Chapter 2 (Lect 2) Canonical and Standard Forms. Standard Form. Other Logic Operators Logic Gates. Sum of Minterms Product of Maxterms

Transcription

1 Chapter 2 (Lect 2) Canonical and Standard Forms Sum of Minterms Product of Maxterms Standard Form Sum of products Product of sums Other Logic Operators Logic Gates Basic and Multiple Inputs Positive and Negative Logic Integrated Circuits

2 Sum of minterms: from truth table 1. Indentify minterms that equate to 1 2. OR them together to form Boolean expression x y z F Minterm Designation F = F(x.y,x) =

3 Sum of minterms: from function In canonical form each minterm contains all variables, primed or unprimed 1. Expand into a sum of AND terms 2. Treat each term separately and add in missing terms by ANDing with (x + x ) = 1 3. Recombine remove duplicates F = A + B C Or create a truth table

4 Product of maxterms: from truth table 1. Identify maxterms that evaluate to 0 2. AND them together to form Boolean expression x y z F Maxterm Designation F = F(x.y,x) =

5 Product of maxterms: from function In canonical form each maxterm contains all variables, primed or unprimed 1. Expand into ANDed -ORed terms using x + yz = (x + y)(x + z) repeatedly 2. Treat each term separately, and add in missing terms by ORing (xx ) = 0 3. Recombine remove duplicates F = AB + B C Or create a truth table

6 Conversion Between Canonical Forms 1. Interchange the symbols and 2. List missing numbers in parenthesizes 3. This is the other canonical form F (x, y, z)= (m 0, m 2,m 4,m 7 ) F = x + yz

7 Does SOP and POS produce the same result? x y F Minterm Maxterm

8 Canonical form: Sum of minterms F = xyz + x yz + xyz Standard form: Sum-of-products SOP F = yz + xy Canonical form: Product of maxterms F = (x + y + z)(x + y + z ) Standard form: Products-of-sums POS F = x(x + y )(y + z) Non-Standard form F = x+x(y + z)

9 Creating circuit from expression F = x y + xy Make note AND-OR implementation F = (x + y ) (x + y) Make note OR-AND implementation

10 Most used Boolean functions and Operators Function F = x (From Table 2.8) Operator Symbol Name/Comments Transfer, buffer F = x x Complement, not x F = xy x y AND, x and y F = (xy) x y NAND, not AND F = x+y x+y OR, x or y F = (x+y) x y NOR, not OR F = xy + x y x y Exclusive OR, x or y not both F = xy + x y (x y) XNOR, x = y

11 Logic Gates Buffer F = x x F Inverter F = x x F

12 Logic Gates AND F = xy x y F OR F = x + y x y F

13 NAND F = (xy) = x y x y F NOR F = (x + y) = x y x y F

14 XOR F = xy + x y = x y x y F XNOR F = xy + x y = (x y)' x y F

15 Extension to Multiple inputs (more than two) Must satisfy Commutative and Associative laws AND and OR gates no problem NOR and NANDs ok if we use (OR).. (x + y + z) (AND). (xyz) Example1

16 Works for XOR, with modified definition, But uncommon in hardware implementation x T1 x y z T1 F1 T2 F2 y z F x y T z F

17 Positive and Negative Logic (The meaning of High and Low) 1 Signal High x y F Signal Low Positive Logic Signal High x y F Negative Logic Signal Low Note signals are not defined as positive or negative but as high and low values Convert from positive to negative logic 0 becomes 1 and 1 becomes 0 = new gate behavior Book uses positive logic

18 Integrated Circuits Logic gates are available as integrated circuits IC or chips. Each IC gas a number printed on it Identifying its function. Vendors provide data sheets that identify the pin configurations and IC characteristics. Texas Instruments Levels of integration complexity and density of gates in single package Small-scale integration (SSI): Fewer than 10 gates, limited by number of pins on IC package Medium-scale integration (MSI): Typically 10 to 1000 gates, elementary digital operations, such as decoders, adders, counters. Large-scale integration (LSI): Contain thousands of gates, digital systems such as processors, memory chips, programmable logic.. Very large-scale integration (VLSI): Contain hundreds of thousands of gates, computer systems, large scale memory arrays, microcomputer chips.

19 What you should know 1. Be able to generate expressions for Sum of Products and Products of Sums, and draw circuit 2. Boolean operations 3. Convert between Canonical forms 4. The 8 listed gates, schematic symbols, and truth tables 5. The difference between positive and negative logic 6. Be familiar with levels of integration and common characteristics listed in manufactures data sheets.