Digital Logic
Definition: Chapter 3 digit, n. any of the Arabic figures of 0 through 9. digital, adj. involving using digits to Boolean Algebra and Combinational represent discretely all variables occurring Networks in a given problem. logic, n. a particular method of reasoning or argumentation.
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Analog Systems Digital vs. Analog
Definition: Digital solution to a problem is analog, adj. that solve a given problem by advantageous in that it allows us to build using physical analogies, such as electric more reliable components and implement voltages or shaft rotations (which are more sophisticated functions. quantitatively continuous), of the numerical variables occurring in the problem.
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Two main types of logic circuit: General form of a combinational circuit:
• Combinational (without memory) - chapters 2-6 x1 • Sequential (with memory) - chapters 7 and 8 f1(x1, x2, … xm)
x2 f2(x1, x2, … xm)
xm fn(x1, x2, … xm)
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1 For example, the behavior Truth table of the combinational circuit x to the right can be described f(x, y) • The behavior of a combinational circuit can by the truth table shown y below: always be specified by using a truth table.
x y f(x, y) A truth table exhaustively 0 0 f(0, 0) lists all possible input-output 0 1 f(0, 1) pairs. 1 0 f(1, 0) 1 1 f(1, 1)
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Example of a truth table Three Basic Logic Devices x y f(x, y)
0 0 0 x x x NOT y OR z AND z 0 1 1 y y
1 0 1 yy OR x NOT x 01AND 0 1 1 1 0 0 1 00 1 00 0 xx 1 0 11 1 10 1
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Logic symbols for the basic gates x y f(x, y) x 0 0 1 f(x, y) 0 1 1 x x x•y x+y y 1 0 0 y y 1 1 1
AND gate OR gate The intended function of this combinational circuit can be directly described by the truth table on the right, which can be represented by the Boolean function below: xx’ f(x, y) = x’y’ + x’y + xy NOT gate which can be realized by the logic circuit shown in the next slide.
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2 x xy y Algebra (Mathematical System) x’y f An algebra (or a mathematical system) consists of
S =
Realization of f(x, y) = x’y’ + x’y + xy where oi's are operations defined on S, ri's are relations defined among elements of S, and ci's are constants, some elements of S having some special significance. x Furthermore, it satisfies a set of axioms or postulates y f that governs its behavior. Minimal realization of f(x, y)
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Boolean Algebras Boolean Algebras (continued)
A Boolean algebra is a mathematical system: 2a. Existence of identity: there exists an element in B, denoted by 0, such that x+0=x for any x ∈ B. • B =
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Boolean Algebras (continued) Boolean Algebras (continued)
4a. Distributive law: for any x, y, z ∈ B, 6. There exists at least two elements in B x•(y+z) = (x•y)+(x•z). These are known as the Huntington postulates. 4b. Distributive law: for any x, y, z ∈ B, x+(y•z) = (x+y)•(x+z). 5. For any element x ∈ B there exists an element x' ∈ B such that (a) x+x'=1 and (b) x•x'=0.
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3 Huntington postulates Theorems
1. (a) x+y ∈ B (b) x•y ∈ B Thm 1: The element x’ in a Boolean algebra is 2. (a) x+0 = 0+x = x (b) x•1 = 1•x = x uniquely determined by x. 3. (a) x+y = y+x (b) x•y = y•x Thm 2: • 4. (a) x+(y•z) = (x+y)•(x+z) (a) x + 1 = 1 (b) x 0 = 0 (b) x•(y+z) = x•y+x•z Thm 3: (involution) 5. (a) x+x' = 1 (b) x•x' = 0 (a) 0’ = 1 (b) 1’ = 0 6. |B| ≥ 2
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Theorems (continued) Theorems (continued)
Thm 4: (both operations are idempotent) Thm 7: (a) x + x = x (b) x • x = x (a) x + x’y = x + y (b) x(x’ + y) = xy Thm 5: (involution) Thm 8: (associativity) (x')' = x (a) (x + y) + z = x + (y + z) Thm 6: (absorption) (b) (x • y) • z = x • (y • z) • (a) x + xy = x (b) x (x+y) = x Thm 9: (De Morgan's Law) (a) (x+y)'=x'•y' (b) (x•y)'=x'+y'
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Theorems (continued) Dual
Thm 10: Given an expression in Boolean algebra, its B = 〈 {0, 1}, +, •, = 〉, where the two dual is obtained by replacing every "+" operations are defined as shown below is a operator with "•" operator, and vice versa, Boolean algrbra. and by replacing every 0 with 1, and vice x+y y x•y y versa. +01 • 01
001 000 x x 111 101
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4 Principle of duality Perfect induction
The dual of any true statement in Boolean • Perfect induction is proof by exhaustion in algebra is also a true statement. which all possibilities are considered.
