Fuw-Yi Yang1 數位系統 Digital Systems Department of Computer Science and Information Engineering, Chaoyang University of Technology 朝陽科技大學資工系 Speaker: Fuw-Yi Yang 楊伏夷 伏夷非征番, 道德經 察政章 (Chapter 58) 伏 者潛藏也 道紀章 (Chapter 14) 道無形象, 視之不可見者曰 夷

Fuw-Yi Yang2 Text Book: Digital Design 4th Ed. Chap 2 Boolean Algebra and Logic Gates 2.1 Introduction 2.2 Basic Definitions 2.3 Axiomatic Definition of Boolean Algebra 2.4 Basic Theorems and Properties of Boolean Algebra 2.5 Boolean Functions 2.6 Canonical and Standard Forms 2.7 Other Logic Operations 2.8 Digital Logic Gates 2.9 Integrated Circuits

Fuw-Yi Yang3 Text Book: Digital Design 4th Ed. Chap 2.1 Introduction Because binary logic is used in all of today’s digital computers and devices, the cost of the circuit that implement it is an important factor addressed by designers. Finding simpler and cheaper, but equivalent, realizations of a circuit can reap huge payoffs in reducing the overall cost of the design.

Fuw-Yi Yang4 Text Book: Digital Design 4th Ed. Chap 2.2 Basic Definitions The most common postulates used to formulate various algebraic structures are as follows: 1. Closure. A set S is closed with respect to a binary operator  if, for every a, b  S, a  b  S. 2. Associative law. (a  b)  c =a  (b  c) for a, b, c  S. 3. Commutative law. a  b = b  a for a, b  S. 4. Identity element. If there exists an element e  S such that e  b = b  e = b for b  S. 5. Inverse. For a  S, if there exists an element b  S such that a  b = e, b is called the inverse of a.

Fuw-Yi Yang5 Text Book: Digital Design 4th Ed. Chap 2.2 Basic Definitions 6. Distributive law. If  and * are two binary operators on a set S, * is said to be distributive over  whenever a * (b  c) = (a * b)  (a * c).

Fuw-Yi Yang6 Text Book: Digital Design 4th Ed. Chap 2.3 Axiomatic Definition of Boolean Algebra In 1854, George Boole developed an algebraic system now called Boolean Algebra. Boolean algebra is an algebraic structure defined by a set of elements, B, together with two binary operators, + and ∙, if the following postulates are satisfied: 1. Closure. With respect to the operators + and ∙. 2. Identity element. The element 0 is an identity element w.r.t. +; i.e. x + 0 = 0 + x = x.

Fuw-Yi Yang7 Text Book: Digital Design 4th Ed. Chap 2.3 Axiomatic Definition of Boolean Algebra The element 1 is an identity element w.r.t. ∙; i.e. x ∙1 =1 ∙x = x. 3. Commutative law. W.r.t. the operators + and ∙. i.e. a + b = b + a; a ∙b = b ∙a. 4. Distributive law. a + (b ∙ c) = (a + b) ∙(a + c); a ∙ (b + c) = (a ∙b) + (a ∙c); 5. Complement. for a  B, there exists an element a'  B such that a + a' = 1, and a ∙a' = There exist at least two elements a, b  B, such that a  b.

Fuw-Yi Yang8 Text Book: Digital Design 4th Ed. Chap 2.3 Axiomatic Definition of Boolean Algebra – Example of an algebraic structure A two-valued Boolean algebra is defined on a set of two elements, B = {0, 1}, with rules for the two binary operators + (OR) and * (AND) as shown in the following tables: xyx*yx*y xyx+yx+y xx'x' 01 10

Fuw-Yi Yang9 Text Book: Digital Design 4th Ed. Chap 2.3 Axiomatic Definition of Boolean Algebra Show that the two-valued Boolean algebra defined above satisfies postulates 1~6.

