Digital Logic CPE231

1. Digital Logic CPE231 Digital Fundamentals, Ninth Edition By Floyd Instructor: Eng. Tuqa Manasrah

2. Ch4: Boolean Algebra and Logic Simplification • Apply the basic laws and rules of Boolean algebra • Apply DeMorgan's theorems to Boolean expressions • Describe gate networks with Boolean expressions • Evaluate Boolean expressions • Simplify expressions by using the laws and rules of Boolean algebra • Convert any Boolean expression into a sum-of-products (SOP) form • Convert any Boolean expression into a product-of-sums (POS) form • Use a Karnaugh map to simplify Boolean expressions • Use a Karnaugh map to simplify truth table functions • Utilize "don't care" conditions to simplify logic functions Digital Logic - Spring 08

3. Term Boolean function Laws and rules of Boolean algebra • Boolean algebra is the mathematics of digital systems. • Boolean function: described by Boolean equation. • Boolean equation: express logical relationship between binary variables Digital Logic - Spring 08

4. Basic laws of Boolean algebra • Commutative law: the order of variables makes no difference • X + Y = Y + X • XY = YX • Associative law: the result is the same regardless of the grouping of the variables. • X + (Y + Z) = (X + Y) + Z • X(YZ) = (XY)Z • Distributive law: express the process of factoring in which a common variable is factored out the product term. • XY + XZ = X( Y + Z) • X + YZ = (X + Y)(X + Z) Digital Logic - Spring 08

5. Rules of Boolean algebra Digital Logic - Spring 08

6. Continue.. • X + XY = X = X(1 + Y) …distributive law = X . 1 … Rule 2 = X … Rule 4 • X + X’Y = X +Y = (X + XY) + X’Y … Rule 10 = XX + XY + X’y … Rule 7 = XX + XY + XX’ + X’Y …Rule 8 = (X + X’) (X + Y)…Factoring = 1 . (X+Y) … Rule 6 = X + Y … Rule 4 • (X + Y)(X + Z) = X + YZ Do it … Digital Logic - Spring 08

7. Demorgan’s Theorem • State the equivalency of NAND & negative-OR: • State the equivalency of NOR & negative-AND: • Prove using truth table. Digital Logic - Spring 08

8. Examples • Apply DeMorgans Theorem: • (AB’+A’B)’ • (XYZ)’ • (X’+Y’+Z’)’ • ((A+B+C)D)’ • ((ABC)’+D+E)’ • (A’B(C+D’)+E)’ Digital Logic - Spring 08

9. Boolean Analysis of Logic Circuits • Each logic function can be expressed with three methods: • Boolean expression • Logic circuit • Truth table • Having a function expressed with one method, you can derive the remaining. Digital Logic - Spring 08

10. Example • The logic circuit shown below implement two logic functions: F1 and F2. • Analyze the circuit by deriving the Boolean expressions of F1 and F2. • Once the Boolean expression for a given logic circuit has been determined, a truth table that shows the output for all values of the input variables can be developed. Derive the truth table of the logic functions F1, F2. Digital Logic - Spring 08

11. Simplification using Boolean algebra. • The goal is to reduce a given expression to its simplest form, or to a more convenient one for implementation. • Use laws, rules and theorems of Boolean algebra to manipulate and simplify an expression Digital Logic - Spring 08

12. Examples • A+AB+AB’C = A • AB’ + A(B+C)’ + B(B+C)’ = AB’ • ABC’+A’B’C+A’BC+A’B’C’ = ABC’+A’C+A’B’ • (AB)’+(AC)’+A’B’C’ = A’ + B’+ C’ • AB’C(BD+CDE)+AC = A(C’ + B’DE) Digital Logic - Spring 08

13. Standard forms for Boolean algebra • Standardization makes the evaluation, simplification and implementation of Boolean expressions much more systematic and easier. • All Boolean expressions, regardless of their form, can be converted into either of two forms: sum-of-product, product-of-sum. Digital Logic - Spring 08

14. Sum-of-Products (SOP) form • A product term consists of the product of literals (variables or their complements). • When two or more products are summed the results is an SOP. • E.g. AB + ABC , A’B + A’BC’+ AC • SOP expression can contain a single variable term. • In SOP a single overbar cannot extend over more than one variable ( ) Digital Logic - Spring 08

