Digital Logic

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Chapter 10_1

• Digital Logic

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Boolean Operators

• NOT
• AND
• OR
• XOR
• NAND
• NOR
• Truth Tables

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Boolean Operators

• NOT A A
• 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

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Boolean Algebra

• Based on symbolic logic, designed by George Boole
• Boolean expressions created from
• NOT, AND, OR

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NOT

• Inverts (reverses) a boolean value
• Truth table for Boolean NOT operator

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AND

• Truth table for Boolean AND operator

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OR

• Truth table for Boolean OR operator

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Operator Precedence

• Examples showing the order of operations

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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
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Truth Tables (2 of 3)

• Example X ? ?Y

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Truth Tables (3 of 3)

• Example (Y ? S) ? (X ? ?S)

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Basic Identities of Boolean Algebra
Basic Postulates Basic Postulates Basic Postulates
A B B A A B B A Commutative Laws
A (B C) (A B) (A C) A (B C) (A B) (A C) Distributive Laws
1 A A 0 A A Identity Elements
A 0 A 1 Inverse Elements
Other Identities Other Identities Other Identities
0 A 0 1 A 1  
A A A A A A  
A (B C) (A B) C A (B C) (A B) C Associative Laws
DeMorgan's Theorem
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De Morgans Theorem

• A NOR B (NOT A) AND (NOT B)
• A NAND B (NOT A) OR (NOT B)

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Basic Logic Gates

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NAND Gates
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NOR Gates

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Sum of products

• F ABC ABC ABC

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Product of sums

• (X?Y?Z) X Y Z (De Morgan)

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Product of sums

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Simplification of Boolean expression

• Algebraic simplification
• Karnaugh maps
• Quine McKluskey Tables

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Algebraic simplification

• Show how to simplify
• F ABC ABC ABC
• To become
• F AB BC
• B(A C)

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Simplified implementation of F ABC ABC
ABC B(A C)
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Karnaugh Maps
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The use of Karnaugh maps
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Overlapping groups F ABC ABC ABC
B(A C)
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The Quine-McKluskey Method
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2nd stageAll pairs that differ in one variable
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Last stage
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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

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NAND

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Multiplexor
S2 S1 F
0 0 D0
0 1 D1
1 0 D2
1 1 D3
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Multiplexor implementation

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Decoder

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Use of decoders

• To address 1K byte memory using four
• 256 x 8 bit RAM chips

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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.

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Programmable Logic Array (PLA)

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Programmed PLA

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Read-only memory

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A 64 bit ROM

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Adders

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4-Bit Adder

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Implementation of an Adder

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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.

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32-Bit Adder using 8-Bit Adders

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Carry look ahead