1. 2 - 1 Chapter 2 Combinational Systems

2. 2 - 2 Chapter 2 Combinational Systems 2.1 The Design Process for Combinational Systems  Continuing Example(CE) CE1. A system with four inputs, A, B, C, and D, and one output, Z, such that Z = 1 iff three of the inputs are 1. CE2. A single light (that can be on or off) that can be controlled by any one of three switches. One switch is the master on/off switch. If it is down, the lights are off. When the master switch is up, a change in the position of one of the other switches (from up to down or from down to up) will cause the light to change state. CE3. A system to do 1 bit of binary addition. It has three inputs (the 2 bits to be added plus the carry from the next lower order bit) and produces two outputs: a sum bit and a carry to the next higher order position.

3. 2 - 3 2.1 The Design Process for Combinational Systems Chapter 2 Combinational Systems CE4. A display driver; a system that has as its input the code for a decimal digit and produces as its output the signals to drive a seven-segment display, such as those on most digital watches and numeric displays (more later). CE5. A system with nine inputs, representing two 4-bit binary numbers and a ca- rry input, and one 5-bit output, representing the sum. (Each input number can range from 0 to 15; the output can range from 0. to 31.)  Continuing Example(CE)

4. 2 - 4 2.1 The Design Process for Combinational Systems Chapter 2 Combinational Systems A two-input truth table  Continuing Example(CE) Step 1 : Represent each of the inputs and outputs in binary. Step 2 : Formalize the design specification either in the form of a truth table or of an algebraic expression. Step 1.5 : If necessary, break the problem into smaller subproblems. Step 3 : Simplify the description. Step 4 : Implement the system with the available components, subject to the design objectives and constraints. A B Y OR gate symbol ABY 000 011 101 111 P. 36, 38

5. 2 - 5 Chapter 2 Combinational Systems  Don’t Care Conditions 2.1 The Design Process for Combinational Systems A truth table with a don’t care. Acceptable truth tables. System one System Two J K A B C Design example with don’t cares. abf 000 011 101 11X abf 1 f 2 0000 0111 1011 1101 P. 38- 39

6. 2 - 6 Chapter 2 Combinational Systems 2.1 The Design Process for Combinational Systems  The Development of Truth Tables Truth table for CE1. ABCDZ1Z1 Z2Z2 Z3Z3 0000000 0001000 0010000 0011000 0100000 0101000 0110000 0111111 1000000 1001000 1010000 1011111 1100000 1101111 1110111 111101X P. 40

7. 2 - 7 Chapter 2 Combinational Systems 2.1 The Design Process for Combinational Systems  The Development of Truth Tables (a)(b) Truth table for CE2. abcf 0000 0010 0100 0110 1000 1011 1101 1110 abcf 0000 0010 0100 0110 1001 1010 1100 1111 P. 40

8. 2 - 8 Chapter 2 Combinational Systems 2.1 The Design Process for Combinational Systems Display Driver abcdefgabcdefg W X Y Z a b c d e f g A seven-segment display.  The Development of Truth Tables P. 41

9. 2 - 9 Chapter 2 Combinational Systems 2.1 The Design Process for Combinational Systems  The Development of Truth Tables A truth table for the seven-segment display driver. DigitWXYZabcdefg 000001111110 100010110000 200101101101 300111111001 401000110011 501011011011 60110X011111 7011111100X0 810001111111 91001111X011 -1010XXXXXXX -1011XXXXXXX -1100XXXXXXX -1101XXXXXXX -1110XXXXXXX -1111XXXXXXX P. 42

10. 2 - 10 Chapter 2 Combinational Systems 2.1 The Design Process for Combinational Systems Input : Condition Output : largest integer meets the input condition a = 0 : odd, a = 1 : even b = 0 : prime, b = 1 : not prime c = 0 : less then 8, c = 1 : greater than or equal to 8 a = 0, b = 0, c = 0 : the largest odd prime number less then 8  7(0111) Example 4.3 abcwxyZ 0000111 0011101 010XXXX 0111111 1000010 101XXXX 1100110 1111110 P. 42

