Boolean Algebra and Logic Gates - KFUPM...Boolean Algebra- Formal Definitions deals with binary...

Boolean Algebra and Logic Gates

COE 202

Digital Logic Design

Dr. Abdulaziz Tabbakh

College of Computer Sciences and Engineering

King Fahd University of Petroleum and Minerals

EE 200– Digital Logic Circuit Design – KFUPM slide 2

Outline

Binary Logic and Gates.

Definition of Boolean Algebra

Axioms, Theorems and Properties of Boolean Algebra

Boolean Functions

Algebraic Manipulation

Canonical & Standard Forms

Minterms, Maxterms, SOP, POS.

Two-Level implementation

Physical Properties of Logic Gates:

Fan-in and Fan-out, Propagation delay, Cost.


Introduction Our objective is to learn how to design digital circuits.

These circuits use binary systems.

Signals in such binary systems may represent only one of 2 possible

values 0 or 1.

Physically, these signals are electrical voltage signals

These signals may assume either a high or a Low voltage value.

The High voltage value typically equals the voltage of the power

supply (e.g. 5 volts or 3.3 volts), and the Low voltage value is

typically 0 volts (or Ground).

When a signal is at the High voltage value, we say that the signal

has a Logic 1 value.

When a signal is at the Low voltage value, we say that the signal

has a Logic 0 value.


Digital Circuits

Generally, the circuit will have a number of input signals

(say n of them) as x1, x2, up to xn, and a number of

output signals (say m ) Z1, Z2, up to Zm.

The value assumed by the ith output signal Zi depends

on the values of the input signals x1, x2, up to xn.

In other words, we can say that Zi is a function of the n

input signals x1, x2, up to xn. Or we can write:

Zi = Fi (x1, x2, ……, xn ) for i = 1, 2, 3, ….m

The m output functions (Fi) are functions of binary

signals and each produces a single binary output signal.

Thus, these functions are binary functions and require

binary logic algebra for their derivation and manipulation.


Boolean Algebra

This binary system algebra is commonly referred to as

Boolean Algebra after the mathematician George Boole.

The functions are known as Boolean functions while the

binary signals are represented by Boolean variables.

To be able to design a digital circuit, we must learn how

to derive the Boolean function implemented by this

circuit.

Systems manipulating Binary Logic Signals are

commonly referred to as Binary Logic systems.

Digital circuits implementing a particular Binary

(Boolean) function are commonly known as Logic

Circuits.


Binary Logic

Binary logic deals with variables (also called literals)

that can takes one of two possible values (1 = True, 0 =

False) and operations that assume logical meaning.

A literal is a binary variable or its complement

Binary Logic is also called (Boolean Algebra)

Binary Logic consists of binary variables and logical

operations

Variables can be designated by letters (A, B, C, x, y, z)

with two possible values (1,0).

Three basic logical operators: AND, OR, NOT

They form logic functions


Binary Logic

Binary logic should not be confused with binary arithmetic

Arithmetic: 1+1=10 (one plus one equals two)

Logic: 1+1=1 (one or one equals one)

Truth Table is a table of all possible combinations of the

variables, showing the relation between the values the

variables may take and the result.


Elements of Boolean Algebra (Binary Logic)

As in standard algebra, Boolean algebra has 3 main

elements:

1. Constants,

2. Variables, and

3. Operators.

Logically

Constant Values are either 0 or 1

Binary Variables ∈{ 0, 1}

3 Possible Operators: The AND operator, the OR operator, and

the NOT operator.


Elements of Boolean Algebra (Binary Logic)

Physically

Constants ⇒

Power Supply Voltage (Logic 1)

Ground Voltage (Logic 0)

Variables ⇒ Signals (High = 1, Low = 0)

Operators ⇒ Electronic Devices (Logic Gates)

1. AND - Gate

2. OR - Gate

3. NOT - Gate (Inverter)


Logic Gates

Logic gates: Circuits that implement logic functions

Boolean Algebra: a useful mathematical system for

specifying and transforming logic functions

We will study Boolean algebra as a foundation for

designing and analyzing digital systems

The three basic logical operations are:

AND is denoted by a dot (·)

OR is denoted by a plus (+)

