plotting functions not in canonical form
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Plotting functions not in canonical form. Plot the function f(a, b, c) = a + bc ab a ab c 00 01 11 10 c 00 01 11 10 0 1 1 0 0 2 6 4 - PowerPoint PPT PresentationTRANSCRIPT
Plotting functions not in canonical form
• Plot the function f(a, b, c) = a + bc
ab a abc 00 01 11 10 c 00 01 11 10 0 1 1 0 0 2 6 4 1 1 1 1 1 1 3 7 5
bThe squares are numbered – derive the
canonical form
5-variable K-maps - alternative
001110 01
00
01
11
10
00
01
11
10
00 11 10010 1 3
4 5
12 13 15
8 9
2
7 6
14
11 10
18 19 17
22 23
30 31 29
26 27
16
21 20
28
25 24
0
1
6-variable K-maps - alternative
40 41 43
44 45
36 37 39
32 33
42
47 46
38
35 34
00
01
11
10
00 11 10010 1 3
4 5
12 13 15
8 9
2
7 6
14
11 10
001110 01
00
01
11
10
18 19 17
22 23
30 31 29
26 27
16
21 20
28
25 24
10
11
01
0000 11 1001 001110 01
10
11
01
00
62 63 61
58 59
54 55 53
50 51
60
57 56
52
49 48
00
10
01
11
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
The Quine-McCluskey minimization method
• Tabular• Systematic• Can handle a large number of variables• Can be used for functions with more than one
output
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