Fuw-Yi Yang10 Text Book: Digital Design 4th Ed. Chap 2.4 Basic Theorems and Properties of Boolean Algebra Table 2.1 lists six theorems of Boolean algebra and four of its postulates. Note that the property of Duality— every algebraic expression deducible from the postulates of Boolean algebra remains valid if the operators and identity elements are interchanged. (part a and part b in Table 2.1)

Fuw-Yi Yang11 Text Book: Digital Design 4th Ed. Chap 2.4 Basic Theorems and Properties of Boolean Algebra Table 2.1 lists six theorems of Boolean algebra and four of its postulates.

Fuw-Yi Yang12 Text Book: Digital Design 4th Ed. Chap 2.4 Basic Theorems and Properties of Boolean Algebra Theorem 1 (a) x + x = x StatementJustification x + x = (x + x) * 1 = (x + x) * (x + x') = x + (x * x') = x + 0 = x Postulate 2(b), identity Postulate 5(a), x + x' = 1 Postulate 4(b), x + (y * z) = (x + y) * (x + z) Postulate 5(b), x * x' = 0 Postulate 2(a), x + 0 = x

Fuw-Yi Yang13 Text Book: Digital Design 4th Ed. Chap 2.4 Basic Theorems and Properties of Boolean Algebra Theorem 1 (b) x * x = x StatementJustification x * x = (x * x) + 0 = (x * x) + (x * x') = x * (x + x') = x * 1 = x Postulate 2(a), identity Postulate 5(b), x * x' = 0 Postulate 4(a), x * (y + z) = (x * y) + (x * z) Postulate 5(a), x + x' = 1 Postulate 2(b), x * 1= x

Fuw-Yi Yang14 Text Book: Digital Design 4th Ed. Chap 2.4 Basic Theorems and Properties of Boolean Algebra Theorem 2 (a) x + 1 = 1 (b) x * 0 = 0 by duality StatementJustification x + 1 = (x + 1) * 1 = (x + 1) * (x + x') = x + (1 * x') = x + x' = 1 Postulate 2(b), identity Postulate 5(a), x + x' = 1 Postulate 4(b), x + (y * z) = (x + y) * (x + z) Postulate 2(b), 1 * x = x Postulate 5(a), x + x' =1

Fuw-Yi Yang15 Text Book: Digital Design 4th Ed. Chap 2.4 Basic Theorems and Properties of Boolean Algebra Theorem 3 (x')' = x Postulate 5: x + x' = 1 and x * x' = 0 together define the complement of x. The complement of x' is x and is also (x')'. Since the complement is unique, we complete the proof.

Fuw-Yi Yang16 Text Book: Digital Design 4th Ed. Chap 2.4 Basic Theorems and Properties of Boolean Algebra Theorem 6 (a) x + x * y = x (b) x * (x + y) = x by duality Absorption StatementJustification x + x * y = x * 1 + x * y = x * (1 + y) = x * 1 = x Postulate 2(b), identity Postulate 4(a), x * (y + z) = (x * y) + (x * z) Postulate 2(a), 1 + x = 1 Postulate 2(b), 1 * x = x

Fuw-Yi Yang17 Text Book: Digital Design 4th Ed. Chap 2.4 Basic Theorems and Properties of Boolean Algebra Theorem 4 Associative x + (y + z) = (x + y) + z x * (y * z) = (x * y) * z Theorem 5 DeMorgan (x + y)' = x' * y' (x * y)' = x' + y‘ Show its validity with truth table!!

Fuw-Yi Yang18 Text Book: Digital Design 4th Ed. Chap 2.4 Basic Theorems and Properties of Boolean Algebra The Operator Precedence for evaluating Boolean expressions is: 1. parentheses 2. NOT 3. AND 4. OR

Fuw-Yi Yang19 Text Book: Digital Design 4th Ed. Chap 2.5 Boolean Functions Boolean algebra is an algebra that deals with binary variables and logic operations. A Boolean function described by an algebraic expression consists of binary variables, the constant 0 and 1, and the logic operation symbols. For a given value of the binary variables, the function can be equal to either 0 or 1. Example: F = x + yz, F is equal to 1 if x is equal to 1 or if both y and z are equal to 1.

