ECE2030 Introduction to Computer Engineering Lecture 7: Simplification using K-map Prof. Hsien-Hsin Sean Lee Prof. Hsien-Hsin Sean Lee School of Electrical and Computer School of Electrical and Computer Engineering Engineering Georgia Tech Georgia Tech
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Lec7 Intro to Computer Engineering by Hsien-Hsin Sean Lee Georgia Tech -- Karnaugh Map
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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
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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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
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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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
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
essential PIs for 0’s• Identify the non-
essential PIs to cover 0’s
• 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
essential PIs for 0’s• Identify the non-
essential PIs to cover 0’s
• 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