NOT is denoted by an overbar ( ¯ ), a single quote mark (') after,

or (~ or #) before the literal, e.g. A, A’, ~A, or #A

yAB

Logic

CircuitC

Input literals Output literal


AND Operator

Represented as:

𝑥. 𝑦 = 𝑧 𝑜𝑟 𝑥𝑦 = 𝑧

read as “X and Y equals to Z”

Z=1 if and only if x=1 and y=1


OR Operator

Represented as:

𝑥 + 𝑦 = 𝑧

read as “X or Y equals to Z”

Z=1 if x=1 or y=1 or if both x=1 and y=1


NOT Operator

Represented as:

ҧ𝑥 = 𝑧 𝑜𝑟 𝑥′ = 𝑧

read as “Not X equals to Z” meaning z is not what x

Also called Negation or complementing

Z=1 if X=0 and Z=1 if X=0


Logic Gates

Logic gates are electronic circuits that operates on one

or more input signals to produce an output signal.

In today’s computers,

Switches are implemented

using transistors, e.g. 0 = Low Voltage, e.g. 0 V

1 = High Voltage, e.g. 3 V


Boolean Algebra- Formal Definitions

deals with binary literals and logic functions

A Boolean Expression (e.g. X+YZ) in Boolean Algebra is formed by:

Binary literals and constants (0,1)

Logic operations (operators) on the literals and constants

Parenthesis

A Boolean Function can be described by :

Boolean Equation: Output = Boolean Expression (not unique)

Logic Diagram (gates)(not unique)

Truth Table (unique) maps each possible combination of the input

literals to the corresponding output literal (n input literals 2n

combinations)

Simplest functions require the smallest number of the smallest gates

and therefore are most economical to implement


Logic Circuits and Boolean Expressions

A Boolean expression (or a Boolean function) is a

combination of Boolean variables, AND-operators, OR-

operators, and NOT operators.

Boolean Expressions (Functions) are fully defined by

their truth tables.

Each Boolean function (expression) can be implemented

by a digital logic circuit which consists of logic gates.

Variables of the function correspond to signals in the logic

circuit,

Operators of the function are converted into corresponding logic

gates in the logic circuit.


Basic Theorem

Existence

Exist. of comp

Idempotence

Existence


Theorem and Properties

1. Duality:

Duality principle states that every algebraic expression remains

valid if the operators and identity elements are interchanged

(i.e. replacing each 1 with a 0, each 0 with a 1, and replacing

each AND (.) with an OR (+), and each OR (+) with an AND(.)).

e.g. (X+Y+Z) and X.Y.Z (are Duals)

x+1=1 and x.0=0 (are Duals)

a·(b+c)=(a·b)+(a·c) and a+(b·c)=(a+b)·(a+c) (are Duals)

The dual of an expression does not equal the expression itself

unless the dual expression is the same as the original expressions.

E.g x’’=x (called self-dual)


Duality Principle

Given a Boolean expression,

its dual is obtained by

replacing each 1 with a 0, each 0

with a 1,

each AND (.) with an OR (+), and

each OR (+) with an AND(.).

The dual of an identity is also

an identity. This is known as

the duality principle.

It can be easily shown that the

AND basic identities and the

OR basic identities are duals.


Useful Theorems (in Dual forms)

Expression Dual

𝑥. 𝑦 + ҧ𝑥. 𝑦 = 𝑦 𝑥 + 𝑦 . ( ҧ𝑥 + 𝑦) = 𝑦 Minimization

𝑥 + 𝑥. 𝑦 = 𝑥 𝑥. 𝑥 + 𝑦 = 𝑥 Absorption

𝑥 + ҧ𝑥. 𝑦 = 𝑥 + 𝑦 𝑥. ( ҧ𝑥 + 𝑦) = 𝑥𝑦 Simplification

𝑥. 𝑦 + ҧ𝑥. 𝑧 + 𝑦. 𝑧= 𝑥. 𝑦 + ҧ𝑥. 𝑧

𝑥 + 𝑦 . ҧ𝑥 + 𝑧 . 𝑦 + 𝑧= 𝑥 + 𝑦 . ( ҧ𝑥 + 𝑧)

