Chapter 3 CS 121Digital Logic Design Gate-Level Minimization
Outline 3.1 Introduction 3.2 The Map Method 3.3 Four-Variable Map 3.4 Product of sums simplification 3.5 Don‘t Care Conditions 3.7 NAND and NOR Implementaion 3.8 Other Two-Level Implementaion 3.9 Exclusive-OR function
3.7 NAND and NOR Implementation (1-15) • Digital circuits are frequently constructed with NAND or NOR gates rather than with AND and OR gates.
3.7 NAND and NOR Implementation (2-15) NAND Implementation • NAND gate: a universal gate. • Any digital system can be implemented with it.
3.7 NAND and NOR Implementation (3-15) NAND Implementation • To facilitate the conversion to NAND logic, there are alternative graphic symbol for it.
3.7 NAND and NOR Implementation (4-15) NAND Implementation Two-Level Implementation • Procedures of Implementation with two levels of NAND gates: • Express simplified function in sum of products form. • Draw a NAND gate for each product term that has at least two literals to constitute a group of first-level gates • Draw a single gate using AND-invert or invert-OR in the second level • A term with a single literal requires an inverter in the first level.
3.7 NAND and NOR Implementation (5-15) NAND Implementation Two-Level Implementation F = AB + CD = [(AB + CD)’]’ = [(AB)’*(CD)’]’
3.7 NAND and NOR Implementation (6-15) NAND Implementation Two-Level Implementation Example (3.10): F(X,Y,Z) = ∑ (1,2,3,4,5,7) X’Y F = XY’ + X’Y + Z y z 11 01 00 10 x 1 1 1 0 1 1 1 1 Z XY’
3.7 NAND and NOR Implementation (7-15) NAND Implementation Multilevel Implementation • Procedures of Implementation with multilevel of NAND gates: • Convert all AND gates to NAND gates with AND-invert graphic symbols • Convert all OR gates to NAND gates with invert-OR graphic symbols • Check all the bubbles in the diagrams. For a single bubble, invert aninverter (one-input NAND gate) or complement the input literal
3.7 NAND and NOR Implementation (8-15) NAND Implementation Multilevel Implementation EXAMPLE 1: F = A(CD + B) + BC’
3.7 NAND and NOR Implementation (9-15) NAND Implementation Multilevel Implementation EXAMPLE 2: F = (AB’ + A’B).(C + D’)
3.7 NAND and NOR Implementation (10-15) NOR Implementation • The NOR operation is the dual of the NAND operation. • The NOR gate is anothar universal gate to implement any Boolean function.
3.7 NAND and NOR Implementation (11-15) NOR Implementation • To facilitate the conversion to NOR logic, there are alternative graphic symbol for it.
3.7 NAND and NOR Implementation (12-15) NOR Implementation Two-Level Implementation • Procedures of Implementation with two levels of NOR gates: • Express simplified function in product of sums form. • Draw a NOR gate for each product term that has at least two literals to constitute a group of first-level gates • Draw a single gate using OR-invert or invert-AND in the second level • A term with a single literal requires an inverter in the first level.