11. 2 - 11 Chapter 2 Combinational Systems 2.2 Switching Algebra  Definition of Switching Algebra OR (written as +) a + b (read a OR b) is 1 if and only if a = 1 or b = 1 or both AND (written as · or simply two variables catenate) a · b = ab (read a AND b) is 1 if and only if a = 1 and b = 1. NOT (written ') a' (read NOT a) is 1 if and only if a = 0 Truth table for OR, AND, and NOT. aba + b 000 011 101 111 abab 000 010 100 111 aa’ 01 10 P. 44

12. 2 - 12 Chapter 2 Combinational Systems 2.2 Switching Algebra  Definition of Switching Algebra commutativeP1a. a + b = b + a P1b. ab = ba associativeP2a. a + (b + c) = (a + b) + c P2b. a(bc) = (ab)c a + b + c + d + · · · is 1 if any of the operands (a, b, c, d,...) is 1 and is 0 only if all are 0 abcd... is 1 if all of the operands are 1 and is 0 if any is 0 Symbols for OR and AND gates. P. 45

13. 2 - 13 Chapter 2 Combinational Systems 2.2 Switching Algebra  Definition of Switching Algebra AND gate implementation of Property 2b. a a'a' a NOT gate. Parentheses are used as in other mathematics; expressions inside the parentheses are evaluated first. Order of precedence : NOT > AND > OR P. 45- 46

14. 2 - 14 Chapter 2 Combinational Systems 2.2 Switching Algebra  Basic Properties of Switching Algebra identityP3a. a + 0 = a P3b. a ∙ 1 = a nullP4a. a + 1 = 1 P4b. a ∙ 0 = 0 complementP5a. a + a ′ = 1 P5b. a ∙ a ′ = 0 combining the commutative property with 3, 4, 5. P3aa. 0 + a = a P3bb. 1 ∙ a = a P4aa. 1 + a = 1 P4bb. 0 ∙ a = 0 P5aa. a ′ + a = 1 P5bb. a ′ ∙ a = 0

15. 2 - 15 Chapter 2 Combinational Systems 2.2 Switching Algebra  Basic Properties of Switching Algebra Idempotency P6a. a + a = a P6b. a ∙ a = a involution P7. (a ′ ) ′ = a distributive P8a. a(b + c) = ab + ac P8b. a + bc = (a + b)(a + c) Truth table to prove Property 8b. abcbcLHSa + ba + cRHS 00000000 00100010 01000100 01111111 10001111 10101111 11001111 11111111 P. 48

16. 2 - 16 Chapter 2 Combinational Systems 2.2 Switching Algebra Example 2.2 f = y’z’ + x’y + x’yz’ g = xy’ + x’z’ + x’y h = (x’+ y’)(x + y + z’) xyzy’z’x’yx’y’zfxy’x’z’x’ygx’+y’x+y+z’h 00010010101111 00100000000100 01001110111111 01101010011111 10010011001111 10100001001111 11000000000010 11100000000010 P. 48

17. 2 - 17 Chapter 2 Combinational Systems 2.2 Switching Algebra  Manipulation of Algebraic Functions literal is the appearance of a variable or its complement. ab ′ + bc ′ d + a ′ d + e ′ → 8 literal product term is one or more literals connected by AND operators. ab ′ + bc ′ d + a ′ d + e ′ → 4 product term minterm (standard product term) is a product term that includes each variable of the problem, either uncomplemented or complemented. function of four variables, w, x, y, and z. → wxyz, w ′ xyz ′ (minterm) → w ′ yz (is not)

18. 2 - 18 Chapter 2 Combinational Systems 2.2 Switching Algebra  Manipulation of Algebraic Functions sum of products expression is one or more product terms connected by OR operators. w ′ xyz ′ + wx ′ y ′ z ′ + wx ′ yz + wxyz(4 product terms) x + w ′ y + wxy ′ z(3 product terms) x ′ + y + z(3 product terms) wy ′ (1 product terms) z(1 product terms) canonical sum (sum of standard product terms) is a sum of products expression where all of the terms are standard product terms.