Consensus

𝑥 + 𝑦 = ҧ𝑥. ത𝑦 𝑥. 𝑦 = ҧ𝑥 + ത𝑦 DeMorgan’sTheroem


Proof of Minimization

Consider the LHS form

x. y + തx. y = 𝑦(x + തx) = y

𝑥. 𝑦 + ҧ𝑥. 𝑦 = 𝑦 𝑥 + 𝑦 . ( ҧ𝑥 + 𝑦) = 𝑦 Minimization


Proof of Absorption


Proof of Simplification

Consider the LHS

x + തx. y = (x + തx)(x + y)

= 1. 𝑥 + 𝑦

= (𝑥 + 𝑦)

𝑥 + ҧ𝑥. 𝑦 = 𝑥 + 𝑦 𝑥. ( ҧ𝑥 + 𝑦) = 𝑥𝑦 Simplification


Proof of Consensus

Consider :

𝑥𝑦 + 𝑥′𝑧 + 𝑦𝑧 = 𝑥𝑦 + 𝑥′𝑧 + 𝑦𝑧. 1

= 𝑥𝑦 + 𝑥′𝑧 + 𝑦𝑧 𝑥 + 𝑥′

= 𝑥𝑦 + 𝑥′𝑧 + 𝑥𝑦𝑧 + 𝑥′𝑦𝑧

= 𝑥𝑦 1 + 𝑧 + 𝑥′𝑧 1 + 𝑦

= 𝑥𝑦 + 𝑥′𝑧

𝑥. 𝑦 + ҧ𝑥. 𝑧 + 𝑦. 𝑧= 𝑥. 𝑦 + ҧ𝑥. 𝑧

𝑥 + 𝑦 . ҧ𝑥 + 𝑧 . 𝑦 + 𝑧= 𝑥 + 𝑦 . ( ҧ𝑥 + 𝑧)

Consensus


Proof of DeMorgan’s Law


Properties, Postulates & Theorems of Boolean Algebra

yx


Operator Precedence

Given the Boolean expression X.Y + W.Z the order of

applying the operators will affect the final value of the

expression.


Operator Precedence

For Boolean Algebra, the precedence rules for various

operators are given below, in a decreasing order of

priority:

Parenthesis

NOT Operator

AND Operator

OR Operator


Boolean Functions

A Boolean function is described by an algebraic

expression that expresses the logical relationship

between binary variables and consists of:

Binary variables

Constants (0,1)

Logical operators

The function can be evaluated to a specific value for

given value of the Boolean variables.

F = x + y’z


Boolean Functions

A Boolean function can be represented by:

Truth Table. (2n rows)

Circuit Diagram.

Function of 3 input variables

23 = 8 input combinations

Truth table has 8 rows

Table lists all possible

combinations of the inputs

and the corresponding output


Boolean Functions Expression

One truth table

Multiple algebraic expressions multiple circuits.

By manipulating a Boolean expression according to the

rules of Boolean algebra, it is sometimes possible to

obtain a simpler expression for the same function and

thus reduce the number of gates in the circuit and the

number of inputs to the gate.

Designers are motivated to reduce the complexity and

number of gates to reduce the cost of the circuit.


Example



A literal is a single variable within a term, in

complemented or uncomplemented form

Simplification means to contain the smallest number of

literals

Less literals and terms simpler circuit.

Use theorem and axioms to simplify functions.

e.g.

F2 = x’y’z + x’yz + xy’

= x’z(y’+y) + xy’

= x’z(1) + xy’

= x’z + xy’


Examples

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 + 𝑥𝑦𝑧 + 𝑥′𝑦𝑧= 𝑥𝑦 1 + 𝑧 + 𝑥′𝑧 1 + 𝑦 = 𝑥𝑦 + 𝑥′𝑧

𝑥 + 𝑦 𝑥′ + 𝑧 𝑦 + 𝑧 = (𝑥 + 𝑦)(𝑥′ + 𝑧)



Example: Simplify the function


Complement of a Function The complement of a function can be derived algebraically using

DeMorgan’s theorems for two or more variables

(a+b+c+d)’ = a’b’c’d’

(abcd)’ = a’+b’+c’+d’

The generalized form of DeMorgan’s theorems states that the

complement of a function is obtained by interchanging AND and

OR operators and complementing each literal.

