Digital Logic Design presentation Boolean Algebra and Logic Gates.pptx

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Dr Yasir Awais Butt Digital Logic Design
BOOLEAN ALGEBRA AND LOGIC
GATES
Chapter 2

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Introduction
 Cost of the digital circuits is an important factor addressed by designers—be
they computer engineers, electrical engineers, or computer scientists.
 Finding simpler and cheaper, but equivalent, realizations of a circuit can reap
huge payoffs in reducing the overall cost of the design.
 Mathematical methods that simplify circuits rely primarily on Boolean algebra.
 Boolean Algebra (formulated by E.V. Huntington, 1904)
 Boolean algebra enables you to optimize simple circuits and to understand the
purpose of algorithms used by software tools to optimize complex circuits
involving millions of logic gates.

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2.1 BASIC DEFINITIONS

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Basıc Defınıtıons
 A set is collection of elements having the same property.
 S: set, x and y: element or event
 For example: S = {1, 2, 3, 4}
 If x = 2, then xÎS.
 If y = 5, then y S.
 A binary operator defines on a set S of elements is a rule that assigns, to each
pair of elements from S, a unique element from S.
 For example: given a set S, consider a*b = c and * is a binary operator.
 If (a, b) through * get c and a, b, cÎS, then * is a binary operator of S.
 On the other hand, if * is not a binary operator of S and a, bÎS, then c  S.

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2.1 Algebras
 What is an algebra?
 An algebra is an algebraic structure consisting of a set together with operations
of multiplication and addition and scalar multiplication by elements of a field
and satisfying the axioms implied by "vector space" and "bilinear"
 Mathematical system consisting of
 Set of elements (example: N = {1,2,3,4,…})
 Set of operators (+, -, ×, ÷)
 Axioms or postulates (associativity, distributivity, closure, identity elements, etc.)
 Why is it important?
 Defines rules of “calculations”
 Note: operators with two inputs are called binary
 Does not mean they are restricted to binary numbers!
 Operator(s) with one input are called unary

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Basıc Defınıtıons
 The common postulates used to formulate algebraic structures are:
 Closure: a set S is closed with respect to a binary operator if, for every pair of
elements of S, the binary operator specifies a rule for obtaining a unique
element of S.
 For example, natural numbers is closed w.r.t. the binary operator + by the rule of
arithmetic addition, since, for any , there is a unique cÎN such that
 But operator – is not closed for , because and , but .
 Associative law: a binary operator * on a set S is said to be associative whenever
 for all
 Commutative law: a binary operator * on a set S is said to be commutative
when
 and

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BASIC DEFINITIONS
 Identity element: a set S is said to have an identity element with respect to a
binary operation * on S if there exists an element eÎS with the property that
 e * x = x * e = x for every
 for every I = {…, -3, -2, -1, 0, 1, 2, 3, …}.
 for every I = {…, -3, -2, -1, 0, 1, 2, 3, …}.
 Inverse: a set having the identity element e with respect to the binary operator
to have an inverse whenever, for every xÎS, there exists an element yÎS such that
 The operator + over I, with e = 0, the inverse of an element a is (-a), since a+(-a) = 0.
 Distributive law: if (*) and (.) are two binary operators on a set S, is said to be
distributive over (.) whenever

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George Boole
 Father of Boolean algebra
 He came up with a type of linguistic algebra, the three most
basic operations of which were (and still are) AND, OR and
NOT. It was these three functions that formed the basis of
his premise, and were the only operations necessary to
perform comparisons or basic mathematical functions.
 Boole’s system was based on a binary approach, processing
only two objects - the yes-no, true-false, on-off, zero-one
approach.
 Surprisingly, given his standing in the academic
community, Boole's idea was either criticized or completely
ignored by the majority of his peers.
 Eventually, one bright student, Claude Shunnon(1916-
2001), picked up the idea and ran with it
George Boole (1815 - 1864)

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2.3 AXIOMATİC DEFİNİTİON OF
BOOLEAN ALGEBRA

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Axiomatic Definition of Boolean Algebra
 Boolean Algebra is based on set B={0,1} and two binary operators + and •
 Huntington postulates
 1. Closure w.r.t. the operator + (•)
 x, y B x+y B; x, y B x•y B
∈ ⇒ ∈ ∈ ⇒ ∈
 2. Associative w.r.t. + (•)
 (x+y)+z = x + (y + z); (x•y)•z = x • (y•z)
 3. Commutative w.r.t. + (•)
 x+y = y+x; x•y = y•x
 4. An identity element w.r.t. + (•)
 0+x = x+0 = x; 1•x = x•1= x
 5. x B, x' B (complement of x)
∀ ∈ ∃ ∈
 x+x'=1; x•x'=0
 6. • is distributive over + : x•(y+z)=(x•y)+(x•z)
 + is distributive over •: x+ (y•z)=(x+ y)•(x+ z)

