Digitl Ciruit Engineering 5 IN UT P MAPS (A) Loop squres with no friens, they n never shre. (B) Loop squres with no rothers, n simutneously loop their friens. Iterte: (A') Loop squres seperte from friens in lst itertion (B') Loop squres seperte from rothers in lst itertion Repet: 2n Distriutive ( + A)( + B) = + AB ( + A)( + B)( + C) = + ABC Simplifition Asorption + = + = + + = Crleton University 22 Most Common Stupi Errors. = + = John Knight ig5morek_mpsn.fm p. Revise; Ferury 3, 22 Slie i
Five-vrile mps Dul 4-vrile mps Vrile-entere mps Split-squre mps (optionl) Illustrtions showing when vrile entere mps re esier Multiple-output mps Exmple: rille eoer Methos of irling multiple-output mps. Do hlf-mps first 2. Loop s with no friens. Loop squres whose friens left for the rk sie. 3. Loop s with no rothers on the sme mp Expn these nite loops on other mps 4. Iterte over 2 n 3 5. Heuristi, result my not lwys work. Appenix exmple: BCD to 7-segment eoer Common Errors Crleton University John Knight ig5morek_mpsn.fm p. 2, Revise; 22 Comment on Slie i
Five-Vrile Krnugh Mps Methos for Five Vriles ) Dul 4-Vrile mps e= e= Think of two mps on top of eh other One n loop squres: on either level, or oth levels. 2) Vrile-Entere Mp top ottom lyer Best for where e hs simple reltions: e uses only few squres or e uses lmost ll the squres. lyer This mp for e This mp for e Enter vriles (letters) on the 6 squre mp e for s on top lyer Cn loop e e Cnnot loop e for s on ottom lyer. e e e e John Knight ig5morek_mpsn.fm p. 3 Revise; 22 Slie 5
Five-Vrile Krnugh Mps 5-Vrile Mps Methos 5-Vrile Mps Methos Dul 4-Vrile Mps Best for messy ptterns. Best with s (on t res) Vrile Entere Mps Best when one vrile, sy x, is use ut x is use little or not t ll. - Exmple, Cler input whih mkes ll outputs zero. Chosing the right vrile s the entere one n mke the looping muh esier. Speil Notes on 5-Vrile Mps Normlly s re left lnk to reue the lutter. Here is speifilly entere on one mp if is on the other mp. See Metho on top of slie. rp Aroun To loop etween lyers, the lyers must hve in the sme position on oth lyers. These re the squres whih iffer y only one input it. As shown on the right, orner squres re onfusing. e= e= OK e= e= rong looping Crleton University John Knight ig5morek_mpsn.fm p. 4, Revise; 22 Comment on Slie 5
Equtions From Five Vrile Mps Metho : Dul 4-Vrile Mps Terms for loops on the e= (top) mp re ANDe with e Terms for loops on oth mps on t mention e. Terms for loops on the e= (ottom) mp re ANDe with e Metho 2: Vrile-Entere Mp e e Loops ontining e re like loops only on the top (e=) mp ove. These re ANDe with e Loops tht ontin only s on t hve e in there expression. Loops ontining e re like loops only on the ottom (e=) mp ove. These re ANDe with e s loope only y e or e F= e( ) + + + e() F= e( ) + + + e() is only hlf loope e= e= e e e e Equivlent loops Compre wht hppens without this loop? John Knight ig5morek_mpsn.fm p. 5 Revise; 22 Slie 6
Equtions From Five Vrile Mps 5 Vriles Plotte on 6 Squre Mp 5 Vriles Plotte on 6 Squre Mp Aove exmple illustrtes: ) Drwing mp from wors 2) Being reful to hve ll s loope in oth the e n the e plne. - If you see only single loop roun, mke sure tht loop oes not ontin e or e. Exmple Using Dul Mps Loop n get n eqution. e= e= e= e= properly e e loope in oth plnes e Not loope in e plne e F= e() + + + + e() e Noting it is the sme prolem, try using vrile entere mp try to loop n fin minimum eqution. e= e e Crleton University John Knight ig5morek_mpsn.fm p. 6, Revise; 22 Comment on Slie 6
Use of Vrile-Entere Mps Exmple: Drw Vrile-Entere Mp for F, n Loop For Min Ciruit F is true when: oth A n B re off or oth C n B re off or oth A n E re on A D B A D B A E E E E D E E E E B Comine on one mp D C oth A n B off C oth C n B off C oth A n E on (OR the 3 mps) A E E E E E E B C D Loop mp Are these loops enough? A E E E E E E B E nees nother loop C D These squres were not loope in e plne Loop e Funtion written A E E E E C E E B F = EA + A B + C B + e plne oth plnes e plne John Knight ig5morek_mpsn.fm p. 7 Revise; 22 Slie 7
Use of Vrile-Entere Mps 5 Vriles Plotte on 6 Squre Mp Choosing the est mp to get n eqution Aove the F vrile hs simple pttern in tht it overs hlf of eh mp in n esy to loop form. If one vrile n e ftore out s in the prolem elow, this is goo guieline on wht to use. 5-. PROBLEM (Dul Mps) e= F = ( + + )e + ( + + )e = ( + )e + ( + )e + Plot F on the 5 vrile mp on the right. e= 5-2. PROBLEM (se on the lst prolem) Loop the 5 vrile mp n reue F to 2 letters, 4 terms of 3 letters eh. There is rumor this n e reue to letters 5-3. PROBLEM rite own the equtions for T L n T R from the slie. Crleton University John Knight ig5morek_mpsn.fm p. 8, Revise; 22 Comment on Slie 7
Vrile Entere Mps, T-Bir Til Lights Til Light Control Inputs Til light Til light LR B T L T R - - - - R- Right turn Flsh LR- 4-wy, Flsh Flsh L- - Left turn, Flsh - - B Brke Soli Soli - R B Right turn & Brke Soli Flsh LR B 4-wy n Brke Flsh Flsh L- B Left turn & Brke Flsh Soli L B Inputs R FLASH F Hz on-off signl 4 inputs on 3 se-input mp The Flsh input signl interts in simple wy. The 3 se-input vrile entere mp is simpler thn the 4-vrile mp. L B B F F R F F F F L F F Til light T L Til light T R ith 5 or 6 vrile prolem, think out using vrile entere mp. Choosing the proper entere vrile is importnt. Unfortuntely there my not e proper vrile. R T L L E26-7A B Til light T L F TR R John Knight ig5morek_mpsn.fm p. 9 Revise; 22 Slie 8
