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Digital Circuits

BLG231E

Instructor: Dr. Fuat Kucuk

1BLG231E DIGITAL CIRCUITS by Fuat Kucuk

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Binary Codes

BLG231E DIGITAL CIRCUITS by Fuat Kucuk 2

A binary code is a match representing each set of element with

binary values.

Since digital systems can only manipulate binary signals, physical

quantities (voltage, temperature, etc) and any other kinds of data

(sound, letter, decimal numbers) must be converted to binary form or

code.

Binary codes only change the symbols of sets of elements that is

used to represent information, not the meaning of information itself.

Each elements must be assigned to a unique binary bit

combination. If two or more elements can have the same value, the

code assignment will be ambiguous.

An n-bit binary code is a group of n-bits with 2ncombinations and

each combination representing one element of the set that is being

coded.

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Binary Codes


For example, four elements of a set can be coded with2-bits as: 00, 01,10,11.

A set of 8 elements requires at least 3-bit code.

Although the minimum number of bits required to code2nelements is n, there is no upper limit for bits for binary

coding.

That is, 8 elements can also be coded with 4-bits or

more. The more bits you use, the more sensitivity you get.

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Binary Codes


As example, seven colors of rainbow can be coded with 3

bits as follows:

Colour Binary Code

Red 000

Orange 001

Yellow 010

Green 011

Blue 101

Indigo 110

Violet 111

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BCD Code


Most people are more accustomed to the decimal system,

To make arithmetic operations in a digital systems, the decimal

numbers is first converted to binary. After performing all arithmetic

calculations in binary, the results is converted back to decimal.

For 10-decimal digits must be coded at least 4 bits although 24-10=6

combinations are not used.

This type of coding is called binary coded decimal (BCD).

symbol code symbol code

0 0000 5 0101

1 0001 6 01102 0010 7 0111

3 0011 8 1000

4 0100 9 1001

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BCD Code


Consider decimal 185 and its corresponding value in BCD and

binary:(185)10= (0001 1000 0101)BCD= (10111001)2

The BCD value has 12 bits, but the equivalent binary number

needs only 8 bits. It is obvious that a BCD number needs more

bits than its equivalent binary value.

Although BCD requires more bits, it presents advantages at

places that people interfere.

0001 1000 0101

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Gray Code


In Gray code, only one bit in the code group changes when

going from one number to the next.

One-bit

code

Two-bit

code

Three-bit

code

0 00 0 0 0

1 01 0 0 111 0 1 1

10 0 1 0

1 1 0

1 1 1

1 0 1

1 0 0

Two-bit code

Two-bit code in

reverse order

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Gray Code and BCD code comparison


Decimal BCD Gray0 000 000

1 001 001

2 010 011

3 011 010

4 100 110

5 101 111

6 110 1017 111 100

The first andlast terms also

obeys the Gray

code rule.

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Binary Logic


Binary logic deals with binary variables operations that has

logical meaning.

The variables are shown by letters such as A,B,C,x,y,z, etc. Eachvarible can take only two possible values: 0 and 1.

There are three basic logical operations: AND, OR, and NOT.

AND: This operation is denoted by a dot or by the absence ofany operator. For example, x y = z or xy = z is read "x AND y is

equal to z. The logical operation AND outputs z=1 if and only if

both input variables are x,y=1; otherwise z = 0.

x y xy0 0 0

0 1 0

1 0 0

1 1 1

AND operation.

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Binary Logic


OR: This operation is denoted by a plus sign. For example, x + y =

z is read "x OR y is equal to z. The logical operation OR outputs

z=0 if and only if both input variables are x,y=0; otherwise z = 1.

NOT: This operation is represented by a prime (sometimes by an

overbar). For example, x' = z is read "not x is equal to z," meaningthat z is what x is not. The logical operation NOT outputs z=0 if

input variable is x=1 and z=1 if input variables are x=0. This is also

referred to as complement operation.

x y x+y

0 0 0

0 1 1

1 0 1

1 1 1

OR operation.

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Binary Logic


Although similarities availeble, binary logic should not be con-

fused with binary arithmetic.

In binary arithmetic, one variable may includes many digits.

However, a logic variable is always either 1 or 0. for example,

In binary arithmetic, 1 + 1 = 10 ("one plus one is equal to two").

In binary logic 1 + 1 =1 ("one OR one is equal to one").

x x0 1

1 0

NOT operation.

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Logic Gates


A logic gate is an electronic circuit that operates with one or

more input signals and produces specific output signal.

Combinations of logic gates form digital circuits. These are

voltage-operated circuits and respond to two separate voltage

levels that represent a binary variable 0 or 1.

