344-521 ก ˘ก ˇˇ ˆ˙˝˛ computer organization and...
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Chapter 4 Digital Logic Circuits
344-521 ��� �ก������������ก ��������� �Computer Organization and Architecture
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Digital Computers
� A digital system that performs various computational tasks
� Use binary number system (0 or 1)� A binary digit is called a bit
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Logic Gates
� Binary information is represented in digital computers by physical quantities called signals
� Digital Signals have two basic states:1 (logic “high”, or H, or “on”)0 (logic “low”, or L, or “off”)
� Digital values are in a binary format. Binary means 2 states.
� A good example of binary is a light (only on or off)
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� Manipulation of binary information is done by logic circuits called gates.
� Gates are blocked of hardware that produce signals of binary 1 or 0.
� Digital logic gates can be described in terms of standard logic symbols and their corresponding truth tables.
Logic Gates (cont’d)
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Boolean Algebra
� An algebra that deals with binary variables and logic operations� Variables: A, B, x, y� Operations: AND (�), OR (+), or ¯
� Purpose: to facilitate the analysis and design of digital circuits
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Boolean Function
� Can be either 1 or 0� Example: F = x+y z
F = 1 if x =1 or if (y and z) = 1
� Relationship between function and binary variables can be represented in truth table� Need a list of the 2n combinations of the n binary
variables.
� A Boolean function can be transformed an algebraic expression into a logic diagram composed of AND, OR, and inverter gates
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Most Basic Identities of Boolean Algebra
1. A+0 = A2. A�1 = A
3. A+1 = 14. A+A = 15. A�0 = 06. A�A = 0
7. A+A = A8. A�A = A
9. A+B= B+A10.AB = BA
11.A+(B+C) = (A+B)+C12.A(BC) = (AB)C13.A+(BC) = (A+B)(A+C)14.A(B+C) = (AB)+ (AC)15.A = (A )16.A+AB = A17.A(A+B) = A18.A+A B = A+B19.A(A +B) = AB20.DeMorgan’s theorem
� (A+B) = A B
� (AB) = A +B
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Digital Logic Gates
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Digital Logic Gates (cont’d)
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Digital Logic Gates (cont’d)
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DeMorgan’s Theorem
� Very important in dealing with NOR and NAND gates.
Figure Two graphic symbols for NOR gate
Figure Two graphic symbols for NAND gate
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DeMorgan’s Theorem
� Example: F = ABC + ABC + A C
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DeMorgan’s Theorem (cont’d)
� F = ABC + ABC + A C= AB(C+C ) + A C = AB + A C
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Circuit Simplification
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1. Use Boolean Algebra
2. Use map called Karnaugh map or K-map3. Use Variable Entered Map4. Use Quine McCluskey - �'��� ��ก ( + &�ก��'"*+
��กก�,� 1 output
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Circuit Simplification: use Boolean Algebra
� Example 1: �# $�&�ก��'" Y = AB+AB +A B��-+*.� Y = AB+AB +A B
=
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Circuit Simplification: use Boolean Algebra (cont’d)
� Example 2: �# $�&�ก��'" Y = (A +AB )(A B )��-+*.�
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� A pictorial arrangement of the truth table which allows an easy interpretation for choosing the minimum number of terms needed to express the function algebraically.
� Each combination of the variables in a truth table is called a minterm.
� In a truth table, a function of n variables will have 2n
minterms� Information in a truth table is in compact form by
listing the decimal equivalent of those minterms that produce a 1 for the function.
Circuit Simplification: use K-map
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� Example: F (x,y,z) = Σ (1,4,5,6,7)
� The letters in parentheses list the binary variables in the orfer that they appear in the truth table.
� The symbol Σ stands for the sum of the minterms that follow in parentheses.� Minterms that produce 1 for the function are listed in
their decimal equivalent; otherwise produce 0.
� The map is a diagram made up of squares, with each square representing one minterm.
Circuit Simplification: use K-map (cont’d)
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Circuit Simplification: use K-map (cont’d)
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� Example 1: F(A,B,C) = Σ (3,4,6,7)� Example 2: F(A,B,C) = Σ (0,2,4,5,6)� Example 3: F(A,B,C,D) = Σ (0,1,2,6,8,9,10)
Circuit Simplification: use K-map (cont’d)
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Don’t Care Condition
� There are occasions when it does not matter if the function produces 0 or 1 for a given minterm.
� Then, we don’t care what the function output is to be for the minterm.
� This minterm is called don’t care condition � Mark with x in the map� Used to provide further simplification of the
algebraic expression.
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� Example: F(A,B,C) = Σ (0,2,6)d(A,B,C) = Σ (1,3,5)
Don’t Care Condition (cont’d)
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Combinational Circuits
� A connected arrangement of logic gates with a set of inputs and outputs
Figure Block diagram of a combinational circuit
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Half-Adder
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Full-Adder
Map for full-adderTruth table for full-adder
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Full-Adder (cont’d)
Full-adder circuit