Vrile Entere Mps, T-Bir Til Lights 4 Vriles Plotte on 3-Bse Vrile 4 Vriles Plotte on 3-Bse Vrile Mp Til Light Control There re two outputs, one for eh til light. Eh hs mp. ou n ignore the 2n mp (T R ) for this isussion. Unlike most rs, there re two seprte swithes for the turn signls, L n R. Turning on oth L n R turns on the emergeny (4-wy) flshers. The left (right) rke light n the left (right) turn signls use the sme light uls or LEDs. If the rke is pushe with turn signl flshing, the other sie stys on ontinuously (soli). If L, R n B re ll on, the 4-wy flshers overrie the rke,. Vrile-Entere mp Choosing the vrile to est to use s the entere vrile is importnt. Often this vrile is little ifferent from the other vriles, like the Cler n Flsh signls in the exmples. A more omplex mp is use in the T-Bir Til Lights l. 6 Vrile Mps If you hve 6 vrile prolem, you just might e le to esily reue it to 4x4 mp with 2 entere vriles. In the T-Bir Til Lights l, 5-vrile prolem is plotte on 4x2 mp. 5-4. PROBLEM: rite the eqution, or rw the iruit, represente y this 6-vrile,y vrile-entere mp. ht ws this iruit lle in the Phone Swith L? 5-5. PROBLEM: Loop the mp for for T R ner the mile of Slie 9, n the fin the lgeri expression for T R. A D B C mp of g Crleton University John Knight ig5morek_mpsn.fm p., Revise; 22 Comment on Slie 8
Five-Vrile Mps ith Inluing in 5-Vrile Mps Using Dul Mps e= e= e= Using Vrile-Entere Mps John s solution e= F= e() + + e() Tom s solution e= e= F= e() + + e / / / / On top On ottom ht mess Tom sys, The intertion of n e is too omplex. Use ul mps for 5-input prolems with s. John Knight ig5morek_mpsn.fm p. Revise; 22 Slie 9
Five-Vrile Mps ith Five-Vrile Mps ith Don t Cres Five-Vrile Mps ith Don t Cres Dul 4-vrile mps The extension of s to 5 vriles is stright forwr. As efore, n e optionlly loope like. 5-6. PROBLEM e= F = + + e + e Further,,,e n never tke on the vlues,,, ( e) n,,, n never tke on the vlues,,, ( ) Plot F on the 5 vrile mp on the right, inluing s. e= 5-7. PROBLEM (se on the lst prolem) Loop the 5 vrile mp n reue F to 2 letters, Compre the nswer with the slie. Using vrile-entere with Vrile-entere mps lone usully nnot hnle. One hs to opt split-squre nottion for t lest some squres. Although there re few speil ses where vrile-entere mps might e useful, like when the s re ll stke on top of eh other in the two lyers, it is usully etter to use ul mps for these prolems. Crleton University John Knight ig5morek_mpsn.fm p. 2, Revise; 22 Comment on Slie 9
Multiple Outputs From Sme Ciruit Two outputs, F n G Sme inputs Fin the iruits for F n G Nee two mps Nee two iruits Often one n shre some gtes e optimize mps iniviully got one ommon gte. iruit size estimtes 29 letters (literls) 3+3+3 37 gte inputs 5+5+2+5 gtes In CMOS (numer trnsistors)/2 Mp of F F u = 3 letters Mp of G F= + G= + +u++ +u++ 3 letters 3 letters 5 AND inputs + 5 OR inputs 2 AND inputs + 5 OR inputs G John Knight ig5morek_mpsn.fm p. 3 Revise; 22 Slie 2
Multiple Outputs From Sme Ciruit Five-Vrile Mps ith Don t Cres Multiple Outputs Ciruits ith Two Outputs The ifferene etween multiple outputs n single outputs ith multiple outputs, one n often fin ommon gtes tht n e use for oth outputs. Often these ommon gtes re not optimum for either iniviul iruit, ut re optimum for the whole iruit. The exmple In this slie the iruits were optimize iniviully with hlf-herte effort to fin ommon terms. In the next slie, ommon terms were ggressively sought out. Ciruit omplexity There re severl wys to estimte the size of the iruit. The sme mesures lso estimte power issiption whih is now likely to e more importnt thn size. Inverters re not usully ounte in the gte ount. This is euse most will e sore when one oes n AND/OR to NAND/NOR onversion. Three methos of estimting iruit size:. The numer of gtes. 2. The numer of gte inputs. This mits tht multi-input gtes re lrger. One gte input usully orrespons to two trnsistors in CMOS logi. 3. The numer of letters on the right hn sie of the expressions. This is esy to o, n some onsiere it the est estimte. Note these re reltive estimtes; use for estimting if one iruit is signifintly igger thn nother. An solute or n urte estimte requires one to know the etils of the implementtion. Crleton University John Knight ig5morek_mpsn.fm p. 4, Revise; 22 Comment on Slie 2
Multiple Outputs Minimiztion Try to hrer to shre terms (gtes) Ientify ommon squres on oth mps Loop ommon terms even if the iniviul mps llow lrger loops Chek, sometimes ommon terms o not help. Here it hnge 3-input AND to 4-input AND n remove nother 3-input. Mp of F Mp of G gte inputs size mesures v= letters Prev This slie slie w= 29 29 letters (literls) 37 33 gte inputs u= 9 gtes G F F=v+w+u G=v+w+u ++ ++ 6 AND inputs + 5 OR inputs 9 letters gte inputs 9 letters John Knight ig5morek_mpsn.fm p. 5 Revise; 22 Slie 2
Multiple Outputs Minimiztion Multiple Outputs Multiple Outputs Colleting the u+v+w terms woul reue the numer of letters n gte inputs, ut will inrese the numer of gtes. However the totl logi is lerly reue. x = u+v+w = + + ( letters, 4 inputs, 4 gtes)) F = + + x (7 letters, 9 inputs, 3 gtes) G = + + x (7 letters, 9 inputs, 3 gtes) Totl: 25 letters, gtes, 32 gte inputs 5-8. PROBLEM Fin the Σ of Π expressions with miniml logi for the two-output iruit E, F. Soln. hs 5 gtes. If it is not pure Σ of Π, it n e one in 5 twoinput gtes, or, with ftoring, 4 gtes. 5-9. PROBLEM ) Fin the Σ of Π equtions using the minimum totl numer of gtes if no gtes re shre etween outputs. ) Fin the wy to reue this to 9 gtes if gtes re shre etween mps. One solution hs: two 5-input gtes, one 4-input gte, five 3-input gtes, n one 2-input gte. D CD AB A E = C E F B E D CD AB A C F = G= B F D CD AB A C MT4B B Crleton University John Knight ig5morek_mpsn.fm p. 6, Revise; 22 Comment on Slie 2
