ece2030 introduction to computer engineering lecture 7: simplification using k-map prof. hsien-hsin...
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ECE2030 Introduction to Computer Engineering
Lecture 7: Simplification using K-map
Prof. Hsien-Hsin Sean LeeProf. Hsien-Hsin Sean Lee
School of Electrical and Computer EngineeringSchool of Electrical and Computer Engineering
Georgia 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
3
Gray Code Construction
01
01
10
00
11
0001111010110100
00001111
000001011010110111101100
100101111110010011001000
00000000
11111111
4
Gray CodeBinary Encoding Gray Code Encoding
Decimal b3 b2 b1 b0 g3 g2 G1 g0
0 0 0 0 0 0 0 0 0
1 0 0 0 1 0 0 0 1
2 0 0 1 0 0 0 1 1
3 0 0 1 1 0 0 1 0
4 0 1 0 0 0 1 1 0
5 0 1 0 1 0 1 1 1
6 0 1 1 0 0 1 0 1
7 0 1 1 1 0 1 0 0
8 1 0 0 0 1 1 0 0
9 1 0 0 1 1 1 0 1
10 1 0 1 0 1 1 1 1
11 1 0 1 1 1 1 1 0
12 1 1 0 0 1 0 1 0
13 1 1 0 1 1 0 1 1
14 1 1 1 0 1 0 0 1
15 1 1 1 1 1 0 0 0
0b e wherbbg 41iii
5
Rotary Position Sensor
6
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
8
Karnaugh Map
Adjacent columns or rows allow grouping of minterms (maxterms) for simplification
9
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 Prime ImplicantImplicant.
– 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 Prime Implicant (PI)(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
12
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
or
CBABBAF
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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
or
DCABCDABCCBF
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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
or
DCAABCBDDBF
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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
22
Don’t Care Example BCD to Gray Code
BCD-coded input Gray Code Output
Decimal b3 b2 b1 b0 g3 g2 g1 g0
0 0 0 0 0 0 0 0 0
1 0 0 0 1 0 0 0 1
2 0 0 1 0 0 0 1 1
3 0 0 1 1 0 0 1 0
4 0 1 0 0 0 1 1 0
5 0 1 0 1 0 1 1 1
6 0 1 1 0 0 1 0 1
7 0 1 1 1 0 1 0 0
8 1 0 0 0 1 1 0 0
9 1 0 0 1 1 1 0 1
10 1 0 1 0 XX XX XX XX
11 1 0 1 1 XX XX XX XX
12 1 1 0 0 XX XX XX XX
13 1 1 0 1 XX XX XX XX
14 1 1 1 0 XX XX XX XX
15 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 Output
Decimal b3 b2 b1 b0 g3 g2 g1 g0
0 0 0 0 0 0 0 0 0
1 0 0 0 1 0 0 0 1
2 0 0 1 0 0 0 1 1
3 0 0 1 1 0 0 1 0
4 0 1 0 0 0 1 1 0
5 0 1 0 1 0 1 1 1
6 0 1 1 0 0 1 0 1
7 0 1 1 1 0 1 0 0
8 1 0 0 0 1 1 0 0
9 1 0 0 1 1 1 0 1
10 1 0 1 0 11 11 11 11
11 1 0 1 1 11 11 11 00
12 1 1 0 0 11 00 11 00
13 1 1 0 1 11 00 11 11
14 1 1 1 0 11 00 00 11
15 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)
25
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
28
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
29
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
31
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 1
0 x 11 0 x
A B
F = F = Ā (SOP)Ā (SOP)F = F = B (POS)B (POS)
F = F = Ā (POS)Ā (POS)
32
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