Combinational LogicPart 2:
Karnaugh maps (quick)
Sum of Minterms Implementation
•OR all of the minterms of truthtable for which the function
value is 1
F = m0 + m2 + m5 + m7
F = X’Y’Z’ + X’YZ’+
XY’Z + XYZ
2
Sum of Products Implementation
•Simplifying sum-of-minterms canyield a sum of products
•Difference is that each term neednot have all variables
•Resulting gates
•ANDs and one OR
F = Y’ + X’YZ’ + XY
3
Two-Level Implementation
•Sum of products has 2 levels of gates
Fig 2-6
4
More Levels of Gates?
•What’s best?
♦Hard to answer
♦More gate delays (more on this later)
♦But maybe we only have 2-input gates
5
Product of Maxterms Implementation
•Can express F as AND of Maxterms forall rows that should evaluate to 0
or
This makes one Maxterm fail each time F should be 0
6
Product of Sums Implementation
•ORs followed by AND
7
Karnaugh Map
•Graphical depiction of truth table
•A box for each minterm
♦So 2 variables, 4 boxes
♦3 variable, 8 boxes
♦And so on
•Useful for simplification
♦by inspection
♦Algebraic manipulation harder
8
K-Map from Truth TableExamples
•There are implied 0s in empty boxes
9
Function from K-Map
•Can generate function from K-map
Simplifies to X + Y (in a moment)
10
In Practice:
•Karnaugh maps were mildly usefulwhen people did simplification
•Computers now do it!
•We’ll cover Karnaugh maps as a wayfor you to gain insight,
♦not as real tool
11
Three-Variable Map
•Eight minterms
•Look at encoding of columns androws
12
Simplification
•Adjacent squares (horizontally orvertically) are minterms that varyby single variable
•Draw rectangles on map to simplifyfunction
•Illustration next
13
Example
instead of
14
Adjacency is cylindrical
•Note that wraps from left edgeto right edge.
15
Covering 4 Squares
is
16
Another Example
•Help me solve this one
17
In General
•One box -> 3 literals
•Rectangle of 2 boxes -> 2 literals
•Rectangle of 4 boxes -> 1 literal
•Rectangle of 8 boxes -> Logic 1 (on3-variable map)
♦Covers all minterms
18
Slight Variation
•Overlap is OK.
•No need to use full m5-- waste of input
19
4-variable map
•At limit of K-map
20
Also Wraps (toroidal topology)
21
Systematic Simplification
A Prime Implicant is a product term obtained by combiningthe maximum possible number of adjacent squares in the mapinto a rectangle with the number of squares a power of 2.
A prime implicant is called an Essential Prime Implicant if it isthe only prime implicant that covers (includes) one or moreminterms.
Prime Implicants and Essential Prime Implicants can bedetermined by inspection of a K-Map.
A set of prime implicants "covers all minterms" if, for eachminterm of the function, at least one prime implicant in theset of prime implicants includes the minterm.
Chapter 2 - Part 2 22
D
B
C
B
1
1
1
1
1
1
B
D
A
1
1
1
1
1
Example of Prime Implicants
Find ALL Prime Implicants
B’D’ and BD are ESSENTIAL Prime Implicants
C
BD
CD
BD
Minterms covered by single prime implicant
D
B
1
1
1
1
1
1
B
C
D
A
1
1
1
1
1
AD
B
A
Chapter 2 - Part 2 23
Prime Implicant Practice
Find all prime implicants for:
Chapter 2 - Part 2 24
Prime Implicant Practice
Find all prime implicants for:
1
1
1
1
1
1
B
D
A
1
1
1
C
1
1
A
C
B
D
B
Chapter 2 - Part 2 25
Algorithm to Find An OptimalExpression for A Function
Find all prime implicants.
Include all essential prime implicants in thesolution
Select a minimum cost set of non-essentialprime implicants to cover all minterms not yetcovered.
The solution consists of all essential prime andthe selected minimum cost set of non-essentialprime implicants
minimum cost
selected minimum cost
Chapter 2 - Part 2 26
The Selection Rule
Obtaining a good simplified solution:Use the Selection Rule
Chapter 2 - Part 2 27
Prime Implicant Selection Rule
Minimize the overlap among primeimplicants as much as possible.
In the solution, make sure that eachprime implicant selected includes atleast one minterm not included in anyother prime implicant selected.
Chapter 2 - Part 2 28
Selection Rule Example
Simplify F(A, B, C, D) given on the K-map.
1
1
1
1
1
1
1
B
D
A
C
1
1
1
1
1
1
1
1
1
B
D
A
C
1
1
Essential
Minterms covered by essential prime implicants
Selected
Chapter 2 - Part 2 29
Don’t Care
•So far have dealt with functionsthat were always either 0 or 1
•Sometimes we have someconditions where we don’t carewhat result is
•Example: dealing with BCD
♦Only care about first 10
30
Mark With an X
•In a K-map, mark don’t care with X
•Simpler implementations
•Can select an X either as 1 or 0
31
Example
or
What would we have if Xs were 0?
32
Selection Rule 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
1
1
x
x
x
x
x
1
B
D
A
C
1
1
Essential
Chapter 2 - Part 2 33
Product of Sums Example
Find the optimum POS solution:
•Hint: Use and complement it to get theresult.
F
34