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Simplification by the K-Map Method, Lecture notes of Engineering

Wayne State CollegeEngineering

System level design of digital logic circuits using hardwired and programmable logic devices. ROMs, PROMs, and PLAs. Synchronous and asynchronous circuit design and analysis.

Typology: Lecture notes

2021/2022

Available from 12/22/2022

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KMap 1
Simplification by the K-Map Method
(Section 3.5 Dueck)
•Karnaugh Map (K-Map) is a tool used in
digital circuit analysis and design in order to
simplify a logic equation (instead of using
Boolean algebra).
•It is a graphical representation of the output
for each input combination for a given logic
function.
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KMap^

Simplification by the K-Map Method

(Section 3.5 Dueck)

•Karnaugh Map (K-Map) is a tool used indigital circuit analysis and design in order tosimplify a logic equation (instead of usingBoolean algebra). •It is a graphical representation of the outputfor each input combination for a given logicfunction.

KMap^

KMap^

KARNAUGH MAPPING

•A K-Map derives either the simplified SOP or POS formof the expression from the truth table. •The simplification is done graphically using the K-map. •The correct labeling of input values on the K-map iscritical. •The labeling is done to assure that there is only oneinput variable that changes between adjacent cells.

KMap^

2-variable Truth table A B^ 0 0^ 0 1^ 1 0^ 1 1

KMap^

2-variable K-map

B A 0 1

A’B’^

A’B

AB’^

AB

KMap^

3-variable K-map C^ A B^0

KMap^

SOP Simplification using K-Maps • Group the adjacent 1’s; maximize the size of the groupsbut minimize the number of groups. • All 1’s in the output must be included in a group; evenif it is only a group of one cell. •Each AND term is written with only the input variablescommon to the group. •Form the final SOP expression by ORing all AND terms.

KMap^

Ex1 (Problem 3.31)

Use the K-Map method to find the maximum SOP simplificationof the logic diagram in Fig. 3.21.

KMap^

13

Ex1 (3.31) F(A,B,C,D) =

Σm ( 0,1,3,8,10,12,13,

KMap^

Ex1 (3.31)

F(A,B,C,D) =

Σ^ m (0,1,3,8,10,12,13,14)

KMap^

Ex1 (3.31)

KMap^

Ex1 (3.31)

F(A,B,C,D) = A’B’D+ABC’+AD’+A’B’C’

SOP simplification

KMap^

Ex2 (3.32c)

Use the K-Map method to reduce the following Booleanexpression to their maximum SOP simplification:^ F(A,B,C) =A’B’C’+A’BC+ABC+AB’C

F(A,B,C) =A’B’C’+BC+AC

KMap^

Ex3 (Problem 3.35)

Use the K-Map method to reduce the following Booleanexpression represented by the following truth table to simplestSOP form.