Multiple Outputs; BCD -> Brille BCD to Brille Conversion BCD (Binry Coe Deiml) Four its n efine numers up to 5. BCD igits use 4 its ut not ll omintions. Binry through 5 re not use Brille Brille symols for igits re 4 rise ots. The ot s position is enote y,,,. is upper left, is lower right Brille Digits 2 3 4 5 6 7 Binry De 2 3 4 5 6 7 8 9 2 3 4 5 Binry Coe Deimls AB \CD 4 2 3 2 8 9 Corresponing eiml numers 5 3 7 5 Binry positions on K-mp 6 4 3 2 4 5 7 6 8 9 8 9 Brille ptterns for eiml igits John Knight ig5morek_mpsn.fm p. 7 Revise; 22 Slie 22
Multiple Outputs; BCD -> Brille Brille Brille The Brille oe onsists of 6 squres, only the upper 4 re use for igits. A B C D E F G H I J K L M N O P Q R S T U V The first ten letters of the lphet re the sme s the igits. Crleton University John Knight ig5morek_mpsn.fm p. 8, Revise; 22 Comment on Slie 22
Multiple Outputs; Brille Mps.BCD to Brille Brille Dot ientifition Design BCD to Brille Converter BCD input ABCD Design iruit to: ept inry-oe eiml input ABCD generte Brille output. Brille ) Drw mps for outputs,, n. Inlue on t res. ) Loop the mps for minimum Σ of Π logi if eh output is trete seprtely. ) Loop the mps to minimize logi inluing gte shring. ) Stte (i) the minimum numer of gtes use (ii) the numer of gte inputs (iii) the numer of letters. AB \CD Not use Not use Not use BCD positions on K-mp 3 2 4 5 7 6 Not 2 use 3 5 4 Not use 8 9 Not use Corresponing eiml numers 3 2 4 5 7 6 8 9 Brille ptterns for eiml igits John Knight ig5morek_mpsn.fm p. 9 Revise; 22 Slie 23
Multiple Outputs; Brille Mps Four Output Minimiztion Four Output Minimiztion The figure ove on the right shows: - the inry positions on Krnugh mp, - the orresponing eiml numers, - n the Brille pttern for the eiml igits Brille symols re pttern of rise ots. The positions of the ots is given y the symols,, n. Design Exmple Design iruit whih will ept inry-oe eiml input ABCD n generte Brille output. Binry-oe eiml igits only go from to 9. The other numers, through 5 will never e reeive s inputs. ) Drw the mp for the output. Be sure to inlue on t res. ) Drw the mps for eh of the,, n outputs. ) Loop the mps to give the minimum Σ of Π logi if eh output is trete seprtely. ) Drw new set of mps. Then loop them to give minimum logi if gtes n e shre etween mps. e) Stte (i) the minimum numer of gtes use n (ii) the numer of gte inputs, (iii) the numer of letters. Crleton University John Knight ig5morek_mpsn.fm p. 2, Revise; 22 Comment on Slie 23
John Knight ig5morek_mpsn.fm p. 2 Revise; 22 ) Drw mps for outputs,, n. Inlue on t res. Mp of Mp of Mp of Mp of Slie 24 BCD--> Brille; Initil Mps
BCD--> Brille; Initil Mps Four Output Minimiztion Drwing the Four Mps is the ot in the upper left orner. Thus the mp onsists of ll the numers whih hve tht ot, nmely, 3, 2, 4, 5, 7,6 n 8. All the other squres on the mp re either or. is the ot in the upper right orner. The mp onsists of ll the numers whih hve tht ot, nmely, 3, 4, 6, 7 n 9. Crleton University John Knight ig5morek_mpsn.fm p. 22, Revise; 22 Comment on Slie 24
BCD--> Brille; No Gte Shring (How not to o it) ) Mps loope s though eh output ws inepenent. Lter this will e use to show svings from shring Mp of = + + + Mp of = + + 6+9+5+4 = 24 letters 3+5+3+3 = 4 gtes ( AND) 8+3+7+6 = 34 inputs Mp of = + + + Mp of = + John Knight ig5morek_mpsn.fm p. 23 Revise; 22 Slie 25
BCD--> Brille; No Gte Shring (How Four Output Minimiztion Cirling the Mps s Thought They ere Completely Inepenent These mps re loope, with no ttempt to shre AND terms; one gets: 24 letters (lso lle literls). Rememer one ounts only the letters on the right hn sie. 4 gtes. AND gtes whih will e the importnt numer when we look t Progrmmle Logi Arrys. 34 gte inputs. One n esily ount them y rwing piture. This lso turns out to e the numer of letters plus for eh AND term.(2 or more inputs AND terms) in the,,, n equtions. 6 letters 8 gte inputs Eh AND terms hs n output into the OR gte tht is not ounte when you ount the letters. = + + + A set of irling guielines to minimize gte the gte ount On the next few slies set of heuristis re given to shre gtes. A heuristi is n lgorithm whih usully gives goo nswer, ut is not gurntee to work, or s in this se give the est nswer. These heuristis re esigne to emphsise smll gte ount, n my tully inrese the numer of letters. These heuristis (exept for rule ) re espeilly goo for Progrmmle Logi Arrys (one lter) where the numer of gtes is ll importnt, n the numer of inputs is onstnt. Crleton University John Knight ig5morek_mpsn.fm p. 24, Revise; 22 Comment on Slie 25
BCD--> Brille; Shring Gtes; Hlf-Mps ) Loop the mps trying to shre gtes Hlf-Mp Rule (Rule ) Loop the hlf-mps first. They o not ny gtes. nees only one AND gte for three loops (on );,, (on ); Rule () my not work on Progrmmle Logi Arrys (to e one lter) Mp of Mp of Mp of Mp of hlf-mp hlf-mp qurter-mp A hlf-mp is loop whih overs hlf the squres John Knight ig5morek_mpsn.fm p. 25 Revise; 22 Slie 26
