Two Analytical Methods for Detection and Elimination of the Static Hazard in Combinational Logic Circuits

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1 Crcuts and Systems,, 4, Publshed Onlne November (http//wwwscrporg/journal/cs) http//ddoorg/46/cs476 Two Analytcal Methods for Detecton and Elmnaton of the Statc Hazard n Combnatonal Logc Crcuts Mha Grgore Tms, Aleandru Valach, Aleandru Barleanu, Andre Stan Automatc Control and Computer Engneerng Faculty, Techncal Unversty GhAsach, Ias, Romana Emal Receved August 8, ; revsed September 8, ; accepted September 6, Copyrght Mha Grgore Tms et al Ths s an open access artcle dstrbuted under the Creatve Commons Attrbuton Lcense, whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded the orgnal work s properly cted ABSTRACT In ths paper, the authors contnue the researches descrbed n [], that conssts n a comparatve study of two methods to elmnate the statc hazard from logcal functons, by usng the form of Product of Sums (POS), statc hazard In the frst method, t used the consensus theorem to determne the cover term that s equal wth the product of the two resdual mplcants, and n the second method t resolved a Boolean equaton system The authors observed that n the second method the dgtal hazard can be earler detected If the Boolean equaton system s ncompatble (doesn t have solutons), the consdered logcal functon doesn t have the statc hazard regardng the coupled varable Usng the logcal computatons, ths method permts to determne the needed transtons to elmnate the dgtal hazard Keywords Combnatonal Crcuts; Statc Hazard; Logc Desgn; Boolean Functons Introducton Under certan condtons, on the output of the logcal sgnals may occur unwanted transtons These transtons are known as gltches The logc gltch s a knd of unwanted nose presentng nthe output sgnal that can ntate an uncontrollable process In the net level there s an nput sgnal [] We can dstngush three types of nose that s ntroduced n CLC (Combnatonal Logc Crcuts), called hazards (Statc, Dynamc and Functon Hazards) In the followng we consder only the statc hazard problem n combnatonal logc systems, called statc hazard Statc hazard, also called SOP (Sum of Products) hazard a gltch that occurs n otherwse steady-state output sgnal from SOP logc; Statc hazard, also called POS (Product of Sums) hazard a gltch that occurs n otherwse steady-state output sgnal from POS logc Statc Hazards n Two-Level Combnatonal Logc Crcuts (Consensus Method []) We wll ntally defne Coupled varable; a varable nput s complemented wthn a term of functon and uncomplemented n another term of the same functon Coupled term; one of two terms contanng only one coupled varable Resdue; the part of a coupled term that remans after removng the coupled varable Hazard cover (or consensus term) The RPI (Redundant Prme Implcant) requred to elmnate the statc hazards AND the resdues of coupled p-term to obtan the SOP hazard cover, OR the resdues of coupled s-term to obtan the POS hazard cover POS eample any logc functon can be descrbed as y e e e y n, n,,,,,, e y,,,,,,, n n Sometmes, the same functon can be descrbes as Usng the () form, we can say e ac e bc () y a b c () ()

