Lecture (09) Karnaugh Maps 2

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function f(x,x2,,xn) is said to cover another function g(x,x2,,xn), if f assumes the value whenever g does.

1 Lecture (09) Karnaugh Maps 2 By: Dr. Ahmed ElShafee ١ Dr. Ahmed ElShafee, ACU : Spring 207, Logic Design I Determination of Minimum Expansions Using Essential Prime Implicants Cover: A switching function f(x,x2,,xn) is said to cover another function g(x,x2,,xn), if f assumes the value whenever g does. Implicant : Given a function F of n variables, a product term P is an implicant of F iff for every combination of values of the n variables for which P=, F is also equal.that is, P= implies F=. ٢ Dr. Ahmed ElShafee, ACU : Spring 207, Logic Design I

2 Prime Implicant: A prime implicant of a function F is a product term implicart which is no longer an implicant if any literal is deleted from it. Essential Prime Implicant: If a minterm is covered by only one prime implicant, then that prime implicant is called an essential prime implicant. ٣ Dr. Ahmed ElShafee, ACU : Spring 207, Logic Design I On a Karnaugh Map Any single or any group of s (2 k s, k=0,,2, ) which can be combined together on a map of the function F represents a product term which is called an implicant of F. ٤ Dr. Ahmed ElShafee, ACU : Spring 207, Logic Design I

3 A product term implicant is called a prime implicant if it cannot be combined with another term to eliminate a variable. ٥ Dr. Ahmed ElShafee, ACU : Spring 207, Logic Design I If a minterm is covered by only one prime implicant, then that prime implicant is called an essential prime implicant. ٦ Dr. Ahmed ElShafee, ACU : Spring 207, Logic Design I

4 Examples f=wx+yz, g=wxy g= (w=,x=,y=0) implies f=.+0.z=, f covers g. g is a product term, g is an implicant of f. g is not a prime implicant. The literal y is deleted from wxy, the resulting term wx is also an implicant of f. ٧ Dr. Ahmed ElShafee, ACU : Spring 207, Logic Design I f=wx+yz, h=wx is a prime implicant. The deletion of any literal (w or x) results a new product (x or w) which is not covered by f. [w= does not imply f= (w=,x=0,y=0,z=0 imply f=0)] ٨ Dr. Ahmed ElShafee, ACU : Spring 207, Logic Design I

5 Example: ٩ Dr. Ahmed ElShafee, ACU : Spring 207, Logic Design I ab cd ١٠ Dr. Ahmed ElShafee, ACU : Spring 207, Logic Design I

6 ab cd ١١ Dr. Ahmed ElShafee, ACU : Spring 207, Logic Design I ab cd ١٢ Dr. Ahmed ElShafee, ACU : Spring 207, Logic Design I

7 The minimum sum of products expression for a function consists of some (but not necessarily all) of the prime implicants of a function. A sum of products expression consisting a term which is not a prime implicant cannot be minimum. The essential prime implicant must be included in the minimum sum of products. In order to find the minimum sum of products from a map, we must find a minimum number of prime implicants which cover all of the s on the map. ١٣ Dr. Ahmed ElShafee, ACU : Spring 207, Logic Design I Example: cd 00 ab ١٤ Dr. Ahmed ElShafee, ACU : Spring 207, Logic Design I

8 ١٥ Dr. Ahmed ElShafee, ACU : Spring 207, Logic Design I ١٦ Dr. Ahmed ElShafee, ACU : Spring 207, Logic Design I

9 ١٧ Dr. Ahmed ElShafee, ACU : Spring 207, Logic Design I Example cd 00 ab ١٨ Dr. Ahmed ElShafee, ACU : Spring 207, Logic Design I

10 ١٩ Dr. Ahmed ElShafee, ACU : Spring 207, Logic Design I Essential prime implicates: BD,B C, AC ٢٠ Dr. Ahmed ElShafee, ACU : Spring 207, Logic Design I

11 ٢١ Dr. Ahmed ElShafee, ACU : Spring 207, Logic Design I Example Simplify the function f (A, B,C,D) = Σm(0,,2,4,5,7,,5). ٢٢ Dr. Ahmed ElShafee, ACU : Spring 207, Logic Design I

12 f (A, B,C,D) = Σm(0,,2,4,5,7,,5). ab cd ٢٣ Dr. Ahmed ElShafee, ACU : Spring 207, Logic Design I ٢٤ Dr. Ahmed ElShafee, ACU : Spring 207, Logic Design I

13 Example Simplify the function f (A, B,C,D) = Σm(4,5,6,8,9,0,3) + Σd(0,7,5) ٢٥ Dr. Ahmed ElShafee, ACU : Spring 207, Logic Design I f (A, B,C,D) = Σm(4,5,6,8,9,0,3) + Σd(0,7,5) ab cd x x x ٢٦ Dr. Ahmed ElShafee, ACU : Spring 207, Logic Design I

14 ٢٧ Dr. Ahmed ElShafee, ACU : Spring 207, Logic Design I Example Find min min terms expansion ab cd ٢٨ Dr. Ahmed ElShafee, ACU : Spring 207, Logic Design I

15 ab cd ٢٩ Dr. Ahmed ElShafee, ACU : Spring 207, Logic Design I ٣٠ Dr. Ahmed ElShafee, ACU : Spring 207, Logic Design I

