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 1 1 1 1 1 1 3 7 5 b The squares are numbered – derive the canonical form
5-variable K-maps - alternative 00 11 10 01 00 11 10 01 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 00 01 11 10 00 01 11 10 1
6-variable K-maps - alternative 00 01 40 41 43 44 45 36 37 39 32 33 42 47 46 38 35 34 00 01 11 10 1 3 4 5 12 13 15 8 9 2 7 6 14 18 19 17 22 23 30 31 29 26 27 16 21 20 28 25 24 62 63 61 58 59 54 55 53 50 51 60 57 56 52 49 48 10 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