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