3.7 NAND and NOR Implementation (13-15) NOR Implementation Two-Level Implementation Example : F = (A+B).(C+D).E E
3.7 NAND and NOR Implementation (14-15) NOR Implementation Multilevel Implementation • Procedures of Implementation with multilevel of NOR gates: • Convert all OR gates to NOR gates with OR-invert graphic symbols • Convert all AND gates to NORgates with invert-AND graphic symbols • Check all the bubbles in the diagrams. For a single bubble, invert aninverter (one-input NAND gate) or complement the input literal
3.7 NAND and NOR Implementation (15-15) NOR Implementation Multi-Level Implementation Example : F = (A B’ + A’B).(C+D’) A B’ A’ B
3.8 Other Two-Level Implementations (1-7) Nondegeneratd forms Implementation • 16 possible combinations of two-level forms with 4 types of gates: AND, OR, NAND, and NOR • 8 are degenerate forms: degenerate to a single operation. • (AND-AND , AND-NAND, OR-OR , OR-NOR , NAND-NAND , NAND-NOR , NOR-AND , NOR-NAND) • 8 are generate forms: • NAND-AND = AND-NOR = AND-OR-INVERT • OR-NAND = NOR-OR = OR-AND-INVERT
3.8 Other Two-Level Implementations (3-7) Nondegeneraetd forms Implementation Discussed before Generated forms Discuss now
3.8 Other Two-Level Implementations (4-7) Nondegeneraetd forms Implementation AND-OR-INVERT • AND-NOR = NAND-AND = AND-OR-INVERT • Similar to AND-OR, AND-OR-INVERT requires an expression in sum of products Example: F = (AB + CD + E) ‘
3.8 Other Two-Level Implementations (5-7) Nondegeneraetd forms Implementation OR-AND-INVERT • AND-NOR = NAND-AND = AND-OR-INVERT • Similar to OR-AND, OR-AND-INVERT requires an expression in product of sums Example: F = [(A+B) . (C+D) . E ] ‘
3.8 Other Two-Level Implementations (6-7) Nondegeneraetd forms Implementation
3.8 Other Two-Level Implementations (7-7) Nondegeneraetd forms Implementation Example (3.11) : F(x,y,z) = ∑ (0,7) AND-OR-INVERT: F’ = x’y + xy’ + z F = ( x’y + xy’ + z ) ‘ ----------------------------- OR-AND-INVERT: F = x’y’z’ + xyz’ F = [ (x’y’z’ + xyz’)’ ] ‘ F = [ (x+y+z) . (x’+y’+z) ] ‘
3.9 Exclusive-OR Function (1-7) • Exclusive-OR (XOR) denoted by the symbol : • x y = xy‘ + x‘y • Exclusive-OR is equal to 1, when the values of x and y are diffrent. • Exclusive-NOR (XNOR): • (x y )‘ = xy + x‘y‘ • Exclusive-NOR is equal to 1, when the values of x and y are same. • Only a limited number of Booleanfunctions can be expressed in terms of XOR operations, but it is particularly useful in arithmetic operations and error-detection and correction circuits.
3.9 Exclusive-OR Function (2-7) • Exclusive-OR principles: • x 0 = x • x 1 = x‘ • x x = 0 • x x‘ = 1 • x y‘ = x‘ y = (x y)‘ • x y = y x • (x y) z = x (y z)
3.9 Exclusive-OR Function (3-7) • Implementaion Exclusive-OR with AND-OR-NOT: • x y = xy‘ + x‘y • Implementaion Exclusive-OR with NAND: • x y = xy‘ + x‘y = x (x‘+y‘) + y (x‘+y‘) = x (xy)‘ + y (xy)‘ = [(x(xy)‘ + y(xy)‘)‘]‘ = [(x(xy)‘)‘ + (y(xy)‘)‘ ]‘
3.9 Exclusive-OR Function (4-7) Odd Function: • The 3-variable XOR function is equal to 1 if only one variable is equal to 1 or if all three variables are equal to 1. • Multiple-variable exclusive OR operation = odd function : odd number of variables be equal to 1. • (A B C) = (AB‘ + A‘B) C‘ + (A‘B‘ + AB) C = AB‘C‘ + A‘BC‘ + A‘B‘C + ABC = ∑ (1,2,4,7)
3.9 Exclusive-OR Function (5-7) Odd Function:
3.9 Exclusive-OR Function (6-7) Odd Function: A B C D= ∑ (1,2,4,7,8,11,13,14)
3.9 Exclusive-OR Function (7-7) Parity Generation and Checking: • Exclusive-OR function is useful in systems requiring error-detection and correction circuits. • A parity bit is used for purpose of detection errors during transmission. • Parity bit : an extra bit included with a binary message to make the number of 1’s either odd or even. • The circuit generates the parity bit in transmitter is called parity generator. • The circuit checks the parity bit in receiver is called parity checker.
3.9 Exclusive-OR Function (8-7) Parity Generation and Checking: Example : Three-bit message with even parity • From the truth table , P constitutes an odd function. • It is equal 1 when numerical value of 1’s in a minterm is odd • P = x y z
3.9 Exclusive-OR Function (8-7) Parity Generation and Checking: Example : Three-bit message with even parity • From the truth table , C constitutes an odd function. • It is equal 1 when numerical value of 1’s in a minterm is odd • C = x y z P
