv e s ec ton s o f

4 th e p p e ln e a lo n g w th th e d a ta n a lo ca l re g s te r tra n s fe r m a nn e r T h s m ea n s th a t th e d a ta n d ff e re n t s ec ton s o f th e p p e ln e h a s a to k e n a ttrbu te d to t th a t e ss e n ta lly ca rr e s th e n fo rm a ton a b ou t ts n ta l qu a d ra n t a nd do m a n A s ng le b t s gn a l, s gn a nd tw o -b t s gn a ls do m a n a nd qu a d a re u s e d to tra n s fe r th s n fo rm a ton b e tw ee n s u cce ss v e ro ta ton a l s ta g e s x 0 y 0 z 0 m o d e d a ta _v a ld x_ n y _ n z _ n x_ o u t y _ou t z _ o u t s gn_d e te c t ( p p e ln e s ta g e s ) q u a d d o m a n m o d e _ n c o rd_p p e (17 p p e ln e s ta g e s ) q u a d d o m a n d a ta _ v a ld _ n m o d e _ ou t d a ta _ v a ld _ o u t ou tpu t c rc u try (1 p p e ln e s ta g e ) s ta g e ) x 1 y 1 z 1 d a ta _v a ld _ o u t F gu r e S tru c tu re o f t h e s y n c h r onou C O RD IC p r o c ess o r s p p e ln e d T h e ou tp u t c rc u try c on s s ts o f a fx e d s ca lng un t o f 1 / a n d a c o up le o f a dd e r/s ub trac to r un ts T h s c rc u t po s t-p ro ce ss e s th e ou tp u t v ec to r e m e rg n g fro m th e c o rd _ p p e s ec ton n ac c o rd a n ce w th th e n fo rm a to n on th e n ta l v ec to r d e rv e d b y th e s g n_d e tec t c rc u try a n d g e n e ra te s th e fn a l ou tpu t A ll th e op e ra to n s a t th e ou tpu t a re c o m p le te d n on e c lo c k c y c le A m od e s gn a l s a ls o p ro v d e d w th th e p ro ce ss o r T h e log c H IGH s ta te o f th e m od e s gn a l m a k e s th e p ro ce ss o r to op e ra te n ro ta ton m od e w h e rea s ; ts log c L O W s ta te m ea n s th e v ec to rn g m o d e o f op e ra ton s ca rr e d ou t A p a rt fr o m th e m o d e s gn a l, th e (s yn c h ron o u s ) p ro ce ss o r s a ls o e qu pp e d w th a n n pu t d a ta _v a ld a nd a n ou tp u t d a ta _v a ld s g n a l th a t nd ca te s th e v a ld ty o f n pu t a n d ou tpu t d a ta re s p ec tv e ly 3 I m p le m e n ta to n S tr a te g e s m p le m e n te d u s n g a d a ta p a th a rc h tec tu re T h e m o s t c o m p le x b lo c k s n th s d a ta p a th a re C O R D IC -P ro ce ss o r, FF T p ro ce ss o r a nd V te rb D ec o d e r It s ou r go a l to m a k e th e d e s gn o f th e c o m p le te b a s e b a n d p ro ce ss o r a s m u c h a s po ss b le tec hno lo g y nd e p e nd e n t T h e re fo re t h a s b ee n d ec d e d to u s e VHD L fo r d e s g n e n try th ro ug h ou t In o rd e r to d e -ce n tra ls e c on tro l, a to k e n -f lo w a p p ro ac h s u s e d T h e v a ld ty o f d a ta to k e n s s nd ca te d u s n g a dd ton a l s gn a ls a t th e n p u t a nd ou tp u t o f f un c to n a l b lo c k s T o a llo w p e rfo rm a n c e c o m p a rs on, th ree d ff e re n t c rc u ts w e re d e s gn e d : C rc u t M IX -C O R D IC ca n op e ra te b o th s yn c h ro n ou s ly a s w e ll a s a s yn c h ro nou s ly, c rc u t A S Y -C O R D IC op e ra te s a s yn c h ro n ou s ly a n d c rc u t S YN -C O R D IC s a no rm a l s yn c h ron o u s d e s gn w th e x tra s gn a ls to nd ca te d a ta v a ld ty a t npu ts a n d ou tpu ts Dual-mode crcut (MIX-CORDIC): A 4 -ph a s e h a nd s h a k e p ro to c o l s d e p loy e d s n