V N l J Jo ur na l o f S c i e n c c , M a t h e m a t i c s - P hy s i cs 27 ( 1 ) 23 - 130 Detection o f the location o f the hazard during and after the design of combinational circuits Nguyen Quy Thuong VNU, Ỉ44 Xuan Thuy, Can Giav, Hanoi, Vietnam Received 14 April 2011 A bstract Delay fault and glitch fault cause hazards that are structure hazard and function hazard During design process we can use many methods to identify and remove structural hazard [1-2], [3] However, with function hazards, determination and remove much more difficult In this paper we introduce a new solution to determine the structure hazard by the Truth table - Matrix mathematics M ethod and method for determining function hazard over how to determine crosstalk fault [4-71 Keywords: Structure hazard, function hazard, truth table, multiplication matrix, hazard - algebra, crosstalk fault, glitch Introduction D etecting, lo c a tin g an d rem oving hazards in a digital circuits is the com pelling w ork o f a designer K arn au g h m ap [ ] w a s u sed very often to design digital circuits that are com binational an d sequence; sy n ch ro n o u s and a sy n ch ro n o u s w ith hazard - free John K n ig h t [1], [3], T h u o n g N Q [2] applied h azard - algebra m ethod for the d esig n o f digital circuits I f a c irc u it h a s a hazard, then function o f the circuit will be reduced to one o f th ese form s XX, X 4- X , x x + x a n d (x + x)x H azard - algebra m ethod can detect and m ask h azard in both co m b in atio n al an d se q u en c e circuits T o in v estig a te h a z ard in com binational circuits w ith E X - O R gates, E c Tan and M H H o [9] b u ilt m atrix m eth o d th a t g enerate a set o f variables o f all nodes in each gate level o f a circuit p ro g re ssiv e ly until it reach es the output o f circuit H ow ever, this m ethod has notyet show exact location o f h a z ard s a n d w hen dynam ic hazard is d ep endent on static logic static logic “ hazard or d ep en d en t on - hazard [4 -7 ], [10,11] sh o w s the test m ethods for crosstalk fault in d u ced glitch fault Through crosstalk fault, we have function Hazards can be determined, that appears only after the circuit was put into use In this p a p e r a n e w so lu tio n is pro p o sed to investigate structure h azards in co m binational circuits that is b a se d on c o m b in a tio n o f truth table, m atrix m athem atics and hazard - algebra to detect structure E-mail: cp4mua@ yahoo.com.vn 123 N.Q T huong/ VNU Journal o f Science, Mathematics - Physics 27 (20Ỉ Ỉ) Ỉ23-Ỉ30 124 hazards T his p ap er also p o in ts out the m ethod o f determ in in g the function hazards via the determ in atio n o f cro sstalk fault T he structure o f this paper is as follow s S ection gives background on h azard algebra, the d ifferen ce betw een B oolean algebra and H azard algebra; relatio n s betw een m a t ì x m athem atics and truth table and circuit equation o f function; the test m eth o d s to determ ine the cro sstalk fault, w hich is to determ in e function hazard in the digital circuit S ection gives the form al p ro b le m statem ent to be solved, and an intuitive overview o f the new m ethod as the rules to delect sừ u c tu re h azards o f so p h isticated form s (in cluding SO P and PO S) S ectio n gives the m ode o f d e te rm in in g the function h azard in the circuit w as put into use Background T h e potential for a glilch in a com binational circuit is called a hazard H azards fall into two classes: function h azards and structural hazards S tructure