Post on 21-Dec-2015
transcript
EE365Adv. Digital Circuit Design
Clarkson University
Lecture #2
Boolean Laws and Methods
Boolean algebra
• a.k.a. “switching algebra”– deals with boolean values -- 0, 1
• Positive-logic convention– analog voltages LOW, HIGH --> 0, 1
• Signal values denoted by variables(X, Y, FRED, etc.)
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Boolean operators
• Complement: X (opposite of X)• AND: X Y• OR: X + Y
binary operators, describedfunctionally by truth table.
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More definitions
• Literal: a variable or its complement– X, X, FRED, CS_L
• Expression: literals combined by AND, OR, parentheses, complementation– X+Y– P Q R– A + B C– ((FRED Z) + CS_L A B C + Q5) RESET
• Equation: Variable = expression– P = ((FRED Z) + CS_L A B C + Q5)
RESET
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Logic symbols
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Theorems
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More Theorems
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Duality
• Swap 0 & 1, AND & OR– Result: Theorems still true– Note duals in previous 2 tables (e.g. T6 and T6’)– Example:
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N-variable Theorems
• Most important: DeMorgan theorems
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DeMorgan Symbol Equivalence
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Likewise for OR
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DeMorgan Symbols
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Even more definitions• Product term
– W•X’•Y
• Sum-of-products expression– (W•X’•Y)+(X•Z)+(W’•X’•Y’)
• Sum term– A+B’+C
• Product-of-sums expression– (A+B’+C)•(D’+A’)•(D+B+C)
• Normal term– No variable appears more than once– (W•X’•Y)+(A•Z)+(B’•C’)
• Minterm (n variables)• Maxterm (n variables)
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Minterm• An n-variable minterm is a normal product term with n
literals• There are 2n possibilities
• 3-variable example: X’•Y’•Z or ΣX,Y,Z(1)
• A minterm is a product term that is 1 in exactly one row of the truth table:
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X Y Z F
0 0 0 0
0 0 1 1
0 1 0 0
0 1 1 0
1 0 0 0
1 0 1 0
1 1 0 0
1 1 1 0
new notation
Maxterm• An n-variable maxterm is a normal sum term with n literals• There are 2n possibilities
• 3-variable example: X’+Y’+Z or ЛX,Y,Z(6)
• A maxterm is a sum term that is 0 in exactly one row of the truth table:
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X Y Z F
0 0 0 1
0 0 1 1
0 1 0 1
0 1 1 1
1 0 0 1
1 0 1 1
1 1 0 0
1 1 1 1
new notation
Truth table vs. minterms & maxterms
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Combinational analysis
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Signal expressions
• Multiply out:
F = ((X + Y) Z) + (X Y Z)
= (X Z) + (Y Z) + (X Y Z)
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New circuit, same function
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F = ((X + Y) Z) + (X Y Z)
= (X Z) + (Y Z) + (X Y Z)
“Add out” logic function
• Circuit:
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Shortcut: Symbol substitution
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Different circuit, same function
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Practice
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Convert the following function into a POS:
F = ((X + Z) • Y) + (X’ • Z’ • Y’)
Convert the following function into a POS:
F = ((X + Z) • Y) + (X’ • Z’ • Y’)F = (X + Z + X’) • (X + Z + Z’) • (X + Z + Y’) • (Y + X’) • (Y + Z’) • (Y + Y’)
F = 1 • 1 • (X + Z + Y’) • (Y + X’) • (Y + Z’) • 1
F = (X + Z + Y’) • (Y + X’) • (Y + Z’)
Practice
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Next Class
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• Building Combination Circuits• Minimization• Karnaugh Maps