ce t a llo w s th e u s e o f s ng le e dg e trgg e re d flp -f lop s T h e h a nd s h a k e c rc u t o f th e M IX -C O R D IC d e s gn s m p le m e n te d u s n g tw o C -e le m e n ts a s s ho w n n F gu re 3 In o rd e r to a vo d c on fu s on b e tw ee n th e typ e o f th e s gn a l ( npu t o r ou tp u t ) a nd th e po rt o f th e h a nd s h a k e c rc u t, n th s p a p e r w e d e fn e th e npu t p o rt o f th e h a n d s h a k e c rc u t a s po rt a a nd th e ou tpu t p o rt s la b e lle d p o rt b P o r t a re q _a C Element C Element re q _b a c k _ b F gu r e 3 H a nd s h a ke c r c u t f o r c r c u t M IX - C O RD IC P o r t b U n fo rtun a te ly o u r ce ll lb ra ry do e s n o t c on ta n a ny C - e le m e n ts R S flp -f lop s a re a ls o no t a v a la b le C on s e qu e n tly th e C -e le m e n ts h a d to b e c on s tru c te d fr o m g a te s W e m od e lle d th e b e h a v o u r o f th e C -e le m e n ts n VHD L a nd s yn th e s ze d th e m T h e y w e re th e n u s e d a s c o m pon e n ts n th e VHD L d e s gn It w a s fo u nd th a t th e S ynop s y s s yn th e s s e r do e s m e rg e th e fun c ton a lty o f a nu m b e r o f C -e le m e n ts n to a s ng le c rc u t d u rng op tm za ton s ta g e S n ce n th s p ro ce ss tm ng c on s tra n ts a re v o la te d, th e C -e le m e n t c o m p on e n ts h a d to b e p ro tec te d fr o m o p tm s a ton u s ng do n t tou c h a ttrbu te s T h e nd v du a l C -e le m e n ts w e re e x te n s v e ly s m u la te d u n d e r a v a re ty o f c ond to n s a nd w e re fo u nd to b e fun c ton n g c o rr ec tly th rou g hou t B e fo re a p p lca ton n th e C O R D IC p ro ce ss o r, o u r h a nd s h a k e c rc u t w a s a ls o e x te n s v e ly te s te d u s ng a n a s yn c h ro n ou s F IF O T h e m a n c o m p o n e n ts o f th e O F D M b a s e b a nd p ro ce ss o r f o r H p e rla n / a nd IEEE a ca n ea s ly b e

5 n p u t o u tp u t C lo c k _g lo b a l s y n c r e q _ a _ t o p F gu r e 4 S ng le s ta g e o f a n a s y mm e le m e n t e tr c d e lay F o r c o m p le to n d e tec ton a bo u nd e d d e la y m e th od w a s u s e d d e p lo y ng a s p ec a l a s y mm e trc d e la y e le m e n t w th s ho rt rec ov e ry tm e A s ng le s ta g e o f th s d e la y e le m e n t s s ho w n n F gu re 4 A lte rn a tv e ly a ca s ca d e o f tw o -npu t AND g a te s ca n b e u s e d w h e re b y th e npu t s gn a l s c onn ec te d n p a ra lle l to a ll AND g a te s H o w e v e r th e n p u t ca p ac ta n ce o f th s s tru c tu re s ca le s w th th e le ng th o f th e ca s ca d e w h c h c rea te s a h g h ca p ac tv e lo a d on th e npu t s gn a l m a k ng th e d e la y h gh ly un p re d c ta b le A ca s ca d e o f te n s ta g e s o f th e s tru c tu re s ho w n n F gu re 4 d e lv e rs n ou r c rc u t a d e la y o f 13 4 n s a nd 4 8 n s fo r a rs ng a nd a fa llng tra n s ton re s p ec tv e ly (m ea s u re d a fte r b ac k a nno ta to n fro m la y o u t) T h e rec ov e ry tm e fo r th e fa lln g tra n s ton o f s u c h a ca s ca d e s e qu v a le n t to th e d e la y o f fou r nv e rte rs o f a s ng le s ta g e e a pp rox m a te ly 0 8 n s n ou r ca s e T h s rec ov e ry tm e s d m e n s on e d s u c h th a t t s w e ll b e