hazard could be d etected and rem oved even d u rin g the design p ro cess b u t function hazard that can delect only the circu it after having taken into use and the rem oval o f function hazaru is m ore difficult than o f structural hazard T h is section focuses on the p ro b le m to o f hazards, h azard algebra, m atrix m athem atics and cro sstalk fault 2.1 Truth table - M a trix M a th e m a tics M eth o d f o r (he detection a n d location o f s tru c tu re h azards in d ig ita l circuits T ru th table - m atrix m ath em atics m ethod w as built to the detection and lo catio n h azards in co m b in atio n al circuits th at is expressed in eith er sum -of- p ro d u cts (S O P ) form or p roduct-of-sum s (P O S ) form or both T h e m ain idea o f this w ork IS to “d ip ” the variables o f fim c iio n on th eir truth ta b le b y m u ltip ly in g these m atrices c o n fo m i to the rules o f m u ltiplication m atrix (m ath em atics) T he result o f the m ultiplication is co m p ared w ith definitions o f hazards in hazard algebra [1], [2], [3] T h at is ^ = ^ (0 ) as static - hazard, ^ ^ t ” -r ( t^ + ^ ( ) as static I - h a za rd aruli^ ^ = ^ e (l) as dynam ic hazard dep endent on static x’ ^o(O) , ^ ^ - hazard and dynam ic hazard dependent on static - hazard, respectively T h e p rin cip le o f th is m eth o d as follow s: firstly w e find the variables X th at can cause hazard, and th en fix value or in variab les Xj X T o realize this problem w e can “d ip ” the variables, the sum facto rs or the p roduct term s o f circu it equation on the truth table n variab les b a sed on m ultiplying eq u a tio n - m atrix w ith truth table - m atrix that conform to the m les o f m u ltip licatio n m atrix (m ath em atics) T h e equation ~ m a trix is a m atrix express circuit equation I f circu it e q u atio n in form SO P, then c irc u it e q u atio n w ill ho ld s sum factors and if circuit equation in form P O S , then circu it equation w ill h o ld s p ro d u ct term s N u m b e r o f sum factors or product te n n s in these circu it eq u atio n s show s n um ber colum ns o f m atnx, that is, matrix with dim ensions Ixn that IS matrix with row and n colum ns T h e M a trix truth tab le is a m atrix express ưxith table o f circu it function In th is m eth o d the fruth tab le is rep u te d to be a m atrix n X 2""', it m eans m atrix w ith n colum ns and " “ ‘ row s T o m ake n u m b er o f co lu m n s in circuit equation - m atrix equal to n u m b er o f ro w s in truth table matrix we can change this matrix into transpose matrix, that is, let A be an n X 2" ' matrix defined by th e n u m b e r aịj, then the tran sp o se o f A as A^, denoted is the 2"'' X n m atrix defin ed b y the n um ber bjj where bji = N.Q Thuong / VNU Journal o f Science, Mathematics - Physics 27 (2011) 123-130 125 The algorithm to d etect structure h azard s m com binational circuit o f this m ethod IS given as follows: Step 1: C onsider the circuit equation I f the circuit equation is co m p licated , then apply De M organ Law to get the sim plest circu it equation that are circuit equations in form s eith er SO P or PO S or both Step 2: C onsider the variables _ - Firstly, find the v a n a b le s that can cause hazards T hey are those variables having b o th X an d X form , in this case x:= x " a n d x ’:= x' are independent Fix X ) values (0,1) by “d ip ” circuit equation n v ariables on the truth table o f circu it function respectively that realized by m u ltip ly in g tw o m atrices that are circuit equation - m atrix and