ECE
171: Introduction to Digital Circuits |
Fall 1999 |
Rev: 11.2.99 |

- 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

- Boolean Algebra - 4 More Rules
- Factoring
- Karnaugh Maps
- Definitions
- Many Examples
- 5 & 6 Variables

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.*

- Use rule 14 to help reduce the
gate count.

14. AB + AC = A(B + C)

- 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

- 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.

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 2
^{n}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.

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**.

- Prime Implicants: 5
- Distinguished 1-Cells: 2
- Essential Prime Implicants: 2
- Minimal Sums: 1

- Prime Implicants: 7
- Distinguished 1-Cells: 2
- Essential Prime Implicants: 2
- Minimal Sums: 1

**Y = B'D'
+ AD' + A'C'D + BCD**

- 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**

- Prime Implicants: 5
- Distinguished 1-Cells: 3
- Essential Prime Implicants: 3
- Minimal Sums: 1
**Y = A'B' + A'C' + ABC + A'D**

- Prime Implicants: 4
- Distinguished 1-Cells: 4
- Essential Prime Implicants: 4
- Minimal Sums: 1

**Y = A'C
+ A'B + BD + CD**

- Prime Implicants: 5
- Distinguished 1-Cells: 3
- Essential Prime Implicants: 3
- Minimal Sums: 1
**Y = B'D + BC' + AB**

- 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'**

- Prime Implicants: 3
- Distinguished 1-Cells: 8
- Essential Prime Implicants: 3
- Minimal Sums: 1

**Y = B'C
+ D + BC'**

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.

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'**