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Boolean Algebra
 Terminology:
 Literal: A variable or its complement
 Product term: literals connected by (·)
 Sum term: literals connected by (+)

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Postulates of Two-Valued Boolean Algebra
 B = {0, 1} and two binary operations, (+) and (.)
 The rules of operations: AND 、 OR and NOT.
1. Closure (+ and )
‧
2. The identity elements
(1) + = 0
(2) · = 1
x y x.y
0 0 0
0 1 0
1 0 0
1 1 1
AND
x y x+y
0 0 0
0 1 1
1 0 1
1 1 1
OR
x x’
0 1
1 0
NOT

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3. The commutative laws x+y = y+x, x.y = y.x
4. The distributive laws
x y z y+z
x ．
(y+z)
x ．
y
x ．
z
(x ． y)+(x ．
z)
0 0 0 0 0 0 0 0
0 0 1 1 0 0 0 0
0 1 0 1 0 0 0 0
0 1 1 1 0 0 0 0
1 0 0 0 0 0 0 0
1 0 1 1 1 0 1 1
1 1 0 1 1 1 0 1
1 1 1 1 1 1 1 1

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5. Complement
 x+x'=1 → 0+0'=0+1=1; 1+1'=1+0=1
 x ． x'=0 → 0 ． 0'=0 ． 1=0; 1 ． 1'=1 ． 0=0
6. Has two distinct elements 1 and 0, with 0 ≠ 1
 Note
 A set of two elements
 (+) : OR operation; (·) : AND operation
 A complement operator: NOT operation
 Binary logic is a two-valued Boolean algebra

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2.4 BASIC THEOREMS AND
PROPERTIES OF BOOLEAN ALGEBRA
DUALITY

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Basic Theorems And Properties Of Boolean Algebra Duality
 The principle of duality is an important concept.
 This says that if an expression is valid in Boolean algebra, the dual of that
expression is also valid.
 To form the dual of an expression, replace all (+) operators with (·) operators, all
(·) operators with (+) operators, all ones with zeros, and all zeros with ones.
 Following the replacement rules…
 a(b + c) = ab + ac
 Form the dual of the expression
 a + (bc) = (a + b)(a + c)
 Take care not to alter the location of the parentheses if they are present.

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Basic Theorems

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Boolean Theorems
 Huntington’s postulates define some rules
 Need more rules to modify algebraic expressions
 Theorems that are derived from postulates
 What is a theorem?
 A formula or statement that is derived from postulates (or other proven theorems)
 Basic theorems of Boolean algebra
 Theorem 1 (a): x + x = x (b): x · x = x
 Looks straightforward, but needs to be proven !
Post. 1: closure
Post. 2: (a) x+0=x, (b) x·1=x
Post. 3: (a) x+y=y+x, (b) x·y=y·x
Post. 4: (a) x(y+z) = xy+xz, (b) x+yz = (x+y)(x+z)
Post. 5: (a) x+x’=1, (b) x·x’=0

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Proof of x+x=x
 We can only use Huntington postulates:
 Show that x+x=x.
 x+x = (x+x)·1 by 2(b)
 = (x+x)(x+x’) by 5(a)
 = x+xx’ by 4(b)
 = x+0 by 5(b)
 = x by 2(a)
 Q.E.D.
 We can now use Theorem 1(a) in future proofs
Post. 2: (a) x+0=x, (b) x·1=x
Post. 5: (a) x+x’=1, (b) x·x’=0

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Proof of x·x=x
 Similar to previous proof
 Show that x·x = x.
 x·x = xx+0 by 2(a)
 = xx+xx’ by 5(b)
 = x(x+x’) by 4(a)
 = x·1 by 5(a)
 = x by 2(b)
 Q.E.D.
Post. 2: (a) x+0=x, (b) x·1=x
Post. 5: (a) x+x’=1, (b) x·x’=0
Th. 1: (a) x+x=x

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Proof of x+1=1
 Theorem 2(a): x + 1 = 1
 x + 1 = 1 ． (x + 1) by 2(b)
 =(x + x')(x + 1) 5(a)
 = x + x' 1 4(b)
 = x + x' 2(b)
 = 1 5(a)
 Theorem 2(b): x ． 0 = 0 by duality
 Theorem 3: (x')' = x
 Postulate 5 defines the complement of x, x + x' = 1 and x x' = 0
 The complement of x' is x is also (x')'
Post. 2: (a) x+0=x, (b) x·1=x
Post. 5: (a) x+x’=1, (b) x·x’=0
Th. 1: (a) x+x=x

