Logic Design Basics: Gates, Truth Tables, Boolean Algebra, and Logic Equations, Exams of Computer Architecture and Organization

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Basics of Logic Design Appendix C

Logic Gate „

Performs a logical operation (e.g., AND,OR, NOT) on one or more inputs andproduces a single logic output

Normally perform Boolean logic

Primarily implemented electronically usingdiodes or transistors

Can be constructed using electromagneticrelays, optics, molecules, or evenmechanical elements

The Basics of Logic Design — 2

Boolean Algebra „

Logic function with logic equations

OR operator, +, AND operator, * NOT operator,

Identity law: A+0=A, and A*1=A

Zero and One laws: A+1=1, and A*0=

Inverse laws: A+

=1, and A*

Commutative laws: A+B=B+A, and AB=BA

Associate laws: A+(B+C)=(A+B)+C, andA(BC)=(AB)C

Distributive laws: A(B+C)=(AB)+(AC), andA+(BC)=(A+B)*(A+C)

DeMorgan’s laws:

The Basics of Logic Design — 4

Logic Equations

D=1 if at least 1 input is true E=1 if exactly 2 inputs are true F=1 only if all 3 inputs are true^ The Basics of Logic Design — 5

Combinational Logic

The Basics of Logic Design — 7 3-to-8 decoder

Multiplexors

The Basics of Logic Design — 8 For n data inputs, need log 2 n selector

Sum of Products

The Basics of Logic Design — 10

Programmable Logic Array (PLA)

Chapter 3 — Arithmetic for Computers — 11 Sum of products

ROM (Read-Only Memory) „

Read-only

A set of input address lines and a set ofoutputs

Height: m input lines => 2

m

addressable

entries

Width: # of bits in each addressable entry

Chapter 3 — Arithmetic for Computers — 13

Don’t Cares

Chapter 3 — Arithmetic for Computers — 14 Without don’t cares With output don’t cares With input don’t cares Logic minimization can be done using Karnaugh maps

1-Bit Arithmetic Logic Unit

Chapter 3 — Arithmetic for Computers — 16 CarryOut =

1-Bit ALU (cont’d)

Chapter 3 — Arithmetic for Computers — 17

32-Bit ALU

Chapter 3 — Arithmetic for Computers — 19 Ripple carry adder

32-Bit ALU

Chapter 3 — Arithmetic for Computers — 20 Subtraction using adder