CPE100: Digital Logic Design I

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1 Chapter 2 Professor Brendan Morris, SEB 3216, brendan.morris@unlv.edu CPE100: Digital Logic Design I Section 1004: Dr. Morris Combinational Logic Design Chapter 2 <1>

2 Chapter 2 :: Topics Introduction Boolean Equations Boolean Algebra From Logic to Gates Multilevel Combinational Logic X s and Z s, Oh My Karnaugh Maps Combinational Building Blocks Timing Chapter 2 <2>

3 Introduction A logic circuit is composed of: Inputs Outputs Functional specification Timing specification inputs functional spec timing spec outputs Chapter 2 <3>

4 Circuits Nodes Inputs: A, B, C Outputs: Y, Z A E1 n1 Internal: n1 B E3 Y Circuit elements C E2 Z E1, E2, E3 Each a circuit Chapter 2 <4>

5 Types of Logic Circuits Combinational Logic (Ch 2) Memoryless Outputs determined by current values of inputs Sequential Logic (Ch 3) Has memory Outputs determined by previous and current values of inputs inputs functional spec timing spec outputs Chapter 2 <5>

6 Rules of Combinational Composition Every element is combinational Every node is either an input or connects to exactly one output The circuit contains no cyclic paths E.g. no connection from output to internal node Example: Chapter 2 <6>

7 Boolean Equations Functional specification of outputs in terms of inputs Example: S = F(A, B, C in ) C out = F(A, B, C in ) A S B C L C C out in A B C in S C out S = A B C in C out = AB + AC in + BC in Chapter 2 <7>

8 Functional specification Goals: Systematically express logical functions using Boolean equations To simplify Boolean equations Chapter 2 <8>

9 Some Definitions Complement: variable with a bar over it A, B, C Literal: variable or its complement A, A, B, B, C, C Implicant: product (AND) of literals ABC, AC, BC Minterm: product that includes all input variables ABC, ABC, ABC Maxterm: sum (OR) that includes all input variables (A+B+C), (A+B+C), (A+B+C) Chapter 2 <9>

10 Canonical Sum-of-Products (SOP) Form All equations can be written in SOP form Each row has a minterm A minterm is a product (AND) of literals Each minterm is TRUE for that row (and only that row) A B Y minterm A B A B A B A B minterm name m 0 m 1 m 2 m 3 Chapter 2 <10>

11 Canonical Sum-of-Products (SOP) Form All equations can be written in SOP form Each row has a minterm A minterm is a product (AND) of literals Each minterm is TRUE for that row (and only that row) Form function by ORing minterms where the output is TRUE A B Y Y = F(A, B) = minterm A B A B A B A B minterm name m 0 m 1 m 2 m 3 Chapter 2 <11>

12 Canonical Sum-of-Products (SOP) Form All equations can be written in SOP form Each row has a minterm A minterm is a product (AND) of literals Each minterm is TRUE for that row (and only that row) Form function by ORing minterms where the output is TRUE Thus, a sum (OR) of products (AND terms) A B Y minterm A B A B A B A B minterm name m 0 m 1 m 2 m 3 Y = F(A, B) = AB + AB = Σ(m 1, m 3 ) Chapter 2 <12>

13 SOP Example Steps: Find minterms that result in Y=1 Sum TRUE minterms A B Y Y = F(A, B) = Chapter 2 <13>

14 Aside: Precedence AND has precedence over OR In other words: AND is performed before OR Example: Y = A B + A B Equivalent to: Y = AB + (AB) Chapter 2 <14>

15 Canonical Product-of-Sums (POS) Form All Boolean equations can be written in POS form Each row has a maxterm A maxterm is a sum (OR) of literals Each maxterm is FALSE for that row (and only that row) A B Y maxterm A + B A + B A + B A + B maxterm name M 0 M 1 M 2 M 3 Chapter 2 <15>

16 Canonical Product-of-Sums (POS) Form All Boolean equations can be written in POS form Each row has a maxterm A maxterm is a sum (OR) of literals Each maxterm is FALSE for that row (and only that row) Form function by ANDing the maxterms for which the output is FALSE Thus, a product (AND) of sums (OR terms) A B Y maxterm A + B A + B A + B A + B maxterm name M 0 M 1 M 2 M 3 Y = M 0 M 2 = A + B ( A + B) Chapter 2 <16>

17 SOP and POS Comparison Sum of Products (SOP) Implement the ones of the output Sum all one terms OR results in one Product of Sums (POS) Implement the zeros of the output Multiply zero terms AND results in zero Chapter 2 <17>

18 Boolean Equations Example You are going to the cafeteria for lunch You will eat lunch (E=1) If it s open (O=1) and If they re not serving corndogs (C=0) Write a truth table for determining if you will eat lunch (E). E Chapter 2 <18>

19 Boolean Equations Example You are going to the cafeteria for lunch You will eat lunch (E=1) If it s open (O=1) and If they re not serving corndogs (C=0) Write a truth table for determining if you will eat lunch (E). E Chapter 2 <19>

20 SOP & POS Form SOP sum-of-products E POS product-of-sums E minterm maxterm Chapter 2 <20>

21 SOP & POS Form SOP sum-of-products E minterm E = OC = Σ(m 2 ) POS product-of-sums E maxterm E = ()()() = Π(M0, M1, M3) Chapter 2 <21>

22 SOP & POS Form SOP sum-of-products E minterm E = OC = Σ(m 2 ) POS product-of-sums E maxterm Chapter 2 <22>

23 SOP & POS Form SOP sum-of-products E minterm E = OC = Σ(m 2 ) POS product-of-sums E maxterm E = ()()() = Π(M0, M1, M3) Chapter 2 <23>

24 Boolean Algebra Axioms and theorems to simplify Boolean equations Like regular algebra, but simpler: variables have only two values (1 or 0) Duality in axioms and theorems: ANDs and ORs, 0 s and 1 s interchanged Chapter 2 <24>

25 Boolean Axioms Chapter 2 <25>

26 Duality Duality in Boolean axioms and theorems: ANDs and ORs, 0 s and 1 s interchanged Chapter 2 <26>

27 Boolean Axioms Chapter 2 <27>

28 Boolean Axioms Dual: Exchange: and + 0 and 1 Chapter 2 <28>

29 Boolean Axioms Dual: Exchange: and + 0 and 1 Chapter 2 <29>

30 Basic Boolean Theorems B = B Chapter 2 <30>

31 Basic Boolean Theorems: Duals Dual: Exchange: and + 0 and 1 Chapter 2 <31>

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