ecen 248: introduction to digital systems...

ECEN 248: INTRODUCTION TO DIGITAL SYSTEMS DESIGN

Week 3 Dr. Srinivas Shakkottai Dept. of Electrical and Computer Engineering

KARNAUGH MAPS

Overview

  K-maps: an alternate approach to representing Boolean functions

  K-map representation can be used to minimize Boolean functions

  Easy conversion from truth table to K-map to minimized SOP representation.

  Simple rules (steps) used to perform minimization   Leads to minimized SOP representation.

 Much faster and more more efficient than previous minimization techniques with Boolean algebra.

x y F 0 0 1 0 1 1 1 0 0 1 1 0

Karnaugh maps

0 1 y

x 0

1

1

0 0

1

0 1 y

x 0

1

x’y’

xy’ xy

x’y

x

y F = Σ(m0,m1) = x’y + x’y’

x y F 0 0 1 0 1 1 1 0 0 1 1 0

Karnaugh maps   Alternate way of representing Boolean function

 All rows of truth table represented with a square   Each square represents a minterm

  Easy to convert between truth table, K-map, and SOP  Unoptimized form: number of 1’s in K-map equals

number of minterms (products) in SOP  Optimized form: reduced number of minterms

0 1 y

x 0

1

1

0 0

1

0 1 y

x 0

1

x’y’

xy’ xy

x’y

x

y F = Σ(m0,m1) = x’y + x’y’

Karnaugh Maps   A Karnaugh map is a graphical tool for assisting in the

general simplification procedure.   Two variable maps.

0 A

1 0

1

B 0 1 0 1

F=AB ʹ′+A’B 0 A

1 1

1

B 0 1 0 1

° Three variable maps.

0 A

1 1

1

00 01 0 1

BC

0

1 1

1

11 10

F=AB’C’ +AB ʹ′C +ABC +ABC ʹ′ + A’B’C + A’BC’

F=AB +Aʹ′B +AB ʹ′

A B C F 0 0 0 0 0 0 1 1 0 1 0 1 0 1 1 0 1 0 0 1 1 0 1 1 1 1 0 1 1 1 1 1

+

Rules for K-Maps   We can reduce functions by circling 1’s in the K-

map   Each circle represents minterm reduction   Following circling, we can deduce minimized and-or

form. Rules to consider 1. Every cell containing a 1 must be included at least

once. 2. The largest possible “power of 2 rectangle” must

be enclosed. 3. The 1’s must be enclosed in the smallest possible

number of rectangles.



0 A

1 0

1

B 0 1 0 1


1 1

1

B 0 1 0 1


0 A

1 1

1

00 01 0 1

BC

0

1 1

1

11 10





0 A

1 0

1

B 0 1 0 1


1 1

1

B 0 1 0 1 F=A+B


0 A

1 1

1

00 01 0 1

BC

0

1 1

1

11 10





0 A

1 0

1

B 0 1 0 1


1 1

1

B 0 1 0 1 F=A+B


F=A+B ʹ′C +BC ʹ′ 0

A

1 1

1

00 01 0 1

BC

0

1 1

1

11 10



Karnaugh maps

  Numbering scheme based on Gray–code  e.g., 00, 01, 11, 10  Only a single bit changes in code for adjacent map

cells  This is necessary to observe the variable transitions

00 01 AB

C 0

1

11 10

C

B

A

F(A,B,C) = Σm(0,4,5,7)

G(A,B,C) = 0 0

0 0

1 1

1 1 C

B

A

1 0

0 0

0 1

1 1 C

B

A

A

= AC + B’C’

More Karnaugh Map Examples

  Examples

g = b'

0 1 0 1

a b

c ab

00 01 11 10 0 1

0 1 0 1

a b

c ab

00 01 11 10 0 1

0 1 0 1 f = a

0 0 1 0 0 1 1 1

cout = ab + bc + ac

1 1 0 0

0 0 1 1 0 0 1 1

f = a

1. Circle the largest groups possible. 2. Group dimensions must be a power of 2. 3. Remember what circling means!

Application of Karnaugh Maps: The One-bit Adder

Adder

Cin

Cout

S B

A

A B Cin S Cout 0 0 0 0 0 0 0 1 1 0 0 1 0 1 0 0 1 1 0 1 1 0 0 1 0 1 0 1 0 1 1 1 0 0 1 1 1 1 1 1

+

S = A’B’Cin + A’BCin’ + AB’Cin’ + ABCin

Cout = A’BCin + A B’Cin + ABCin’ + ABCin

= A’BCin + ABCin + AB’Cin + ABCin + ABCin’ + ABCin

= BCin + ACin + AB

= (A’ + A)BCin + (B’ + B)ACin + (Cin’ + Cin)AB = 1·BCin + 1· ACin + 1· AB

How to use a Karnaugh Map instead of the

Algebraic simplification?