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Absorption Property (Covering)
x y xy x+xy
0 0 0 0
0 1 0 0
1 0 0 1
1 1 1 1
 Theorem 6(a): x + xy = x
 x + xy = x ． 1 + xy by 2(b)
= x (1 + y) 4(a)
= x (y + 1) 3(a)
= x ． 1 Th 2(a)
= x 2(b)
 Theorem 6(b): x (x + y) = x by duality
 By means of truth table (another way to proof )
Post. 2: (a) x+0=x, (b) x·1=x
Post. 5: (a) x+x’=1, (b) x·x’=0
Th. 2: (a) x+1=1

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DeMorgan’s Theorem
x y x’ y’ x+y (x+y)’ x’y’ xy x’+y' (xy)’
0 0 1 1 0 1 1 0 1 1
0 1 1 0 1 0 0 0 1 1
1 0 0 1 1 0 0 0 1 1
1 1 0 0 1 0 0 1 0 0
 Theorem 5(a): (x + y)’ = x’y’
 Theorem 5(b): (xy)’ = x’ + y’
 By means of truth table

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Consensus Theorem
 xy + x’z + yz = xy + x’z
 (x+y)•(x’+z)•(y+z) = (x+y)•(x’+z) -- (dual)
 Proof:
 xy + x’z + yz
 = xy + x’z + 1.yz 2(a)
 = xy + x’z + (x+x’)yz 5(a)
 = xy + x’z + xyz + x’yz 3(b) &4(a)
 = (xy + xyz) + (x’z + x’zy) Th4(a)
 = x(y + yz) + x’ (z + zy) 4(a)
 = xy + x’z Th6(a)
 QED (2 true by duality).

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Operator Precedence
 The operator precedence for evaluating Boolean Expression is
 Parentheses
 NOT
 AND
 OR
 Examples
 x y' + z
 (x y + z)'

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2.5 Boolean Functions
 A Boolean function
 Binary variables
 Binary operators OR and AND
 Unary operator NOT
 Parentheses
 Examples
 F1= x y z'
 F2 = x + y'z
 F3 = x' y' z + x' y z + x y'
 F4 = x y' + x' z

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Boolean Functions
 The truth table of 2n entries (n=number of variables)
 Two Boolean expressions may specify the same function
x y z F1 F2 F3 F4
0 0 0 0 0 0 0
0 0 1 0 1 1 1
0 1 0 0 0 0 0
0 1 1 0 0 1 1
1 0 0 0 1 1 1
1 0 1 0 1 1 1
1 1 0 1 1 0 0
1 1 1 0 1 0 0

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Boolean Functions
 Implementation with logic gates
 F4 is more economical
F4 = x y' + x' z
F3 = x' y' z + x' y z + x y'
F2 = x + y'z

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Algebraic Manipulation
 When a Boolean expression is implemented with logic gates, each term requires
a gate and each variable (Literal) within the term designates an input to the
gate. (F3 has 3 terms and 8 literal)
 To minimize Boolean expressions, minimize the number of literals and the
number of terms → a circuit with less equipment
 It is a hard problem (no specific rules to follow)
 Example 2.1
 x(x'+y) = xx' + xy = 0+xy = xy
 x+x'y = (x+x')(x+y) = 1 (x+y) = x+y
 (x+y)(x+y') = x+xy+xy'+yy' = x(1+y+y') = x
 xy + x'z + yz = xy + x'z + yz(x+x') = xy + x'z + yzx + yzx' = xy(1+z) + x'z(1+y) = xy
+x'z
 (x+y)(x'+z)(y+z) = (x+y)(x'+z), by duality from function 4. (consensus theorem
with duality)

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Complement of a Function
 An interchange of 0's for 1's and 1's for 0's in the value of F
 By DeMorgan's theorem
 (A+B+C)' = (A+X)' let B+C = X
 = A'X' by theorem 5(a) (DeMorgan's)
 = A'(B+C)' substitute B+C = X
 = A'(B'C') by theorem 5(a) (DeMorgan's)
 = A'B'C' by theorem 4(b) (associative)
 Generalization: a function is obtained by interchanging AND and OR operators
and complementing each literal.
 (A+B+C+D+ ... +F)' = A'B'C'D'... F'
 (ABCD ... F)' = A'+ B'+C'+D' ... +F'