lo w a ny p o ss b le re s pon s e tm e o f th e h a n d s h a k e c rc u try T h e ac h e v e d ra to b e tw ee n rs e tm e a n d fa ll tm e o f n ea rly 3 :1 y e ld s a s ub s ta n ta l s p ee dup o f th e h a nd sh a k e c rc u t w h e n c o m p a re d to u s ng a n o rm a l n v e rte r c h a n T h e c rc u t do e s no t s u ff e r f ro m h gh n pu t ca p ac ta n ce w h c h a ls o k ee p s ts po w e r d ss p a ton lo w T o a llo w s w tc h ng b e tw ee n s yn c h ro nou s a n d a s yn c h ro n ou s m o d e, fo r th e du a l-m od e c rc u t a dd ton a l log c s n trodu c e d to e th e r e n a b le th e a s yn c h ron o u s h a nd s h a k e c rc u t o r th e s yn c h ro n ou s c lo c k to d rv e th e p p e ln e s ta g e s a s s ho w n n F gu re 5 If th e s yn c s gn a l s n log c h g h s ta te, th e c lo c k s gn a l C lo c k_g lo b a l s e n a b le d v a th e NAND g a te a t th e n p u t T h e s tru c tu re s c on fg u re d to w o rk n s yn c h ro n ou s m od e A t th e s a m e tm e r e q_ a _ top s d s a b le d v a th e NO R g a te d rv n g r e q_ a T h e c o m p le te h a nd s h a k e c rc u t s d s a b le d by fo rc ng r e q _ a to log c lo w O n th e c on tra ry f s yn c s n lo g c lo w s ta te, th e g lo b a l c lo c k C lo c k_g lob a l s d s a b le d a nd th e c rc u t s c on fg u re d fo r a s yn c h ro nou s o p e ra ton by e n a b lng r e q_ a _ top to d rv e s gn a l r e q _ a C u rre n tly th e re s no dyn a m c s w tc h ng o f m o d e c o n s d e re d e s gn a l s yn c s a llo w e d to c h a n g e s ta te on ly w h e n C lo c k _ g lob a l s lo w a nd th e p p e ln e s c o m p le te ly s e ttle d T h s p re v e n ts po te n ta l h aza rd s d u rn g m o d e -c h a n g e o p e ra ton s w h c h c ou ld re s u lt n und e fn e d s ta te s o f th e s y s te m x 0 y0 z P p e l n e s ta g e 1 P p e l n e s ta g e P p e l n e s ta g e x 1 y1 z 1 r e q _ a r e q r e q r e q H a n d s h a k e 1 a c k H a n d s h a k e a c k a c k H a n d s h a k e 0 r e q _ b F gu r e 5 S c h e m a tc d a g r a m o f du a l-m od e C O RD IC a c k _ b A s y n c h r o n o u s c r c u t u s n g la tc h e s (A S Y -C O R D I C ): A s ec ond v e rs o n u s e s la tc h e s n s tea d o f f lp -f lop s fo r bu ff e rng th e d a ta S n ce la tc h e s a re s m a lle r a n d c on s u m e le ss po w e r, th s ca n po te n ta lly re du ce s lc on a rea a nd po w e r d ss p a ton F o r ou r a pp lca ton, th e s e m -d ec o up le d la tc h c on tro lle r do e s re p re s e n t a rea s on a b le c o m p ro m s e b e tw ee n h a rd w a re c o m p le x ty a n d p e rfo rm a n c e [1 1 ] T h e a s y mm e trc C -e le m e n ts u s e d n th s c on tro lle r w h e re d e s gn e d n th e s a m e w a y a s d e s c rb e d a bov e T h e re s u ltng h a nd s h a k e c rc u try w a s a g a n te s te d n a n a s yn c h ro n ou s F IF O P e rf o rm a n ce m ea s u re m e n ts b a s e d o n s m u la ton s u s ng po s t-la y o u t e x trac te d p a ra m e te rs a re g v e n n T a b le 1 S y n c h r o n o u s c r c u t (S Y N -C O R D I C ): T h e s yn c h ron o u s m p le m e n ta ton w a s d on e u s n g e x ac tly th e s a m e p p e ln e a rc h tec tu re a s fo r th e o th e r tw o d e s gn s In o rd e r to s up p