Adder

Cin

Cout

S B

A

Karnaugh Map for Cout

Now we have to cover all the 1s in the Karnaugh Map using the largest rectangles and as few rectangles as we can.

A

B

Cin

0

0

0

1 1 1

0 1

A

B

Cin


Adder

Cin

Cout

S B

A

0

0

0 0 1

1 1 1



Cout = ACin

A

B

Cin


Adder

Cin

Cout

S B

A

0

0

0 0 1

1 1 1



Cout = Acin + AB

A

B

Cin


Adder

Cin

Cout

S B

A

0

0

0 0 1

1 1 1



Cout = ACin + AB + BCin

A

B

Cin


Adder

Cin

Cout

S B

A

0

1

1 1 0

0 1 0

Karnaugh Map for S

S = A’BCin’

A

B

Cin


Adder

Cin

Cout

S B

A

0

1

1 1 0

0 1 0

Karnaugh Map for S

S = A’BCin’ + A’B’Cin

A

B

Cin


Adder

Cin

Cout

S B

A

0

1

1 1 0

0 1 0

Karnaugh Map for S

S = A’BCin’ + A’B’Cin + ABCin

A

B

Cin


Adder

Cin

Cout

S B

A

0

1

1 1 0

0 1 0

Karnaugh Map for S

S = A’BCin’ + A’B’Cin + ABCin + AB’Cin’

No Possible Reduction!

Can you draw the circuit diagrams?

Summary

  Karnaugh map allows us to represent functions with new notation

  Representation allows for logic reduction.   Implement same function with less logic

  Each square represents one minterm   Each circle leads to one product term   Not all functions can be reduced   Each circle represents an application of:

  Distributive rule -- x(y + z) = xy + xz

  Complement rule – x + x’ = 1

MORE KARNAUGH MAPS

Overview

  Karnaugh maps with four inputs   Same basic rules as three input K-maps

  Understanding prime implicants  Related to minterms

  Covering all implicants   Using Don’t Cares to simplify functions

 Don’t care outputs are undefined

  Summarizing Karnaugh maps

Karnaugh Maps for Four Input Functions

  Represent functions of 4 inputs with 16 minterms   Use same rules developed for 3-input functions   Note bracketed sections shown in example.

  F(A,B,C,D) = Σm(0,2,3,5,6,7,8,10,11,14,15) F = C + B’D’

Karnaugh map 4-variable example

D

A

B

1 0

0 1

0 1

0 0

1 1

1 1

1 1

1 1 C

+ A’BD

0 4

1 5

12   8

13   9

3   7

2 6

15 11

14 10

Implicants

  An implicant is a rectangle of 1, 2, 4, 8, . . . (any power of 2) 1’s. May not include any 0’s.

Prime Implicant

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

  An essential prime implicant is a prime implicant that includes at least one 1 that is not included in any other prime implicant. Prime implicant, but not essential

prime implicant

A' B' D + A' C + B' C D

B C' D' + A C' + A B D'

LT = EQ = GT =

K-map for LT K-map for GT

Design examples

0 0

1 0

0 0

0 0 D

A

1 1

1 1

0 1

0 0 B

C

K-map for EQ

1 0

0 1

0 0

0 0 D

A

0 0

0 0

1 0

0 1 B

C

0 1

0 0

1 1

1 1 D

A

0 0

0 0

0 0

1 0 B

C

Can you draw the truth table for these examples?

A'B'C'D' + A'BC'D + ABCD + AB'CD’

A B C D

EQ

Physical Implementation

° Step 1: Truth table

° Step 2: K-map

° Step 3: Minimized sum-of-products

° Step 4: Physical implementation with gates

K-map for EQ

1 0

0 1

0 0

0 0 D

A

0 0

0 0

1 0

0 1 B

C

Karnaugh Maps

  Four variable maps.