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Examples
 Example 2.2
 F1' = (x'yz' + x'y'z)' = (x'yz')' (x'y'z)' = (x+y'+z) (x+y+z')
 F2' = [x(y'z'+yz)]' = x' + (y'z'+yz)' = x' + (y'z')' (yz)‘
 = x' + (y+z) (y'+z')
 = x' + yz‘+y'z
 Example 2.3: a simpler procedure
 Take the dual of the function and complement each literal
 F1 = x'yz' + x'y'z.
 The dual of F1 is (x'+y+z') (x'+y'+z).
 Complement each literal: (x+y'+z)(x+y+z') = F1'
 F2 = x(y' z' + yz).
 The dual of F2 is x+(y'+z') (y+z).
 Complement each literal: x'+(y+z)(y' +z') = F2'

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2.6 Canonical and Standard Forms
 Minterms and Maxterms
 A minterm (standard product): an AND term consists of all literals in their
normal form or in their complement form.
 For example, two binary variables x and y,
 xy, xy', x'y, x'y'
 It is also called a standard product.
 n variables can be combined to form 2n minterms.
 A maxterm (standard sums): an OR term
 It is also call a standard sum.
 2n maxterms.

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Minterms and Maxterms
 Each maxterm is the complement of its corresponding minterm, and vice versa.

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 An Boolean function can be expressed by
 A truth table
 Sum of minterms for each combination of variables that produces a (1) in the
function.
 f1 = x'y'z + xy'z' + xyz = m1 + m4 +m7 (Minterms)
 f2 = x'yz+ xy'z + xyz'+xyz = m3 + m5 +m6 + m7 (Minterms)

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 The complement of a Boolean function
 The minterms that produce a 0
 f1' = m0 + m2 +m3 + m5 + m6 = x'y'z'+x'yz'+x'yz+xy'z+xyz'
 f1 = (f1')'
 = (x+y+z)(x+y'+z) (x+y'+z') (x'+y+z')(x'+y'+z) = M0 M2 M3 M5 M6
 f2 = (x+y+z)(x+y+z')(x+y'+z)(x'+y+z)=M0M1M2M4
 Any Boolean function can be expressed asterms).
 A product of maxterms (“product” meaning the ANDing of terms).
 A sum of minterms (“sum” meaning the ORing of Both boolean functions are said
to be in Canonical form.

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Sum of Minterms
 Sum of minterms: there are 2n minterms and 22n combinations of functions
with n Boolean variables.
 Example 2.4: express F = A+B’C as a sum of minterms.
 F = A+B'C = A (B+B') + B'C = AB +AB' + B'C = AB(C+C') + AB'(C+C') + (A+A')B'C
= ABC+ABC'+AB'C+AB'C'+A'B'C
 F = A'B'C +AB'C' +AB'C+ABC'+ ABC = m1 + m4 +m5 + m6 + m7
 F(A, B, C) = S(1, 4, 5, 6, 7)
 or, built the truth table first

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Product of Maxterms
 Product of maxterms: using distributive law to expand.
 x + yz = (x + y)(x + z) = (x+y+zz')(x+z+yy') = (x+y+z)(x+y+z')(x+y'+z)
 Example 2.5: express F = xy + x'z as a product of maxterms.
 F = xy + x'z = (xy + x')(xy +z) = (x+x')(y+x')(x+z)(y+z) = (x'+y)(x+z)(y+z)
 x'+y = x' + y + zz' = (x'+y+z)(x'+y+z')
 F = (x+y+z)(x+y'+z)(x'+y+z)(x'+y+z') = M0M2M4M5
 F(x, y, z) = P(0, 2, 4, 5)

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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.
 F(A, B, C) = S(1, 4, 5, 6, 7)
 Thus, F‘ (A, B, C) = S(0, 2, 3)
 By DeMorgan's theorem
 F(A, B, C) = P(0, 2, 3)
 F'(A, B, C) =P (1, 4, 5, 6, 7)
 mj' = Mj
 To convert from one canonical form to another: interchange the symbols S and
P and list those numbers missing from the original form
 S of 1's
 P of 0's

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 Example
 F = xy + xz
 F(x, y, z) = S(1, 3, 6, 7)
 F(x, y, z) = P (0, 2, 4, 6)

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Standard Forms
 In canonical forms each minterm or maxterm must contain all the variables
either complemented or uncomplemented, thus these forms are very seldom the
ones with the least number of literals.
 Standard forms: the terms that form the function may obtain one, two, or any
number of literals, .There are two types of standard forms:
 Sum of products: F1 = y' + xy+ x'yz'
 Product of sums: F2 = x(y'+z)(x'+y+z')
 A Boolean function may be expressed in a nonstandard form
 F3 = AB + C(D + E)
 But it can be changed to a standard form by. using. The distributive law
 F3 = AB + C(D + E) = AB + CD + CE

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Implementation
 Two-level implementation
 Multi-level implementation
F1 = y' + xy+ x'yz' F2 = x(y'+z)(x'+y+z')

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2.7 Other Logic Operations
 2n rows in the truth table of n binary variables.
 22n functions for n binary variables.
 16 functions of two binary variables.
 All the new symbols except for the exclusive-OR symbol are not in common use
by digital designers.