Be careful with operators precedence!!! Start with the outermost

operation; complement the operation and the operands, then the next

outermost operation and so on

The only difference between the dual of an expression and the

complement of that expression is that

in the dual variables are not complemented while in the complement

expression, all variables are complemented.


Example

F=(x’yz’ + x’y’z)

F’=(x’yz’ + x’y’z)’ = (x’yz’)’ (x’y’z)’

= (x+y’+z)(x+y+z’)

take the dual of the function and complement each literal

Fdual = (x’+y+z’)(x’+y’+z)

F’ = (x+y’+z)(x+y+z’)

G = (a’ + bc)d’ + e

G’ = [(a’ + bc)d+ e]’ = [(a + bc) . d’]’. e’

= [(a’ + bc)’+ d’’]. e’

= [a’’. (b.c)’ + d]. e’

= [a. (b’+c’) + d]. e’


Canonical and Standard Forms Minterms or standard products (AND operation)

For a Boolean function of n variables, a product term (ANDedterm) in which each variable appears once (in either its true or complemented form) is called a minterm.

There are 2n different Minterms.

Maxterms or standard sums (OR operation)

For a Boolean function of n variables, a sum term (ORed term) in which each variable appears once (in either its true or complemented form) is called a maxterm.

There are 2n different maxterms

Each maxterm is the complement of its corresponding minterm

A term that is a minterm or a max term SHOULD INCLUDE ALL INPUT VARIABLES


Minterms and Maxterms – variables order

Minterms and maxterms are designated with a subscript

The subscript is a decimal number that represents the

binary number of input literals in the straight binary

The bits in the pattern represent the complemented or

normal state of each literal listed in a standard fixed

order (MSB…LSB)

All literals will be present in a minterm or maxterm and

will be listed in the same order (usually alphabetically)

Examples of Standard forms: For 3 variables: a, b, c

Maxterms: (a+b’+c) = M010= M2, (a’+b+c’) = M101 =M5

Minterms: abc’ = m110 = m6, a’b’c = m001 = m1


Minterms Consider a system of 3 input signals (variables) x, y, & z.

A term which ANDs all input variables, either in the true or

complement form, is called a minterm.

Thus, the considered 3-input system has 8 minterms, namely:

Each minterm equals 1 at exactly one particular input combination

and is equal to 0 at all other combinations

Thus, for example, is always equal to 0 except for the input

combination xyz = 000, where it is equal to 1.

Accordingly, the minterm is referred to as m0.

In general, minterms are designated mi, where i corresponds the

input combination at which this minterm is equal to 1.


Minterms

For the 3-input system under consideration, the number

of possible input combinations is 23, or 8. This means

that the system has a total of 8 minterms as follows:


Maxterms

Consider a circuit of 3 input signals (variables) x, y, & z.

A term which ORs all input variables, either in the true or

complement form, is called a Maxterm.

With 3-input variables, the system under consideration

has a total of 8 Maxterms, namely:

Each Maxterm equals 0 at exactly one of the 8 possible

input combinations and is equal to 1 at all other

combinations.

For example, (x + y + z) equals 1 at all input combinations

except for the combination xyz = 000, where it is equal to 0.

Accordingly, the Maxterm (x + y + z) is referred to as M0.


Maxterms

In general, Maxterms are designated Mi, where i

corresponds to the input combination at which this

Maxterm is equal to 0.

For the 3-input system, the number of possible input

combinations is 23, or 8. This means that the system has

a total of 8 Maxterms as follows:


Minterms and Maxterms

mi

AND that gives 1

Mi

OR that gives 0The Index represents

the Input combination

in decimalNote: mi is the complement of Mi and vice versa

A product that gives 1 A sum that gives 0Input Combination


Miterms and Maxterms

A Boolean function can be expressed algebraically from

a given truth table by forming a minterm for each

variable combination that produce a ‘1’ in the output, and

then ORing them together. (Sum Of Products) (SOP)

Similarly, we can express the function by forming a

maxterm for each variable combination that produce a ‘0’

in the output, and then ANDing them together. (Product

Of Sums) (POS).