0 AB

1 1

0

00 01 00 01

CD

0

0 1

1

11 10 F=Aʹ′BC ʹ′+Aʹ′CD ʹ′+ABC +AB ʹ′Cʹ′D ʹ′+ABC ʹ′+AB ʹ′C

1

1 0

1 11 10

1

1 1

1

° Need to make sure all 1’s are covered

° Try to minimize total product terms.

° Design could be implemented using NANDs and NORs

Karnaugh Maps

  Four variable maps.

F=BC ʹ′+CD ʹ′+ AC+ AD ʹ′

0 AB

1 1

0

00 01 00 01

CD

0

0 1

1

11 10 F=Aʹ′BC ʹ′+Aʹ′CD ʹ′+ABC +AB ʹ′Cʹ′D ʹ′+ABC ʹ′+AB ʹ′C

1

1 0

1 11 10

1

1 1

1

° Need to make sure all 1’s are covered

° Try to minimize total product terms.

° Design could be implemented using NANDs and NORs

Product of Sums

SoP from PoS

Sum of Products

SoP and PoS

DON’T CARES (ISFs)

Karnaugh maps: Don’t cares   In some cases, outputs are undefined   We “don’t care” if the logic produces a 0 or a 1   This knowledge can be used to simplify functions.

0 0

1 1

X 0

X 1 D

A

1 1

0 X

0 0

0 0

B

C

CD AB

00

01

11

10

00 01 11 10

- Treat X’s like either 1’s or 0’s -  Very useful -  OK to leave some X’s uncovered

+ C’D

Karnaugh maps: Don’t cares

  f(A,B,C,D) = Σ m(1,3,5,7,9) + d(6,12,13)  without don't cares

  f =

0 0

1 1

X 0

X 1 D

A

1 1

0 X

0 0

0 0

B

C

A’D

CD AB

00

01

11

10

00 01 11 10

C f 0 0 0 1 1 0 1 1 0 0 0 1 1 X 1 0 0 1 1 0 0 1 1

D 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1

1 0 1 0 0 X X 0 0

A 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1

+

B 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1

+

Don’t Care Conditions   In some situations, we don’t care about the value of a

function for certain combinations of the variables.  these combinations may be impossible in certain contexts  or the value of the function may not matter in when the

combinations occur

  In such situations we say the function is incompletely specified and there are multiple (completely specified) logic functions that can be used in the design.  so we can select a function that gives the simplest circuit

 When constructing the terms in the simplification procedure, we can choose to either cover or not cover the don’t care conditions.

Prime Implicants

  A prime implicant is a rectangle of 1, 2, 4, 8, …1’s or X’s not included in any one larger rectangle. Thus, from the point of view of finding prime implicants, i.e., X’s (don’t cares) are treated as 1’s.

  An essential prime implicant is 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.

Map Simplification with Don’t Cares

F=Aʹ′Cʹ′D+B+AC 0

AB

x x

1

00 01 00 01

CD

0

x 1

0

11 10

1

x 0

1 11 10

1

1 1

x

0 AB

x x

1

00 01 00 01

CD

0

x 1

0

11 10

1

x 0

1 11 10

1

1 1

x

F=Aʹ′Bʹ′Cʹ′D+ABCʹ′+BC+AC

° Alternative covering.

Karnaugh maps: don’t cares (cont’d)

  f(A,B,C,D) = Σ m(1,3,5,7,9) + d(6,12,13)   f = A'D + B'C'D without don't cares   f = with don't cares

don't cares can be treated as 1s or 0s

depending on which is more advantageous

0 0

1 1

X 0

X 1 D

A

1 1

0 X

0 0

0 0 B

C

A'D

by using don't care as a "1" a 2-cube can be formed rather than a 1-cube to cover this node

+ C'D

FIVE-VARIABLE K-MAPS

50

Five-variable K-maps – f(V,W,X,Y,Z)

V= 0 V= 1

Y m 16 m 17 m 19 m 8 m 20 m 21 m 23 m 22 m 28 m 29 m 31 m 30

X W

m 24 m 25 m 27 m 26 Z

51

Simplify f(V,W,X,Y,Z)=Σm(0,1,4,5,6,11,12,14,16,20,22,28,30,31)

V= 0 V= 1

1 1

1 1 1

1

1 1

1

1 1

1 1 1

f = XZ’ Σm(4,6,12,14,20,22,28,30) + V’W’Y’ Σm(0,1,4,5) + W’Y’Z’ Σm(0,4,16,20) + VWXY Σm(30,31) + V’WX’YZ m11

Example 1

ecen 248: introduction to digital systems...

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