Other forms of 5 Variable Karnaugh Maps ٣١

de 00 00 0 0 bc de 00 00 0 0 0 0 0 0 A=0 A=

16 Other forms of 5 Variable Karnaugh Maps ٣١ Dr. Ahmed ElShafee, ACU : Spring 207, Logic Design I Example F(A,B,C,D,E) = Σm(0,,4,5,3,5,20,2,22,23,24,26,28,30,3) bc de bc de A=0 A= ٣٢ Dr. Ahmed ElShafee, ACU : Spring 207, Logic Design I

17 ٣٣ Dr. Ahmed ElShafee, ACU : Spring 207, Logic Design I Example F(A,B,C,D,E) = Σm(0,,3,8,9,4,5,6,7,9,25,27,3) bc de bc de A=0 A= ٣٤ Dr. Ahmed ElShafee, ACU : Spring 207, Logic Design I

18 ٣٥ Dr. Ahmed ElShafee, ACU : Spring 207, Logic Design I Quine McCluskeyMethod Determination of Prime Implicants the function must be given as a sum of minterms. all of the prime implicants of a function are systematically formed by combining minterms To reduce the required number of comparisons, the binary minterms are sorted into groups according to the number of s in each term ٣٦ Dr. Ahmed ElShafee, ACU : Spring 207, Logic Design I

19 ٣٧ Dr. Ahmed ElShafee, ACU : Spring 207, Logic Design I Two terms in any two groups can be combined as they differ in exactly one variable. First, we will compare the term in group 0 with all of the terms in group. Terms 0000 and 000 can be combined to eliminate the fourth variable, which yields 000. Similarly, 0 and 2 combine to form 00 0 (a b d ), and 0 and 8 combine to form 000 (b c d ). The resulting terms are listed in Column II the corresponding decimal numbers differ by a power of 2 (, 2, 4, 8, etc.). A term may be used more than once because X + X = X. ٣٨ Dr. Ahmed ElShafee, ACU : Spring 206, Logic Design

20 Example Find all of the prime implicants of the function f (a,b,c,d) = Σm(0,,2,5,6,7,8,9,0,4) ٣٩ Dr. Ahmed ElShafee, ACU : Spring 207, Logic Design I ٤٠ Dr. Ahmed ElShafee, ACU : Spring 206, Logic Design

21 ٤١ Dr. Ahmed ElShafee, ACU : Spring 206, Logic Design ٤٢ Dr. Ahmed ElShafee, ACU : Spring 206, Logic Design

22 The terms which have not been checked off because they cannot be combined with other terms are called prime implicants. ٤٣ Dr. Ahmed ElShafee, ACU : Spring 206, Logic Design ٤٤ Dr. Ahmed ElShafee, ACU : Spring 207, Logic Design I

23 All prim implicante Minimum form??? ٤٥ Dr. Ahmed ElShafee, ACU : Spring 207, Logic Design I The Prime Implicant Chart The minterms of the function are listed across the top of the chart, and the prime implicants are listed down the side. If a prime implicant covers a given minterm, an X is placed at the intersection of the corresponding row and column. If a minterm is covered by only one prime implicant, then that prime implicant is called an essential prime implicant ٤٦ Dr. Ahmed ElShafee, ACU : Spring 207, Logic Design I

24 ٤٧ Dr. Ahmed ElShafee, ACU : Spring 207, Logic Design I A minimum set of prime implicants must now be chosen to cover the remaining columns ٤٨ Dr. Ahmed ElShafee, ACU : Spring 206, Logic Design

25 A minimum set of prime implicants must now be chosen to cover the remaining columns ٤٩ Dr. Ahmed ElShafee, ACU : Spring 206, Logic Design Example A prime implicant chart which has two or more X s in every column is called a cyclic prime implicant chart. The following function has such a chart ٥٠ Dr. Ahmed ElShafee, ACU : Spring 206, Logic Design

26 Derivation of prime implicants: ٥١ Dr. Ahmed ElShafee, ACU : Spring 206, Logic Design Select P first. ٥٢ Dr. Ahmed ElShafee, ACU : Spring 206, Logic Design

27 Select P2 first. ٥٣ Dr. Ahmed ElShafee, ACU : Spring 207, Logic Design I Simplification of Incompletely Specified Functions ٥٤ Dr. Ahmed ElShafee, ACU : Spring 207, Logic Design I

28 Example Simplify F(A,B,C,D) = Σm(2,3,7,9,,3) + Σ d(,0,5) ٥٥ Dr. Ahmed ElShafee, ACU : Spring 207, Logic Design I F(A,B,C,D) = Σm(2,3,7,9,,3) + Σ d(,0,5) Treat the don t cares (,0,5) as required minterms ٥٦ Dr. Ahmed ElShafee, ACU : Spring 207, Logic Design I

29 F(A,B,C,D) = Σm(2,3,7,9,,3) + Σ d(,0,5) The don t cares are not list at the top of the table ٥٧ Dr. Ahmed ElShafee, ACU : Spring 207, Logic Design I Thanks,.. See you next week (ISA), ٥٨ Dr. Ahmed ElShafee, ACU : Spring 207, Logic Design I

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