BCD--> Brille; Shring Gtes; Hlf-Mps Heuristi Rules for Looping Orer in Heuristi Rules for Looping Orer in Multiple Output Mps These rules will e followe in the next few pges. Hlf Mp Rule: Loop the hlf-mps first. (like n on the mp ove) They o not ny AND gtes. 2. No Friens Rule: Loop squres tht pper on only one mp. Suh squres n never e shre. 2).My Friens Are Gone Rule: Their friens re loope or on ll the other mps. No useful shring. 3. Prorstente Deisions: ith severl looping hoies for squre, o nother squre first if ville. 4. No Brothers Rule: Loop s with no neighours on the sme mp (the fmily mp), exept loope neighours or. Loop the sme squre on other mps tht hve n unloope there. 4).Expn the fmily mp loops provie you n lso expn the loops on ny frien s mps. Keep the loops if they re mx size, on the fmily mp. If they reh mx size on nother mp first, these rules hve no further vise. 5. ou re on your own. Fortuntely it is usully esy. Looping mps trying to shre gtes () A loop tht enloses hlf the mp, oes not use n AND gte; it is only piee of wire. Alwys loop them, there is no gin from shring piee of wire. However there is n exeption. In PLA (to e overe lter) piee of wire oes tke n AND gte. In tht se o not utomtilly loop hlf mps. (2) The s tht pper on only one mp nnot shre n AND gte with nother mp. Hene loop them with the lrgest possile loop immeitely. The only tht ppers on only one mp. Mp of Mp of Mp of Mp of Crleton University John Knight ig5morek_mpsn.fm p. 26, Revise; 22 Comment on Slie 26
BCD--> Brille; Fmily n Friens Cirling the mps trying to shre gtes Fmily n Friens; mening A squre is one of the 6 squres on the mp. e only look t squres tht ontin. For on one mp: s in sme squre in other mps re friens s in jent squres on the sme mp re rothers No friens See white sq on other mps No rothers Fmily Friens from high shool Mp of Mp of Friens Friens from Crleton Hs 2 rothers Mp of Friens from the street gng Mp of John Knight ig5morek_mpsn.fm p. 27 Revise; 22 Slie 27
BCD--> Brille; Fmily n Friens Heuristi Rules for Looping Orer in Fmily n Friens Preten you re in squre on one mp Fmily re other s on the sme mp. The importnt fmily memers re rothers. Brothers re in jent squres to you. Friens re s in the sme position on other mps. One frien is loope, one sys he/she hs gone to the rk sie. There is no point in trying to shre gtes with tht is lrey loope. Crleton University John Knight ig5morek_mpsn.fm p. 28, Revise; 22 Comment on Slie 27
BCD--> Brille; Shring Gtes, No Friens Rule (Rule2) No Friens Rule (for ht to Loop Next) For s with no friens ( ppers on only one mp). There is no wy to shre them. Loop them with s mny rothers s possile. (ou n inlue other s n s in the loop to mke it lrger) (on ) est loope y (on ) No friens 2 rothers Mp of Mp of Mp of Mp of John Knight ig5morek_mpsn.fm p. 29 Revise; 22 Slie 28
BCD--> Brille; Shring Gtes, No Friens Heuristi Rules for Looping Orer in Cirling the mps trying to shre gtes Rehsh of Rules ( n 2) e hve now loope hlf-mps n s tht only pper on one mp. These lter loops will never e shre, hene we ignore them when we look for further shring. These loops will e gry she to mke them esy to ignore from now on. Rule (2) Now, we look for unloope s tht re loope (gry) on ll the other mps. These s re now frienless; they will never e usefully shre etween mps euse the potentil shring prtners re lrey loope. Rule() n Progrmmle Logi Arrys (PLAs n PALs) In these rrys, whih will e stuie lter, the AND gtes re preuilt. It tkes the sme resoures to mke -input AND gte s it oes 5-input one. In suh ses shring is ll importnt. Hlf-mps shoul e trete like ny other loops, n re likely not est euse they re hr to shre. Crleton University John Knight ig5morek_mpsn.fm p. 3, Revise; 22 Comment on Slie 28
BCD--> Brille; Shring; My Friens Are Gone ) Looping mps trying to shre gtes, (Rule 2) My Friens Are Gone Rule. (Loope into the rk sie) ith no friens left, we nnot usefully shre loops. Loop these new frienless s. This former frien is loope, leving this with no friens to shre gte with Loop him with rothers n s. loop (on ) Frien went to the rk sie Mp of Mp of frien is gone frien is gone Loop him His frien is gone. Mp of Mp of John Knight ig5morek_mpsn.fm p. 3 Revise; 22 Slie 29
BCD--> Brille; Shring; My Friens Are Heuristi Rules for Looping Orer in Squres whose friens were loope erlier, n re now on the rk sie. e re going to hve to loop these squres some time, n there is no point in trying to shre gtes with s tht re lrey inlue in nother loop. Squres where the friens re on t res re not worth shreing. Crleton University John Knight ig5morek_mpsn.fm p. 32, Revise; 22 Comment on Slie 29
BCD--> Brille; Shring; My Friens Are Gone ) Looping mps trying to shre gtes (Rule 2 ) My Friens Are Gone Rule. (They went over to the rk sie) Fin s tht nnot e usefully shre; friens re lrey loope. Loop them with rothers. in nnot e usefully shre. loop (on ) in nnot e usefully shre. loop (on ) Mp of Mp of Loop me My only frien is gone. Mp of Loop me My only frien is gone Mp of John Knight ig5morek_mpsn.fm p. 33 Revise; 22 Slie 3
BCD--> Brille; Shring; My Friens Are Heuristi Rules for Looping Orer in Looping the mps trying to shre gtes There is no point in trying to shre loops if ll your friens hve gone over to the rk sie (lrey loope). Loop these squres now. Here, how to loop them is ler. In some ses it in not, for exmple: If the the friens re loope or, n the loop my inlue other squres. No friens Fmily Frien Here the smller n e shre n is useful on the other mp. The lrger nnot e shre n is poor hoie. Crleton University John Knight ig5morek_mpsn.fm p. 34, Revise; 22 Comment on Slie 3
