Lec7 Intro to Computer Engineering by Hsien-Hsin Sean Lee Georgia Tech -- Karnaugh Map

ECE2030 Introduction to Computer Engineering

Lecture 7: Simplification using K-map

Prof. Hsien-Hsin Sean LeeProf. Hsien-Hsin Sean LeeSchool of Electrical and Computer EngineeringSchool of Electrical and Computer EngineeringGeorgia TechGeorgia Tech

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Hamming Distance• The count of bits different in two binary

patterns• Examples:

– Dh(1001, 0101) = 2– Dh(0xADF4, 0x9FE3) = ??

• Unit-Distance Codes– Reduce errors during transmission such as

rotary positional sensor– E.g. Gray Code

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Gray Code Construction

01

0110

0011

0001111010110100

00001111

000001011010110111101100100101111110010011001000

0000000011111111

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Gray CodeBinary Encoding Gray Code Encoding

Decimal b3 b2 b1 b0 g3 g2 G1 g00 0 0 0 0 0 0 0 01 0 0 0 1 0 0 0 12 0 0 1 0 0 0 1 13 0 0 1 1 0 0 1 04 0 1 0 0 0 1 1 05 0 1 0 1 0 1 1 16 0 1 1 0 0 1 0 17 0 1 1 1 0 1 0 08 1 0 0 0 1 1 0 09 1 0 0 1 1 1 0 1

10 1 0 1 0 1 1 1 111 1 0 1 1 1 1 1 012 1 1 0 0 1 0 1 013 1 1 0 1 1 0 1 114 1 1 1 0 1 0 0 115 1 1 1 1 1 0 0 0

0b e wherbbg 41iii

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Rotary Position Sensor

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Karnaugh Map (K-Map)• A graphical map method to simplify

Boolean function up to 6 variables• A diagram made up of squares• Each square represents one minterm

(or maxterm) of a given Boolean function

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Karnaugh Map Examples

Note that the Hamming DistanceHamming Distance between adjacent columnsadjacent columns or adjacent rows adjacent rows (including cyclic ones) (including cyclic ones) must be 1 for simplification purposes

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Karnaugh Map

Adjacent columns or rows allow grouping of minterms (maxterms) for simplification

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Implicant• Definition

– A product term is an ImplicantImplicant of a Boolean function if the function has an output 1 for all minterms of the product term.

• In K-map, an ImplicantImplicant is – bubble covers only 1

(bubble size must be a power of 2)

00 01 11 10

00 1 1 0 0

01 0 0 1 0

11 0 1 1 1

10 1 1 0 0

ABCD

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Prime Implicant• Definition

– If the removal of any literal from an implicant II results in a product term that is not an implicant of the Boolean function, then II is an Prime ImplicantPrime Implicant.

– Examples• BCDBCD is an implicant, but CDCD or BDBD or BCBC

do not imply a 1 in this function; BCDBCD is a PIPI

• B’C’DB’C’D is an implicant, but B’C’B’C’ is not an implicant, thus B’C’DB’C’D is not a PI

• In K-map, a Prime Implicant (PI)Prime Implicant (PI) is – bubble that is expanded as big as

possible (bubble size must be a power of 2)

00 01 11 10

00 1 1 0 0

01 0 0 1 0

11 0 1 1 1

10 1 1 0 0

ABCD

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Essential Prime Implicant• Definition

– If a minterm of a Boolean function is included in only one PI, then this PI is an Essential Prime Essential Prime ImplicantImplicant.

• In K-map, an Essential Essential Prime ImplicantPrime Implicant is – Bubble that contains a 1

covered only by itself and no other PI bubbles

00 01 11 10

00 1 1 0 0

01 0 0 1 0

11 0 1 1 1

10 1 1 0 0

ABCD

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Non-Essential Prime Implicant• Definition

– A Non-Essential Prime Non-Essential Prime ImplicantImplicant is a PI that is not an Essential PI.