15. Continue… • Implementation: • AND/OR • NAND/NAND • Conversion from general expressions: • Apply Boolean algebra techniques. • E.g. AB + B(CD+EF), ((A+B)’+C)’ • The domain of a Boolean expression is the set of variables contained in the expression either complemented or not. • SOP may not include the domain of variable in all products. (A’BC + ABD’ + AB’CD) • A standard SOP expression is a one in which all the variables in the domain appear in each product term. (A’BC + AB’C + ABC’ + A’B’C’) • A non standard SOP can be converted to the standard one by using the Boolean algebra rule (x + x’ =1). • Convert wx’y + x’yz’ + wxy’ to the standard sop form. Digital Logic - Spring 08

16. The Product-of-Sum (POS) from • A sum term consists of the sum of literals. • When two or more sum terms are multiplied the results is an POS. • E.g. (A+B)(A+B+C), (A’+B)(A’+B+C’)(A+C) • POS expression can contain a single variable term. • In POS a single overbar cannot extend over more than one variable ( ) Digital Logic - Spring 08

17. Continue… • Implementation: • OR/AND • NOR/NOR • POS may not include the domain of variable in all terms. (A’+B+C)(A+B+D’)(B’+C+D) • A standard SOP expression is a one in which all the variables in the domain appear in each product term. (A’+B+C)(A+B+C’)(A+B+C’)(A’+B’+C’) • A non standard POS expression can be converted to the standard form by using the Boolean algebra rule 8 (xx’=0) and rule12((X+Y)(X+Z)= X+YZ). • Convert (a+b’)(b+c) to the standard POS form. Digital Logic - Spring 08

18. Boolean expressions and Truth Table • A SOP expression is equal to 1 if at least one of the product terms is equal to 1. • A POS expression is equal to 0 if at least one of the sum terms is equal to 0. • E.g. determine the truth table for the expressions: • F1= A’BC’ + AB’C • F2= (A+B’+C)(A+B+C’)(A’+B’+C’) • Determine the standard expression from the truth table: • SOP • List the binary values of the inputs for which the output is 1. • Convert the binary values to the corresponding product term.(1  corresponding variable, 0corresponding variable complement) • POS • List the binary values of the inputs for which the output is 0. • Convert the binary values to the corresponding sum term.(0  corresponding variable, 1corresponding variable complement) Digital Logic - Spring 08

19. Definition: Minterm • Product term in which all variables appear once (complemented or not). • For the variables X, Y and Z example minterms: X’Y’Z’, X’Y’Z, X’YZ’, …., XYZ • Each minterm represents exactly one combination of the binary variables in a truth table. • For n variables, there will be 2n minterms. • Minterms are labeled from minterm 0, to minterm 2n-1 m0 , m1 , m2 , … , m2n-2 , m2n-1 • For n = 3, we have m0 , m1 , m2 , m3 , m4 , m5 , m6 , m7 Digital Logic - Spring 08

20. Definition: Maxterm • Sum term in which all variables appear once (complemented or not) • For the variables X, Y and Z the maxterms are: X+Y+Z , X+Y+Z’ …. , X’+Y’+Z’ • Minterms and maxterms with the same subscripts are complements: • Example: Digital Logic - Spring 08

21. Standard Form of F:Sum of Minterms • The same as standard SOP • OR all of the minterms of truth table for which the function value is 1 • F = m0 + m2 + m5 + m7 Digital Logic - Spring 08

22. Complement of F • Not surprisingly, just sum of the other minterms • In this case F’ = m1 + m3 + m4 + m6 Digital Logic - Spring 08

23. Product of Maxterms • The same as standard POS • F = m0 + m2 + m5 + m7 • Can express F as AND of all Maxterms of rows that should evaluate to 0 or Digital Logic - Spring 08

24. The Karnaugh Map • A systematic method for simplifying Boolean expressions. • If properly used, will produce the simplest SOP or POS expression. • Graphical representation of truth table. • A box for each minterm • So 2 variables, 4 boxes • 3 variable, 8 boxes • And so on Digital Logic - Spring 08

25. K-Map from Truth TableExamples • There are implied 0s in empty boxes Digital Logic - Spring 08

26. Function from K-Map • Can generate function from K-map Simplifies to X + Y (in a moment) Digital Logic - Spring 08

27. A 3-variable K-Map • Eight minterms • Look at encoding of columns and rows Digital Logic - Spring 08

28. A 4-variabel K-Map • At limit of K-map Digital Logic - Spring 08

29. Cell Adjacency • Cells are arranged so that there is only a single variable change between adjacent cells. • Each cell is adjacent to the cells that are immediately next to it on any of its four sides. • A cell is not adjacent to the cells that are diagonally touch any of its corners. • Adjacency is cylindrical. • Note that Z’ wraps from left edge to right edge.  Digital Logic - Spring 08

