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# Class Note for ECE 380 at UA-Digital Logic(21)

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This 4 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Alabama - Tuscaloosa taught by a professor in Fall. Since its upload, it has received 16 views.

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Date Created: 02/06/15

Terminology ECE380 Digital Logic Optimized Implementation of Logic Functions Strategy for Minimization Minimum Product of Sums Forms Incompleter Specified Functions mama a cumma snunaang m n i imam Levine n 1 o For a given term each appearance of a variable in true or complemented form is called a literal xyz39 gt three literals abc39d gt four literals 0 Any 1 or group of 1 s that can be combined on a K map represents an implicant of a function 0 An implicant is a prime implicant if it cannot be combined with another implicant to remove a variable 0 A collection of implicants that account of all valuations for which a given function is 1 is called a cover of that function 0 Cost is the number of gates plus the total number of inputs to all gates in the circuit mama a comma Ensquotaqua m n i imam Lam n 2 Terminology example Prime implicants distinctions b Cf 00 01 11 10 fabcdZm01457911 00 1 0 0 01 Example Implicanls all single 139s a39c39 a39b39c39 11 0 a39bd ab39d 100 Prime Implicanls a39c39 a39bd ab39d b39c39d fabcdmm a39c39a39bdab39d Thus a minimum SOP form contains onlx but not necessarily all prime implicants mama a cumma snunaang m n i imam Levine n i Essential needed to form a minimum solution Nonessential not necessarily needed to form a minimum solution ab Cd 00 01 11 10 All prime implicanls b39d a39bc39 abc 00 0 1 o o a39c39d39 55quot Essential primes b39d a39bc39 abc 01 1 Nonessential primes a39c39d acd 11 0 fabcdmm b39da39bc39abc 100 0 0 inimum contains aH essential and possiblz some nonessential primes mama a comma Ensquotaqua m n i imam Lam n a Prime implicants example Prime implica nts example ab 5 00 01 11 1o 0 0 0 Essential primes a39c ac39d Nonessential prims a39bd bc39d V One of these must be included to form a minimum solution a39bd fabcdmm a39cac39d bcd 0 1 1 Identify all prime implicants for the given truth table Which are essential and which are nonessential What is a minimum SOP expression for this function ab 5 00 01 11 1o 00 0 0 0 0 01 0 1 1 1 11 0 1 1 0 10 1 1 0 1 mama a cumma Enunmng m n i imam Levine n 5 mama a comma Ensquotaqua m n i imam Lam n s Minimization of POS expressions Minimization of POS example POS minimization using K maps proceeds exactly as does SOP form except that groupings of 0 s in the K map are used to form POS terms K map can be constructed directly from HM expression for a function 0 Place 0 s in the K map for every maxterm in the HM expression fabc a b39c39a39bc39a39b39ca39b39c39 fabcHM3567 Cabooo1111o o 1 1 0 1 1 1 0 fa b39b c a39c39 mama a cumma Enunmng m n i imam Levine n 1 mama a comma Ensquotaqua m n i imam Lam n n Minimization of POS example Kmap groupings example fabcd139IM014810121415 ab 5 00 01 11 1o Hd abc 00 01 o 1 1 1 11 1 1 o o a c 101 1 o o fabcdmm abca c cd Draw the K map and give the minimized POS logic expression for the following fabcHM02357 Show the groupings made in the K map mama a cannula snunaang m n i imam Levine n a mama a comma Ensquotaqua m n i imam Luna 1 1n Incompletely specified functions In digital systems it often happens that some input conditions ie some input valuations can never happen 0 An input combination that can never happen is referred to as a don t care condition 0 As a circuit is designed a don t care condition can be ignored ie the output for that condition can be treated as 0 or 1 in the truth table 0 A function that has don t care conditions is said to be incompletely speci ed Example function with don t ca res Assume fora three variable function fxyz that the input combination xy01 never occurs otherwise the function is Zm0145 fXvz 2m0145D23 Or fXi liz HW517 39 D2i3 HHHHOOOOX HHOOHHOOK oowwnnwwm HOHOHOHON mama a cannula snunaang m n i imam Levine n 11 mama a comma Ensquotaqua m n i imam Luna 1 12 Example function with don t cares fxyz 2mo145D23 fxyz HM67 D23 zxyoo 01 11 1o zxyoo 01 11 1o o39jdol f 01d01 1 do1 11 1 fXvz v39 fXvz 139 Minimum SOP form 1 Choose a minterm a 1 in the Kmap which is not yet covered don39t consider d39s 2 Find all adjacent 1 s and d39s check the n adjacent cells for an nvariable Kmap 3 If a single term Le a single looping covers the 1 and all adjacent 1 s and d39s then the looping forms an essential prime implicant Loop the essential prime 4 Repeat steps 13 until all essential prime implicants are located 5 Find a minimum set of nonessential prime implicants to cover loop the remaining 1 s If more than 1 set is possible choose the set with the minimum number of literals the largest grouping lemon a cannula Enunmng m n J Jclam Latin 1 n z lmnul a calming Enunml39 Minimum POS form 1 Choose a maxterm a 0 in the Kmap which is not yet covered don39t consider d s 2 Find all adjacent 039s and d39s check the n adjacent cells for an nvariable Kmap 3 If a single term Le a single looping covers the 0 and all adjacent O s and d39s then the looping forms an essential prime implicant Loop the essential prime 4 Repeat steps 13 until all essential prime implicants are located 5 Find a minimum set of nonessential prime implicants to cover loop the remaining O s If more than 1 set is possible choose the set with the minimum number of literals the largest grouping lemon a cannula Enunmng m n J Jclam Latin 1 15 nrnJJclsnn 1mm

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