1 - Sources TSR Katz Boriello& Vahid 1 CSE140 Components...

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Unformatted text preview: Sources: TSR, Katz, Boriello & Vahid 1 CSE140: Components and Design Techniques for Digital Systems Representation of logic functions Tajana Simunic Rosing Sources: TSR, Katz, Boriello & Vahid 2 Canonical Form -- Sum of Minterms • Truth tables are too big for numerous inputs • Use standard form of equation instead – Known as canonical form – Regular algebra: group terms of polynomial by power • ax 2 + bx + c (3x 2 + 4x + 2x 2 + 3 + 1 --> 5x 2 + 4x + 4) – Boolean algebra: create a sum of minterms • Minterm : product term with every literal (e.g. a or a’) appearing exactly once Determine if F(a,b)=ab+a’ is same function as F(a,b)=a’b’+a’b+ab, by converting the first equation to canonical form Sources: TSR, Katz, Boriello & Vahid 3 A B C F F’ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 F = F’ = A’B’C’ + A’BC’ + AB’C’ Sum-of-products canonical forms • Also known as disjunctive normal form • Minterm expansion: F = 001 011 101 110 111 + A’BC + AB’C + ABC’ + ABC A’B’C Sources: TSR, Katz, Boriello & Vahid 4 short-hand notation for minterms of 3 variables A B C minterms A’B’C’ m0 1 A’B’C m1 1 A’BC’ m2 1 1 A’BC m3 1 AB’C’ m4 1 1 AB’C m5 1 1 ABC’ m6 1 1 1 ABC m7 F in canonical form: F(A, B, C) = Σ m(1,3,5,6,7) = m1 + m3 + m5 + m6 + m7 = A’B’C + A’BC + AB’C + ABC’ + ABC canonical form ≠ minimal form F(A, B, C) = A’B’C + A’BC + AB’C + ABC + ABC’ = (A’B’ + A’B + AB’ + AB)C + ABC’ = ((A’ + A)(B’ + B))C + ABC’ = C + ABC’ = ABC’ + C = AB + C Sum-of-products canonical form (cont’d) • Product minterm – ANDed product of literals – input combination for which output is 1 – each variable appears exactly once, true or inverted (but not both) Sources: TSR, Katz, Boriello & Vahid 5 A B C F F’ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 F = 000 010 100 F = F’ = (A + B + C’) (A + B’ + C’) (A’ + B + C’) (A’ + B’ + C) (A’ + B’ + C’) Product-of-sums canonical form • Also known as conjunctive normal form • Also known as maxterm expansion • Implements “zeros” of a function (A + B + C) (A + B’ + C) (A’ + B + C) Sources: TSR, Katz, Boriello & Vahid 6 A B C maxterms A+B+C M0 1 A+B+C’ M1 1 A+B’+C M2 1 1 A+B’+C’ M3 1 A’+B+C M4 1 1 A’+B+C’ M5 1 1 A’+B’+C M6 1 1 1 A’+B’+C’ M7 short-hand notation for maxterms of 3 variables F in canonical form: F(A, B, C) = Π M(0,2,4) = M0 • M2 • M4 = (A + B + C) (A + B’ + C) (A’ + B + C) canonical form ≠ minimal form F(A, B, C) = (A + B + C) (A + B’ + C) (A’ + B + C) = (A + B + C) (A + B’ + C) (A + B + C) (A’ + B + C) = (A + C) (B + C) Product-of-sums canonical form (cont’d) • Sum term (or maxterm) – ORed sum of literals – input combination for which output is false – each variable appears exactly once, true or inverted (but not both) Sources: TSR, Katz, Boriello & Vahid 7 Mapping between canonical forms • Minterm to maxterm conversion – use maxterms whose indices do not appear in minterm expansion...
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1 - Sources TSR Katz Boriello& Vahid 1 CSE140 Components...

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