o rt th e to k e n -f lo w a p p ro ac h d a ta v a ld s gn a ls w h e re n trodu ce d a t n p u t a nd ou tpu t po rts o f th e p p e ln e A p e rf o rm a n ce c o m p a rs on o f th e th ree d ff e re n t m p le m e n ta ton s s g v e n n T a b le 1 In te rf ac n g o f th e a s yn c h ronou s ou tp u ts w th th e s yn c h ro n ou s e n v ron m e n t s don e u s ng a n a dd ton a l (s yn c h ro nou s ) D -typ e flp -f lo p fo r th e re q u e s t s gn a l a t th e ou tpu t ( r e q _b n F gu re 5 ) U s ng a typ ca l c lo c k

6 fr e qu e n c y o f 0 M H z fo r th e s yn c h ro n ou s c rc u t w ll -55 re s u lt n a v e ry lo w p rob a b lty o f m e ta s ta b lty o f P 10 o r a n M T B F o f 3x y ea rs [1 ] F o r ty p ca l a pp lca ton s th s w ll no t re qu re a n y a d d ton a l m ea s u re s to re du ce m e ta s ta b lty p rob a b lty fu rth e r 4 D e s g n -F lo w Iss u e s A ll v e rs on s o f th e C O R D IC p ro ce ss o r w e re m od e le d n VHD L a nd a re fu lly s yn th e s ze d F o r s m u la ton a n d fun c ton a l v e rfca ton w e u s e d M od e ls m fro m M e n to r G ra ph c s S yn th e s s w a s ca rr e d ou t w th S y n op s y s D e s gn A n a ly ze r a n d th e la yo u t w a s g e n e ra te d u s ng S lc on E n s e m b le fr o m C a d e n ce T h e d e s gn w a s m a p p e d to ou r n -h o u s e 0 5 m m S G e :C B C M O S tec hno logy u s ng th e IH P d g ta l d e s g n k t T h e no m n a l s u pp ly vo lta g e fo r c o re ce lls s 5 V F ou r m e ta l la y e rs a re a v a la b le fo r n te rc o n n ec t T h e C O R D IC p ro ce ss o r u tlze s on ly th e C M O S p a rt o f o u r B C M O S tec hno logy B o th, fo r s yn th e s s a nd la you t s o m e a ttrbu te s a nd c on s tra n ts w e re a dd e d to s up p o rt th e a s yn c h ro nou s m p le m e n ta ton A fte r s y n th e s s th e g a te le v e l V e rlog n e tls t g e n e ra te d by th e D e s gn A n a ly ze r w a s re -s m u la te d n c lud ng th e S ta nd a rd D e la y F o rm a t (S D F ) f le T h e c o rr ec t tm ng b e h a v o r o f th e p ro ce ss o r n b o th th e ro ta ton a n d v ec to rng m od e w a s ob s e rv e d F o r d rv n g th e lo ca l c lo c k s gn a ls o f th e nd v d u a l s ta g e s w e ftte d th e c lo c k s gn a l o f e v e ry p p e ln e s ta g e w th a n a pp rop ra te c lo c k b u ff e r T h s bu ffe r w a s s yn th e s ze d s e p a ra te ly a nd p ro tec te d w th th e h e lp o f don t tou c h a ttrbu te s T h e a rea o f a tra n s p a re n t la tc h u s e d n th e A S Y - C O R D IC d e s gn s 118 m m a nd th e a rea o f th e D -ty p e flp - flop s u s e d n M IX -C O R D IC a nd S YN -C O R D IC s 180 m m (acc o rd n g to IH P ce ll lb ra ry ) T h e la tc h b a s e d v e rs on ca n th e re fo re po te n ta lly re s u lt n s m a lle r s lc on a rea T h e d e s g n flo w fo r d e s g n A S Y -C O R D IC s d e s c rb e d n d e ta l b e lo w In p rn c p a l, th e s a m e a p p ro ac h w a s u s e d fo r th e M IX -C O R D IC c rc u t T h e no rm a l s yn c h ro n ou s m p le m e n ta ton d o e s no t re q u re a n y e x tra m ea s u re s a nd s th e re fo re g e n e ra te d u s n g th e s ta nd a rd s yn th e s s flo w w th c lo c k -tree g e n e ra ton du rn g la y o u t ph a