Plotting functions not in canonical form
• Plot the function f(a, b, c) = a + bc
a c 00 01 11 10 0 1 1 1 1 1 1
ab
ab
c 00 01 11 10 0 0 2 6 4 1 1 3 7 5
b The squares are numbered – derive the canonical form
5-variable K-maps - alternative 0
00 00 01 11 10
0 4
01
1 5
11
3 7
10
2 6
10
18 22
11
19 23
01
17 21
00
16 20
00 01 11 10
12
8
13
9
15
11
14
10
30
26
31
27
29
25
28
24
1
6-variable K-maps - alternative
00 00 01 11 10 10 11 01 00 01 11 10
0 4 12 8 40 1 5 13 9 41 3 7 15 11 43 2 6 14 10 42
10 11 01 00
18 22 30 26 62 58 54 50 19 23 31 27 63 59 55 51 17 21 29 25 61 57 53 49 16 20 28 24 60 56 52 48
01 00 01
11 10
10 11 01 00 11
44
36 32
45
37 33
47
39 35
46
38 34
00
10
00 01 11 10
10 11 01 00
Simplifying functions using K-maps
• Why is simplification possible
– Logically adjacent minterms are physically adjacent on the K-map – Adjacent minterms can be combined by eliminating the common variable
• abc and ābc are adjacent • abc + ābc = bc variable a eliminated
• Done by drawing on the map a ring around the terms that can be combined
Simplifying functions using K-maps
Simplifying functions using K-maps
Simplifying functions using K-maps
• Definition of terms
– Implicant product term that can be used to cover minterms – Prime implicant implicant not covered by any other implicant – Essential prime implicant a prime implicant that covers at least one minterm not covered by any other prime implicant – Cover set of prime implicants that cover each minterm of the function
• Minimizing a function finding the minimum cover
Simplifying functions using K-maps
• Definition of terms
– Implicants:
Simplifying functions using K-maps
• Definition of terms
– Prime implicants: only B and AC – Essential prime implicants: B and AC – Cover: { B, AC }
Simplifying functions using K-maps
• Definition of terms
– Implicate sum term that can be used to cover maxterms (0’s on the K-map) – Prime implicate implicate not covered by any other implicate – Essential prime implicate a prime implicate that covers at least one maxterm not covered by any other prime implicate – Cover set of prime implicates that cover each maxterm of the function
Simplifying functions using K-maps
• Algorithm 1:
– Fast and easy, not optimal
Simplifying functions using K-maps
• Algorithm 2:
– More work than the first – Can give better results, because all prime implicants are considered – Still not optimal
Simplifying functions using K-maps
• Algorithm 2:
1: Identify all PIs
Simplifying functions using K-maps
• Algorithm 2:
2: Identify EPIs
Simplifying functions using K-maps
• Algorithm 2:
3: Select cover
• Tabular • Systematic • Can handle a large number of variables • Can be used for functions with more than one output
The Quine-McCluskey minimization method
The Q-M minimization method
The Q-M minimization method
The Q-M minimization method
The Q-M minimization method
– Combine minterms from List 1 into pairs in List 2
• Take pairs from adjacent groups only, that differ in 1 bit
– Combine entries from List 2 into pairs in List 3
The Q-M minimization method
The Q-M minimization method
The Q-M minimization method
The Q-M minimization method