19. 2 - 19 Chapter 2 Combinational Systems 2.2 Switching Algebra  Manipulation of Algebraic Functions (1) x ′ yz ′ + x ′ yz + xy ′ z ′ + xy ′ z + xyz5 terms, 15 literals (2) x ′ y + xy ′ + xyz3 terms, 7 literals (3) x ′ y + xy ′ + xz3 terms, 6 literals (4) x ′ y + xy ′ + yz3 terms, 6 literals minimum sum of products expression is one of those SOP expression for a function that has the fewest number of product terms. Each of the following expressions are equal. ((1) = (2) = (3) = (4)) (minima) x ′ yz ′ + x ′ yz + xy ′ z ′ + xy ′ z + xyz = (x ′ yz ′ + x ′ yz) + (xy ′ z ′ + xy ′ z) + xyz associative = x ′ y(z ′ + z) + xy ′ (z ′ + z) + xyz distributive = x ′ y ∙ 1 + xy ′ ∙ 1 + xyz complement = x ′ y + xy ′ + xyzidentity

20. 2 - 20 Chapter 2 Combinational Systems 2.2 Switching Algebra  Manipulation of Algebraic Functions adjacencyP9a. ab + ab ′ = a P9b. (a + b)(a + b ′ ) = a x ′ yz ′ + x ′ yz + xy ′ z ′ + xy ′ z + xyz + xy ′ z = (x ′ yz ′ + x ′ yz) + (xy ′ z ′ + xy ′ z) + (xyz + xy ′ z) = x ′ y(z ′ + z) + xy ′ (z ′ + z) + xz(y + y ′ ) = x ′ y ∙ 1 + xy ′ ∙ 1 + xz ∙ 1 = x ′ y + xy ′ + xz simplificationP10a. a + a ′ b = a + b P10b. a(a ′ + b) = ab a + a ′ b = (a + a ′ )(a + b) distributive = 1 ∙ (a + b)complement = a + bidentity a(a + b) = aa + ab = 0 + ab = ab

21. 2 - 21 Chapter 2 Combinational Systems 2.2 Switching Algebra a’b’c’ + a’bc’ + a’bc + ab’c’  a’c’ + a’bc + ab’c’ (using P9a)  a’c’ + a’b + ab’c’ (using P10a)  a’c’ + a’b + b’c’ (Not minterm) a’b’c’ + a’bc’ + a’bc + ab’c’  b’c’ + a’b (Minterm) Example 2.3

22. 2 - 22 2.2 Switching Algebra  Manipulation of Algebraic Functions sum term is one or more literal connected by OR operators. maxterm(standard sum term) is a sum term that includes each variable of the problem, either uncomplemented or complemented. function of four variables, w, x, y, and z. → w ′ + x + y + z ′ (maxterm) → w + y ′ + z (is not) POS(product of sums expression) is one or more sum terms connected by AND operators. (w + x)(w + y)(2 terms) w(x + y)(2 terms) w(1 term) w + x(1 term) (w + x ′ + y ′ + z ′ )(w ′ + x + y + z ′ )(2 terms) Chapter 2 Combinational Systems

23. 2 - 23 canonical product (product of standard product terms) is a POS expression in where all of the terms are standard sum terms. 2.2 Switching Algebra  Manipulation of Algebraic Functions Chapter 2 Combinational Systems Minimum is defined the same way for both POS and SOP. Minimum is the expressions with the fewest number of terms (same number of terms).

