XOR Operator

1
XOR Operator

• A short digression
• to introduce another Boolean operation
  exclusive-OR (XOR)

A B A B
0 0 0
0 1 1
1 0 1
1 1 0
2
XOR Operator

• Also referred to as an odd function since it
  returns a 1 only when an odd number of 1s are
  input

A B C ABC
0 0 0 0
0 0 1 1
0 1 0 1
0 1 1 0
1 0 0 1
1 0 1 0
1 1 0 0
1 1 1 1
3
Simplification

• Using the axioms to prove that a simplified
  version of a circuit is equivalent to the complex
  version takes a special kind of person
• of which Im not one
• Fortunately, theres another way

4
Karnaugh Maps

• Also known as K-Map
• Recall that an expression can be written in the
  form
• F(A,B,C) S(0,2,4,5,6)
• Which means the functional value is 1 at binary
  input patterns 0, 2, 4, 5, 6 and 0 at all other
  input patterns
• What does the truth table look like?

5
K-Maps

• F(A,B,C) S(0,2,4,5,6) is called a sum of
  minterms representation
• The expression for such a representation is
• F(A,B,C) ABC ABC ABC ABC ABC
• We could simplify this via the axioms, right?
  (assuming we were that special kind of person)
• Its painful!!!

6
K-Maps

• A K-Map is a grid (map) where each square
  corresponds to a minterm

Note the ordering here is Gray code, not binary
7
K-Maps

• Notice how neighboring squares (minterms) differ
  by a single bitthis is the key to the whole
  thing
• Consider minterms 1 and 3
• 1 ABC
• 3 ABC
• If we were to OR these together
• (ABC ABC) would simplify to AC via the
  axioms

8
K-Maps

• Great, now what do we do with them?
• Place 1s on the squares that correspond to
  minterms in the truth table
• Place 0s on all other squares
• Group adjacent 1s into the largest group whose
  size is a power of 2

9
K-Maps

• Notes
• Adjacencies wrap top-to-bottom and left-to-right
• 1s can be part of more than one group
• When you are grouping adjacent squares youre
  essentially applying axiom 4 (x x 1) so the
  variable that is being spanned can be removed
  from the minterm

10
Simplification via Axioms(aka Proofs)

• Heres a little insight that no one ever taught
  me
• F(x, y, z) xyz xyz xyz
• Notice how the middle term shares two elements
  with each of the others
• Using association, distribution, and inverse
• F(x, y, z) yz xyz
• One more application of distribution
• F(x, y, z) z(y xy)
• We could have arrived at a similar solution by
  grouping the 2nd two terms

11
Simplification via Axioms(aka Proofs)

• But, can we do better?
• Notice that we use the minterm xyz in two
  groupings
• What does that mean in terms of an axiomatic
  proof?

F(x, y, z) yz xz z(x y)
12
Simplification via Axioms(aka Proofs)

• It means exactly this
• F(x, y, z) xyz xyz xyz
• F(x, y, z) xyz xyz xyz xyz
• by idempotence over OR
• Now we can form two associative groupings and
  arrive at the same answer that the Karnaugh Map
  gave us

13
Karnaugh Maps

• What is the truth-table?
• What is the expression in sum-of-minterms form?
• What is the simplified expression?
• What is the (schematic) logic gate implementation?

14
Sum-of-Products

• This is what we previously called the
  sum-of-minterms
• Form the largest power-of-two groupings of 1s on
  the K-map
• Create the schematic

15
Product-Of-Sums

• Instead of forming large adjacent groups of 1s
  (on the K-map), form large adjacent groups of 0s
• What does this mean in terms of the original
  expression/truth-table?
• It means you have simplified F, instead of F
• To fix what youve done you need only negate
  the final result them apply De Morgans theorem

16
Example Sum-of-Products

• F(A,B,C,D) S(0,1,2,5,8,9,10)
• Form the truth-table
• Form the K-map
• Simplify the K-map using sum-of-products
• Formulate the boolean expression
• Draw the schematic diagram

17
Example Sum-of-Products
B
D
F
C
A
D
18
Example Product-of-Sums

• F(A,B,C,D) S(0,1,2,5,8,9,10)
• Form the truth-table
• Form the K-map
• Simplify the K-map using product-of-sums
• Formulate the boolean expression
• Negate, apply De Morgans
• Draw the schematic diagram

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Example Product-of-Sums
B
D
A
F
C
D
20
So What?

• As it turns out, the sum-of-products can be
  easily implemented with NAND gates
• Similarly, the product-of-sums can be easily
  implemented with NOR gates
• This may greatly simplify the design thus saving
  us money!

21
NAND/NOR Implementations
22
Combinational Circuits

• Definition A connected arrangement of logic
  gates with a set of inputs and outputs
• Specifically, they have no memory!
• Basically, its the stuff weve been working on
  so far

23
Combinational Circuit Design

• Design a Half-Adder
• A combinational circuit that adds 2 bits
• Input 1 is call the Augend
• Input 2 is called the Addend
• Output 1 is called the Sum
• Output 2 is called the Carry

24
Combinational Circuit Design

• Design a Full-Adder
• A combinational circuit that adds 3 bits
• Input 1 is call the Augend
• Input 2 is called the Addend
• Input 3 is call the Carry-in
• Output 1 is called the Sum
• Output 2 is called the Carry-out

25
Homework

• Pages 37, 38 1-8, 1-9, 1-10, 1-12, 1-13
• Due Thursday (next lecture)