• In K-map, an Non-Non-Essential Prime Essential Prime ImplicantImplicant is – A 1 covered by more than

one PI bubble

00 01 11 10

00 1 1 0 0

01 0 0 1 0

11 0 1 1 1

10 1 1 0 0

ABCD

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Simplification for SOP• Form K-Map for the given Boolean function

• Identify all Essential Prime Implicants for 1’s in the K-map

• Identify non-Essential Prime Implicants in the K-map for the 1’s which are not covered by the Essential Prime Implicants

• Form a sum-of-products (SOP) with all Essential Prime Implicants and the necessary non-Essential Prime Implicants to cover all 1’s

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Example for SOP• Identify all the

essential PIs for 1’s• Identify the non-

essential PIs to cover 1’s

• Form an SOP based on the selected PIs

7) 6, 4, 1, m(0,F

00 01 11 10

0 1 1 0 0

1 1 0 1 1

ABC

CAABBAF

orCBABBAF

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15) 14, 13, 9, 8, 7, 1, m(0,F

00 01 11 10

00 1 1 0 0

01 0 0 1 0

11 0 1 1 1

10 1 1 0 0

ABCD

ABDBCDABCCBF

orDCABCDABCCBF

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12) 11, 6, 4, 3, M(1,F

00 01 11 10

00 1 0 0 1

01 0 1 1 0

11 0 1 1 1

10 1 1 0 1

ABCD

CBAABCBDDBF

orDCAABCBDDBF

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Prime Implicants• All the prior definitions apply to ‘0’ (or

maxterm) as well• Consider these implicants imply a ‘0’

output

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Simplification for POS• Form K-Map for the given Boolean function

• Identify all Essential Prime Implicants for 0’s in the K-map

• Identify non-Essential Prime Implicants in the K-map for the 0’s which are not covered by the Essential Prime Implicants

• Form a product-of-sums (POS) with all Essential Prime Implicants and the necessary non-Essential Prime Implicants to cover all 0’s

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Example for POS• Identify all the



• Form an POS based on the selected PIs

)CBA)(B(AF

00 01 11 10

0 1 1 0 0

1 1 0 1 1

ABC

5) 3, M(2,F

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Example for POS• Identify all the



• Form an POS based on the selected PIs

D)B)(ADC)(BDBD)(ACB(F

00 01 11 10

00 1 0 0 1

01 0 1 1 0

11 0 1 1 1

10 1 1 0 1

ABCD

12) 11, 6, 4, 3, M(1,F

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Don’t Care Condition X• Don’t care (X)

– Those input combinations which are irrelevant to the target function (i.e. If the input combination signals can be guaranteed never occur)

– Can be used to simplify Boolean equations, thus simply logic design

• In K-map– Use XX to express Don’t Care in the map– Don’t care can be bubbled as 11 or 00 depending

on SOP or POS simplification to result into bigger bubble

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Don’t Care Example BCD to Gray Code

BCD-coded input Gray Code OutputDecimal b3 b2 b1 b0 g3 g2 g1 g0

0 0 0 0 0 0 0 0 01 0 0 0 1 0 0 0 12 0 0 1 0 0 0 1 13 0 0 1 1 0 0 1 04 0 1 0 0 0 1 1 05 0 1 0 1 0 1 1 16 0 1 1 0 0 1 0 17 0 1 1 1 0 1 0 08 1 0 0 0 1 1 0 09 1 0 0 1 1 1 0 1

10 1 0 1 0 XX XX XX XX11 1 0 1 1 XX XX XX XX12 1 1 0 0 XX XX XX XX13 1 1 0 1 XX XX XX XX14 1 1 1 0 XX XX XX XX15 1 1 1 1 XX XX XX XX

• BCD stands for Binary Coded Binary Coded DecimalDecimal

• Each digit of a decimal number is represented by one codecode

• BCD only encodes from 0 to 9– Require 4 bits– Only use 10 out of

16 encoding space

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BCD to Gray Code (Do not use Don’ Care)