truth table - m atrix Step Ỉ: C o n sid er the resu lt o f m ultiplication A fter the variablesX are fixed value or by “ d ip ” circuit eq u atio n n variables on the truth table so we get the result o f m ultiplication that either the sum factors t ” t ’’ = ^ ( ), the c irc u it tain s static - hazard, or the product term s t" + t ‘- = ^(1), that IS the circuit tain s static - hazard, or dynam ic hazard dependent on static logic - hazard h azard dependent on static logic - hazard ^ = (x"+ ) t" = Ị, = t" x‘' = ^o(O) and dynam ic o(l) or not at all, that is the free h a z ard circuit Step 4: Investigate to rem a in in g variables T o find the rem aining v ariables X jthat can cause hazards G o to Step 2, Step until last variab le Xi is considered 2.2 To detect crosstalk fa u lt in d u ce d fu n c tio n hazards A fter the digital circuit is d esig n ed and built, it is alw ays desirable to know w h eth er the c irc u it is co n sừ u c te d w ithout any faults Is it IS properly constructed and in use, it m ay be d isable by alm ost any internal failure T he process o f applying test and determ ining w h eth er a digital circu it is fault free or not is know n as fault detection I f w e know n relationship exists betw een the v arious possible faults and deviations o f output pattern s, IS term ed as fault location [12] as fu n c tio n hazard T he in creased design density in deep - subm icron designs leads to m ore significant in terference b etw een the sig n als b ecause o f capacitive co u p lin g or crosstalk C rosstalk can induce bo th B oolean e o rs and delay faults C ro sstalk - induced pulses are likely to cause errors on hazard - sensitive lines such as inpu ts to dynam ic gates, clock, set/reset and data inputs to flip - flops C ro sstalk p u lses m ig h t result in logic e o rs o r degraded voltage levels, w h ich increase propagation delays [6 ] Studies show that increased co u p lin g effects betw een signals can cause signal delay to in crease (slow dow n) or decrease (speed up) significantly B oth conditions can cause errors Signal slow d o w n can cause delay fa u lts if a transition is propagated along paths with small slacks Signal speed - up can cause race (glitch) conditions i f a tran sitio n s are propagated along short p aths [ ], C rosstalk g litch o ccu rs w hen there is a sw itch for the signal at one line and the signal at the o th er line is driven steady, in w h ich case a glitch is form ed at the output o f the steady line T h e condition for cro sstalk d e la y is that the signal at both line sw itch es to the opposite direction T he resu lt is an increase in fransition tim e [5] F or tw o line in a circuit, i f the signal ữ an sitio n o f to o r to on a line produces co u p lin g effects on an o th er line, then the signal line is called an aggressor line, and the o th er line is c a lled a v ictim line F or instance, i f the victim line and aggressor line are driven resp ectiv ely b y a static a n d a 126 N.Q Thuong / VNU Journal o f Science, Mathematics Physics 27 (20Ỉ I) 123-130 - fast - rising ( to ) transition, then the crosstalk p o sitive glitch is g en erated in th e v ic tim ’s output signal If the height o f cro sstalk glitch happens to be larger than the u p p e r - th re sh o ld value o f logic lo w voltage for the give technology, it w ill produce logic failures fu n c tio n a lity p r o b le m ) [ ], W e c o n sid e r the function hazard in digital circuit, w as put into use, as d etect the c ro ssta lk faults H ere w e d efin e crosstalk fault on digital