30. Also Wraps (toroidal topology) Digital Logic - Spring 08

31. K-Map SOP minimization • A minimized SOP expression contains the fewest possible terms with the fewest possible variables per term. And can be implemented with fewer logic gates than a standard expression. • Mapping • From truth table • From standard SOP (Sum of Minterms) • From a nonstandard SOP • simplification Digital Logic - Spring 08

32. Mapping Examples • Map the following standard SOP expressions: • A’B’C + A’BC’ + ABC’ + ABC • A’BC + AB’C +AB’C’ • A’B’CD + A’BC’D’+ ABC’D + A’B’C’D + AB’CD’ • A’BCD’ + ABCD’ + ABC’D’ + ABCD • Map the following nonstandard SOP expressions: • X’ + XY’ + XYZ’ • YZ + X’Z’ • X’Y’ + WX’ + WXY’ + WX’YZ’ + W’X’Y’Z + WX’YZ • W + Y’Z + WYZ’ + W’XYZ’ Digital Logic - Spring 08

33. Simplification of SOP expressions • The minimum SOP expression is obtained by grouping the 1’s and determining the minimum expression from the map. • Rules of grouping: • A group must contain a power of 2 cells, i.e. either 1,2,4,8, or 16. • Each cell must be adjacent to one or more cells in the group, but all cells not need to be adjacent. • Include the largest possible number of 1’s in a group. • Each 1 on the map must be included in at least one group. Digital Logic - Spring 08

34. Continue.. • Determining the minimum product term of each group: • For a 3-variable map • One cell -> 3 literals • Rectangle of 2 cell s -> 2 literals • Rectangle of 4 cell s -> 1 literal • Rectangle of 8 cell s -> Logic 1 • For a 4-variable map • One cell -> 4 literals • Rectangle of 2 cell s -> 3 literals • Rectangle of 4 cell s -> 2 literal • Rectangle of 8 cell s -> 1 literal • Rectangle of 16 cell s -> Logic 1 Digital Logic - Spring 08

35. Examples instead of Solve this one Digital Logic - Spring 08

36. Overlap is OK. • No need to use full m5-- waste of input Digital Logic - Spring 08

37. K-map POS minimization • A 0 is placed in k-map for each sum term. • E.g. for A+ B’ + C, a 0 is placed in cell 010. • Cell with no zeros are for which the expression is 1. • Mapping procedure: • Determine the binary value for each sum term in standard SOP. • In the k-map place zeros in the corresponding cells. Digital Logic - Spring 08

38. Simplification of POS expressions • Group 0s instead of 1s. • Follow the same rules as grouping 1s. • Conversion between POS and SOP for the sake of implementation with fewer gates. • Use the k-map to optimize a SOP expression: (W+X’+Y+Z’)(W’+X+Y’+Z’)(W’+X’+Y’+Z)(W’+X’+Z’) Digital Logic - Spring 08

39. Examples: • Map the following expressions on a k-map: • A’BC + AB’C + AB’C’ • A’BCD’+ABCD’+ABC’D’+ABCD • BC+A’C’ • A+ C’D+ACD’+A’BCD’ • (A+B’+C’+D)(A+B+C+D’)(A+B+C+D)(A’+B+C’+D) • Use the k-map to minimize the following expressions: • XY’Z+XYZ’+X’YZ+X’YZ’+XY’Z’+XYZ • W’X’Y’Z’+WX’YZ+WX’Y’Z+W’YZ+WX’Y’Z’ • (X+Y’+Z)(X+Y’+Z’)(X’+Y’+Z)(X’+Y+Z) Digital Logic - Spring 08

40. Don’t care conditions • Some input combinations are not allowed never occurred treated as don’t care. • For each don’t care term, X is placed in the cell. • The Xs can be treated as 1s in case of grouping. Digital Logic - Spring 08

41. Examples • Optimize the expression with don’t care conditions: • F(a,b,c)=∑m(1,2,4), d(a,b,c)= ∑m(0,3,6,7) • F(a,b,c,d)=∑m(0,1,7,13,15), d(a,b,c,d)= ∑m(2,6,8,9,10) • F(a,b,c,d)=∑m(2,4,9,12,15), d(a,b,c,d)= ∑m(3,5,6,13) • F(w,x,y,z)=∏M(3,11,13,15), d(w,x,y,z)= ∑m(0,2,5,8,10,14) Digital Logic - Spring 08