s e T h e la tc h e s w th n eac h p p e ln e s ta g e a re c o n tro lle d w th a s gn a l lt w h c h ca u s e s th e la tc h to b e tra n s p a re n t w h e n lo w a n d n h o ld s ta te w h e n h gh S n ce th e d a ta p a th s n to ta l 50 b t w d e, th e lt s gn a l m u s t h a v e a rea s on a b le d rv e s tre ng th T h e re fo re a bu ff e r s u s e d fo r eac h p p e ln e s ta g e to d rv e s gn a l lt T o k ee p th e s lc on a rea a s s m a ll a s po ss b le, w e u s e d la tc h e s w th a n a s yn c h ro n ou s re s e t s gn a l T h s s ac h e v e d by n s e rtn g th e S yn th e s ze r s p ec fc a ttrbu te a s yn c _ s e t_ re s e t n th e VHD L s ou rce c od e A fte r s e p a ra te ly s yn th e s z ng ea c h o f th e 0 p p e ln e s ta g e s, th e y w e re c onn ec te d w th th e r re s p ec tv e h a nd s h a k e c rc u t E a c h h a n d s h a k e c rc u t h a d a d e la y e le m e n t (tu n e d to th e c rtca l p a th o f th e c o m b n a ton a l lo g c ) n s e rte d n to th e n pu t re qu e s t s gn a l A tm ng m a rg n o f 30 % w a s a dd e d to gu a ra n tee re la b le op e ra ton a fte r la y o u t T h e n th e to p le v e l d e s gn w a s s yn th e s ze d s e p a ra te ly a nd th e n e tls ts o f th e nd v du a l p p e ln e s ta g e s w h e re n te g ra te d n to th e top -le v e l n e tls t T h e la you t w a s ca rr e d ou t u s ng th e s ta nd a rd ce ll a pp ro a c h o f S lc on E n s e m b le w th 8 0 % ro w u tlza ton P o s t la you t p a ra s tc p a ra m e te rs w e re e x trac te d a n d a V e rlog n e tls t tog e th e r w th th e po s t la you t S D F fle w e re e xpo rte d T h s V e rlog n e tls t w a s a g a n s m u la te d u s ng th e s a m e VHD L te s t b e n c h th a t w a s u s e d fo r th e b e h a v o ra l m od e l T h s s m u la ton c o n frm e d th e c o rr ec t op e ra ton a nd fo rm s th e b a s s fo r th e la te n c y a nd th rou g hpu t m ea s u re m e n ts re p o rte d n th e fo llo w n g S ec ton 5 R e s u lts a nd C o m p a r s o n F o r m ea s u rng th e p e rfo rm a n ce o f th e a s yn c h ro n ou s c rc u ts th e v a rou s h a nd s h a k e s gn a ls a re g e n e ra te d fr o m th e te s t b e n c h u s ng a n d ea l d a ta s o u rce a n d a n d ea l d a ta s nk T h e n p u t d a ta s s upp le d n th e te s t b e n c h fr o m a fle F o r th e a s yn c h ro n ou s m o d e o f op e ra to n, d a ta s b e ng s upp le d to th e e n tty w th th e rs ng e dg e o f n p u t re q u e s t s gn a l r e q_ a _ top s h o w n n F g u re 5 W th th e d e la y n th e h a nd s h a k e c rc u t b e n g n s e rte d n th e p a th o f s gn a l r e q_ a _ top, th e ac k no w le dg e s gn a l ac k_ a _ top go e s h g h a fte r th e c o m b n a ton a l lo g c b lo c k fn s h e s t s c o m pu ta ton T h e lo ca l c lo c k s gn a l n ea c h o f th e c o m pon e n ts s c onn ec te d to s gn a l ac k _ a o f eac h re s p ec tv e h a nd s h a k e c rc u t H e n ce, w th th e rs ng tra n s ton o f a c k_ a, th e c o m p u te d d a ta s la tc h e d to th e d a ta re g s te rs T h e la te n c y fo r a s y n c h ronou s op e ra to n s m ea s u re d a s th e tm e n te rv a l fro m th e rs ng e dg e o f n p u t s gn a l r e q_ a _ top to th e rs ng e dg e o f o u