A function is said to be in Canonical form if it is

represented as a SOMinterms or POMaxterms.


Example

𝑓1(𝑥, 𝑦, 𝑧) = 𝑥′𝑦′𝑧 + 𝑥𝑦′𝑧′ + 𝑥𝑦𝑧 = 𝑚1 +𝑚4 +𝑚7 =

σ (1,4,7)

𝑓2(𝑥, 𝑦, 𝑧) = 𝑥′𝑦𝑧 + 𝑥𝑦′𝑧 + 𝑥𝑦𝑧′ + 𝑥𝑦𝑧 = 𝑚3 +𝑚5 +𝑚6 +

𝑚7 = σ (3,5,6,7)


Example


σ (1,4,7)

X Y Z i m1 + m4 + m7 f1

0 0 0 0 0 + 0 + 0 0

0 0 1 1 1 + 0 + 0 1

0 1 0 2 0 + 0 + 0 0

0 1 1 3 0 + 0 + 0 0

1 0 0 4 0 + 1 + 0 1

1 0 1 5 0 + 0 + 0 0

1 1 0 6 0 + 0 + 0 0

1 1 1 7 0 + 0 + 1 1

Function is 1 at each of

its specified minterms

So, given a truth table,

How to determine the

function?

As the sum of all

minterms for which the

function is 1 !….


Example


σ (1,4,7)

𝑓1(𝑥, 𝑦, 𝑧) = 𝑥 + 𝑦 + 𝑧 . 𝑥 + 𝑦′ + 𝑧 . 𝑥 + 𝑦′ + 𝑧′ . (𝑥′ +

X Y Z i M0 . M2 . M3 . M5 . M6 f1

0 0 0 0 0 . 1 . 1 . 1 . 1 0

0 0 1 1 1 . 1 . 1 . 1 . 1 1

0 1 0 2 1 . 0 . 1 . 1 . 1 0

0 1 1 3 1 . 1 . 0 . 1 . 1 0

1 0 0 4 1 . 1 . 1 . 1 . 1 1

1 0 1 5 1 . 1 . 1 . 0 . 1 0

1 1 0 6 1 . 1 . 1 . 1 . 0 0

1 1 1 7 1 . 1 . 1 . 1 . 1 1

Function is 0 at each of

its specified maxterms

So, given a truth table,

How to determine the

function?

As the product of all

maxterms for which the

function is 0 !….


Operations on Functions

The AND operation on two functions corresponds to the

intersection of the two sets of minterms of the functions

The OR operation on two functions corresponds to the

union of the two sets of minterms of the functions

Example

Let F(A,B,C)=Σm(1, 3, 6, 7) and G(A,B,C)=Σm(0,1, 2, 4,6, 7)

F . G = Σm(1, 6, 7)

F + G = Σm(0,1, 2, 3, 4,6, 7)

F’ . G = ?

F’ = Σm(0, 2, 4, 5)

F’ . G = Σm(0, 2, 4)


Minterms of Function Complement

Forming a minterm for each combination that produces a

0 in the function and then ORing them.

𝑓1 = 𝑚1 +𝑚4 +𝑚7 ⇒ 𝑓′1 = 𝑚0 +𝑚2 +𝑚3 +𝑚5 +𝑚6𝑓′1 = 𝑥

′𝑦′𝑧′ + 𝑥′𝑦𝑧′ + 𝑥′𝑦𝑧 + 𝑥𝑦′𝑧 + 𝑥𝑦𝑧′

Take the complement of f’1 to get f1

𝑓1 = 𝑥 + 𝑦 + 𝑧 𝑥 + 𝑦′ + 𝑧 𝑥 + 𝑦′ + 𝑧′ 𝑥′ + 𝑦 + 𝑧′ 𝑥′ + 𝑦′ + 𝑧

𝑓1 = 𝑀0𝑀2𝑀3𝑀5𝑀6 =ෑ(0,2,3,5,6)

𝑓1 = 𝑥′𝑦′𝑧 + 𝑥𝑦′𝑧′ + 𝑥𝑦𝑧 = 𝑚1 +𝑚4 +𝑚7 =𝑚(1,4,7)


Conversion to Minterm

It is sometimes convenient to express a Boolean function

in its sum of minterms (SOP) form.