circuits by using B inary D ecision D iag ram (B D D ) o f [ ], So if w e w ant to detect all form s o f hazard in the circuit so, then w e need to d e te rm in e structure h a z ard s w ithin the design process and function hazard by determ in in g the cro sstalk fault Detection structure hazard in com binational circuits F rom definitions o f hazard and the algorithm to detect hazard o f this m eth o d in sectio n now w e can find hazards in circuits for sum - o f - products im plem entation, or for p ro d u c t - o f - sum s im plem entation, or co m p licated circu it that is not only in forni P O S o r SO P b u t also ho ld all POS and SO P L et us consider an EX - O R gate [9] (Fig 5) as com plex circuit X -w Q Y Fig Circuit with EX - OR gate S tep 1: F rom this circuit w e have circuit equation: Q =X Y +X Y +X + W +X Y +X Y +Z U se the B oolean relatio n s to change circuit equation, w e get: Q- + XỸ)X w + (X + Y)(X + Y)Z S tep 2: T he C ircuit equation has tw o variables X ( x ^ ',T ^ ) and Y can cause hazard F irstly , consider for X: X := (Y,Z,W): = (OJ) ^ T h e equation Q has tw o sum fa c to r s that are (XY + XY)XW and (X + Y )(X + Y )Z (in form SO P), b u t in one sum factor hold p ro d u ct term s (PO S) E xam ple: sum facto r (3Õ" + p ro d u c t term s (X Y + X Y ) and x w (PO S and SO P) So w e Can XỸ)XW hold tw o create from circu it e q u a tio n Q to one m afrix M w ith tw o p ro d u ct term s (X Y + X Y ) , X w and one sum factor (X + Y )(X + Y )Z ‘0 1 1 M.A^ =[(3Ò^ + XỸ) (XW) (X + Y)(X + Y )z ] 0 1 0 1 0 0 1 11 N.Q- T huong/ VNU Journal o f Science, Mathematics - 127 Physics 27 (20Ĩ1) Ỉ23-Ỉ30 Also resu lt o f m u ltip licatio n in a colum n is defined by addition (A oC ) for sum factors and m u ltip ly (M oC ) for p ro d u ct term s E xam ple, result o f m ultiplication in first colum n o f b elo w M A ) is S o w e get: (x«) (T ^ ) M A ^ = ( 0) (t") L L _L (T^) (t^^) (0) xL (x » ) (t^ ) Y (0 ) (0 ) z 0 w -1 II ( t L ) + ^ t: ( L ).( t t L ) + t L L t I] Y -1 Y=1 Y=0 z =0 ệ (l)in^ z =0 z=0 ^( ) in - W=1 w =0 w =0 C om pare w ith D efinition and w e find out one static - hazard ậ (0 ) in Y- w - 1, z - 0, one static - h a z ard ^ ( l ) i n Y = Z = W = and one dynam ic hazard d ep en d en t static - hazard ^ ( ) in Y = l,z = w = Step 4: G o to Step 2, Step to co n sid er variable Y : Y ; = (T»,x^) (X,Z,W): = (0,1) M A ' =[M 1 0 1 0 0 0 1 0 1 (x“ ) (x") (T*-) (x^-) (t '-) (x^-) X (x " ) (0 ) (0 ) (t “ ) (t" ) (0 ) (0 ) T*- z w (t“ ) (t" ) (t» )(:-'■)+ x =0 x -0 z =0 ệ (0) in • z =0 w =0 0 Ji u ^ (l)in < u u - W =1 r'- (T” )(r^ ) u X=1 m in ■z =0 W =1 128 we N.Q Thuong / VNU Journal o f Science, Mathematics - Physics 27 (2011) 123-130 have identified h azard s ^ (1 ) eỊx=Z=W=oỊ, ^ (0 ) eíX=W=l,Z=0)Ị and ^Q(0)e(x = l, z = w = 0) T hus, the circuit functio n Q has not only dynam ic hazard [9], but also sta tic - h a z ard and static hazard Detection Crosstalk induced function hazard T o determ ine the glitch in the circuit than we need to identify the c ro s s ta lk fau lt In p rin cip le to d eterm in e the stuck at fault or cro sstalk fault is to create the test v ector I f th e re IS a fau lt in a circuit th en the test vectors o f the fault are the input assignm ents that cause th e fau lty c irc u it and norm al c irc u it (fault - free circuit) to produce different output values T he test v e c to r d istin g u ish betw een the g oo d m achine and the faulted m achine So the test vecto r is built, w h ic h is th e X O R o p e tio n o f the fau lt - free circuit and faulty circuit