24. 2 - 24 Chapter 2 Combinational Systems 2.3 Implementation of Functions with AND, OR, and NOT Gates f = x ′ yz ′ + x ′ yz + xy ′ z ′ + xy ′ z + xyz  Block diagram Block diagram of f in sum of standard products form. P. 54

25. 2 - 25 Chapter 2 Combinational Systems 2.3 Implementation of Functions with AND, OR, and NOT Gates  Block diagram Minimum SOP expression f = x ′ yz ′ + x ′ yz + xy ′ z ′ + xy ′ z + xyz = x ′ y + xy ′ + xz Minimum sum of product implementation of f. Circuit with only uncomplemented inputs. P. 55

26. 2 - 26 Chapter 2 Combinational Systems 2.3 Implementation of Functions with AND, OR, and NOT Gates  Block diagram Minimum POS expression f = x ′ y + xy ′ + xz = (x + y)(x ′ + y ′ + z) Minimum product of sums implementation of f. P. 55

27. 2 - 27 Chapter 2 Combinational Systems 2.3 Implementation of Functions with AND, OR, and NOT Gates  Block diagram Multilevel circuit h = z ′ + wx ′ y + v(xz + w ′ ) P. 56

28. 2 - 28 Chapter 2 Combinational Systems 2.3 Implementation of Functions with AND, OR, and NOT Gates f = x’y + x(y’ + z) x y x’ y’ z f Example 2.4 P. 56

29. 2 - 29 Chapter 2 Combinational Systems 2.3 Implementation of Functions with AND, OR, and NOT Gates  Block diagram a. High/Low b. Positive logic c. Negative logic abf LLL LHH HLH HHH abf 000 011 101 111 abf 111 100 010 000 P. 58

30. 2 - 30 Chapter 2 Combinational Systems 2.4 The Complement  DeMorgan’s theorem DeMorganP11a. (a + b) ′ = a ′ b ′ P11b. (ab) ′ = a ′ + b ′ Proof of DeMorgan’s theorem. P11aa. (a + b + c...) ′ = a ′ b ′ c ′... P11bb. (abc...) ′ = a ′ + b ′ + c ′ +... Extended DeMorgan’s theorem ab a + b (a + b)’ a’b’a’b’ab(ab)’a’ + b’ 0001111011 0110100011 1010010011 1110000100 11a 11b P. 58

31. 2 - 31 Chapter 2 Combinational Systems 2.4 The Complement f = wx’y + xy’ + wxz f’ = (wx’y + xy’ + wxz)’ = (wx’y)’(xy’)’(wxz)’[P11a] = (w’ + x + y’)(x’ + y)(w’ + x’ + z’)[P11b] Example 2.5

32. 2 - 32 Chapter 2 Combinational Systems 2.4 The Complement Set of rules.  DeMorgan’s theorem 1. Complement each variable (that is, a to a ′ or a ′ to a). 2. Replace 0 by 1 and 1 by 0. 3. Replace AND by OR and OR by AND, being sure to preserve the order of operations. That sometimes requires additional parentheses.

33. 2 - 33 Chapter 2 Combinational Systems 2.4 The Complement f = ab’(c + d’e) + a’bc’ f’ = [ab’(c + d’e) + a’bc’]’using [P11a], [P11b] = [ab’(c + d’e)]’[a’bc’]’ = [a’ + b + (c + d’e)’][a + b’ + c] = [a’ + b + c’(d’e)’][a + b’ + c] = [a’ + b + c’(d + e’)][a + b’ + c] Example 2.6

34. 2 - 34 Chapter 2 Combinational Systems 2.5 From The Truth Table to Algebraic Expressions  Truth table A two-variable truth table. Verbal descriptions of systems can most easily be translated into the truth table. => need the ability to go from the truth table to an algebraic expression. To understand the process, consider the two-variable truth table. Minterms. Each row of the truth table corresponds to a product term. abf 000 011 101 111 ABCMintermNunber 000A’B’C’0 001A’B’C1 010A’BC’2 011A’BC3 100AB’C’4 101AB’C5 110ABC’6 111ABC7 P. 60 - 61