BCD--> Brille; Shring; Friens Are All Gone ) My Friens Are Gone Rule, rule 2 (ont) Two frienless. n tht n t shre loops It mkes ifferene whih is one first Choie (4 orners), Choie 2 or (lst olumn) Mp of frien me first Mp of No loop me first De Loop Mp of De frien De frien Mp of Mp of Mp of One Choie loop (right entre) Mp of John Knight ig5morek_mpsn.fm p. 35 Revise; 22 Slie 3
BCD--> Brille; Shring; Friens Are All Heuristi Rules for Looping Orer in Cirling the mps trying to shre gtes (Rule 2) (ontinue) Continue to look for s tht re unloope on only one mp. Loop them. Prorstente if one squre hs severl loop hoies (Rule 3). Sometimes they n e loope in severl wys. Look t wht else is inlue in the loop. Look t, in the upper right orner. Choie : Here four orner loop will lso over one extr squre. Choie 2: A vertil olumn loop will lso over one extr squre. Look t, lote out here. Single hoie: It n e loope to over one more unovere fmily squres. Tht loop n lso e shre so his rother overs nother frien squre. Its loop lso removes the vertil olumn hoie for, the upper right orner. Conlusion: Loop first. Then loop using the four orners. Crleton University John Knight ig5morek_mpsn.fm p. 36, Revise; 22 Comment on Slie 3
BCD--> Brille; Prorstente ) My Friens Are Gone Rule, rule 2 (ont) Two frienless s. Loop first? Two hoies Loop first? One hoie n It mkes ifferene whih is one first Choie Choie 2 or (lst olumn) Single Choie loop (right entre) Use Rule 3 Prorstente ou must o oth squres, ut - tht n t shre loops (4 orners), hih to o first? Do the single hoie one first. It my stop you from mking poor hoie on the other one. Mp of Mp of Mp of Do first. John Knight ig5morek_mpsn.fm p. 37 Revise; 22 Slie 32
BCD--> Brille; Prorstente Heuristi Rules for Looping Orer in hy Prorstente? There ws question of whih squre to loop next. One squre oul e loope in only one wy. The other h two possile loops to hoose from. Usully the single hoie squre is the one to o next. It puts off mking hoie whih might not e the right one. In this se it remove one of the hoies from the other squre, mking the next loop no-riner. Crleton University John Knight ig5morek_mpsn.fm p. 38, Revise; 22 Comment on Slie 32
BCD--> Brille; Prorstention my help ) Looping the mps trying to shre gtes Result of Looping using Prorstention It removes one hoie from Mp of Mp of Mp of Mp of Loop first. (in mp ) Expn loop to over unovere squre in mp This lso removes hoie from Must loop four orners John Knight ig5morek_mpsn.fm p. 39 Revise; 22 Slie 33
BCD--> Brille; Prorstention my help Heuristi Rules for Looping Orer in Finl results of Rule 2. My friens re gone. Note tht y looping just s tht one is sure nnot e shre, the size of the prolem hs een gretly reue. ht to Loop Next rules in summry. Hlf Mp Rule: Loop the hlf-mps first. They o not ny AND gtes. 2. No Friens Rule: Loop squres tht pper on only one mp. Suh squres n never e shre. 2).My Friens Are Gone Rule: Loop squres whos friens re loope or on ll other mps. No useful shring. 3. Prorstente Deisions: ith severl looping hoies for squre, o nother squre first if ville. 4. No Brothers Rule: Loop s with no neighours on the sme mp (the fmily mp), exept loope neighours or. Loop the sme squre on other mps tht hve n unloope. 4).Expn the fmily mp loops provie you n lso expn the loops on ny frien s mps. Keep the loops if they re mx size, on the fmily mp. If they reh mx size on nother mp first, these rules hve no further vise. 5. Use originl thought. Fortuntely it is usully esy. Crleton University John Knight ig5morek_mpsn.fm p. 4, Revise; 22 Comment on Slie 33
John Knight ig5morek_mpsn.fm p. 4 Revise; 22 ) Looping the mps trying to shre gtes Loops from the previous slie re now she. Mp of Mp of Mp of Mp of Summry of wht is loope up to now. Left to o Slie 34 BCD--> Brille; Shring
BCD--> Brille; Shring Heuristi Rules for Looping Orer in Cirling the mps strting with the fmily Rule (4): Squres with The oviously unshrele loops hve een foun. e will now look for loops tht n e shre. To o this is to strt smll. Look for single squres on one mp tht re lso n unloope on one or more other mps. Cll the originl mp the fmily mp. e know we will hve to loop tht, n we will try to shre its loop. Then expn the fmily squre s loop to inlue more squres. Do the sme expnsion on the other mps. Eventully you will hve to stop. If you hve to stop euse limittions on the fmily mp this squre must e the est. ou know tht must e loope, n this is the lrgest squre tht n o it. If you hve to stop euse of limittions on the other mps, you will hve to evlute the enefits of shring versus the enefits of hving the lrger squre on the fmily mp. This requires ifferent kin of thinking thn just following the rules. Rememer Lrger loops mke smller gtes. Shring loops elimintes gte(s). Coul the friens squre e inlue in nother loop? Goo luk. Crleton University John Knight ig5morek_mpsn.fm p. 42, Revise; 22 Comment on Slie 34
John Knight ig5morek_mpsn.fm p. 43 Revise; 22 ) looping the mps trying to shre gtes. Mp of Mp of Mp of Mp of Mp of Mp of Mp of Mp of Loop s with NO rothers on its fmily mp (4) Expn the loops on the fmily mp - Stop expning when loop Don t ount loope rothers or s. Try to lso expn on the Then loop friens on other mps Single hil friens mp(s). frien (Rule 4) No Brothers Rule. on fmily mp is mx size. - It my e est to stop when the friens loop is mx size. Use your jugement. Slie 35 BCD--> Brille; Loop Single Chilren