2 M G TIMIS ET AL 467 Usng the algorthm descrbed n [], f cab, the epresson from (), () doesn t present statc hazard n relaton wth the nput, and f a = b =, then results c = The condton to have statc hazard n relaton wth the nput, s when a = b = and c = The consensus method [4] conssts of determnaton of coupled terms, then by removng the coupled varables we obtan resdual values That meanng the () equaton can be wrtten lke y a b c ab () It can be observed that the epresson of the functon s multpled by the sum of resdual values, the new epresson presents statc hazard n relaton wth the nput We proposed as eample the 4 nputs logc functon y y 4,,,, R,,6,7,9,,4,7,9,,5,7,8,5,,6,, 4, 9, R Usng the Qune-McCluskey mnmzaton method we obtan the equaton from (5) and also the resdual values determned by nput y a b (4) (5) a b The epresson of no statc hazard n relaton wth nput y Method of Resolvng of Boolean Equatons [5] In ths paragraph we apply the consensus method [5] and the method of solvng some specfc Boolean equatons If y abc, by resolvng the net system equatons t can be determned the vectors nput values whch presents statc hazard a b (7) c If the (7) system has no soluton, the functon doesn t presents statc hazard n relaton wth Therefore, the epresson of the functon becomes (6) 4 4 a b c 4 Therefore, mposes the reducton of the system to 4 or 4 So, the soluton s 4 (8) So, the functon wll have hazard at commutaton So, n the POS relaton wll be added the multpled prme mplcant 4 The functon wll have the same epresson lke n (7) Statc Hazards n Two-Level Combnatonal Logc Crcuts We wll consder two analytcal methods to detect and elmnate ths type of hazard (A) Consensus method [] We wll ntally defne Coupled varable; a varable nput s complemented wthn a term of functon and uncomplemented n another term of the same functon Coupled term; one of two terms contanng only one coupled varable Resdue; the part of a coupled term that remans after removng the coupled varable Hazard cover (or consensus term) The RPI (Redundant Prme Implcant) requred to elmnate the statc hazards AND the resdues of coupled p-term to obtan the SOP hazard cover, OR the resdues of coupled s-term to obtan the POS hazard cover Eample Lets consder the logc functon f R,,5,7 a) SOP eample wll be determned the prme mplcants usng Vetch-Karnaugh or Qune-McCluskey methods, as A B C One of the mnmal equatons s (8),, 7 (9) 5, 7

3 468 M G TIMIS ET AL y AC () we have coupled varable coupled terms, resdues, consensus term Therefore, the logc epresson that has no statc hazard n relaton to varable s y () b) POS eample wll be determned the prme mplcants usng Vetch-Karnaugh or Qune-Mc Cluskey methods, as a b c One of the mnmal equatons s,, 4 4,6 () y () we have coupled varable coupled terms, resdues, consensus term The equaton () shows no statc hazard Eample Let s consder the functon of four varables y f R,,,5,6,7,8,9,,4 SOP hazard wll be determned the prme mplcants usng Qune-McCluskey method, as A B C D E F G H, 5,6 5, 7 6,7,4,,8, 9,,8,, 6,,4 (4) Applyng the Patrck method [6], gong from prme mplcants table wll be determned all SOP solutons Let s consder the logcal p varables attached to the prme mplcants as follows p A, f p, the A prme mplcant s present n the logcal functon epresson, otherwse p (A prme mplcant s not present n the logcal functon epresson), etc Therefore, consderng the correspondence p B, p C, p D, p4 E, p5 F, p6 G, p H, n the table llustrated n Table s shown the 7 p Table The SOP coverage table dec equv p p p p p 4 p 5 p 6 p 7 Patrck coverage It wrtes the coverage equaton p5 p6 p p5 p p6 p7 p pp p p7p p p p p p p p p p (5) Smplfcatons are made by usng the laws of Boolean algebra the redundance law, the dentty law and the dstrbutve law p5 p p6 p7 p p p7 p4 p7p pp p or p5p7 p p p6p7 p4 p p p or p5p7 p p4 p p4 p6 p p p or p p p p p p p p p p p p (6) A verson of the optmal soluton corresponds to p p p trplet, e 5 7 y f,,, F H C The cost of ths functon n SOP mplementaton s (7) C y C C C (It was consdered the varables,, avalable at nput) It can verfy that any other coverage has a hgher cost For eample, the coverage p5 p7 p p whch corresponds to y F H AD (8)

4 M G TIMIS ET AL 469 has the cost C y 4 4 The Statc Hazard Elmnaton (B) The consensus method We apply the same method as n [7], only that t has a strong computng nature Any logc functon can be wrtten as y e e (), e y n, n,,,,,, e y,,,,,,, n n Obvously, f y a b c (), then e ac, e bc If we add the term e e to relaton (), the functon presents no hazard towards In terms of the consensus method, the term that covers the statc hazard s e e ac bc ca b (9), therefore for the form () wll be e e, and for the form (), ab Consderng the second eample, we wll have hazard n relaton to the nput y F H C (), e, e, e e F D By addng F D term to relaton (4), t obtans y F H CF D () hazard n relaton to the nput e e e e G, () () Therefore, the epresson of the functon becomes y F H CDG (4) hazard n relaton to the nput e e Therefore, e e H A The epresson of the functon becomes y F H CDG A t hazard n relaton to the nput e t, t e t (5) (6) e e t t t (7) so that remans the same epresson (), whch has no hazards n relaton to From the relaton (), t sees that the epresson of the functon wthout SOP hazards contans all prme mplcants wthout B and E (C) The method of solvng of some Boolean equatons [8] A logc functon can be wrtten as y a b c (8) a c f,,,,,,, n n n n bc f,,,,,,, Accordng to a theorem from [8], a logc functon epressed as SOP, presents a statc hazard n the stuaton, a stuaton deducted by solvng the followng system of logcal equatons a b (9) c We return to the same functon, (4) y F H C hazard n relaton to the nput y (),