tpu t s gn a l r e q_ b _ to p T h s m ea s u re m e n t s ca rr e d o u t w th a n e m p ty p p e ln e a s a n n ta l s ta te a nd h e n ce a ss u m e s op e ra ton n th e d a ta lm te d re g o n a s d e fn e d n [1 3 ] T h e m a x m u m th ro ug h pu t s m ea s u re d a s th e nu m b e r o f d a ta s a m p le s e m e rg ng a t th e ou tp u t w h e n th e npu t s d rv e n by a n d ea l s ou rce a nd th e ou tp u t s c onn ec te d to a n d ea l d a ta s nk F o r th e s yn c h ro n ou s d e s gn, th e la te n c y s d e p e nd e n t on th e c lo c k fr e qu e n c y T h e c o re a rea fo r a ll th ree d e s gn s s m ea s u re d fr o m th e la y o u t a n d re p o rte d n T a b le 1 tog e th e r w th p o w e r d ss p a ton, la te n c y a nd th ro u gh p u t

7 T a b le 1 P e r fo r m a n c e c o m p a r s on o C O RD IC m p le m e n ta ton s C o re a rea (n mm ) P o w e r (n m W ) L a te n c y (n n s ) T h ro u gh - pu t (M S /s ) f t h r ee A S Y - M IX -C O R D IC S YN - C O R D IC S yn c h m od e A s yn c h m od e C O R D IC (16 ) (9 4 ) (7 ) (9 8 ) It s s ho w n n T a b le 1 th a t th e M IX -C O R D IC d e s gn re qu re s on ly 13 % m o re s lc on a rea th a n th e S YN - C O R D IC A s ub s ta n ta l a dv a n ta g e s th e re du c e d la te n c y In p a rtc u la r fo r tra n s m ttng ac k no w le dg e m e n t fr a m e s w h c h h a v e to b e g e n e ra te d a t a fx e d tm e n te rv a l o f 1 6 m s a fte r r ece p to n o f a fr a m e, a ny re du c to n n la te n c y n th e d a ta p a th s w e lc o m e T h e d a ta th rou g hpu t o f M IX - C O R D IC a nd S YN -C O R D IC a re s m la r T h e d ff e re n ce o f 45 M S /s v e rs u s 40 M s /s s p rob a b ly du e to th e p a rtc u la r la you t F u rth e rm o re, th e le v e l o f m a n u a l n te rac ton fo r th e M IX -C O R D IC d e s gn w a s h gh e r, w th b u ff e rs b e ng e xp lc tly n s e rte d n to e v e ry p p e ln e s ta g e C rc u t A S Y -C O R D IC h a s s gn fca n tly s m a lle r s lc on a rea e % le ss th a n S YN -C O R D IC T h s s p a rtly du e to th e s m a lle r s ze o f tra n s p a re n t la tc h e s c o m p a re d to e d g e trgg e re d flp -f lo p s F u rth e rm o re, n c on tra s t to S YN - C O R D IC a nd M IX -C O R D IC, th e A S Y -C O R D IC d e s gn do e s no t re qu re a ny c lo c k tree T h e u s e o f lo ca l n te rc onn ec t on ly, re s u lts n c on s d e ra b le s a v n g s n s lc on a rea T h e h gh e r la te n c y o f c rc u t A S Y -C O R D IC s du e to th e m o re c o m p le x h a nd s h a k e c rc u t fo r th s d e s gn T h e tra n s p a re n t p h a s e o f th e la tc h e s re qu re s c o u p lng o f c on s ec u tv e s ta g e s th a t s re flec te d by a dd ton a l d e la y s T h e po w e r d ss p a ton fgu re s g v e n n T a b le 1 w e re d e rv e d fro m th e s yn th e s s too l F ro m p re v ou s e xp e re n ce th e s e fgu re s a re v e ry u n re la b le T h e d s tn c tly h gh e r po w e r d ss p a ton o f A S Y -C O R D IC m a y re s u lt fr o m th e S ynop s y s s o ftw a re fa ls e ly e s tm a tng h g h g ltc h n g ac tv ty n th e do w n s trea m c rc u ts du rng th e tra n s p a re n t ph a s e o f th e la tc h e s C u rre n tly w o rk s u n d e rw a y to g e n e ra