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Chapter 3: Karnaugh Maps. Uchechukwu Ofoegbu Temple University. Riddle.

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Chapter 3: Karnaugh Maps

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Chapter 3 karnaugh maps

Chapter 3: Karnaugh Maps

Uchechukwu Ofoegbu

Temple University


Chapter 3 karnaugh maps

Riddle

Three people check into a hotel. They pay $30 to the manager and go to their room. The manager suddenly remembers that the room rate is $25 and gives $5 to the bellboy to return to the people. On the way to the room the bellboy reasons that $5 would be difficult to share among three people so he pockets $2 and gives $1 to each person. Now each person paid $10 and got back $1. So they paid $9 each, totalling $27. The bellboy has $2, totalling $29. Where is the missing $1?


Chapter 3 karnaugh maps

Introduction to Karnaugh Maps

  • If implemented correctly, they almost always produce a minimum solution.

  • They are more straightforward that algebraic manipulations

  • They generally produce SOPs, but POS can be generated from their complements if required.


Chapter 3 karnaugh maps

Two-variable Karnaugh maps

A

0

1

B

B’

A’

A

0

A’B’

AB’

B

1

A’B

AB

A

0

1

B

0

m0

m2

1

m1

m3


Chapter 3 karnaugh maps

Three-variable Karnaugh maps

Product terms corresponding to groups of two.


Chapter 3 karnaugh maps

Four-variable Karnaugh maps


Chapter 3 karnaugh maps

Implicants

  • An implicant of a function is a product term.

  • From the point of view of the map, an implicant is a rectangle of 1, 2, 4, 8, . . . (any power of 2) 1’s. That rectangle may not include any 0’s.

  • Example:

    • f = A’B’C’D’+A’B’CD+A’BCD+AB’CD+ABC’D’+ABC’D+ABCD

AB

00

01

11

10

CD

00

1

1

01

1

1

1

1

11

1

10


Chapter 3 karnaugh maps

Implicants

  • The implicants of f are:

Minterms (1 implicant)

ABCD

ABCD

ABCD

ABCD

ABCD

ABCD

ABCD

Groups of 2

ACD

BCD

ACD

BCD

ABC’

ABD

Groups of 4

CD


Chapter 3 karnaugh maps

Prime and Essential Prime Implicants

  • Prime Implicant:

    • an implicant that (from the point of view of the map) is not fully contained in any one other implicant.

  • Essential Prime Implicant:

    • a prime implicant that includes at least one 1 that is not included in any other prime implicant.

AB

00

01

11

10

CD

00

1

1

01

1

1

1

1

11

1

10


Chapter 3 karnaugh maps

Minimum SOP Expressions From Karnaugh Maps


Chapter 3 karnaugh maps

Minimum SOP Expressions From Karnaugh Maps

  • Find all essential prime implicants.

    • Circle them on the map and mark the minterm(s) that make them essential with an asterisk (*).

  • 2.Find enough other prime implicants to cover the function. Do this using two criteria:

  • a.Choose a prime implicant that covers as many new 1’s (that is, those not already covered by a chosen prime implicant).

  • b.Avoid leaving isolated uncovered 1’s.

  • The main idea is

  • To Have all ones covered

  • To Have as few terms as possible

  • To have several rectangles with more 1’s and few rectangles with less 1’s


Chapter 3 karnaugh maps

Example

f = w’x’y’z’+w’xy’z’+ w’xy’z+ w’xyz+ wx’y’z’+ w’xyz+ wxy’z’+ wxyz

AB

00

01

11

10

CD

* * *

00

1

1

1

1

unnecessary

1

01

1

1

*

11

1

10

f = y’z’+wyz+w’xz


Chapter 3 karnaugh maps

In Groups

f = b + a' c


Chapter 3 karnaugh maps

Don’t Cares

  • Prime implicant

    • A rectangle of 1, 2, 4, 8, . . . 1’s or X’s not included in any one larger rectangle.

    • From the point of view of finding prime implicants, X’s (don’t cares) are treated as 1’s.

  • Essential prime implicant

    • A prime implicant that covers at least one 1 not covered by any other prime implicant (as always).

    • Don’t cares (X’s) do not make a prime implicant essential.


Chapter 3 karnaugh maps

Example

f = Σm(1,7,10,11,13) + Σd(5,8,15)

AB

00

01

11

10

CD

00

x

Use don’t cares to get as many minterms in each tem as possible

1

x

01

1

1

x

1

11

10

1

F = BD + ACD + ABC


Chapter 3 karnaugh maps

In groups

  • For the following problem, find the minimum SOP expression within the options given

  • h(a,b,c) = Σm(0,1,5) + d(3,4,6,7)

  • h = a'b' + c + a

  • h = a + c + b’

  • h = c + b’

  • h = b’

  • h = c


Chapter 3 karnaugh maps

Implementation of Two Functions


Chapter 3 karnaugh maps

Example

F = A’B’C’+A’BC’+ABC’+ABC; G = A’B’C+A’BC+ABC’+ABC

AB

AB

00

01

11

10

00

01

11

10

C

C

0

0

1

1

1

1

1

1

1

1

1

1

F = A’C’+AB

G = A’C+AB


Chapter 3 karnaugh maps

Example

F = AB+ABC

G = AB +BC

F = AB+ABC

G = AB +ABC


Chapter 3 karnaugh maps

Example

f = ab + bc g = ab + ac

f = ab + abc g = ab + abc


Chapter 3 karnaugh maps

Example

F = AC +ACD+ABCG = AC+ACD+ABC


Chapter 3 karnaugh maps

Try


Chapter 3 karnaugh maps

NAND, NOR

  • Many electronic systems automatically invert gates

  • Easier to fabricate with electronic components

  • Basic gates used in integrated circuits (IC) digital logic families.

  • NAND gate

    • universal gate

    • Could be used to construct any logic gate


Chapter 3 karnaugh maps

NAND gates.

Alternate symbol for NAND.

Symbols for NOR gate.


Nand gate implementation

NAND Gate Implementation

When we have a circuit consisting of AND and OR gates such that

the output of the circuit comes from an OR,

the inputs to all OR gates come either from a system input or from the output of an AND, and

the inputs to all AND gates come either from a system input or from the output of an OR.

All gates are replaced by NAND gates, and any input coming directly into an OR is complemented.


Example

Example

Try: g = wx(y+z)+x’y


Nor gate implementation

NOR Gate Implementation

When we have a circuit consisting of AND and OR gates such that

the output of the circuit comes from an AND,

the inputs to all OR gates come either from a system input or from the output of an AND, and

the inputs to all AND gates come either from a system input or from the output of an OR.

All gates are replaced by NOR gates, and any input coming directly into an AND is complemented.


Example1

Example

Try: g = (x+y’)(x’+y)(x’+z)


Chapter 3 karnaugh maps

XOR and XNOR

A xor B is 1 if a = 1 or b is 1 and 0 if both are 1 or 0;

Develop a truth table for XOR


Chapter 3 karnaugh maps

Homework

  • 1-17

  • 20

  • 21

  • 22

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