F igures [10] tells us m ore about th is in F u n c tio n a l E quivalence a n d F unctional D om inance (F unctional C ollapsing): F or an input v ector, V , to b e a test for a fault, we have: F ,( V ) F ,( V ) = w h ere Fo is the fault - free function and F | is the faulty function, resp e c tiv e ly C o n s id e r a second fault th a t p ro d u ces a fault function p A ccording lo the definition o f fault e q u iv a le n t fau lts h a v e exactly the sam e tests T herefore, for tw o faults to be equivalent, w e have [F„ ( V ) © F, (V )] e F, (V )] = => F, (V )] © R ( V ) = Fig \ ’iewing fault Equivalence In [ ] test vector is called test B D D (T est B inary D ecisio n D iag ram ), n o rm a l c irc u it a re know n as n o rm a l B D D and faulty c irc u it is faulty B D D , so w e have test BD D; T est B D D = norm al B D D • faulty B D D + norm al B D D • fa u lty B D D =1 In the test BD D , each inp u t assig n m en t w ith attribute value is a test v e c to r o f th e fault T h e crosstalk fault is one o f the interference effects b e in g cau sed b y p a sitic c ap acitan ce and in d u ctan ce coupling F o r tw o line in circuit, if the signal tran sit o f to o r to o n e line produces c o u p lin g effe c ts on an o th er line, then the line is called an ag g resso r line, th e o th e r line F ig u re show s the relatio n sh ip betw een aggressor line and v ictim lin e [11] IS c a lle d a victim N.Q Thuong / VNU Journal o f Science, Mathematics ~ Physics 27 (201Ỉ) 123-Ỉ 30 Y1 Yi-1 Yi victim Y1 Yi Yi-1 Y Yi-1 victim Y1 victim Yi-1 victim Y1 Y1 Yi+I Yi+1 Yi+1 Yi+1 Yn Yn Yn Yn i i victim 129 A V victim victim Negative glitch Positive glitch victim Slow to fall Slow to rise Fig Maximal aggressor fault model T he P ositive g litc h a n d N e g a tiv e glitch in Fig are function hazards T h e se H azards can n o t be rem oved d u rin g th e d e sig n p ro cess, b ecau se they appear only after having taken into use H ere for c irc u it C [ ] sh o w n in Fig 4, w e give an exam ple for test g en eratio n w hen th ere is a crosstalk fau lt b e tw e e n sig n a l lines C3 and C4 T h e task o t test g en eration is to se arc h for the inputs vectors o f circ u it c 17 in o rd e r to d etect the cro sstalk fault For exam ple, a test v ecto r o f the c ro sstalk fault is made up o f circuit input vectors Vi = (Xi, X , X , X , X ) = (0, 0, 0, 0, 0) and V = (X|, X2 , X3 , X , X5 ) = (0, 0, 0, 0, 1) A p p ly V i and V to the circuit sequentially I f the circu it o u u ts are yi = an d y i = for V i, y, = a n d y = for V , then there is not crosstalk I f the circu it o u u ts are yi = a n d yz = for V ,, y, = a n d y = for V ,, then there is crosstalk T herefore, th is te st v e c to r can d e te c t the crosstalk fau lt b e tw e e n 63 and 64 H ere, assum e that C4 ÌS a ag g resso r line and that a d ow n tra n sitio n (1 to ) in signal line 64 is a victim lin e, and produces a glitch (1 to 0) in signal line e3, th at is, th ere is a function h azard ei XI yi X3 X2 G3 X4 X5 63 o64 Fig C17 Cừcuit y2 130 N.Q T h u o n g / VNU Journal o f Science, M athematics - Physics 27 (2 ỈĨ) Ỉ2 -Ỉ3 5, C onclusion T h e detection, lo c a te a n d re m o v e th e H a z a rd s o f th e d ig ita l c irc u its is v e ry c ritic a l fo r circuit d esig n ers Structure h a z a rd a re d e te c te d an d re m o v e d e v e n d u rin g th e d e s ig n p ro c e s s and th ere w ere so m e m eth o d s to so lv e th is T ru th ta b le - M a trix M a th e m a tic s M e th o d p re s e n te d here is a new so lu tio n to in v estig