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The concept of a function Truth table and function
• It is a one-to-one or many-to-one mapping. • A truth table, therefore, describes a • The preceding truth table describes a function. mapping of a set of pairs of 0’s and 1’s into • A truth table can be represented by using a a set consisting of 0 and 1, viz., Boolean expression, i.e., an expression → → <0, 0> 0 <0, 1> 1 written in terms of the language of Boolean <1, 0> → 1 <1, 1> → 0 algebra. • That mapping is a function because it is many-to-one.
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Truth table and Boolean function Literals and product/sum terms
• An output of a combinational circuit, which •A literal is defined as each occurrence of can be exhaustively listed on a column in either a complemented or uncomplemented the truth table, can also be described by a variable in a Boolean expression. Boolean expression. •A product term is a literal or a product (i.e., • The Boolean expression can be constructed AND together, or conjunction) of literals. as explained below. •A sum term is a literal or a sum (i.e., OR together, or disjunction) of literals.
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5 Minterms of 2 variables Minterms minterm that is x y m-notation • A minterm is a product term in which all evaluated to 1 variables occur exactly once, either 0 0 x’y’ m0 complemented or uncomplemented. • A minterm has the property that it is 0 1 x’y m1 evaluated to 1 on one and only one row of 1 0 xy’ m the truth table. 2
1 1 xy m3
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Minterms of 3 variables minterm that is → x y z m-notation Truth table Boolean expression evaluated to 1 0 0 0 x’y’z’ m 0 • The function described by a truth table can 0 0 1 x’y’z m1 also be described by a sum (i.e., OR 0 1 0 x’yz’ m2 together, or disjunction) of minterms. 0 1 1 x’yz m 3 • The minterms involved are those, and only 1 0 0 xy’z’ m4 those, associated with the rows on which 1 0 1 xy’z m5 the minterm is evaluated to 1. 1 1 0 xyz’ m6
1 1 1 xyz m7
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Thus the following function can be described as f(x, y, z) = x’yz’ + xy’z + xyz f(x, y) = x’y’ + xy’ x y z f(x, y z) 0 0 0 0 x y f(x, y) 0 0 1 0 0 0 1 0 1 0 1 0 1 1 0 0 1 0 1 0 0 0 1 0 1 1 1 0 1 1 1 0 0 1 1 0 1 1 1 1
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6 Properties of minterms Row indices of a truth table
• For a function of n variables, there are 2n • It is convenient to refer to a row in a truth possible minterms. table by its index, which is equal to the • The sum of all possible minterms is equal to binary equivalent of the values assigned to 1. the variables on that row. • The product of any two minterms is equal to • Thus, for a truth table of a function of n 0 unless they are identical. variables, its rows are indexed by integers 0 through 2n from top to bottom.
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m-Notation Maxterms
• A minterm can be denoted by mi for short, • A maxterm is a sum term in which all where i is the index of the row on which variables occur exactly once, either that minterm is evaluated to 1. complemented or uncomplemented. • Thus f(x, y, z) = x’y’z’ + x’yz + xyz’ + xyz • A maxterm has the property that it is can be alternatively expressed as evaluated to 0 on one and only one row of the truth table. f(x, y, z) = m0 + m3 + m6 + m7 = Σm(0, 3, 6, 7)
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Maxterms of 3 variables Maxterms of 2 variables maxterm that is x y z M-notation maxterm that is evaluated to 0 x y M-notation evaluated to 0 0 0 0 x + y +z M0 0 0 1 x + y + z’ M 0 0 x + y M0 1 0 1 0 x + y’ + z M2 0 1 x + y’ M1 0 1 1 x + y’ + z’ M3
1 0 0 x’ + y + z M4 1 0 x’ + y M2 1 0 1 x’ + y + z’ M5 1 1 x’ + y’ M 3 1 1 0 x’ + y’ + z M6
1 1 1 x’ + y’ + z’ M7
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7 Thus the following function can be described as f(x, y) = (x + y’)(x’ + y’) Truth table → Boolean expression
• The function described by a truth table can x y f(x, y) also be described by a product (i.e., AND 0 0 1 together, or conjunction) of maxterms. • The maxterms involved are those, and only 0 1 0 those, associated with the rows on which 1 0 1 the maxterm is evaluated to 0. 1 1 0
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f(x, y, z) = (x + y + z)(x + y + z’)(x + y’ + z’)(x’ + y + z)(x’ + y’ + z) x y z f(x, y z) Properties of maxterms 0 0 0 0 0 0 1 0 • For a function of n variables, there are 2n 0 1 0 1 possible maxterms. 0 1 1 0 • The product of all possible maxterms is 1 0 0 0 equal to 0. 1 0 1 1 • The sum of any two maxterms is equal to 1 1 1 0 0 unless they are identical. 1 1 1 1
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M-Notation Normal forms
• A maxterm can be denoted by Mi for short, • A Boolean function expressed in sum-of- where i is the index of the row on which products form, also known as disjunctive that maxterm is evaluated to 0. normal form • Thus f(x, y, z) = (x+y+z)(x’+y+z)(x’+y’+z) • A Boolean function expressed in product- can be alternatively expressed as of-sums form, also known as conjunctive normal form f(x, y, z) = M0M4M6 = ΠM(0, 4, 6)
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8 Canonical forms Complementing a canonical formula
• Two types of expressions are directly • The complement of a canonical formula is obtainable from a truth table: another canonical formula of the same type minterm canonical formula consisting of all missing terms. (standard sum-of-products) maxterm canonical formula (standard product-of-sums)
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Transformation of canonical formula Double-rail logic
• A minterm canonical formula can be •In double-rail logic, it is assumed that all transformed into an equivalent maxterm variables and their complements are always canonical formula by available as inputs. 1. complementing it by using the property of • Otherwise, it is known as single-rail logic, a canonical formula, and then and NOT gates have to be used to obtain the complements. 2. complementing it again by applying the De Morgan’s Law. • Unless otherwise indicated, the use of double-rail logic is assumed.
