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Designing Combinational Systems: Examples and Switching Algebra

This chapter explores the design process for combinational systems, including examples such as binary addition and display drivers. It also covers the basics of switching algebra and its properties.

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Designing Combinational Systems: Examples and Switching Algebra

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  1. Chapter 2 Combinational Systems

  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. Chapter 2 Combinational Systems 2.1 The Design Process for Combinational Systems • Continuing Example(CE) 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.)

  4. Chapter 2 Combinational Systems 2.1 The Design Process for Combinational Systems • Continuing Example(CE) Step 1 : Represent each of the inputs and outputs in binary. • A two-input truth table 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. • OR gate symbol Step 4 : Implement the system with the available components, subject to the design objectives and constraints. A Y B P. 36, 38

  5. Chapter 2 Combinational Systems 2.1 The Design Process for Combinational Systems • Don’t Care Conditions • A truth table with a don’t care. • Acceptable truth tables. • Design example with don’t cares. A J System one System Two B K C P. 38- 39

  6. Chapter 2 Combinational Systems 2.1 The Design Process for Combinational Systems • The Development of Truth Tables • Truth table for CE1. P. 40

  7. Chapter 2 Combinational Systems 2.1 The Design Process for Combinational Systems • The Development of Truth Tables • Truth table for CE2. (a) (b) P. 40

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

  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. P. 42

  10. Chapter 2 Combinational Systems 2.1 The Design Process for Combinational Systems • Example 4.3 • 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) P. 42

  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. P. 44

  12. Chapter 2 Combinational Systems 2.2 Switching Algebra • Definition of Switching Algebra commutative P1a. a + b = b + a P1b. ab = ba associative P2a. 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. Chapter 2 Combinational Systems 2.2 Switching Algebra • Definition of Switching Algebra • AND gate implementation of Property 2b. • a NOT gate. a' a • Parenthesesare used as in other mathematics; expressions inside • the parentheses are evaluated first. • Order of precedence : NOT > AND > OR P. 45- 46

  14. Chapter 2 Combinational Systems 2.2 Switching Algebra • Basic Properties of Switching Algebra identity P3a. a + 0 = a P3b. a ∙ 1 = a null P4a. a + 1 = 1 P4b. a ∙ 0 = 0 complement P5a. 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. 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. P. 48

  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’) P. 48

  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. 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. Chapter 2 Combinational Systems 2.2 Switching Algebra • Manipulation of Algebraic Functions • minimum sum of products expression is one of those SOP expression for • a function that has the fewest number of product terms. (1) x′yz′ + x′yz + xy′z′ + xy′z + xyz 5 terms, 15 literals (2) x′y + xy′ + xyz 3 terms, 7 literals (3) x′y + xy′ + xz 3 terms, 6 literals (4) x′y + xy′ + yz 3 terms, 6 literals (minima) (minima) Each of the following expressions are equal. ((1) = (2) = (3) = (4)) 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′ + xyz identity

  20. Chapter 2 Combinational Systems 2.2 Switching Algebra • Manipulation of Algebraic Functions adjacency P9a. 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 simplification P10a. a + a′b = a + b P10b. a(a′ + b) = ab a + a′b = (a + a′)(a + b) distributive = 1 ∙ (a + b) complement = a + b identity a(a + b) = aa + ab = 0 + ab = ab

  21. Chapter 2 Combinational Systems 2.2 Switching Algebra • Example 2.3 • 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)

  22. Chapter 2 Combinational Systems 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)

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

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

  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 • Circuit with only uncomplemented • inputs. • Minimum sum of product • implementation of f. P. 55

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

  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 P. 58

  30. Chapter 2 Combinational Systems 2.4 The Complement • DeMorgan’s theorem DeMorgan P11a. (a + b)′ = a′b′ P11b. (ab)′ = a′ + b′ • Proof of DeMorgan’s theorem. • Extended DeMorgan’s theorem P11aa. (a + b + c . . .)′ = a′b′c′ . . . P11bb. (abc . . .)′ = a′ + b′ + c′ + . . . P. 58

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

  32. Chapter 2 Combinational Systems 2.4 The Complement • DeMorgan’s theorem • Set of rules. 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. Chapter 2 Combinational Systems 2.4 The Complement • Example 2.6 • 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]

  34. Chapter 2 Combinational Systems 2.5 From The Truth Table to Algebraic Expressions • 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. • Each row of the truth table corresponds to a product term. • A two-variable truth table. • Minterms. P. 60 - 61

  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’) P. 61

  36. Chapter 2 Combinational Systems 2.5 From The Truth Table to Algebraic Expressions • Example 2.8 • f(a, b, c) = ∑m(1, 2, 5) + ∑d(0, 3) P. 63

  37. Chapter 2 Combinational Systems 2.5 From The Truth Table to Algebraic Expressions • Example 2.9 • Using Z2 for CE • Z2 = A’BCD + AB’CD + ABC’D + ABCD’ + ABCD • ABCD can be combined with each of the others(using P10a) • Z2 = BCD + ACD + ABD + ABC (minimum SOP) • Z1= A’BCD + AB’CD + ABC’D + ABCD’ (can’t minimize)

  38. Chapter 2 Combinational Systems 2.5 From The Truth Table to Algebraic Expressions • Example 2.10 • 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)

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

  40. Chapter 2 Combinational Systems 2.5 From The Truth Table to Algebraic Expressions • Truth table • To find a minimum POS expression Manipulate the previous POS expression to obtain • f = (A + B + C)(A’ + B’) 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. • Number of functions of n variables. • All two-variable functions. P. 64

  41. Chapter 2 Combinational Systems 2 - 41 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. 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. Chapter 2 Combinational Systems 2 - 43 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. Chapter 2 Combinational Systems 2 - 44 2.6 NAND, NOR, AND EXCLUSIVE-OR GATES • Example 2.12 • f = wx(y + z) + x’y • AND and OR Gates f • First Version P. 67

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

  46. Chapter 2 Combinational Systems 2 - 46 2 - 46 2.6 NAND, NOR, AND EXCLUSIVE-OR GATES • Three other gates • NOR gates. • Circuit consisting of AND and OR gates such that • the output of the circuit comes from an AND, • the inputs to OR gates come either from a system input or from the output of an AND, and • 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. Chapter 2 Combinational Systems 2 - 47 2.6 NAND, NOR, AND EXCLUSIVE-OR GATES • Example 2.13 • g = (x + y’)(x’ + y)(x’ + z’) P. 68

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

  49. Chapter 2 Combinational Systems 2 - 49 2.6 NAND, NOR, AND EXCLUSIVE-OR GATES • Example 2.14 • With AND and OR Gates : 4 packs • With NAND Gates : 3 packs P. 70

  50. Chapter 2 Combinational Systems 2 - 50 2 - 50 2 - 50 2 - 50 2.7 SIMPLIFICATION OF ALGEVRAIC EXPRESSIONS • Process of simplifying algebraic expressions absorption P12a. 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 consensus P13a. at1 + a’t2 + t1t2 = at1 + a’t2 P13b. (a + t1)(a’ + t2)(t1 + t2) = (a + t1)(a’ + t2) • Consensus. at1 + a’t2 = (at1 + at1t2) + (a’t2 + a’t1t2) = at1 + a’t2 + (at1t2 + a’t1t2) = at1 + a’t2 + t1t2 t1t2 is the consensus term. P. 74

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