Fuw-Yi Yang20 Text Book: Digital Design 4th Ed. Chap 2.5 Boolean Functions A Boolean function can be represented in a truth table. The number of rows in the truth table is 2 n, where n is the number of variables in the function. Table 2.2 shows the truth table for the function F 1 = x + y'z. Can we derive the Boolean function described by the column F 2 ? Can we draw the gate implementation of F 1 and F 2 ?

Fuw-Yi Yang21 Text Book: Digital Design 4th Ed. Chap 2.5 Boolean Functions

Fuw-Yi Yang22 Text Book: Digital Design 4th Ed. Chap 2.5 Boolean Functions

Fuw-Yi Yang23 Text Book: Digital Design 4th Ed. Chap 2.5 Boolean Functions When a Boolean expression is implemented with logic gates, each term requires a gate and each variable within the term designates an input to the gate. We define a literal to be a single variable within a term, in complemented or un-complemented form. Example 2.1 Simplify the following Boolean functions to a minimum number of literals. 1. x (x' + y) 2. x + x' y 3. (x + y) (x + y ') 4. x y + x' z + y z 5. (x + y) (x' + z) (y+ z)

Fuw-Yi Yang24 Text Book: Digital Design 4th Ed. Chap 2.5 Boolean Functions The complement of a function F is F' and is obtained from an interchanges of 0’s for 1’s and 1’s for 0’s in the value of F. It can be derived algebraically through DeMorgan’s theorem. (A + B)' = A' B' ; (A B)' = A' + B' Example 2.2 Find the complement of the functions F 1 = x' y z' + x' y' z; F 2 = x (y' z' + y z)

Fuw-Yi Yang25 Text Book: Digital Design 4th Ed. Chap 2.5 Boolean Functions Example 2.3 Find the complement of the functions F 1 = x' y z' + x' y' z and F 2 = x (y' z' + y z) by taking their duals and complementing each literal. F 1 = x' y z' + x' y' z The dual of F 1 is (x' + y + z')(x' + y' + z) Complement each literal: F' 1 = (x + y' + z)(x + y + z')

Fuw-Yi Yang26 Text Book: Digital Design 4th Ed. Chap 2.6 Canonical and Standard forms A binary variable may appear either in its normal form (x) or in its complement form (x'). For two binary variables x and y combined with an AND Operation, we have four possible combinations: x y, x' y, x y', x' y'. Each of these four AND terms is called a minterm, or a standard product. In a similar manner, n variables can be combined to form 2 n minterms.

Fuw-Yi Yang27 Text Book: Digital Design 4th Ed. Chap 2.6 Canonical and Standard forms A binary variable may appear either in its normal form (x) or in its complement form (x'). For two binary variables x and y combined with an OR Operation, we have four possible combinations: x + y, x' + y, x + y', x' + y'. Each of these four OR terms is called a maxterm, or a standard sum. In a similar manner, n variables can be combined to form 2 n maxerms.

Fuw-Yi Yang28 Text Book: Digital Design 4th Ed. Chap 2.6 Canonical and Standard forms

Fuw-Yi Yang29 Text Book: Digital Design 4th Ed. Chap 2.6 Canonical and Standard forms

Fuw-Yi Yang30 Text Book: Digital Design 4th Ed. Chap 2.6 Canonical and Standard forms A Boolean function can be expressed algebraically from a given truth table by forming a minterm for each combination of the variables that produces a 1 in the function and then taking the OR of all those terms. See Table 2.3, 2.4 f 1 = x'y'z + xy'z' + xyz =m 1 + m 4 + m 7 f 2 = x'yz + xy'z + xy'z' + xyz =m 3 + m 5 + m 6 + m 7

Fuw-Yi Yang31 Text Book: Digital Design 4th Ed. Chap 2.6 Canonical and Standard forms Now consider the complement of a Boolean function. It can be expressed algebraically from a given truth table by forming a minterm for each combination of the variables that produces a 0 in the function and then taking the OR of all those terms. f 1 = x'y'z + xy'z' + xyz =m 1 + m 4 + m 7 f ' 1 = x'y'z' + x'yz' + x'yz + xy'z + xy'z' = m 0 +m 2 +m 3 +m 5 +m 6 If we take the complement of f ' 1 we obtain the function f 1 : f 1 = (x+y+z)(x+y'+z)(x+y'+z')(x'+y+z')(x'+y+z) =M 0 M 2 M 3 M 5 M 6