35. 2 - 35 Chapter 2 Combinational Systems 2.5 From The Truth Table to Algebraic Expressions  Truth table For a specific function, those terms for which The function is 1 are used to form an SOP expression for f. The function is 0 are used to form an SOP expression for f’. Complement f’ to form a POS expression for f. Example 2.7 1. f (A, B, C) = Σm(1, 2, 3, 4, 5) = A’B’C + A’BC’ + A’BC + AB’C’ + AB’C 2. f’(A, B, C) = Σm(0, 6, 7) = A’B’C’ + ABC’ + ABC 1-1. f = A’B’C + A’BC’ + A’BC + AB’C’ + AB’C = A’B’C + A’B + AB’ [P9a] = A’C + A’B + AB’ = B’C + A’B + AB’ 2-1. f’ = A’B’C’ + AB 2-2. f = (f’)’ = (A + B + C)(A’ + B’ + C)(A’ + B’ + C’) ABCff’ 00001 00110 01010 01110 10010 10110 11001 11101 P. 61

36. 2 - 36 Chapter 2 Combinational Systems 2.5 From The Truth Table to Algebraic Expressions f(a, b, c) = ∑m(1, 2, 5) + ∑d(0, 3) Example 2.8 abcf 000X 0011 0101 011X 1000 1011 1100 1110 P. 63

37. 2 - 37 Chapter 2 Combinational Systems 2.5 From The Truth Table to Algebraic Expressions Using Z 2 for CE Z 2 = A’BCD + AB’CD + ABC’D + ABCD’ + ABCD ABCD can be combined with each of the others(using P10a) Z 2 = BCD + ACD + ABD + ABC (minimum SOP) Z 1 = A’BCD + AB’CD + ABC’D + ABCD’ (can’t minimize) Example 2.9

38. 2 - 38 Chapter 2 Combinational Systems 2.5 From The Truth Table to Algebraic Expressions For CE2 f = ab’c + abc’ or f = ab’c’ + abc  No simplification is possible f’ = a’b’c’ + a’b’c + a’bc’ + a’bc + ab’c’ + abc = a’b’ + a’b + ab’c’ + abc[P9a, P9b] = a’ + ab’c’ + abc = a’ + b’c’ + bc[P9a, P10a] f = (a + b + c)(a + b + c’)(a + b’ + c)(a + b’ + c’)(a’ + b + c)(a’ + b’ + c’) = a(b + c)(b’ + c’)(minimum POS) Example 2.10

39. 2 - 39 Chapter 2 Combinational Systems 2.5 From The Truth Table to Algebraic Expressions For Full Adder CE3 C out = a’bc + a’bc + abc’ + abc s = a’b’c + a’bc’ + ab’c’ + abc C out = bc + ac + ab(like Example 2.9) s = a’b’c + a’bc + ab’c + abc Example 2.11

40. 2 - 40 Chapter 2 Combinational Systems 2.5 From The Truth Table to Algebraic Expressions  Truth table To find a minimum POS expression 1. Manipulate the previous POS expression to obtain f = (A + B + C)(A’ + B’) 2. Simplify the SOP expression for f’ and then use DeMorgan to convert it to a POS expression. Both approaches(1, 2) produce the same result. How many different functions of n variables are there? For two variables, there are 16 possible truth tables, resulting in 16 different functions. All two-variable functions. Number of functions of n variables. abf0f0 f1f1 f2f2 f3f3 f4f4 f5f5 f6f6 f7f7 f8f8 f9f9 f10f10 f11f11 f12f12 f13f13 f14f14 f15f15 000000000011111111 010000111100001111 100011001100110011 110101010101010101 VariablesTerms 14 216 3256 465,536 54,294,967,296 P. 64

41. 2 - 41 Chapter 2 Combinational Systems 2.6 NAND, NOR, AND EXCLUSIVE-OR GATES  Three other gates NAND gates. Note that DeMorgan’s theorem states that (ab)’ = a’ + b’ P. 65

42. 2 - 42 Chapter 2 Combinational Systems 2.6 NAND, NOR, AND EXCLUSIVE-OR GATES  Three other gates Functional completeness of NAND. NAND gate implementation. P. 66

