07 KM

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Karnaugh Maps
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What are Karnaugh Maps?

• A simpler way to handle most (but not all) jobs
  of manipulating logic functions.

Hooray !!
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Karnaugh Map Advantages

• Minimization can be done more systematically
• Much simpler to find minimum solutions
• Easier to see what is happening (graphical)
• Almost always used instead
• of boolean minimization.

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Gray Codes

• Gray code is a binary value encoding in which
  adjacent values only differ by one bit

2-bit Gray Code
00
01
11
10
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Truth Table Adjacencies
These are adjacent in a gray code sense -
they differ by 1 bit We can apply XY XY
X AB AB A(BB) A(1) A
F A
F B
Same idea AB AB B
Key idea Gray code adjacency allows use of
simplification theorems
Problem Physical adjacency in truth table does
not indicate gray code adjacency
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2-Variable Karnaugh Map
A B F
0 0
0 1
1 0
1 1
A1, B0
A0, B0
A1, B1
A0, B1
A different way to draw a truth table by folding
it
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Karnaugh Map

• In a K-map, physical adjacency does imply gray
  code adjacency

F AB AB B
F AB AB A
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2-Variable Karnaugh Map
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2-Variable Karnaugh Map
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2-Variable Karnaugh Map
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2-Variable Karnaugh Map
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2-Variable Karnaugh Map
F AB AB A
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2-Variable Karnaugh Map
A 0
F A
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Another Example
F AB AB AB (AB AB) (AB AB)
A B
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Another Example
A 1
B 1
F A B
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Yet Another Example
F 1
Groups of more than two 1s can be combined
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3-Variable Karnaugh Map Showing Minterm Locations
Note the order of the B C variables 0 0
0 1 1 1 1 0
ABC 101
ABC 010
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3-Variable Karnaugh Map Showing Minterm Locations
Note the order of the B C variables 0 0
0 1 1 1 1 0
ABC 101
ABC 010
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Adjacencies

• Adjacent squares differ by exactly one variable

There is wrap-around top and bottom rows are
adjacent
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Truth Table to Karnaugh Map
0
0
1
0
1
1
0
1
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Minimization Example
ABCABC AC
ABCABC AB
F AB AC
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Another Example
ABCABC AC
ABCABC AC
F AC AC A ? C
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Minterm Expansion to K-Map
F ?m( 1, 3, 4, 6 )
0
1
1
0
1
0
0
1
Minterms are the 1s, everything else is 0
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Maxterm Expansion to KMap
F ?M( 0, 2, 5, 7 )
0
1
1
0
1
0
0
1
Maxterms are the 0s, everything else is 1
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Yet Another Example
2n 1s can be circled at a time 1, 2, 4, 8,
OK 3 not OK
ABCABCABCABC B
ABCABC AC
F B AC
The larger the group of 1s the simpler the
resulting product term
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Boolean Algebra to Karnaugh Map

• Plot abc bc a

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Boolean Algebra to Karnaugh Map

• Plot abc bc a

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Boolean Algebra to Karnaugh Map

• Plot abc bc a

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Boolean Algebra to Karnaugh Map

• Plot abc bc a

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Boolean Algebra to Karnaugh Map

• Plot abc bc a

Remaining spaces are 0
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Boolean Algebra to Karnaugh Map
Now minimize . . .

• F BC BC A

This is a simpler equation thanwe started
with. Do you see how we obtained it?
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Mapping Sum of Product Terms

• The 3-variable map has 12 possible groups of 2
  spaces
• These become terms with 2 literals

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Mapping Sum of Product Terms

• The 3-variable map has 6 possible groups of 4
  spaces
• These become terms with 1 literal

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4-Variable Karnaugh Map
ABC
D
ABC
F ABC ABC D
Note the rowand column orderings. Required for
adjacency
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Find a POS Solution
BC
CD
F CD BC ABCD F (CD)(BC)(ABCD
)
ABCD
Find solutions to groups of 0s to find F Invert
to get F then use DeMorgans
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Dealing With Dont Cares
F ?m(1, 3, 7) ?d(0, 5)
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Dealing With Dont Cares
F ?m(1, 3, 7) ?d(0, 5)
ABCABCABCABC C
F C
Circle the xs that help get bigger groups of 1s
(or 0s if POS) Dont circle the xs that dont
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Minimal K-Map Solutions

• Some Terminology
• and
• An Algorithm to Find Them

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Prime Implicants

• A group of one or more 1s which are adjacent and
  can be combined on a Karnaugh Map is called an
  implicant.
• The biggest group of 1s which can be circled to
  cover a given 1 is called a prime implicant.
• They are the only implicants we care about.

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Prime Implicants
Prime Implicants
Non-prime Implicants
Are there any additional prime implicants in the
map that are not shown above?
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All The Prime Implicants
Prime Implicants
When looking for a minimal solution only
circle prime implicants A minimal solution will
never contain non-prime implicants
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Essential Prime Implicants
Not all prime implicants are required
A prime implicant which is the only cover of some
1 is essential a minimal solution requires it.
Essential Prime Implicants
Non-essential Prime Implicants
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A Minimal Solution Example
Not required
F AB BC AD
Minimum
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Another Example
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Another Example
AB is not requiredEvery one one of
itslocations is covered by multiple
implicants After choosing essentials, everything
is covered
F AD BCD BD
Minimum
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Finding the Minimum Sum of Products

• 1. Find each essential prime implicant and
  include it in the solution.
• 2. Determine if any minterms are not yet covered.
  
• 3. Find the minimal of remaining prime
  implicants which finish the cover.

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Yet Another Example(Use of non-essential primes)
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Yet Another Example(Use of non-essential primes)
AD
CD
AC
Essentials AD and ADNon-essentials AC and
CD Solution AD AD AC
or AD AD CD
AD
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K-Map Solution Summary

• Identify prime implicants
• Add essentials to solution
• Find a minimum non-essentials required to cover
  rest of map

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5- and 6-Variable K-Maps
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5-Variable Karnaugh Map
This is the A0 plane
This is the A1 plane
The planes are adjacent to one another (one is
above the other in 3D)
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Some Implicants in a 5-Variable KMap
DE
ABCDE
A0
A1
ABCD
ABCD
BCDE
Some of these are not prime
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5-Variable KMap Example
Find the minimum sum-of-products for F ? m
(0,1,4,5,11,14,15,16,17,20,21,30,31)
A0
A1
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5-Variable KMap Example
Find the minimum sum-of-products for F ? m
(0,1,4,5,11,14,15,16,17,20,21,30,31)
A0
A1
F BD BCD ABDE
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6-Variable Karnaugh Map
AB00
AB10
AB01
AB11
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AB00
AB10
AB01
AB11
Solution ACD CDEF
CDEF
ACD
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KMap Summary

• A Kmap is simply a folded truth table
• where physical adjacency implies logical
  adjacency
• KMaps are most commonly used hand method for
  logic minimization
• KMaps have other uses for visualizing Boolean
  equations
• you may see some later.