Fuw-Yi Yang32 Text Book: Digital Design 4th Ed. Chap 2.6 Canonical and Standard forms Sum of minterms Boolean functions expressed as a sum of minterms or product of maxterms are said to be in canonical form. Sum of minterms: Example 2.4 Express the Boolean function F = A + B'C as a sum of minterms. F = A + B'C = A(B + B') (C + C') + (A + A') B'C = ABC+ABC'+AB'C+AB'C'+AB'C+A'B'C = ABC+ABC'+AB'C+AB'C'+A'B'C = m 7 + m 6 + m 5 + m 4 + m 1 Or F(A, B, C) =  (1, 4, 5, 6, 7)

Fuw-Yi Yang33 Text Book: Digital Design 4th Ed. Chap 2.6 Canonical and Standard forms Sum of minterms Example 2.4' Deriving the minterms of a Boolean function directly from the given truth table. See next page

Fuw-Yi Yang34 Text Book: Digital Design 4th Ed. Chap 2.6 Canonical and Standard forms Sum of minterms

Fuw-Yi Yang35 Text Book: Digital Design 4th Ed. Chap 2.6 Canonical and Standard forms Product of maxterms Product of maxterms: Example 2.5 Express the Boolean function F = xy + x'z as a product of maxterms. F = xy + x'z = (xy + x')(xy + z) distributive law = (x + x')(y + x')(x + z)(y + z) distributive law = (y + x' + zz' )(x + z + yy' )(y + z + xx' ) = (y+x'+z') (y+x'+z) (x+z+y') (x+z+y) (y+z+x') (y+z+x) = (y+x'+z') (y+x'+z) (x+z+y')(x+z+y) = M 5 M 4 M 2 M 0 Or F(x, y, z) =  (0, 2, 4, 5)

Fuw-Yi Yang36 Text Book: Digital Design 4th Ed. Chap 2.6 Canonical and Standard forms Conversion between Canonical Forms The complement of a function expressed as the sum of minterms equals the sum of minterms missing from the original function. Example: F(A, B, C) =  (1, 4, 5, 6, 7) F'(A, B, C) =  (0, 2, 3) = m 0 + m 2 + m 3 Now, take the complement of F' by DeMorgan’s theorem, we obtain F in a different form: F(A, B, C) = (m 0 + m 2 + m 3 )' = m' 0 m' 2 m' 3 = M 0 M 2 M 3 =  (0, 2, 3)

Fuw-Yi Yang37 Text Book: Digital Design 4th Ed. Chap 2.6 Canonical and Standard forms Conversion between Canonical Forms A Boolean function can be expressed as product of maxterms or sum of minterms directly from its truth table. F(x, y, z) =  (1, 3, 6, 7) F(x, y, z) =  (0, 2, 4, 5) Table 2.6 F = xy + x'z xyzF

Fuw-Yi Yang38 Text Book: Digital Design 4th Ed. Chap 2.6 Canonical and Standard forms Standard Forms The two canonical forms of Boolean algebra are basic forms that one obtains from reading a given function from the truth table. These forms are very seldom the ones with the least number of literals, because each minterm or maxterm must contain, by definition, all the variables, either complemented or un-complemented.

Fuw-Yi Yang39 Text Book: Digital Design 4th Ed. Chap 2.6 Canonical and Standard forms Standard Forms Another way to express Boolean functions is in standard form. In this configuration, the terms that form the function may contain one, two, or any number of literals. There are two type of standard forms: the sum of products (SOP) and products of sums (POS). Both POS and SOP are referred to as two-level implementation. See next page

Fuw-Yi Yang40 Text Book: Digital Design 4th Ed. Chap 2.6 Canonical and Standard forms Standard Forms — two level ckts

Fuw-Yi Yang41 Text Book: Digital Design 4th Ed. Chap 2.6 Canonical and Standard forms Standard Forms — two level ckts

Fuw-Yi Yang42 Text Book: Digital Design 4th Ed. Chap 2.6 Canonical and Standard forms Non-standard Forms A Boolean functions may be expressed in a nonstandard form. The implementation may requires three levels of gating in this circuit. See next page

