Appendix B Digital Logic

Irvine, Kip R. Assembly Language for Intel-Based Computers, NOT AND OR XOR NAND NOR Truth Tables Boolean Operators

Irvine, Kip R. Assembly Language for Intel-Based Computers, NOT A = Ā A AND B = A  B A OR B = A + B A XOR B= 1 if and only if one of A or B is 1 A NAND B = NOT ( A AND B) NOR= NOT (A OR B) Truth Tables Boolean Operators

Irvine, Kip R. Assembly Language for Intel-Based Computers, Boolean Algebra Based on symbolic logic, designed by George Boole Boolean expressions created from: – NOT, AND, OR

Irvine, Kip R. Assembly Language for Intel-Based Computers, NOT Inverts (reverses) a boolean value Truth table for Boolean NOT operator: Digital gate diagram for NOT:

Irvine, Kip R. Assembly Language for Intel-Based Computers, AND Truth table for Boolean AND operator: Digital gate diagram for AND:

Irvine, Kip R. Assembly Language for Intel-Based Computers, OR Truth table for Boolean OR operator: Digital gate diagram for OR:

Irvine, Kip R. Assembly Language for Intel-Based Computers, Operator Precedence Examples showing the order of operations:

Irvine, Kip R. Assembly Language for Intel-Based Computers, Truth Tables (1 of 3) A Boolean function has one or more Boolean inputs, and returns a single Boolean output. A truth table shows all the inputs and outputs of a Boolean function Example:  X  Y

Irvine, Kip R. Assembly Language for Intel-Based Computers, Truth Tables (2 of 3) Example: X   Y

Irvine, Kip R. Assembly Language for Intel-Based Computers, Truth Tables (3 of 3) Example: (Y  S)  (X   S) Two-input multiplexer

Basic Identities

De Morgan’s Theorem A NOR B = (NOT A) AND (NOT B) A NAND B = (NOT A) OR (NOT B)

Basic Logic Gates

NAND Gates

NOR Gates

Sum of products F = ABC + ABC + ABC

Product of sums (X  Y  Z) = X + Y + Z (De Morgan) 

Product of sums

Simplification of Boolean expression Algebraic simplification Karnaugh maps Quine McKluskey Tables

Algebraic simplification Show how to simplify F = ABC + ABC + ABC To become F = AB + BC = B(A + C)

Simplified implementation of F = ABC + ABC + ABC = B(A + C)

Karnaugh Maps

The use of Karnaugh maps

Overlapping groups F = ABC + ABC + ABC = B(A + C)

The Quine-McKluskey Method

2 nd stage All pairs that differ in one variable

Last stage

Final stage Circle each x that is alone in a column. Then place a square around each X in any row in which there is a circled X. If every column now has either a squared or a circled X, then we are done, and those row elements whose Xs have been marked constitute the minimal expression. ABC + ACD + ABC + ACD

NAND

Multiplexor S2S1F 00D0 01D1 10D2 11D3

Multiplexor implementation

Decoder

Use of decoders To address 1K byte memory using four 256 x 8 bit RAM chips

Small-scale integration Early integrated circuit provided from one to ten gates on a chip. The next slide shows a few examples of these SSI chips.

Programmable Logic Array (PLA)

Programmed PLA

Read-only memory

A 64 bit ROM

Adders

4-Bit Adder

Implementation of an Adder

Multi-output adder The output from each adder depends on the output from the previous adder. Thus there is an increasing delay from the least significant to the most significant bit. For larger adders the accumulated delay can become unacceptably high.

32-Bit Adder using 8-Bit Adders

Carry look ahead

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