te m o re acc u ra te fgu re s u s n g th e p o w e r s m u la to r P o w e r M ll T h e la you t o f th e M IX -C O R D IC p ro ce ss o r, ftte d w th a n a log te s t c rc u ts fo r c ro ss ta lk nv e s tg a ton, s s ho w n n F gu re 6 In o rd e r to p re v e n t d rec t c oup ln g fro m n p u t s gn a ls n to th e a n a log te s t c rc u ts a C -s h a p e d p lace m e n t o f th e d g ta l p a d s w a s und e rta k e n F o u r d e n tca l o s c lla to rs a re p lace d a t th e ea s t s d e o f th s la you t, eac h ftte d w th a d ff e re n t s tru c tu re fo r p ro tec ton fr o m s ub s tra te no s e F ro m to p to bo tto m th e d e p loy e d p ro tec ton s tru c tu re s a re : 1 ) GND rn g a n d n -w e ll rn g, ) bu re d la y e r s h e ld, 3 ) n o p ro tec ton, a nd 4 ) G ND rng on ly A d d ton a lly th re e s ub s tra te c on tac ts fo r d rec t m ea s u re m e n t o f s ub s tra te no s e a re p lac e d a t top e nd, m dd le a nd bo tto m e nd o f th e rgh t e d g e re s p ec tv e ly W e hop e th a t th s d e s g n w ll d e lv e r v a lu a b le n fo rm a ton on th e e ff c e n c y o f d ffe re n t m ea s u re s fo r p ro tec to n fr o m s ub s tra te no s e, bo th fo r s y n c h ro nou s a n d a s yn c h ro n ou s op e ra ton o f th e C O R D IC p ro ce ss o r C O R D I C c o r e s u b s t r a te c o n ta c ts a n a lo g o s c ll a to r s F gu r e 6 L ay ou t o f c r c u t M IX -C O RD IC w th a n a log tes t c r c u ts a t r gh t s d e 6 C o n c lu s o n s a nd F u tu re Wo r k A p p e ln e d C O R D IC p ro ce ss o r w h c h ca n o p e ra te b o th s yn c h ro n ou s ly a s w e ll a s a s yn c h ro nou s ly h a s b ee n d e s gn e d n VHD L a n d m p le m e n te d u s ng a 0 5 m m S G e :C B C M O S p ro ce ss W th n a s o ftw a re d e fn e d ra d o fra m e w o rk, th e op p o rtun ty to op e ra te k e y h a rd w a re c o m pon e n ts e th e r n s yn c h ro n ou s o r a s yn c h ro n ou s m od e ca n o ff e r a dd to n a l fr ee d o m fo r th e s y s te m d e v e lop e r U s ng th s fea tu re, a m o re o p tm a l c o m p ro m s e b e tw ee n po w e r- d ss p a ton, c ro ss ta lk, th ro u gh p u t, la te n c y a nd (s e lf-) te s ta b lty ca n b e fo und A s a n e x a m p le, to re du ce c ro ss ta lk, th e c rc u t ca n b e s w tc h e d n to a s yn c h ro n ou s m od e w h e n th e rece v e s gn a l le v e l s v e ry lo w F o r rea s on a b le s gn a l s tre ng th a nd c on tn u ou s op e ra ton th e m o re p o w e r e ff c e n t s y n c h ro n ou s m od e ca n b e c h o s e n A n a dd ton a l a pp lca to n o f ou r M IX -C O R D IC c rc u t s to s e rv e a s a n v e h c le fo r d e ta le d nv e s tg a tng o f th e n te rf e re n ce c a u s e d by s y n c h ro n o u s a nd a s yn c h ro nou s d g ta l c rc u ts on S G e :C b po la r a nd C M O S a n a log c rc u ts F u rth e r w o rk w ll fo c u s o n e x te n d ng ou r ce ll lb ra ry a nd d e s gn k t by a nu m b e r o f ce lls c ru c a l fo r a s yn c h ro n ou s d e s gn S u c h ce lls a re, fo r n s ta n ce, v a rou s C -e le m e n ts, c o m p le te h a nd s h a k e c rc u ts a nd lo w -po w e r

