Lecture Digital logic design - Lecture 8: More Karnaugh Maps and Don’t Cares

Lecture 8 More Karnaugh Maps and Don’t Cares. The main contents of the chapter consist of the following: 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.

Lecture 8

More Karnaugh Maps and Don’t Cares

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° 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.

A B

C D 0000

A'B'C'D'  +  A'BC'D  +  ABCD  +  AB'CD’

Can you draw the truth table for these examples?

° Need to make sure all 1’s are covered

° Try to minimize total product terms

° Design could be implemented using NANDs and NORs

Don’t cares

° In digital systems it often happens that certain

input conditions can never occur

interlocked switches such that both switches

cannot be closed at the same time Thus the only three possible states of the switches are that both switches are open or that one switch is open and the other switch is closed.

Namely, the input valuations (x 1 , x 2 ) = 00, 01, and

10 are possible, but 11 is guaranteed not to occur Then we say that (x 1 , x 2 ) = 11 is a don’t-care

condition , meaning that a circuit with x 1 and x 2 as inputs can be designed by ignoring this condition.

° A function that has don’t-care condition(s) is said

to be incompletely specified.

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 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.

Map Simplification with Don’t Cares

1s or 0s depending on which is more 

A'D

by using don't care as a "1"

a 2­cube can be formed  rather than a 1­cube to cover this node

+ C'D

Some You Group, Some You Don’t

B A

This don’t care condition was treated as a (1)

This allowed the grouping of a single one to become a grouping of two, resulting in a simpler term.

There was no advantage in treating this don’t

care condition as a (1), thus it was treated as a

(0) and not grouped.

° Essential prime implicant

• Prime implicant is essential if it alone covers a minterm in the K-map

• Remember that all squares marked with 1 must be covered

° Objective:

• Grow implicant into prime implicants (minimize literals per term)

• Cover the K-map with as few prime implicants as possible

(minimize number of product terms)

A 1

1 1

F, and which ones are prime implicants of F?

(a) AC’D’

(b) BD (c) A’B’C’D’

(d) AC’

(e) B’C’D’

Implicants: (a),(c),(d),(e) Prime Implicants:

(d),(e)

Essential Prime Implicants

• A product term is an essential prime implicant if

there is a minterm that is only covered by that prime implicant.

• The minimal sum-of-products form of F must include

all the essential prime implicants of F

Example of Prime Implicants

D

B

C B

ESSENTIAL Prime Implicants

B C

D A

Prime Implicant Practice

° Find all prime implicants for:

13,14,15) ,10,11,12,

(0,2,3,8,9  

  D) C, B,

B C

Don’t Care Conditions Ex:

Ex: Don’t Care Conditions

After labeling and transferring the truth table data into the K-Map, write the simplified sum-of-

products (SOP) logic expression for the logic

SRT

S R

S R

S R

U

T T U T U TU

T R

S R

Ex: Don’t Care Conditions

After labeling and transferring the truth table data into the K-Map, write the simplified sum-of-

products (SOP) logic expression for the logic

care conditions.

B

D A

C 1

1

B

D A

C 1

1 Essential

Minterms covered by essential prime implicants

Selected

Example with Don't Cares

° Simplify F(A, B, C, D) given on the K-map

Selected

Minterms covered by essential prime implicants

1

1 x

x

x x x

1

B

D A

C

1

1 x

x

x x x

1

B

D A

C

1

1 Essential

Don’ t-care Conditions

SOP implementation

POS implementation

Ex:

° K-maps of four literals considered

• Larger examples exist

° Don’t care conditions help minimize functions

• Output for don’t cares are undefined

° Result of minimization is minimal sum-of-products

° Result contains prime implicants

° Essential prime implicants are required in the

implementation