BCD--> Brille; Loop Single Chilren Heuristi Rules for Looping Orer in The rules (Guielines) hve finishe The rest is: Common sense. Low unning. Luk Here the wy to loop the other squres is firly ler. 5-. PROBLEM Assume the term,,, =,,, is use for some other purpose, like eiml point, n is no longer on t re input. Fin the new set of mps, n the new equtions. ou will inrese the numer of gte inputs y two (I think), ut you shoul not hve to ny new gtes. Crleton University John Knight ig5morek_mpsn.fm p. 44, Revise; 22 Comment on Slie 35
John Knight ig5morek_mpsn.fm p. 45 Revise; 22 ) Looping the mps trying to shre gtes. Use Originl Thought Here the solution is very ler. Mp of Mp of Mp of Mp of Tke wht is left n try to loop n shre Slie 36 BCD--> Brille; Finl Clen Up
BCD--> Brille; Finl Clen Up Finl Equtions n Gte Count Finl Equtions n Gte Count The finl equtions show: Letters 7.7% loss Gtes 4.3% sving AND gtes 2% sving Gte inputs 8.9% sving. Not too impressive? For more imressive result see Comment on Slie 46. ht ommonly hppens is tht shring oes not sve too muh on the letter ount, ut hs etter sving on the gte ount. Unfortuntely, omputer-ie esign softwre often ounts letters. AND gtes my e the most importnt For mny CPLD (omplex progrmmle logi evies) the internl logi is with PALs (Progrmmle Arry Logi) in whih the importnt prmeter is the numer of AND gtes. Shring gtes is then muh more importnt thn the letter ount. ho res how mny gtes there re? ith ten million gtes on piee of silion tht osts $3. why shoul one worry. If you hve smll iruit whih my e replite, times on piee of silion. It mtters. Also power usge inreses with gte ount. This mens your ell phone runs own more quikly, or your lp top nees igger fn. Dely in iruit often inreses with more gtes. Crleton University John Knight ig5morek_mpsn.fm p. 46, Revise; 22 Comment on Slie 36
BCD--> Brille; Finl Equtions n Gte Count ) Finl equtions n gte ount, Mp of G H Mp of N N L L Mp of J J L Mp of J J H L = G on mp H = = + H + + = + L + N on mp This solution with gte shring 26 letters 2 gtes (8 ANDs) 3 gte inputs J = = J + + L + = J + + H ith no shring 24 4 ( AND) 34 Letters like H re put on the mp to help mth loops with equtions. John Knight ig5morek_mpsn.fm p. 47 Revise; 22 Slie 37
BCD--> Brille; Finl Equtions n Gte Minimiztion of prolems of this omplexity woul e one using omputer-ie esign progrm. However the initil setting up of the prolem, proly with the i of Krnugh mps, woul hve to e one mnully. Also you nee to know wht the progrm is trying to o. For exmple, oes it give hlf-mps omplete priority? this is unesirle for rry logi implementtions. Crleton University John Knight ig5morek_mpsn.fm p. 48, Revise; 22 Comment on Slie 37
Appenix: Another Multiple Output Exmple The Common Errors Setion Is At the En The 7-Segment Disply Another Exmple with mny Multiple Outputs An exmple muh like the BCD to Brille exmple My opinion is tht setting up the equtions is the most useful thing to lern here. It hs 7 outputs inste of 4. Before rule 2 ws importnt n rule 3 ws simple. Here rule 2 oes nothing, n rule 3 is omplex. This gives more prtie rther thn giving new onepts. John Knight ig5morek_mpsn.fm p. 49 Revise; 22 Slie 38
Appenix: Another Multiple Output Seven-Segment Disply Driver Seven-Segment Disply Driver Design Exmple The slie ove, shows the rs in seven segment isply suh s is use in mny utomotive shor isplys, or other right isplys. All the igits from through nine n e shown y lighting the proper rs. Design iruit whih tkes inry-oe-eiml (BCD) igit in on les,, n n sens out the signls to light the 7-segment isply on les,,,, e, f, n g. Binry-oe eiml (BCD) igits only go from to 9. The other numers, through 5 will never e reeive s inputs. Utilize this ft in your solution. 5-. PROBLEM The igits on the right hve revise form for, 7, 6 n 9. Derive the mps for the isply rivers. \ 5-2. PROBLEM Fin the minimum iruit with the three outputs efine y the mps elow. This is hr prolem. ou shoul re over the exmple for the 7-segment isply rivers efore ttempting it. Revise isply F = G = H =. The right isplys, suh s on loks n lk/don t-lk signls use light-emitting ioes. The immer wth n ontrol pnel isplys re usully liqui rystl n hve more omplex river logi. Crleton University John Knight ig5morek_mpsn.fm p. 5, Revise; 22 Comment on Slie 38
ExmpleOutput Minimiztion; 7-Segment Disply 7-Segment Disply Driver, BCD Digits in inry Design Driver Logi 4 inputs, 7 outputs 7 mps, eh with 6 on t res Generte Mps Choose segment fin ll the squres where is lit. Repet for,,... LOGIC f g e Design this logi f e g Deiml igits isplye Digits with lit Digits with lit Digits with lit Digits with lit Digits with e lit Digits with f lit Digits with g lit John Knight ig5morek_mpsn.fm p. 5 Revise; 22 Slie 39
