|
ECE
171: Introduction to Digital Circuits |
Fall 1999 |
Rev: 11.2.99 |
Lecture Notes 10
Last Time
- Project 1 Assigned
- 4 More BA Rules
- TT
POS Simplify w/BA
NOR LD
- Karnaugh Maps
- Introduction
- 2,3, & 4 Variables
- Review of Exam 1
This Time
- Boolean Algebra - 4 More Rules
- Factoring
- Karnaugh Maps
- Definitions
- Many Examples
- 5 & 6 Variables
Boolean Algebra
29. A XOR A = 0
30. A XOR A' = 1
31. (A + B)(B' + C)(A + C) = (A + B)(B' + C)
32. AB + B'C + AC = AB + B'C
Derive rule 31 by a truth table
proof.
Derive rule 32 by a karnaugh
map.
Factoring
- Use rule 14 to help reduce the
gate count.
14. AB + AC = A(B + C)
Example
- Y = A' D' + A' C'
- A straightforward implementation
requires:
- 3 2-Input NAND's
- 3 Inverters
- After factoring, we have
Y = A' (C'+D')
- This can be implemented more
efficiently as shown below
- This requires
- 2 2-Input NAND's
- 2 Inverters
Example
- Y = A' C' + A' D' + AEF + A
B'
- A straightforward implementation
requires:
- 3 2-Input NAND's
- 1 3-Input NAND
- 1 4-Input NAND (to combine all
four products)
- 4 Inverters (for A, B, C, and
D)
- This would take 3-4 IC's to
implement depending on whether you used one of the spare NAND's
to implement the inverter.
- After factoring, we have
Y = A' (C' + D') + A (EF + B')
- This can be implemented as follows
- This requires
- 6 2-Input NAND's
- 1 Inverter
- Assuming you used standard IC's
that have 4 2-Input NAND's per an IC, this would take 2 IC's
to implement if you used one of the spare NAND's to implement
the inverter.
Karnaugh Maps
Today we will discuss Karnaugh
maps more formally than last time and discuss a more orderly method
for optaining the minimal sum.
- Literal
- A variable or complement of
a variable.
Examples: A, B', C', D
- Normal Product Term
- A product term in which no variable
appears more than once.
- Examples: ABC, AB'C', A'B'C'
Counter Examples: AA'BC, AB'B'C', ABCC
- n-variable Minterm
- A normal product term with n
literals. There are 2n such terms.
- Example:
|
A |
B |
|
|
0 |
0 |
A'B' |
|
0 |
1 |
A'B |
|
1 |
0 |
AB' |
|
1 |
1 |
AB |
- Each of the product terms corresponding
to a row in the truth table is a minterm.
- Minimal Sum
- A sum of products (SOP) expression
such that no SOP expression for Y has fewer product terms and
any SOP expression with the same number of product terms has
at least as many literals.
This is what we are trying to produce through the use of Karnaugh
maps.
- Implicant
- A normal product term that implies
Y.
- Example: For the function Y
= AB + ABC + BC, the implicants are AB, ABC, and BC because if
any one of those terms are true, then Y is true.
- Prime Implicant
- An implicant of Y such that
if any variable is removed from the implicant, the resulting
term does not imply Y.
- Example: Y = AB + ABC + BC
Prime Implicants: AB, BC
- Not a prime implicant: ABC
- ABC is not a prime implicant
because the literal A can be removed to give BC and BC still
implies Y. Conversely AB is not a prime implicant because you
can't remove either A or B and have the remaining term still
imply Y.
- In truth tables the prime implicants
are represented by the largest rectangular groups of ones that
can be circled. If a smaller subgroup is circled, the smaller
group is an implicant, but not a prime implicant.
- PI Theorem
- A minimal sum is a sum of prime
implicants.
- Distinguished 1-Cell
- An input combination that is
covered by 1 prime implicant. In terms of Karnaugh maps, distinguished
1-cells are 1's that are circled by only 1 prime implicant.
