chapter 10_1 digital logic. irvine, kip r. assembly language for intel-based computers, 2003. 2 not...
TRANSCRIPT
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Chapter 10_1
Digital Logic
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Irvine, Kip R. Assembly Language for Intel-Based Computers, 2003.
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• NOT• AND• OR• XOR• NAND• NOR• Truth Tables
Boolean Operators
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Irvine, Kip R. Assembly Language for Intel-Based Computers, 2003.
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• NOT A = Ā• A AND B = AB• 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
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Irvine, Kip R. Assembly Language for Intel-Based Computers, 2003.
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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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Irvine, Kip R. Assembly Language for Intel-Based Computers, 2003.
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NOT• Inverts (reverses) a boolean value• Truth table for Boolean NOT operator:
NOT
Digital gate diagram for NOT:
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Irvine, Kip R. Assembly Language for Intel-Based Computers, 2003.
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AND• Truth table for Boolean AND operator:
AND
Digital gate diagram for AND:
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Irvine, Kip R. Assembly Language for Intel-Based Computers, 2003.
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OR• Truth table for Boolean OR operator:
OR
Digital gate diagram for OR:
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Irvine, Kip R. Assembly Language for Intel-Based Computers, 2003.
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Operator Precedence• Examples showing the order of operations:
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Irvine, Kip R. Assembly Language for Intel-Based Computers, 2003.
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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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Irvine, Kip R. Assembly Language for Intel-Based Computers, 2003.
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Truth Tables (2 of 3)• Example: X Y
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Irvine, Kip R. Assembly Language for Intel-Based Computers, 2003.
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Truth Tables (3 of 3)• Example: (Y S) (X S)
Two-input multiplexer
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Basic Identities of Boolean AlgebraBasic 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
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
A A
A B A B A B A B
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De Morgan’s 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
(XYZ) = 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 simplifyF = 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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MultiplexorS2 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