ate stru c tu re h a z a rd T h is m e th o d n o t o n ly d e te c te d all k in d s h azards in combinational circuits but also point out location o f hazards with high accuracy The Truth table M a trix M ath em atics ca n d e te c t h a z a rd in all c irc u it fu n c tio n s th a t c a n e x p re s s e d b y tru th table The rem o v in g structure h a z a rd e rro rs n o d iffic u lty i f w e u se K a rn a u g h m a p [ ] or h a z a rd a lg e b [1-3] to su p p ly red u n d an t term s c o rre s p o n d in g e a c h k in d o f h a z ard T h e s e fu n c tio n h a z a rd can n o t b e rem oved d u rin g the design p ro c e ss, b e c a u s e th ey a p p e a r o n ly a fte r h a v in g ta k e n in to use D u tio n o f function h a z ard can p e rm a n en t, te m p o ry o r in te rm itte n t, th u s re m o v in g it is n o t easy W e can determ ine fu n ctio n hazard, for e x a m p le th ro u g h th e id e n tific a tio n o f c ro s s ta lk fa u lt as d e s c rib e d above R eferences [1] John Knight, Asynchronous Circuits Races Cycles, and Effect o f Hazards, Electronics Departm ent Carleton University, April 1, 2006, [2]N Q Thuong, Race and hazard algebra in asynchronous system, K//Ơ Journal o f Science, MathematicsPhysics, VoL24, N o l (2008) 47 [3] John Knight, Glitches and Hazards in D igital Circuits, Electronics D epartm ent, Carĩeton University April 1, 2006 [4] Shehzad Hasan (advisor: Prof w Anheier) (hasan, anheier), Test Pattern Generation and Compaction fo r Crosstalk induced Gỉitch Fault, ITEM, University o f Bremen, Otto - H ahn - Allee 1, 28359 Bremen, Germany [5] Xiaoyun sun, seonki Kim, Bapiraju Vinnakoda, Crosstalk fa u lt detection by dynam ic Idd, Department of Electrical and Com puter Engineering U niversity o f M innesola, M inneapolis, M M , 55455 [6 ] Zhong - Liang Pan, Ling Chen, Guang - Zhao Zhang, Cultural A lgorithm for M inim ization of Binary Decision Diagram and its A pplication in Crosstalk Fault Detection, International Journal o f Automation and Computing, 7(1) February (2010) 70 [7] Kwang - Ting Cheng, C urrent Directions in Autom atic Test - Pattern Generation^ U niversity of California, Santa Barbara [8] M Karnaugh, A M ap M ethod for synthesis o f com binational logic circuit, Transactions o f the AỈEE, Communications and Electronics, V ol72:l (1953) 593 [9] E c Tan, M.H Ho, M atrix m ethod to detect logic hazard in com binational circuit with EX OR gate, Journal o f Universal Computer Science, vol 5, 1 (1999) 765 [10] Raja K K R Sandừeddy, Vishwanti D Agrawal, Diagnostic and D etection Fault CoUapsing fo r Multiple Output Circuits, D epartm ent o f Electrical and Com puter Engineering A uburn University, AL 36849, USA [11] Jin Fu Li, Transistor Stuck - Open Fault, Advanced Reliable systems (ARES) Lab [12] Thamarai, S.M Kuppusam y, K M eyyappan, T Enhancing test pattern com pacrion algorithm s for simple two stage cừcuits, International Journal o f Current Research, v.)l (2010) 015 ... ig ita l circuits T ru th table - m atrix m ath em atics m ethod w as built to the detection and lo catio n h azards in co m b in atio n al circuits th at is expressed in eith er sum -of- p ro... ltiplication m atrix (m ath em atics) T he result o f the m ultiplication is co m p ared w ith definitions o f hazards in hazard algebra [1], [2], [3] T h at is ^ = ^ (0 ) as static - hazard, ... ’’ = ^ ( ), the c irc u it tain s static - hazard, or the product term s t" + t ‘- = ^(1), that IS the circuit tain s static - hazard, or dynam ic hazard dependent on static logic - hazard h azard