Post on 29-Dec-2019
ENEE244-010x Digital Logic Design
Lecture 6
Announcements
β’ Homework 2 due today
β’ Homework 3 up on webpage (tonight), due Wed, 9/30.
β’ Class is canceled on Wed, 9/23
β’ Substitute on Mon, 9/28
Agenda
β’ Last time:
β Manipulations of Boolean Formulas (3.6)
β’ This time:
β Gates and Combinational Networks (3.7)
β Incomplete Boolean Functions and Donβt Care Conditions (3.8 )
β Universal Gates (3.9.3)
Digital Logic Gates
β’ AND π π₯, π¦ = π₯π¦
β’ OR π π₯, π¦ = π₯ + π¦
β’ NOT (Inverter) π π₯ = π₯
β’ Buffer (Transfer) π π₯ = π₯
β’ NAND π π₯, π¦ = (π₯π¦) = π₯ + π¦
β’ NOR π π₯, π¦ = π₯ + π¦ = π₯ π¦
β’ XOR π π₯, π¦ = π₯ π¦ + π₯ π¦ = π₯ β π¦
β’ X-NOR (Equivalence) π π₯, π¦ = π₯π¦ + π₯ π¦
Some Terminology
β’ System specification: A description of the function of a system and of other characteristics required for its use.
β A function (table, algebraic) on a finite set of inputs.
β’ Double-rail logic*: both variables and their complements are considered as primary inputs.
β’ Single-rail logic: only variables are considered as primary inputs. Need inverters for their complements.
Gates and Combinational Networks β’ Synthesis Procedure
β’ Example: Truth table for parity function on three variables
X Y Z f
0 0 0 0
0 0 1 1
0 1 0 1
0 1 1 0
1 0 0 1
1 0 1 0
1 1 0 0
1 1 1 1
Synthesis Procedure
X Y Z f
0 0 0 0
0 0 1 1
0 1 0 1
0 1 1 0
1 0 0 1
1 0 1 0
1 1 0 0
1 1 1 1
Minterm Canonical Form: π π₯, π¦, π§ = π₯ π¦π§ + π₯π¦ π§ + π₯π¦ π§ + xyz
Two-level Gate Network Minterm Canonical Form:
π π₯, π¦, π§ = π₯ π¦π§ + π₯π¦ π§ + π₯π¦ π§ + π₯π¦π§
π₯ π¦ π§
π₯ π¦ π§
π₯ π¦ π§
π₯ π¦ π§
Incomplete Boolean Functions and Donβt Care Conditions
Incomplete Boolean Functions and Donβt Care Conditions
β’ n-variable incomplete Boolean function is represented by a truth table with n+1 columns and 2π rows.
β’ For those combinations of values in which a functional value is not to be specified, a symbol, --, is entered.
β’ The complement of an incomplete Boolean function is also an incomplete Boolean function having the same unspecified rows of the truth table.
Describing Incomplete Boolean Functions
X Y Z F
0 0 0 1
0 0 1 1
0 1 0 0
0 1 1 --
1 0 0 0
1 0 1 --
1 1 0 0
1 1 1 1
Minterm canonical formula: π π₯, π¦, π§ = βπ 0,1,7 + ππ(3,5)
Describing Incomplete Boolean Functions
X Y Z F
0 0 0 1
0 0 1 1
0 1 0 0
0 1 1 --
1 0 0 0
1 0 1 --
1 1 0 0
1 1 1 1
Maxterm canonical formula: π π₯, π¦, π§ = Ξ π 2,4,6 + ππ(3,5)
Describing Incomplete Boolean Functions
β’ Manipulating Boolean equations derived from incomplete Boolean functions is a very difficult task.
β’ In the next chapter, there are procedures for obtaining minimal expressions that can handle the donβt care conditions.
β’ Can leverage donβt care conditions to get simplified expressions for functions (smaller gate networks).
{π₯π¦, π₯π¦} {π₯π¦, π₯ π¦}
Universal Gates
β’ A gate or set of gates is called universal if it can implement all Boolean functions.
β’ Standard universal gates:
β AND, OR, NOT
β Proof?
β’ Theorem:
β A set of gates πΊ1, β¦ , πΊπ is universal if it can implement AND, OR, NOT.
NAND is Universal β’ Recall ππ΄ππ· π₯, π¦ = π₯ π¦ = π₯ + π¦
β’ πππ π₯ = π₯ = π₯ + π₯ = ππ΄ππ· π₯, π₯
β’ π΄ππ· π₯, π¦ =
π₯π¦ = (π₯π¦) = πππ (π₯π¦) = πππ ππ΄ππ· π₯, π¦
= ππ΄ππ· ππ΄ππ· π₯, π¦ , ππ΄ππ· π₯, π¦
β’ ππ π₯, π¦ =
π₯ + π¦ = (π₯ + π¦) = (π₯ π¦) = ππ΄ππ· πππ π₯ , πππ π¦
= ππ΄ππ· ππ΄ππ· π₯, π₯ , ππ΄ππ· π¦, π¦
NOR is Universal β’ Recall πππ = π₯ + π¦ = π₯ π¦
β’ πππ π₯ = π₯ = π₯ π₯ = πππ π₯, π₯
β’ π΄ππ· π₯ =
π₯π¦ = (π₯π¦) = (π₯ + π¦) = πππ πππ π₯ , πππ π¦
= πππ πππ π₯, π₯ , πππ π¦, π¦
β’ ππ π₯ =
π₯ + π¦ = (π₯ + π¦) = πππ π₯ + π¦
= πππ πππ π₯, π¦
= πππ πππ π₯, π¦ , πππ π₯, π¦
Class Exercise