Expanding the expression into a sum of AND terms

Inspect each term

If it contains all the variables (good)

If it misses one or more variables, it is ANDed with an

expression such as (x + x’)


Example

Express the Boolean function 𝐹 = 𝐴 + 𝐵′𝐶 as a sum of minterms.

The first term A is missing two variables

𝐴 = 𝐴 𝐵 + 𝐵′ = 𝐴𝐵 + 𝐴𝐵′ = 𝐴𝐵 𝐶 + 𝐶′ + 𝐴𝐵′(𝐶 + 𝐶′)𝐴 = 𝐴𝐵𝐶 + 𝐴𝐵𝐶′ + 𝐴𝐵′𝐶 + 𝐴𝐵′𝐶′

The second term B’C is missing one variables

𝐵′𝐶 = 𝐵′𝐶 𝐴 + 𝐴′ = 𝐴𝐵′𝐶 + 𝐴′𝐵′𝐶

Combine all terms

𝐹 = 𝐴 + 𝐵′𝐶 = 𝐴𝐵𝐶 + 𝐴𝐵𝐶′ + 𝐴𝐵′𝐶 + 𝐴𝐵′𝐶′ + 𝐴′𝐵′𝐶𝐹 = 𝑚1 +𝑚4 +𝑚5 +𝑚6 +𝑚7

𝐹(𝐴, 𝐵, 𝐶) =(1,4,5,6,7)


Example

An alternative procedure for deriving the minterms of a

Boolean function is to obtain the truth table of the

function directly from the algebraic expression and then

read the minterms from the truth table.

𝐹 = 𝐴 + 𝐵′𝐶𝐹 = 𝑚1 +𝑚4 +𝑚5 +𝑚6 +𝑚7


Conversion to Maxterms

Express it in form of OR terms (use distributive law)

Any missing variable x in each term is ORed with xx’

Express the Boolean function 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)

Each OR term is missing one variable.

𝑥′ + 𝑦 = 𝑥’ + 𝑦 + 𝑧𝑧’ = (𝑥’ + 𝑦 + 𝑧)(𝑥’ + 𝑦 + 𝑧’)𝑥 + 𝑧 = 𝑥 + 𝑧 + 𝑦𝑦’ = (𝑥 + 𝑦 + 𝑧)(𝑥 + 𝑦’ + 𝑧)𝑦 + 𝑧 = 𝑦 + 𝑧 + 𝑥𝑥 = (𝑥 + 𝑦 + 𝑧)(𝑥’ + 𝑦 + 𝑧)

Combine.

𝐹 = 𝑥 + 𝑦 + 𝑧 𝑥 + 𝑦’ + 𝑧 𝑥’ + 𝑦 + 𝑧 𝑥’ + 𝑦 + 𝑧’ = 𝑀0𝑀2𝑀4𝑀5

𝐹(𝑥, 𝑦, 𝑧) =ෑ(0,2,4,5)


Conversion between Canonical Forms

The minterms of F’ are the minterms missing from F

𝐹 𝐴, 𝐵, 𝐶 = σ 1,4,5,6,7 ⇒ 𝐹’ 𝐴, 𝐵, 𝐶 = σ(0,2,3)

Now, if we take the complement of F‘ by DeMorgan’s

theorem:

𝐹’ ’ = 𝐹 = 𝑚0+𝑚2+𝑚3 ’ = 𝑚0′ . 𝑚2

′ . 𝑚3′ = 𝑀0𝑀2𝑀3

=ෑ(0,2,3)

To convert from one canonical form to another,

interchange the symbols ∑ and ∏ and list those numbers

missing from the original form.

Maxterm with subscript j is a complement of the Minterm with the

same subscript j and vice versa.


Conversion between Canonical Forms

𝐹 =(1,3,6,7) =ෑ(0,2,4,5)


Standard Form In Canonical forms, minterms and maxterms MUST contain

all variables.

In Standard forms, Sum of Products (SOP) and Product of Sums (POS) contain any number variables.