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Number of levels These two circuits implement the same function.
• The largest number of gates a signal must pass through from input to output is called the number of level of logic.
f(w, x, y, z) = w’x + x’(y + z) = w’x + x’y + x’z
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9 2-level logic implementation of an odd-parity bit generator. The logic diagram of a BCD odd-parity bit generator.
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All 16 functions of 2 variables Universal set of operations
fi(x, y) = • A set of operations is said to be universal if i=0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 any function can be described by using that set of operations alone. y)' y)' y)' y)' y (XNOR) y (XNOR) y (XOR) y (XOR) y (IMPL.) y y (NAND) y (NAND) y (NOR) ⊂ ⊃ y (AND) y (AND) ↑ ⊃ ⊂ ⊕ ↓ ~ • x 1 x' x y' x y x x+y (OR) x x x (x •Examples: x y 0 x (x 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 {AND, OR, NOT} 0 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 1 0 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 {AND, NOT} {OR, NOT} {NAND} {NOR}
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NAND and NOR functions NAND and OR realization
• By using current technology, the simplest • A two-level NAND logic works exactly the logic gates are NAND and NOR gates. same as a two-level AND-OR logic. • The system with operations of NAND and • A two-level NOR logic works exactly the NOR does not constitute a Boolean algebra. same as a two-level OR-AND logic. • A procedure to do NAND or NOR • These two operations are not associative. realization: Write the function in sum of products (or product of sums), and then use a two-level logic.
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10 x1 x1 x 1 x2 x2 x 1 x 1 x 2 x 2 x3 x3 x 2 x4 x4 x5 x5 (a) x 1 x 2 = x 1 + x 2
x 1 x1 x 1 x 1 x2 x 2 x 2 x 2 x3 x4 (b) x 1 + x 2 = x 1 x 2 x5
DeMorgan’s theorem in terms of logic gates Using NAND gates to implement a sum-of-products
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Nand-gate symbol in general x 1 x 1 x 2 x 2
x 3 x 3 x 4 x 4 x 5 x 5
x 1 x 2
x 3 x 4 x 5
Using NOR gates to implement a product-of-sums
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Nor-gate symbol in general The universal property of nand-gates
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11 The universal property of nor-gates Steps involved in realizing f(w,x,y,z) = w'z + wz'(x+y')
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Steps involved in realizing f(x,y,z) = x(y'+z) Two equivalent not-gate symbols
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Steps involved in the realization of f(w,x,y,z) = w'z + wz'(x+y') Steps involved in realizing f(w,x,y,z) = w'z + wz'(x+y')
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12 Steps involved in realizing f(w,x,y,z) = w'z + wz'(x+y) Exclusive-OR function
• It is commutative and associative •x ⊕ 0 = x and x ⊕ 1 = x’ •x ⊕ x = 0 and x ⊕ x’ = 1 • It is also defined as “odd function” (meaning equal to one when the odd number of arguments are set to 1) for three or more variables.
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Exclusive-OR
= x’y + xy’
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Exclusive NOR Gate Properties
• Voltage representation of 0 and 1 • Positive and negative logic • Noise margins
= x’y’ + xy • Fan-out = x ≡ y • Propagation delays • Power dissipation
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13 The working of a gate is commonly described by a table like the one shown below. What gate is described by this table? It depends on the type of logic used. If positive logic is used, Power dissipation i.e., 0 ← L and 1 ← H then it is an AND gate. If negative logic is used, i.e., 0 ← H and 1 ← H, then it is an OR gate. • Low power dissipation and high switch speed are desired. The law of physics, input however, requires high power to achieve output x y fast switch. L L L • A common measure of gate performance is L H L the product of the propagation delay and the H L L power dissipation of the gate. H H H
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