Appenix: Another Multiple Output Rules for Multi-mp Minimiztion Rules for Multi-mp Minimiztion These re the sme suggeste rules s use in the BCD--> Brille exmple. If you hve not thoroughly stuie tht, the rules will e hr to unerstn. However, the rules re expline in more etil in the next few pges. The rules here re stte little more formlly thn in the BCD--> Brille exmple. The rules re firly utomti up to 4. At rule 4 one hs to use more jugment, intuition n low unning. After oing these rules, irling ny remining squres is usully firly esy. If the minimiztion is for rry logi, whih will e stuie lter. The first rule (loop the hlf mps) is not one. In rry logi AND n OR gtes re preuilt, so -input AND n one with more inputs use the sme resoures. Solution to Pro 5-2. L=+ + F= +L G=+L H=++ 2 letters, 27 gte inputs, gtes Crleton University John Knight ig5morek_mpsn.fm p. 52, Revise; 22 Comment on Slie 39
Mps for 7-Segment Disply Drivers f e g Minimiztion Digits with lit Trnsfer lit segment mps to Krnugh mps Loop s in on one mp, who s squre is either loope, or on ll other m e Do fin loops ommon to two or more mps using these suggeste rules introue in the Brille exmple: f () Look for hlf-mp loops (2) Loop s on one mp who s squres on other mps re. (No friens rule) (2) Loop s on one mp, who s friens on other mps, re loope, or. (3) ith severl loop next nites, put off the ones with hoies of loops, until the en. (4) Loop s with rothers on the sme mp, exept loope neighours or. (4) Expn the nite loops, rete in (3), provie you n lso expn on the other mps. - Keep loops tht re mx size on the fmily mp. (4) Otherwise think out it! Some expnsions my intert with (4) g John Knight ig5morek_mpsn.fm p. 53 Revise; 22 Slie 4
Mps for 7-Segment Disply Drivers Rules for Multi-mp Minimiztion Mps for 7-Segment Disply Driver Minimiztion (ontinue) Loops tht over hlf the mp These re represente y single letter n re prtiulrly goo. Sine they only ontin single letter, they o not nee n AND gte. The input n fee iretly into the OR gte. There is no vntge to shring these terms etween mps euse there is no hrwre to shre. 5. Lote ll loops whih, with s if neee, over hlf of mp. There re some ten of them in this exmple. 6. It is esy to overo this rule Two of these loops over no s tht re not overe y other loops. The only new squres they over ontin s n hene re useless. Remove suh loops. The n mps hve suh useless loops. e f g Crleton University John Knight ig5morek_mpsn.fm p. 54, Revise; 22 Comment on Slie 4
7-Segment Disply; Loop Hlf-Mps Minimiztion Rule, Hlf-Mp Rule Exmple: nees only one AND gte (Rule ) Look for hlf-mp loops (one letter terms) for three loops These o not require n AND gte. Hene they n lwys e loope without loss of potentil gte shring. e f g These loops re ll hlf-mp loops. A ommon error: The she loops, on the n mps, re reunnt. John Knight ig5morek_mpsn.fm p. 55 Revise; 22 Slie 4
7-Segment Disply; Loop Hlf-Mps Rules for Multi-mp Minimiztion Minimiztion (ontinue) Rule (2) The No Friens Rule A whih ppers on only one mp, n never e shre. These shoul e loope on the one mp with the lrgest possile loop. Unfortuntely there re no frienless s in the seven-segment eoer. Rule (2) All My Friens Hve Gone to The Drk Sie Rule These s might s well e loope now, euse there is no sving from shring the loop with nother mp where the squre is lrey loope or. Unfortuntely, ll the s hve friens on one one or more mps. hih mps use ny prtiulr squre is summrize in the mp in the upper right orner ove. Crleton University John Knight ig5morek_mpsn.fm p. 56, Revise; 22 Comment on Slie 4
7-Segment Disply; No Friens Minimiztion Rule (2) n (2) No Friens Rule (2) Loop s on one mp who s squre on other mps re (Hve no frien on friens mps) There re none here. My Friens Hve Gone to the Drk Sie (2) Loop s on one mp, who s squre is either, loope or on ll other mps. These nnot e usefully shre with other mps. Exmples: Unloope, ut on more thn one mp, There re none here. e f g Loop s in on one mp, who s squre is either loope, or on ll other mps. Mps with unloope squres,,, e,f g e,g,g,,,e g g,e This is one John Knight ig5morek_mpsn.fm p. 57 Revise; 22 Slie 42
7-Segment Disply; No Friens Rules for Multi-mp Minimiztion Minimiztion (ontinue) Rule (4) Loop single s whih hve no rothers on the sme mp exept loope s or s. Loop the single s. There hve to e s in the orresponing squres in other mp(s), or they woul hve een loope in rule 2. These s hve gret potentil for shring. Further, y strting with the smllest loop, we o not lose shring opportunities. In this exmple mps, n f, hve single squres. The ft tht the loops oul expn into s or into lrey loope res oes not mtter. This is hnle in the next rule. These single squre loops re nite loops. f Rule (4)Expn the loops on the fmily mp n on the friens mps. Stop when ny of the loops nnot e further expne. Stop for goo if the fmily mp loop is oing the limiting. Otherwise see Rule (4) Exmple; the originl fmily mp loop in n e expne to two squres in, n g. This is the mximum expnsion in. n lso on the frien s mps n g, Similrly, the fmily mp expnsion shown in n e expne in. It is limite on the fmily mp, n not further limite y the other mps, so we expn to the full 4 squres. g Crleton University John Knight ig5morek_mpsn.fm p. 58, Revise; 22 Comment on Slie 42
7-Segment Disply; Single Chil Rule Minimiztion (4) Loop s with no rothers on the sme mp, exept loope rothers or. Single loop s in the sme squre on other mps. e f g This is one (4) Expn these fmily mp loops provie you n lso expn on ll the other mps. fmily mp loops fmily mp loop Expn e f Expn Expn Expn fmily mp loop g This is one John Knight ig5morek_mpsn.fm p. 59 Revise; 22 Slie 43
