plotting functions not in canonical form

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