8 tun a b le d e la y e le m e n ts T h e ce lls w ll b e d e s gn e d on tra n s s to r le v e l, n c lud e d n to th e ce ll lb ra ry a n d m a d e a v a la b le a s c o m p o n e n ts n VHD L a nd V e rlo g T h e M IX -C O R D IC d e s gn re po rte d h e re, n c lu d ng th e a n a log te s t c rc u ts, h a s b ee n ta p e d o u t fo r fa b rca to n o n ou r n -hou s e 0 5 m m S G e :C B C M O S p ro ce ss n D ece m b e r 0 01 F rs t re s u lts, n c lud n g m ea s u re m e n ts on n te rf e re n ce o f th e a n a log c rc u ts fo r s yn c h ro nou s a n d a s yn c h ro n ou s o p e ra ton o f th e M IX -C O R D IC d e s gn w ll b e a v a la b le b y th e m d d le o f M a rc h 00 A c kn o w le d g e m e n ts W e w ou ld lk e to ac k n o w le dg e th e s upp o rt fro m U lrc h J a gdho ld fo r g e n e ra tn g th e d g ta l la y ou t F u rth e rm o re, w e a re g ra te fu l o f th e s u p po rt fr o m W o lfg a ng W nk le r fo r d e s g n a nd la you t o f th e a n a logu e te s t c rc u ts fo r th e du a l-m od e C O R D IC m p le m e n ta ton R e fe re n c e s [1 ] S ta nd a rd ET S I D T S /BR AN , B ro a d b a n d R a d o A cce ss N e tw o rk s (BR AN ); H IP E R L AN T yp e F un c ton a l S p ec fca to n s P a rt 1 P hy s ca l (P HY ) la y e r ; J un e [ ] S ta nd a rd IEEE P a /D 7 0, P a rt 1 1 : W re le ss L AN M e d u m A cce ss C on tro l (M A C ) a nd P h y s ca l L a y e r (P HY ) s p ec fca ton s : H gh S p ee d P h y s ca l L a y e r n th e 5 GH z B a nd ; J u ly [3 ] J M C a s a lta, X A ro g on e s a n d A R ub o, S ub s tra te C ou p ln g e v a lu a to n n B C M O S T e c h n o lo gy, IEEE J ou rn a l o f S o ld S ta te C rc u ts, vo l 3, n o 4, p p , A p rl [5 ] D S C o c h ra n, A lgo rth m s a n d acc u rac y n th e H P 35, H e w le tt P ac k a rd J, pp , J u n e [6 ] A M D e s p a n, V e ry fa s t F ou re r tra n s fo rm a lg o rth m s fo r m p le m e n ta to n, IE E E T ra n s C o m p u t, v o l C 8, no 5, p p , M a y [7 ] D T L ee a n d M M o rf, G e n e ra lze d C O R D IC fo r d g ta l s gn a l p ro ce ss ng, P ro c IC A SSP, v o l 3, p p , M a y [8 ] M C M a nd a l, A S D h a r a n d S B a n e rjee, M u ltp le rle ss a rr a y a rc h tec tu re fo r c o m pu tn g d s c re te c o s n e tra n s fo rm, C o m p u te rs E lec E ng, v o l 1, n o 1, p p , [9 ] A S D h a r a nd S B a n e rjee, A n a rr a y a rc h tec tu re fo r fa s t c o m pu ta to n o f d s c re te H a rtle y tra n s fo rm, IEE E T ra n s C rc u ts a n d S y s t, vo l 3 8, n o 9, p p , [10 ] I S u th e rla n d, M c ro p p e ln e s, C o mm u n c a to n s o f th e A C M, 3 (6 ), pp , Jun e [11 ] P D a y, J V W o o d s, In v e s tg a to n n to M c rop p e ln e L a tc h D e s gn S ty le s, IE E E T r a n s O n V L S I S y s te m s, 3 ( ), pp 64-7, [1 ] J N S e z o v c, P p e ln e S yn c h ro n s a ton, P ro c In te rn a to n a l S y m po s u m o n A d v a n ce d R e s ea rc h n A s yn c h ro n o u s C rc u ts a n d S y s te m s, pp , N o v e m b e r [13 ] T E W lla m s, P e rf o rm a n ce o f I te ra tv e C o m p u ta to n n S e lf- T m e d R n g s J ou rn a l o f V L S I S g n a l P ro ce ss ng, (7 ), pp , [4 ] J B u c k a n d E A L ee, T h e to k e n flo w m o d e l, D a ta F lo w W o rk sh o p, H a m lto n Is la n d, A u s tra la, p p , M a y 1 9 9

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