ECE U322 Digital Logic Design

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ECE U322Digital Logic Design
Sept 15, 2005

• Lecture 5
• Boolean Algebra
• NOTs, NANDs, and NORs
• Reading Marcovitz 2.2, 2.3, 2.4

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N 4
Number Represented
Unsigned 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Binary 0000 0001 0010 0011 0100 0101 0110 0111 1
000 1001 1010 1011 1100 1101 1110 1111
Signed Mag 0 1 2 3 4 5 6 7 -0 -1 -2 -3 -4 -5 -6 -
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One's Comp 0 1 2 3 4 5 6 7 -7 -6 -5 -4 -3 -2 -1 -
0
Two's Comp 0 1 2 3 4 5 6 7 -8 -7 -6 -5 -4 -3 -2 -
1
Twos complement Most important
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• Ex 537 using 16-bit unsigned integers is
• 16-bit unsigned integers can represent a range of
  integers from
• 0 to ________
• or 0 to 65,535
• 16-bit unsigned fraction with binary point to the
  left of the most significant digit can represent
  fractions from
• 0 to_________
• What is the range for signed numbers? 8-bit

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Boolean Algebra

• Operators
• AND ?
• OR v
• NOT A ?A
• Values
• 1 (true) 0 false

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Associative and Distributive

• Associative rules
• a (b c) (a b) c
• a ? (b ? c)
• Distributive rules
• a (b ? c) (a b) ? (a c)
• a ? (b c)
• Note these rules look like algebra !

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Complement (NOT)

• a
• a a 1
• a 0
• Additional rules
• a a a
• a ? a a

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Properties of 0 and 1

• a 0 a
• a ? 1
• 0
• 1

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Basic Identities of Boolean Algebra
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Venn Diagram

• You can think of Boolean equations as sets 1 is
  everything, and 0 is nothing.

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Absorption Theorem

• a ab a
• a ( a b) a
• Venn Diagram

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An expression is true

• A valid expression is true
• true true
• false false
• x x
• false true

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Absorption Theorem

• a ab a
• Proof
• a ab apply distributive law (14)
• a ab a (1b) apply 3 1 b 1
• a ab a ? 1 apply 2
• a ab

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Boolean Algebra

• Defined as

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Laws of Boolean Algebra
Any law that is true for an expression is also
true for its dual. Operations with 0 and 1 1.
x 0 x x 1 x 2. x 1 1 x 0
0 Idempotent Law 3. x x x x x
x Involution Law 4. (x) x Laws of
Complementarity 5. x x 1 x x
0 Commutative Law 6. x y y x x y y x
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Laws of Boolean Algebra (cont.)

• Associative Laws
• (x y) z x (y z) x y z x (y
  z)
• Distributive Laws
• x (y z) (x y) (x z) x (y z)
• (x y)(x z)
• Simplification Theorems
• x y x y x (x y) (x y) x
• x x y x x (x y) x
• (x y) y x y (x y) y x y
• DeMorgans Law
• (x y z ) (x y z )
• x y z x y z
• Theorems for Multiplying and Factoring
• (x y) (x z) x y x z
• x z x y (x z) (x y)
• Consensus Theorem
• x y y z x z (x y) (y z)
  (x z)
• x y x z (x y) (x z)

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Dual

• The dual of an algebraic expression is obtained
  by interchanging OR and AND operations, and
  replacing 1s by 0s and 0s by 1s.
• Replace
• 1
• 0

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Duality Principle

• A Boolean equation remains valid (true) if we
  take the dual of the equation.
• If an expression is true, its dual is true
• To take the dual
• Replace
• ?
• 1
• 0

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DeMorgans Laws

• (a ? b) a b
• (a b) a ? b
• Replace AND with OR and OR with AND.
• Remove complement from the entire expression and
  place over each variable instead.
• These laws are duals of one another.
• NOTE The LHS of each equation is NOT the dual
  of the RHS.

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DeMorgans Laws in Pictures
A
A
B
B
A
A
B
B
A
A
B
B
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• Truth tables can be used to verify expressions.
• Example, verify DeMorgans Theorem

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Algebraic Manipulation

• Ex F XYZ XYZ XZ
• Boolean algebra is a useful tool for simplifying
  digital circuits.
• Literal single variable within a term that may
  or may not be complemented.

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• Simplify
• F XYZ XYZ XZ

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• Truth Table
• Truth table for both expressions are equivalent.
• By reducing the number of terms and number of
  literals, it is possible to obtain a simpler
  circuit.

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Consensus Theorem

• XY XZ YZ XY XZ
• Note Y and Z are associated with X and X, and
  appear together in the term that is eliminated.
• The dual of the consensus theorem is

redundant
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Proof

• XY XZ YZ

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Complement of a Function

• F can be obtained by interchanging 1s to 0s and
  0s to 1s for values of F in the truth table.
• Can apply DeMorgans theorem as many times as
  necessary to find F.
• F1 XYZ XYZ
• Obtain F1
• F1 XYZ XYZ

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Example

• F2 X(YZYZ)
• Obtain F2

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Negative Logic

• Positive Logic
• true 1
• false 0
• Negative Logic
• true 0
• false 1
• I can think of negated inputs as negative logic.

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Inverted Values

• Bubbles represent inverted values on inputs or
  outputs
• Two bubbles cancel
• X

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NOR and NAND gates
NOR gates
NAND gates
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OR and AND gates


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Circuit Analysis
F
F
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All you need is NAND gates

• Can build AND, OR, NOT from NAND gates
• NAND(X,X) NOT (X)

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All you need is NAND gates

• AND from NAND gates
• AND(X,Y) NAND followed by NOT (X)

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All you need is NAND gates

• OR from NAND gates
• Use Demorgans
• NAND(X,Y) X ? Y
• How to invert inputs?

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Why Do Everything With NANDs

• Because you can
• can only use NORs too.
• The technology of most digital circuits (CMOS) is
  naturally inverting.
• To build non-inverting logic, build
• AND NAND NOT
• OR
• Dont do that too many NOT gates.

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LAB 1 XOR gate
Truth Table

• XOR
• Exclusive-OR gate
• Symbol to designate its operation.
• Similar to the OR gate, but excludes (has the
  value 0 for) the combination with both X and Y
  equal to 1.
• 1 if _______________ variable is equal to 1.

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Lab1 XOR from NAND gates