43. 2 - 43 Chapter 2 Combinational Systems 2.6 NAND, NOR, AND EXCLUSIVE-OR GATES  Three other gates Better NAND gate implementation. Double NOT gate approach. A multilevel NAND implementation P. 67

44. 2 - 44 Chapter 2 Combinational Systems 2.6 NAND, NOR, AND EXCLUSIVE-OR GATES f = wx(y + z) + x’y AND and OR Gates First Version f Example 2.12 P. 67

45. 2 - 45 Chapter 2 Combinational Systems 2.6 NAND, NOR, AND EXCLUSIVE-OR GATES f = wx(y + z) + x’y Second Version Example 2.12(cont.) P. 68

46. 2 - 46 Chapter 2 Combinational Systems 2.6 NAND, NOR, AND EXCLUSIVE-OR GATES  Three other gates NOR gates. Circuit consisting of AND and OR gates such that 1.the output of the circuit comes from an AND, 2.the inputs to OR gates come either from a system input or from the output of an AND, and 3.the inputs to AND gates come either from a system input or from the output of an OR. Then all gates can be converted to NOR gates, and, if an input comes directly into an AND gate, that input must be complemented. Note that DeMorgan’s theorem states that (a + b)’ = a’b’

47. 2 - 47 Chapter 2 Combinational Systems 2.6 NAND, NOR, AND EXCLUSIVE-OR GATES g = (x + y’)(x’ + y)(x’ + z’) Example 2.13 P. 68

48. 2 - 48 Chapter 2 Combinational Systems 2.6 NAND, NOR, AND EXCLUSIVE-OR GATES  Three other gates Exclusive-OR gates. Exclusive-OR gate implements the expression (a b) = a’b + ab’ An Exclusive-OR gate An Exclusive-NOR gate (a b)’ = a’b’ + ab P. 69

49. 2 - 49 Chapter 2 Combinational Systems 2.6 NAND, NOR, AND EXCLUSIVE-OR GATES With AND and OR Gates : 4 packs With NAND Gates : 3 packs Example 2.14 GatesPacks InputsANDORANDORNANDPacks 232151 3211131 41111 Total633193 P. 70

50. 2 - 50 Chapter 2 Combinational Systems 2.7 SIMPLIFICATION OF ALGEVRAIC EXPRESSIONS  Process of simplifying algebraic expressions absorptionP12a. a + ab = a P12b. a(a + b) = a a + ab = a · 1 + ab = a(1 + b) = a · 1 = a a(a + b) = a · a + ab = a + ab consensusP13a. at 1 + a’t 2 + t 1 t 2 = at 1 + a’t 2 P13b. (a + t 1 )(a’ + t 2 )(t 1 + t 2 ) = (a + t 1 )(a’ + t 2 ) at 1 + a’t 2 = (at 1 + at 1 t 2 ) + (a’t 2 + a’t 1 t 2 ) = at 1 + a’t 2 + (at 1 t 2 + a’t 1 t 2 ) = at 1 + a’t 2 + t 1 t 2 Consensus. t 1 t 2 is the consensus term. at1t1 t2t2 at 1 a’t 2 RHSt1t2t1t2 LHS 00000000 00101101 01000000 01101111 10000000 10100000 11010101 11110111 P. 74

51. 2 - 51 Chapter 2 Combinational Systems xyz + x’y + x’y’ = xyz + x’[P9a] = x’ + yz[P10a] where a = x’, a’ = x, b = yz wx + wxy + w’yz + w’y’z + w’xyz’ = (wx + wxy) + (w’yz + w’y’z) + w’xyz’ = wx + w’z + w’xyz’ [P12a, P9a] = wx + w’(z + xyz’) = wx + w’(z + xy)[P10a] = wx + w’z + w’xy = w’z + x(w + w’y) = w’z + w(w + y)[P10a] = w’z + wx + wy 2.7 SIMPLIFICATION OF ALGEVRAIC EXPRESSIONS Example 2.15 Example 2.16