7-Segment Disply; Single Chil Rule Rules for Multi-mp Minimiztion Minimiztion (ontinue) Rule (4n 4) Expn the fmily mp loops ut hek efore over expning. Previously the expnsion of the loop on the fmily mp ws limite on tht mp Here we re expning loops tht re limite y the frien mps. Thus squre n expn to or to on mp. f g g One oul ignore shring gte with g n expn the strting loop to, the full olumn. However look t f. There n e use to over the strting squre in the orner. This sves gte, provie there is goo wy (4 orners?) to loop tht orner squre on the other mps, nmely, n e. However one nnot similrly expn the loop ll the wy on the g mp. g Four orners is goo wy, s shown on the next slie, ut it turns out not to e quite s goo. e Crleton University John Knight ig5morek_mpsn.fm p. 6, Revise; 22 Comment on Slie 43
7-Seg Disply; Don t Overexpn On Fmily Mp Minimiztion (4) Consier your expnsions: - Keep loops tht re mx size on fmily mp n re mx size on fmily mp; they shoul e kept. - Others expnsions re less ler: Expning on mp, removes for shring on mp g, ut it overs two more squres in f. However, shre with,, & e n over them, ut see next pge. Remove mx size e e mx size f f Remove g This is one This is one g John Knight ig5morek_mpsn.fm p. 6 Revise; 22 Slie 44
7-Seg Disply; Don t Overexpn On Rules for Multi-mp Minimiztion Minimiztion (ontinue) Rule (4) Expn the fmily mp loops n ll the orresponing loops on other mps. Two prtil solutions emerge: Solution () expns nite squre on mp, to Solution () overexpns it to. Solution () loops 2 orners. Solution () loops 4 orners. It is not yet ler whih will e etter. Crleton University John Knight ig5morek_mpsn.fm p. 62, Revise; 22 Comment on Slie 44
John Knight ig5morek_mpsn.fm p. 63 Revise; 22 Minimiztion Solution () Solution () g e This is one g e This is one f f Then ing (4 orners) to tke the ple of shring on,, e & f. expning. Slie 45 7-Seg Disply; Cheking Expnsion
7-Seg Disply; Cheking Expnsion Minimiztion (the en) Try the est tril solutions. It is only t the en tht we n tell whih is the est solution. Here the g mp eies the nswer: The () solution g mp looks like it hs more gtes. However every AND gte is shre with nother mp. g The () solution hs 4-squre loop,, whih is only use on the g mp, n thus osts n extr gte. g Crleton University John Knight ig5morek_mpsn.fm p. 64, Revise; 22 Comment on Slie 45
John Knight ig5morek_mpsn.fm p. 65 Revise; 22 Finl Fill In Solution () using (2 orners) g e This is one f Solution () using (4 orners) g e This is one f Nee two more loops n Nee three more loops, n Slie 46 7-Seg Disply; Finl Fill In
7-Seg Disply; Finl Fill In Disply Drivers, Forming Equtions Disply Drivers, Forming Equtions One wy of forming equtions is to use letter like J, H, L.. for eh term n put the letter in the squres overe y the loop for the term.thus Q = is written in the two left-hn orners on four mps. To voi onfusion, leve in squres whih re overe y severl loops. One writes the equtions for the segments s the OR of these letters. Thus e = Q + P Terms whih hve only one input like, o not require speil letter, n we give them the nme of the input vrile. Finl Results ith no shring of gtes 46 letters (literls) 23 gtes (6 AND) 58 gte inputs Mximum Shring Solution () Less Shring Solution () 4 letters 37 letters 3 gtes (7 AND) 4 gtes (6 AND) 4 gte inputs 38 gte inputs Using outputs in Solution () s inputs for other outputs. 38 letters 3 gtes (7 AND) 38 gte inputs Here shring sves muh more thn it i for the Brille prolem. As is generlly oserve, shring minly reues the numer of gtes. It hs less effet on letters or gte inputs, in ft it my even inrese these. For exmple Solution () hs more shring n fewer gtes, ut it hs more letters n more gte inputs. 5-3. PROBLEM (Do 5-. efore oing this.) Minimize the equtions for multiple outputs, using the revise isply for numers, 7, 6 n 9. Follow the methos use in the lst few pges. Keep the sme pitl letters for ll expressions tht o not hnge. If you nee new expressions, the letters K, Q, R, S, T, U n V hve not een use Crleton University John Knight ig5morek_mpsn.fm p. 66, Revise; 22 Comment on Slie 46
7-Seg Disply; Finl Equtions Form Equtions (Solution ) Lel AND terms with letters If only one term overs squre reple y letter. If severl terms over squre leve s. H M N J L J H M N P H H H P L N H e f g M P Size mesures J = M = H = N = L = P = 4 letters (literls) 3 gtes (6 ANDs) 4 gte inputs = H + J + M + N + = J + L + = + + = H + N + M + P e = H + P f = H + + g = L+ M + N + P + All terms reuse. Using: = f + = e + N + M 38 letters 3 gtes 38 gte inputs John Knight ig5morek_mpsn.fm p. 67 Revise; 22 Slie 47
7-Seg Disply; Finl Equtions Finl Results Chek for wrprouns Chek for 4-orners Common Errors ith Krnugh Mps Chek for wrproun on opposite sie Chek for wrproun top-to-ottom Chek for expnsion of loop to 4 squres or 8 squres Lern to spot squres with no friens. Crleton University John Knight ig5morek_mpsn.fm p. 68, Revise; 22 Comment on Slie 47
John Knight ig5morek_mpsn.fm p. 69 Revise; 22 Common Errors Mp of F Mp of G Chek our Mp Entries If you put one vrile in the wrong squres, you re tost! Don t Tret Multiple Output Prolems Like Unrelte Ciruits Mp of F Mp of G Do Hlf-Mps First (Exept for PLAs in lter setion) 2 gtes 9 gtes, 3 shre No friens, n other rules re heuristis, they often, ut not lwys, work. Don t Tret Two-Output Prolems Like 5-Vrile Prolems Chek for rproun, n ie rp Aroun Common Mp Errors
John Knight ig5morek_mpsn.fm p. 7 Revise; 22 Slie i
Common Mp Errors Finl Results Crleton University John Knight ig5morek_mpsn.fm p. 7, Revise; 22 Comment on Slie 48