1 - Sources TSR Katz Boriello& Vahid 1 CSE140 Components and Design Techniques for Digital Systems Representation of logic functions Tajana

Info iconThis preview shows pages 1–8. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

This note was uploaded on 11/10/2010 for the course CSE 140 taught by Professor Rosing during the Spring '06 term at UCSD.

Page1 / 47

1 - Sources TSR Katz Boriello& Vahid 1 CSE140 Components and Design Techniques for Digital Systems Representation of logic functions Tajana

This preview shows document pages 1 - 8. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online