𝐹 = (𝑥 + 𝑦 + 𝑧)(𝑥 + 𝑦’ + 𝑧)(𝑥’ + 𝑦 + 𝑧)(𝑥’ + 𝑦 + 𝑧’)𝐹 = 𝑥𝑦 + 𝑥’𝑧

The sum of products is a Boolean expression containing

AND terms, called product terms, with one or more literals

each. The sum denotes the ORing of these terms.

A product of sums is a Boolean expression containing OR

terms, called sum terms. Each term may have any number

of literals. The product denotes the ANDing of these terms.


Standard Forms

These standard forms results in a two‐level structure of gates.

𝐹1 = 𝑦’ + 𝑥𝑦 + 𝑥’𝑦𝑧’ 𝐹2 = 𝑥(𝑦’ + 𝑧)(𝑥’ + 𝑦 + 𝑧)


Standard vs. Non-Standard Form

a two‐level implementation is

preferred :

it produces the least

amount of delay through

the gates

However, the number of

inputs to a given gate

might not be practical.


Two-Level Implementations of Standard Forms

Sum of Products Expressions (SOP):

Any SOP expression can be implemented in 2-levels of

gates.

The first level consists of a number of AND gates which

equals the number of product terms in the expression.

Each AND gate implements one of the product terms in

the expression.

The second level consists of a SINGLE OR gate whose

number of inputs equals the number of product terms in

the expression.



Example: Implement the following SOP function

F = XZ + Y`Z + X`YZ



Product of Sums Expression (POS):

Any POS expression can be implemented in 2-levels of

gates.

The first level consists of a number of OR gates which

equals the number of sum terms in the expression.

Each gate implements one of the sum terms in the

expression.

The second level consists of a SINGLE AND gate whose

number of inputs equals the number of sum terms.



Example: Implement the following POS function

F = (X+Z )(Y`+Z)(X`+Y+Z )


VLSI Technology Parameters Propagation Delay : The time required for a change in the value of

a signal to propagate from an input to an output

Fan-out: number of standard loads a gate output can drive

Fan-in: the number of inputs available on a gate. Usually limited by

speed considerations to 4-5 inputs. Larger input gates are

implemented from a multiple of such smaller gates

Cost of a gate: a measure of the contribution by the gate to the cost

of the integrated circuit (its area)

Power Dissipation: the amount of power drawn from the power

supply and consumed by the gate

Noise Margin: the maximum external noise voltage that will not

cause an undesirable change in the logic output of a gate output

when superimposed on a normal input level


Propagation Delay

Consider the shown inverter with

input X and output Z.

A change in the input (X) from 0 to

1 causes the inverter output (Z) to

change from 1 to 0.

The change in the output (Z),

however is not instantaneous.

Rather, it occurs slightly after the

input change.

This delay between an input signal

change and the corresponding

output signal change is what is

known as the propagation delay.


Propagation Delay

A signal change on the input of some IC takes a finite

amount of time to cause a corresponding change on the

output.

This finite delay time is known as Propagation Delay.

Faster circuits are characterized by smaller propagation

delays.

Higher performance systems require higher speeds

(smaller propagation delays).


Timing Diagrams

A timing diagram shows the logic values of signals in a

circuit versus time.

A signal shape versus time is typically referred to as

Waveform.

Example:


Computing Longest Delay

Each gate has a given propagation delay.

We start at the inputs and compute the delay at the

output of each gate as follows:

The delay at the output of a gate = gate propagation delay +

maximum delay at its inputs

Maximum propagation delay from any input to any output

is called the Critical Path.

The critical path determines the minimum clock period

(T) and the maximum clock frequency (f).

Clock frequency (f) = 1 / T


Computing Longest Delay

Example: Assume that delay of each gate is related to

number of its inputs i.e. delay of 1 input gate is 1 ns,

delay of 2-input gate is 2 ns. Compute longest

propagation delay and maximum frequency.

Longest propagation delay = 7 ns

Maximum frequency = 1 / 7ns= 143 MHZ.


Exercises

Date post:	14-Feb-2021
Category:	Documents
Upload:	others
View:	13 times
Download:	3 times

Boolean Algebra and Logic Gates - KFUPM...Boolean Algebra- Formal Definitions deals with binary...

Documents