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# Class Note for ECE 474A with Professor Lysecky at UA

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This 9 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Arizona taught by a professor in Fall. Since its upload, it has received 12 views.

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Date Created: 02/06/15

ECE 474A57A ComputerAided Logic Design Lecture 9 SUPPLEMENT Espresso Definitions and Functions ECE 474a575a Susan Lysecky Cubes and covers Compact Form A cube in the Boolean nspace associated with a multipleoutput function can be specified by it s vertices and by an index indicating to which components of f it belongs The information can also be given in compact form Let p be a product term associated with an algebraic sum of products with n inputs and m outputs The cube p is specified by a row vector c c1 cn c c where n1 nm 0 if xi appears complemented in p 1 if xi appears not complemented in p i1 n 2 if xi does not appear in p 3 if p is not present in the algebraic representation of n in1 nm 4 if p is present in the algebraic representation of fn ECE 474a575a 2 of 9 Susan Lysecky Cubes and covers Intersection The intersection or and or product of two cubes c and d written c n d or cd is a cube e The entries of ei are obtained from the entries of c and d accordingly di di n 0 1 2 n 3 4 ci 0 0 D 0 1ltltn ci 3 3 3 nltiSnm 1 D 1 1 4 3 4 2 0 1 2 When an index i such that ci and di give D the cube is said to be an empty cube If the output part of e has all 3 s e is also the empty cube ECE 474a575a 3 of 9 Susan Lysecky Cubes and covers Union The union or sum or product of two cubes c and d written c u d or cd is the set of verticies covered by the input part of either c or d If we use matrix representation of the cubes c u d is the matrix formed by the two rows corresponding to c and d respectively ECE 474a575a 4 of 9 Susan Lysecky Unate Complement UNATECOMPLEMENTF begin M MF compute personality matrix of F V MONOTONEF determine if monotone increasingdecreasing M PERSUNATECOMPLEMENTM complement M R TRANSLATEM V complement of F is determined by M and V end Personality Matrix Let F be the matrix representation of a single output unate cover with k cubes and n input variables The matrix M is the Boolean matrix de ned as follows lifFij00r1 fori1kj1n own2 ECE 474a575a 5 of 9 Susan Lysecky Personality Matrix Complement PERSU NATECOMPLEMENTM begin M D initialize M T SPECIALCASESM look for special cases if T1 return M found a special case j UCOMPSELECTM select a splitting variable M1 M0 PERSCOFACTORSM j computes M of the cofactors wrt x W PERSUNATECOMPLEMENT M1 one branch of recursion We PERSUNATECOMPLEMENT MO another branch of recursion Return MERG E W W merging process essentially concatenates Mland M0 end SPECIALCASES Result Return There is a row Of a 0395 Function is a tautology and the complement of the function is empty T 1 M CD M is empty The complement is a tautology T 1 M 0 0 M has only 1 term The complement is computed by DeMorgan s Law M has one row T 1 M DEMORG None of the above Return indication of this T0 M CD ECE 474a575a 6 of 9 Susan Lysecky Cubes and covers Shannon Expansion Given a set of cubes G C1 cL and a cube p all with n entries in the input part and m entries in the output part the cofactor of G with respect to p Gp is a set of cubes possibly empty obtained by computing the cofactor of each of the cubes in the cover G The cofactor of ci with respect to p as the cube with components G if Ci 0 p D ifa 10 combination it s empty Ci 2 if pk 0 0r 1 pk variable is what we are factoring out p k 4 if pk 3 ignore we are dealing with single output functions CL otherwise copy whatever was in cL ECE 474a575a 7 of 9 Susan Lysecky Complement COMPLEMENTF D begin R D for i1 m for each output extract single output function begin Fi Di EXTRACTF D i Ri COMP1Fi u D find complement of single output function R R R merge results end Reutrn R End ECE 474a575a 8 of 9 Susan Lysecky Comp1 Single Output Function Complementor COMP1F begin if row of all 2 s Return R d9 special case contain universe complement is empty if F unate Return R UNATECOMPLEMENTF special case unate c F1 special case column of all 1 s extract variable c forj1 n begin fori2 F ifcj at Fji then cj 2 end R UNATECOMPLEMENTc variable c unate determine complement of c F Fc get remaining function Fc j BINATESELECTF choose splitting variable R R MERGEWITHCONTAINMENTCOMP1FXj COMP1FXT call recursively on cofactors Return end ECE 474a575a 9 of 9 Susan Lysecky

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