- Essential Prime Implicant
- A prime implicant that that
includes one or more distinguished one cells. Essential prime
implicants are important because a minimal sum contains all essential
prime implicants.
Karnaugh Map Examples
In the following examples the
distinguished 1-cells are marked in the upper left corner of the
cell with an asterisk (*). The essential prime implicants are
circled in blue, the prime implicants are circled in
black, and the non-essential prime implicants included
in the minimal sum are shown in red.
Example 1
- Prime Implicants: 5
- Distinguished 1-Cells: 2
- Essential Prime Implicants:
2
- Minimal Sums: 1
Y = A'CD'
+ AC'D + BCD
Example 2
- Prime Implicants: 7
- Distinguished 1-Cells: 2
- Essential Prime Implicants:
2
- Minimal Sums: 1
Y = B'D'
+ AD' + A'C'D + BCD
Example 3
- Prime Implicants: 6
- Distinguished 1-Cells: 2
- Essential Prime Implicants:
2
- Minimal Sums: 3
Y = AB'C' +
A'CD' + AC'D + BCD
Y = AB'C' +
A'CD' + ABD + A'BC
Y = AB'C'
+ A'CD' + ABD + BCD
Example 4
- Prime Implicants: 5
- Distinguished 1-Cells: 3
- Essential Prime Implicants:
3
- Minimal Sums: 1
- Y = A'B'
+ A'C' + ABC + A'D
Example 5
- Prime Implicants: 4
- Distinguished 1-Cells: 4
- Essential Prime Implicants:
4
- Minimal Sums: 1
Y = A'C
+ A'B + BD + CD
Example 6
- Prime Implicants: 5
- Distinguished 1-Cells: 3
- Essential Prime Implicants:
3
- Minimal Sums: 1
- Y = B'D
+ BC' + AB
Example 7
- Prime Implicants: 8
- Distinguished 1-Cells: 0
- Essential Prime Implicants:
0
- Minimal Sums: 2
Y = A'B'C
+ A'BD + ABC' + AB'D'
Y = B'CD'
+ A'CD + BC'D + AC'D'
Example 8
- Prime Implicants: 3
- Distinguished 1-Cells: 8
- Essential Prime Implicants:
3
- Minimal Sums: 1
Y = B'C
+ D + BC'
5-Variable Karnaugh Maps
For these you must circle the
prime implicants on each map individually and then the prime implicants
on the joint map. The joint essential prime implicants are shown
in green.
- Prime Implicants: 7
- Distinguished 1-Cells: 7
- Essential Prime Implicants:
4
- Minimal Sums: 2
Y = A'B'C'
+ BE + ABC' + ACE + A'DE
Y = A'B'C'
+ BE + ABC' + ACE + CDE
Note that the joint map can help
you identify the joint prime implicants.
6-Variable Karnaugh Maps
The prime implicants unique to
each map are shown in black.
The prime implicants shared between maps 0 and 1 (A=0) are shown
in aqua.
The prime implicants shared between maps 0 and 2 (B=0) are shown
in violet.
The prime implicants shared between maps 1 and 3 (B=1) are shown
in olive.
The prime implicants shared between maps 2 and 3 (A=1) are shown
in brown.
The prime implicants shared between all 4 maps are shown in orange.
To find the prime implicants shared
among maps it may help to draw out each of the 5 joint maps.
- Distinguished 1-Cells: 10
- Essential Prime Implicants:
5
- Minimal Sums: 2
-
- Essential Prime Implicants
- A'EF (Maps 0 & 1)
BCD' (Maps 1 & 3)
B'D'F' (Maps 0 & 2)
ACE' (Maps 2 & 3)
ABDE' (Map 3)
Y = A'EF
+ BCD' + B'D'F' + ACE' + ABDE' + B'DE'F + A'B'C'F
Y = A'EF
+ BCD' + B'D'F' + ACE' + ABDE' + B'DE'F + A'B'C'D'