SUPLEMENTARY CHAPTER 1: An Introduction to Digital Logic

SUPLEMENTARY CHAPTER 1:

An Introduction to Digital Logic

The Architecture of Computer Hardware and Systems Software:

An Information Technology Approach 3rd Edition, Irv Englander

John Wiley and Sons 2003

Linda Senne, Bentley College

Wilson Wong, Bentley College

Integrated Circuits

 The building blocks of computers

 Designed for specialized functions

 Examples: the CPU, bus interface, memory management unit

 Transistors: primary components of ICs

 Motorola MPC 7400 PowerPC modules:

6.5 million transistors in less than ½ ⁱⁿ

Transistors

 Boolean algebra: basis for computer logic design

 Transistors: means for implementing Boolean algebra

 Switches: on/off to represent the 0’s and 1’s of binary digital circuits

 Combined to form logic gates

Digital Circuits

 Combinatorial logic

 Results of an operation depend only on the present inputs to the operation

 Uses: perform arithmetic, control data movement, compare values for decision making

 Sequential logic

 Results depend on both the inputs to the operation and the result of the previous operation

 Uses: counter

Boolean Algebra

 Rules that govern constants and variables that can take on 2 values

 True/false; on/off; yes/no; 0/1

 Boolean logic

 Rules for handling Boolean constants and variables

 3 fundamental operations:

AND , ^OR and ^NOT

 Truth Table: specifies results for all possible input

combinations

Boolean Operators

 AND

 Result TRUE if and only if both input operands are true

 C = A ^ B

 INCLUSIVE-OR

 Result TRUE if any input operands are true

 C = A + B

A B C 0 0 0 0 1 0 1 0 0 1 1 1

A B C

0 0 0

0 1 1

1 0 1

1 1 1

Boolean Operators

 NOT

 Result TRUE if single input value is FALSE

 C = A

A C

0 1

1 0

Boolean Operators

 EXCLUSIVE-OR

 Result TRUE if either A or B is TRUE but not both

 C = A ⊕ B

 Can be derived from

INCLUSIVE-OR, AND and NOT



A xor B equals A or B but not both A and B



A xor B = either A and not B or B and not A

A B C 0 0 0 0 1 1 1 0 1 1 1 0 A ⊕ B = (A + B) ^ ( A ^ B )

A ⊕ B = (A ^ B ) + ( B ^ A )

Boolean Algebra Operations

 Valid for INCLUSIVE-OR, AND, XOR

 Associative



 Distributive



 Commutative



 DeMorgan’s Theorems



A + ( B + C ) = ( A + B ) + C

A ^ ( B + C ) = A ^ B + A ^ C

A + B = B + A

A + B = A ^ B

Gates and Combinatorial Logic

 Many computer functions defined in terms of Boolean equations

 Example: sum of 2 single binary digit numbers

 Truth table for sum Truth table for carry

XOR AND

A B C

0 0 0

0 1 0

1 0 0

1 1 1

A B C

0 0 0

0 1 1

1 0 1

1 1 0

Computer Implementation

 Gates or logical gates

 Integrated circuits constructed from transistor switches and other electronic components

 VLSI: very large-scale integration

Boolean Algebra Implementation

 Single type of gate appropriately combined

 2 possibilities

 NAND gate: AND operation followed by a NOT operation

 NOR gate: INCLUSIVE-OR followed by a NOT operation

Note:

indicates a NOT operation

Selector or Multiplexer

 Switch input back and forth between inputs

 Logic circuits that make up a computer

 are relatively simple but

 look complicated because many circuits required

Half-Adder

Full Adder

 Handles possible carry from previous bit

 Half adder shown as block to simplify

( portion of half adder in Fig. S1.11 enclosed in dotted line)

 2-bit adder contains 32 circuits

 Also called ripple adder because the carry

ripples through 32 bits

Sequential Logic Circuits

 Output depends on

 Input

 Previous state of the circuit

 Flip-flop: basic memory element

 State table: output for all combinations of input and previous states



Cf. Truth Table

Flip-Flop Types with State Tables

Register COPY Operation

 Uses both

sequential and

combinatorial

logic

Steps in a LOAD Instruction

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