Today's Outline

controlled inverter (repeat)
Boolean Relations
Duality Relations
Sum of Products
Algebraic Simplifications
Karnaugh Maps:
- Pairs, Quads, Octets
- Simplifications:
  - Overlapping
  - Redundant
  - Rolling
Don't care outputs

Controlled inverter

also called a programmed inverter
a regular inverter always inverts
we want an inverter that will only invert when its control line is high
solution: use an (array of) XOR
when control is low, word passes through unchanged: Y = A
when control is high, word passes through inverted: Y = (not A)
example: 4 bit controlled inverter
inverting is different from negation! For negation, you must add 1
in-class exercise: try adding a 4-bit number (first bit is 0, second bit is 1) and its inverse, then the same four-bit number and its inverse+1. Then repeat with an 8-bit number.

Boolean Relations

Commutative, Associative, and Distributive
OR:
- A + 0 = ?
- A + A = ?
- A + 1 = ?
- A + (not A) = ?
AND:
- A . 0 = ?
- A . A = ?
- A . 1 = ?
- A . (not A) = ?
double negation is the identity function
De Morgan's theorems

Duality Relations

change ORs to ANDs
change ANDs to ORs
complement constants (0's and 1's)
if the original logic relation holds, so does the dual

example: apply to A(B + C) = AB + AC
the result is A + BC = (A + B)(A + C)

Sum of Products

given a truth table, we want to design a circuit
four fundamental products of two inputs:
1. Y = (not A) (not B)
2. Y = (not A) B
3. Y = A (not B)
4. Y = A B
truth table ("1-output" rows) gives us the fundamental products that produce a high output -- OR these
example: XOR = A (not B) | (not A) B
works for any truth table, of any size

in-class exercise: produce a logic function for the following truth table:

  A  B  C   |  Out
------------+-------
  0  0  0   |   1
  0  0  1   |   0
  0  1  0   |   0
  0  1  1   |   0
  1  0  0   |   1
  1  0  1   |   0
  1  1  0   |   0
  1  1  1   |   1

Algebraic Simplification

minimize the number of gates
example: Y = A (not B) + A B
example: Y = A B + A C + B D + C D
in-class exercise: simplify Y = (not A) (not B) C D + (not A) B C D + A (not B) (not C) D

Karnaugh Maps

Complement and direct on rows, column

Two variables:

	(not B)	B
(not A)	out1	out2
A	out3	out4

Four variables:

	(not C) (not D)	(not C) D	C D	C (not D)
(not A) (not B)	out1	out2	out3	out4
(not A) B	out5	out6	out7	out8
A B	out9	out10	out11	out12
A (not B)	out13	out14	out15	out16

Write 1 for the high outputs, 0 for the rest
change only one bit at a time at top and bottom: a Gray code (see also http://www.nist.gov/dads/HTML/graycode.html )

Pairs, Quads, Octets

try and group pairs of ones -- the corresponding (direct and complement) variable can be eliminated
try and group quads of ones -- the corresponding (direct and complement) variables can be eliminated
try and group octets of ones

	(not C) (not D)	(not C) D	C D	C (not D)
(not A) (not B)	0	1	1	1
(not A) B	0	0	0	1
A B	1	1	0	1
A (not B)	1	1	0	1

Y = (not A) (not B) D + A (not C) + C (not D)

Simplifications

overlapping (two pairs, quads, octets using the same "1") is fine
can always non-overlap, but may lead to more complicated equations
can go "over the edge" of the map ("rolling" the map)
eliminate redundant groups, e.g. the pairs that are also in a quad

Don't care Outputs

X marks a "don't care" output
they would normally come from a truth table
what groupings would you do for this map? (bigger groups are always better).
how would you implement it using gates?

	(not C) (not D)	(not C) D	C D	C (not D)
(not A) (not B)	1	0	1	0
(not A) B	1	1	1	0
A B	X	X	X	X
A (not B)	X	X	X	X

Homework 3

Due Wed, Sep 17nd, at 1:30pm
email answers to esb@hawaii.edu, or hand in in class
Questions 3-1, 3-2, 3-4, 3-5, 3-7 (if by email, write your answers using AB, A+B, and NOT A), 3-10, 3-14, 3-26, 3-27.