52. 2 - 52 Chapter 2 Combinational Systems (x + y)(x + y + z’) + y’ = (x + y) + y’[P12b] = x + (y + y’)[P5a] = x + 1 = 1[P4a] (a + b’ + c)(a + c’)(a’ + b’ + c)(a + c + d) = (b’ + c)(a + c’)(a + d)[P9b, P10b] (a + c’)(a + c + d) = a + c’(c + d) = a + c’d = (a + c’)(a + d) 2.7 SIMPLIFICATION OF ALGEVRAIC EXPRESSIONS Example 2.17 Example 2.18

53. 2 - 53 Chapter 2 Combinational Systems ab’c ¢ a’d = b’cd ab’c ¢ a’cd = b’cd abc’ ¢ bcd’ = abd’ b’c’d’ ¢ b’cd’ = b’d’ abc’ ¢ bc’d = undefined – no such variable a’bd ¢ ab’cd = undefined – two variables, a and b In Example 2.3 f = a’b’c’ + a’bc’ + a’bc + ab’c’ f 1 = a’c’ + a’b + b’c’[P9a] f 2 = b’c’ + a’b[P10a] 2.7 SIMPLIFICATION OF ALGEVRAIC EXPRESSIONS Example 2.19 Example 2.20

54. 2 - 54 Chapter 2 Combinational Systems g = bc’ + abd + acd The only consensus term defined is bc’ ¢ acd = abd g = bc’ + acd[Property 13] A’BCD + A’BC’D + B’EF + CDE’G + A’DEF + A’B’EF = A’BD + B’EF + CDE’G + A’DEF[P12a, P9a] But A’BD ¢ B’EF = A’DEF A’BD + B’EF + CDE’G + A’DEF = A’BD + B’EF + CDE’G 2.7 SIMPLIFICATION OF ALGEVRAIC EXPRESSIONS Example 2.21 Example 2.22

55. 2 - 55 Chapter 2 Combinational Systems 2.8 MANIPULATION OF ALGEBRAIC FUNCTIONS AND NAND GATE IMPLEMENTATIONS  Process of manipulate the algebraic An SOP expression expand to sum of minterms. 1. Use a truth table. 2. Use P9a to add variables to a term. f = bc + ac + ab = bca + bca’ + ac + ab Repeat the process on the other two terms f = bca + bca’ + acb + acb’ + abc + abc’ = abc + a’bc + abc + ab’c + abc + abc’ = a’bc + ab’c + abc’ + abc Example 2.23

56. 2 - 56 Chapter 2 Combinational Systems 2.8 MANIPULATION OF ALGEBRAIC FUNCTIONS AND NAND GATE IMPLEMENTATIONS g = x’ + xyz = x’y + x’y’ + xyz = x’yz + x’yz’ + x’y’z + x’y’z’ + xyz g(x,y,z) = ∑m(0,1,2,3,7) f = (A + B + C)(A’ + B’) = (A + B + C)(A’ + B’ + C)(A’ + B’ + C’) Example 2.24 Example 2.25

57. 2 - 57 Chapter 2 Combinational Systems 2.8 MANIPULATION OF ALGEBRAIC FUNCTIONS AND NAND GATE IMPLEMENTATIONS  Process of manipulate the algebraic P14a. ab + a’c = (a + c)(a’ + b) (a + c)(a’ + b) = (a + c)a’ + (a + c)b = aa’ + a’c + ab + bc aa’ = 0 and bc = a’c ¢ ab and thus, using P3aa and P13a (a + c)(a’ + b) = aa’ + a’c + ab + bc = a’c + ab This property is particularly useful in converting POS expression to SOP and vice versa. Another application of P14a and this type of algebraic manipulation comes when we wish to implement functions using only two-input NAND or NOR gate. P. 78