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Boolean Expressions

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2.8 Digital Logic Gates
 Boolean expression: AND, OR and NOT operations
 Constructing gates of other logic operations
 The feasibility and economy;
 The possibility of extending gate's inputs;
 The basic properties of the binary operations (commutative and associative);
 The ability of the gate to implement Boolean functions.

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Standard Gates
 Consider the 16 functions in Table 2.8
 Two functions produce a constant : (F0 and F15).
 Four functions with unary operations: complement and transfer: (F3, F5, F10 and
F12).
 The other ten functions with binary operators
 Eight function are used as standard gates : complement (F12), transfer (F3),
AND (F1), OR (F7), NAND (F14), NOR (F8), XOR (F6), and equivalence (XNOR)
(F9).
 Complement: inverter.
 Transfer: buffer (increasing drive strength).
 Equivalence: XNOR.

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Summary of Logic Gates

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Summary of Logic Gates

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Multiple Inputs
 Extension to multiple inputs
 A gate can be extended to multiple inputs.
 If its binary operation is commutative and associative.
 AND and OR are commutative and associative.
 OR
– x+y = y+x
– (x+y)+z = x+(y+z) = x+y+z
 AND
– xy = yx
– (x y)z = x(y z) = x y z

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Multiple Inputs
 Multiple NOR = a complement of OR gate, Multiple NAND = a complement of
AND.
 The cascaded NAND operations = sum of products.
 The cascaded NOR operations = product of sums.
Multiple-input and cascated NOR and NAND gates

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Multiple Inputs
 The XOR and XNOR gates are commutative and associative.
 Multiple-input XOR gates are uncommon?
 XOR is an odd function: it is equal to 1 if the inputs variables have an odd
number of 1's.
Figure 2.8 3-input XOR gate

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Positive and Negative Logic
 Positive and Negative Logic
 Two signal values <=> two logic values
 Positive logic: H=1; L=0
 Negative logic: H=0; L=1
 Consider a TTL gates
 A positive logic AND gate
 A negative logic OR gate
Signal assignment and logic polarity

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Positive and Negative Logic
Demonstration of positive and negative logic

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2.9 Integrated Circuits
 Level of Integration
 An IC (a chip)
 Examples:
 Small-scale Integration (SSI): < 10 gates
 Medium-scale Integration (MSI): 10 ~ 100 gates
 Large-scale Integration (LSI): 100 ~ xk gates
 Very Large-scale Integration (VLSI): > xk gates
 VLSI
 Small size (compact size)
 Low cost
 Low power consumption
 High reliability
 High speed

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Digital Logic Families
 Digital logic families: circuit technology
 TTL: transistor-transistor logic (dying?)
 ECL: emitter-coupled logic (high speed, high power consumption)
 MOS: metal-oxide semiconductor (NMOS, high density)
 CMOS: complementary MOS (low power)
 BiCMOS: high speed, high density

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Digital Logic Families
 The characteristics of digital logic families
 Fan-out: the number of standard loads that the output of a typical gate can drive.
 Power dissipation.
 Propagation delay: the average transition delay time for the signal to propagate
from input to output.
 Noise margin: the minimum of external noise voltage that caused an undesirable
change in the circuit output.

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CAD
 CAD – Computer-Aided Design
 Software programs that support computer-based representation of circuits of
millions of gates.
 Automate the design process
 Two design entry:
 Schematic capture
 HDL – Hardware Description Language
– Verilog, VHDL
 Simulation
 Physical realization

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Home Work (4)
Digital Design (4th
)- Morris Mano-Page 66-
Problems:
2.3 d,f
2.4 d,e
2.6 Only for (2.3 d,f)
2.7 Only for (2.4 d,f)
2.9
2.20
2.22

59
Home Work (5)
Digital Design (4th
)- Morris Mano-Page 66-
Problems:
2.13
2.14
2.15
2.27
2.28

Digital Logic Design presentation Boolean Algebra and Logic Gates.pptx

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