Fuw-Yi Yang43 Text Book: Digital Design 4th Ed. Chap 2.6 Canonical and Standard forms Non-standard Forms

Fuw-Yi Yang44 Text Book: Digital Design 4th Ed. Chap 2.6 Canonical and Standard forms Non-standard Forms

Fuw-Yi Yang45 Text Book: Digital Design 4th Ed. Chap 2.7 Other logic operations When the binary operators AND and OR are placed between two variables, x and y, they form two Boolean functions, xy and x+y, respectively. Previously, we stated that there are 2 2 n functions for n binary variables. see next pages xyf:16 combinations 000/1, two possible values

Fuw-Yi Yang46 Text Book: Digital Design 4th Ed. Chap 2.7 Other logic operations

Fuw-Yi Yang47 Text Book: Digital Design 4th Ed. Chap 2.7 Other logic operations

Fuw-Yi Yang48 Text Book: Digital Design 4th Ed. Chap 2.8 Digital logic gates Since Boolean functions are expressed in terms of AND, OR, and NOT operations, it is easier to implement a Boolean function with these type of gates. Still, the possibility of constructing gates for the other logic operations is of practical interest.

Fuw-Yi Yang49 Text Book: Digital Design 4th Ed. Chap 2.8 Digital logic gates Factors to be weighed in considering the construction of the other types of logic gates are: 1. the feasibility and economy of producing the gate 2. the possibility of extending the gate to more inputs 3. the basic properties of the binary operator, such as commutativity and associativity 4. the ability of the gate to implement Boolean functions alone or in conjunction with other gates See next pages

Fuw-Yi Yang50 Text Book: Digital Design 4th Ed. Chap 2.8 Digital logic gates

Fuw-Yi Yang51 Text Book: Digital Design 4th Ed. Chap 2.8 Digital logic gates

Fuw-Yi Yang52 Text Book: Digital Design 4th Ed. Chap 2.8 Digital logic gates

Fuw-Yi Yang53 Text Book: Digital Design 4th Ed. Chap 2.8 Digital logic gates

Fuw-Yi Yang54 Text Book: Digital Design 4th Ed. Chap 2.8 Digital logic gates

Fuw-Yi Yang55 Text Book: Digital Design 4th Ed. Chap 2.8 Digital logic gates

Fuw-Yi Yang56 Text Book: Digital Design 4th Ed. Chap 2.8 Digital logic gates

Fuw-Yi Yang57 Text Book: Digital Design 4th Ed. Chap 2.8 Digital logic gates ---positive and negative logic The binary signal at the inputs and outputs of any gate has one of two values, except during transition. One signal value represents logic 1 and the other logic 0. Since two signal values are assigned to two logic values, there exist two different assignments of signal level to logic value. See next pages

Fuw-Yi Yang58 Text Book: Digital Design 4th Ed. Chap 2.8 Digital logic gates ---positive and negative logic

Fuw-Yi Yang59 Text Book: Digital Design 4th Ed. Chap 2.8 Digital logic gates ---positive and negative logic

Fuw-Yi Yang60 Text Book: Digital Design 4th Ed. Chap 2.8 Digital logic gates ---positive and negative logic

Fuw-Yi Yang61 Text Book: Digital Design 4th Ed. Chap 2.8 Digital logic gates ---positive and negative logic

Fuw-Yi Yang62 Text Book: Digital Design 4th Ed. Chap 2.9 Integrated circuits An integrated circuit (IC) is a silicon semiconductor crystal, called a chip, containing the electronic components for constructing digital gates. Levels of Integration: Small-scale integration (SSI) Medium-scale integration (MSI) Large-scale integration (LSI) Very large-scale integration (VLSI)

Fuw-Yi Yang63 Text Book: Digital Design 4th Ed. Chap 2.9 Integrated circuits Digital logic Families: Transistor-transistor logic (TTL) Emitter-coupled logic (ECL) Metal-oxide semiconductor (MOS) Complementary Metal-oxide semiconductor (CMOS) Computer-Aided Design