58. 2 - 58 Chapter 2 Combinational Systems 2.8 MANIPULATION OF ALGEBRAIC FUNCTIONS AND NAND GATE IMPLEMENTATIONS f = (A + B + C)(A’ + B’) = AB’ + A’(B + C) [P14a] = AB’ + A’B + A’C f = AA’ + AB’ + BA’ + BB’ + CA’ + CB’ = AB’ + A’B + A’C + B’C (A + B’ + C)(A + B + D)(A’ + C’ + D’) = [A + (B’ + C)(B + D)](A’ + C’ + D’)[P8b] = (A + B’D + BC)(A + C’ + D’)[P14a] = A(C’ + D’) + A’(B’D + BC)[P14a] = AC’ + AD’ + A’B’D + A’BC[P8a] Example 2.26 Example 2.27

59. 2 - 59 Chapter 2 Combinational Systems 2.8 MANIPULATION OF ALGEBRAIC FUNCTIONS AND NAND GATE IMPLEMENTATIONS wxy’ + xyz + w’x’z = x(wy’ + yz) + w’x’z’[P8a] = x(y’ + z)(y + w) + w’x’z’ [P14a] = (x + w’z’)[x’ + (y’ + z)(y + w)] [P14a] = (x + w’)(x + z’)(x’ + y’ + z)(x’ + y + w)[P8b] Example 2.28

60. 2 - 60 Chapter 2 Combinational Systems 2.8 MANIPULATION OF ALGEBRAIC FUNCTIONS AND NAND GATE IMPLEMENTATIONS G = DE’ + A’B’C’ + CD’E + ABC’E = DE’ + A’B’C’ + E(CD’ + ABC’) - No way of eliminating 3- input gate G = C’(A’B’ + ABE) + DE’ + CD’E = C’(A’ + BE)(A + B’) + DE’ + CD’E [P14a] or G = C’(B’ + AE)(B + A’) + DE’ + CD’E Reduce 3-input gates to 2-input gates G = (C’ + D’E)[C + (B’ + AE)(B + A’)] + DE’ Example 2.29 P. 80

61. 2 - 61 Chapter 2 Combinational Systems 2.8 MANIPULATION OF ALGEBRAIC FUNCTIONS AND NAND GATE IMPLEMENTATIONS G = C’(A’ + BE)(A + B’) + DE’ + CD’E = C’(A’ + BE)(A + B’) + (D + CE)(D’ + E’) = (A’ + BE)(AC’ + B’C’) + (D + CE)(D’ + E’) Example 2.29(cont.) P. 80

62. 2 - 62 Chapter 2 Combinational Systems 2.8 MANIPULATION OF ALGEBRAIC FUNCTIONS AND NAND GATE IMPLEMENTATIONS G = C’D’ + ABC’ + A’C + B’C = C’(D’ + AB) + C(A’ + B’) G Example 2.30 P. 80

63. 2 - 63 Chapter 2 Combinational Systems 2.8 MANIPULATION OF ALGEBRAIC FUNCTIONS AND NAND GATE IMPLEMENTATIONS s = a’b’c + a’bc’ + ab’c + abc c out = bc + ac + ab To reduce the gate requirements, s = c(a’b’ + ab) + c’(ab’ + a’b) c out = c(a + b) + ab Returning to the expression for sum, note that s = c(a b)’ + c’(a b) = c (a b) c out = c(a b) + ab Example 2.31 (for CE3) P. 82

64. 2 - 64 Chapter 2 Combinational Systems 2.8 MANIPULATION OF ALGEBRAIC FUNCTIONS AND NAND GATE IMPLEMENTATIONS Each Exclusive-OR with 4 two-input NAND gates, without requiring complemented inputs. Example 2.31 (cont.) P. 82

65. 2 - 65 Chapter 2 Combinational Systems 2.8 MANIPULATION OF ALGEBRAIC FUNCTIONS AND NAND GATE IMPLEMENTATIONS Note that the two green NAND gates have the same inputs and the two light green ones also have the same inputs. Only one copy of each is necessary, yielding the final circuit with only nine NAND gates. Example 2.31 (cont.) P. 83