5 47 M G TIMIS ET AL The functon wll present SOP hazard, f () Therefore,,,, whch mposes a hazard at commutaton, whch mposes the addng of the prme mplcant D 6,7 to functon The functon becomes y F H CD hazard n relaton to the nput a b or c () () Therefore, we wll have hazards n the followng cases, 8, The prevous commutatons are equvalent to the mplcant G,,8, The functon becomes y F H CDG (4) hazard n relaton to the nput a b and c, We wll have the soluton The correspondng commutaton s (5) (5),5 Therefore, the term A, 5 s added to the functon And therefore y F H CDG A (6) hazard n relaton to the nput y (7) a b (8) c Because one of the terms ab, s zero, we have no hazards n relaton to that varable 5 Conclusons The contrbuton of the authors conssts n that by analyss of two methods of detecton/elmnaton of the statc hazard, nsstng of the POS method for the logc functon whch wasn t analyzed n [] The boolean equaton [,], presents some advantages nstead the consensus methods, the most mportant to determne the transactons whch causes statc hazard It concludes that the classcal method of the 7s, the method of solvng some specfc Boolean equatons [4], presents some advantages compared to consensus method [5], whch has a strong heurstc nature In the frst method t used the consensus theorem to determne the cover term that s equal wth the product of the two resdual mplcants [6], and n the second method t resolved a Boolean equaton system [7] The authors observed that n the second method the dgtal hazard can be earler detected If the Boolean equaton system s ncompatble (doesn t have solutons), the consdered logcal functon doesn t have the statc hazard regardng the coupled varable Usng the logcal computatons, ths method permts to determne the needed transtons to elmnate the dgtal hazard From the both methods, we can observe that statc hazard can be removed by addng the prme mplcants step by step The same method wth the same conclusons was appled to the statc hazard (POS), usng the dualty theorem [8,9] The authors observed that n the second method the dgtal hazard can be earler detected If the Boolean equaton system s ncompatble (doesn t have solutons), the consdered logcal functon doesn t have the statc hazard regardng the coupled varable Usng the logcal com-

6 M G TIMIS ET AL 47 putatons, ths method permts to determne the needed transtons to elmnate the dgtal hazard REFERENCES [] The Comparatve Study of Two Analytcal Methods for Detecton and Elmnaton of the Statc Hazard n Combnatonal Logc Crcuts, 5th Internatonal Conference on System Theory, Control and Computng (ITCC), Snaa, 4-6 Octomber, pp -4 [] R F Tnder, Engneerng Dgtal Desgn, Academc Press, Waltham, [] Ch Roth, Fundamentals of Logc Desgn, West Publshng Company, Eagan, 999 [4] J P Perrn, M Denouette and E Dacln, Systems Lo- gques, Tome Dunord, Pars, 997 [5] E T Rngkjob, A Method for Detecton and Elmnaton of Statc Hazards n Factored Combnatonal Swtchng Crcuts, Syracuse Unversty, New York, [6] J A McCormck, Detecton and Elmnaton of Statc Hazards n Multlevel XOR-SOP/EQV-POS Functons, Washngton State Unversty, Pullman, [7] K Raj, Dgtal Systems Prncples and Desgn, Pearson Educaton Inda, Upper Saddle Rver, [8] Al Valach, R Slon, V Onofre and Fl Hoza, Analyss, Synthess and Verfcaton of Dgtal Logc Systems, Ed Nord-Est, Ias, 99 [9] O Ursaru and C Aghon, Multlevel Inverters wth Imbrcated Swtchng Cells, PWM and DPWM-Controlled, Electroncs and Electrcal Engneerng, Kaunas,

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