BCD-coded input Gray Code OutputDecimal b3 b2 b1 b0 g3 g2 g1 g0

0 0 0 0 0 0 0 0 01 0 0 0 1 0 0 0 12 0 0 1 0 0 0 1 13 0 0 1 1 0 0 1 04 0 1 0 0 0 1 1 05 0 1 0 1 0 1 1 16 0 1 1 0 0 1 0 17 0 1 1 1 0 1 0 08 1 0 0 0 1 1 0 09 1 0 0 1 1 1 0 110 1 0 1 0 11 11 11 1111 1 0 1 1 11 11 11 0012 1 1 0 0 11 00 11 0013 1 1 0 1 11 00 11 1114 1 1 1 0 11 00 00 1115 1 1 1 1 11 00 00 00

• BCD stands for Binary Coded Binary Coded DecimalDecimal

• Each digit of a decimal number is represented by one codecode

• BCD only encodes from 0 to 9– Require 4 bits– Only use 10 out of

16 encoding space

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What g2 is?• g2 = F(b3, b2, b1, b0)

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Example: g2 of BCD-to-Gray

3223320 bbbbbbg

00 01 11 10

00 0 0 0 0

01 1 1 1 1

11 X X X X

10 1 1 X X

b3b2

b1b000 01 11 10

00 0 0 0 0

01 1 1 1 1

11 0 0 0 0

10 1 1 1 1

b3b2

b1b0

Without using Don’t Care Take Don’t Care into account

230 bbg

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Another Example of Don’t Care (SOPSOP)

14) 13, d(10,12) 11, 6, 4, 3, m(2,D) C, B, F(A,

00 01 11 10

00 0 0 1 1

01 1 0 0 1

11 1 X 0 X

10 0 0 1 X

ABCD

CBDBF

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Another Example of Don’t Care (POSPOS)

14) 13, d(10,12) 11, 6, 4, 3, m(2,D) C, B, F(A,

00 01 11 10

00 0 0 1 1

01 1 0 0 1

11 1 X 0 X

10 0 0 1 X

ABCD

C))(BDB(F

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Use Karnaugh Map in BB5 5 or BB66

• In BB55

– 2 K-maps are to be constructed (2 submaps)– Consider one submap is on top of the other – Cells that occupy the same relative position in 2

maps are considered adjacent– Bubble can be constructed in vertical dimension

when stacking up 2 maps• In BB66

– 4 K-maps are to be constructed (4 submaps)– Similar to BB55 , yet another dimension needs to be

considered

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Karnaugh Map Example in BB55

27) 24, 20, 19, 18, 16, 11, 8, 7, 6, 4, 3, 2, m(0,D)C,B,A, F(E,

00 01 11 10

00 1 0 1 1

01 1 0 1 1

11 0 0 0 0

10 1 0 1 0

ABCD

00 01 11 10

00 1 0 1 1

01 1 0 0 0

11 0 0 0 0

10 1 0 1 0

ABCD

E = 0 E = 1

CDBDCBDCACBAECAF

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Karnaugh Map Example in BB66

)45(61) 53, 37, 31, 29, 23, 21, 13, 9, 5, m(1,Z)Y,X,W,V,F(U, d

00 01 11 10

00 1

01 1

11 1 1

10 1

WXYZ

U,V=0,0

00 01 11 10

00

01 1

11 X

10

WXYZ

U,V=1,0

00 01 11 10

00

01 1 1

11 1 1

10

WXYZ

U,V=0,1

00 01 11 10

00

01 1

11 1

10

WXYZ

U,V=1,1

VXZUZYVUWXZUZYXF

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Minimal SOP and POS w/ Don’t care• Could lead to different

function if the same x is treated as 1 in SOP but as 0 in POS.

• Even though the final Boolean functions could be different, but for those ones with non-don’t care square, correct functionality was maintained.

0 10 x 11 0 x

A B

F = F = Ā (SOP)Ā (SOP)F = F = B (POS)B (POS)F = F = Ā (POS)Ā (POS)

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Use Karnaugh Map in BB5 5 or BB66

• It is getting hard and complicated • Eye-ball simplification on several

submaps is error-prone, leading to sub-optimal results

• Do we have a better algorithm?– Quine-McCluskey Method

Lec7 Intro to Computer Engineering by Hsien-Hsin Sean Lee Georgia Tech -- Karnaugh Map

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