As an example, logic 0 and 1 may be defined as 0Vand 5V for aparticular digital systems.

Range of Logic 1

Range of Logic 0

volts

5V

3V

1V

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Symbols of the logic gates


Two-input AND gate

Two-input OR gate

NOT gate or inverter

Four-input OR gate

Three-input AND gate

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Representation of input-output signals of gates


x 1 1 0 1 0 1

y 0 1 1 1 0 0

AND: xy 0 1 0 1 0 0

OR: x+y 1 1 1 1 0 1

NOT: x 0 0 1 0 1 0

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Boolean Algebra


Boolean algebra is the algebra of truth values 0 and 1.

It was developed in 1854 by an English mathematician

and philosopher, Goerge Boole.

Boolean algebra has constructed on the set B={0,1}

Boolean algebra is an algebraic structure defined by a set

of elements, B, together with two binary operators AND ()

and OR (+)

Although there are some similarities, boolean algebra

should not be confused with ordinary arithmetic.

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Huntington Postulates for Boolean Algebra (1)


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Huntington Postulates for Boolean Algebra (2)


The axioms are independent; none can be proved

from the others.Associativity is not included, since it can be derived

(for both operators) from the given axioms.

In ordinary algebra, +is not distributive over .No subtraction or division operations in Boolean

algebra.

Complement is not available in ordinary algebra.Set B is defined as {0,1}, which means two-valued

Boolean Algebra.

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Theorems and Properties of Boolean Algebra

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Duality of Boolean Algebra


A dual of logic expression can be obtained by replacing

() with (+), (+) with () , (0) with (1) and (1) with (0). Thisproperty is called duality.

For example,

x y+1x+y 0

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Precedence order of operators


From higher to lower precedence

1)Parenthesis

2) Complement (NOT)

3) AND

4) OR

h d f l l b

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Theorem x + 1 = 1 x 0 = 0

x + 1= 1 . (x + 1)

= (x + x')(x + 1)

= x + x' 1

= x + x'

=1

by duali ty.

Theorem x + x = x x x = x

x + x = (x + x) 1

= (x + x)(x + x')

= x + xx'

=x

xx=xx+0=xx+xx'

= x(x + x')

= x 1

=x

h d i f l l b

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Theorem x + xy = x x(x + y) = x

x+xy=x1+xy

= x(1 + y)

= x(y + 1)

=x 1

=x

by duali ty.

Theorem x + 1 = 1 x 0 = 0

x + 1= 1 . (x + 1)

= (x + x')(x + 1)

= x + x' 1

= x + x'

=1

by duali ty.

Th d P i f B l Al b

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Theorem (x+y)'=x'y' (xy)'=x'+y'

De Morgan

They can also proved by the truth table.

x y x y' (x+y) xy' (xy) x+y0 0 1 1 1 1 1 1

0 1 1 0 0 0 1 1

1 0 0 1 0 0 1 1

1 1 0 0 0 0 0 0

B l F i

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Boolean Functions


A Boolean function described by an algebraic

expression consists of binary variables (0 and 1) and thelogic operation symbols.

For a given value of the binary variables, the function

can be equal to either 1 or 0.

A boolean function expresses the logical relationship

between the variables.

It can be evaluated by determining all possible values

of variables.

A boolean function can be represented in a truth table.

B l F ti

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Boolean Functions


The number of rows in a truth table is found by calculconsidering

all possible values of variables: 2n.

For a given function F1= x + y'z and F2= x'y' z + x'yz + xy'

x y z F1 F2

0 0 0 0 0

0 0 1 1 1

0 1 0 0 0

0 1 1 0 1

1 0 0 1 1

1 0 1 1 1

1 1 0 1 0

1 1 1 1 0

F ti i l t ti b l i t (1)

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Function implementation by logic gates (1)


A Boolean function can be transformed from an algebraic

expression into a circuit diagram composed of logic gates.

The logic-circuit diagram for F1,

A literal is single variable within a term that may becomplemented or not.

F1 function has 3 literals.


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Function implemenetation by logic gates (2)


For the case of secon function F2, one of possible logic

implementations is


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Function implemenetation by logic gates (3)


If we consider possible simplification in F2

F2= x'y'z + x'yz + xy' = x'z(y' + y )+ xy= x'z+ xy'

Therefore, F2 function has reduced from 3 terms, 8

literals to 2 terms 4 literals.

Si lifi ti f b l i

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Simplification of boolean expression


There is often a possibility to reduce number of terms

and the literals in a logical function .

The aim of simplification of a boolean expression is to

obtain a simpler logic circuit.

This process offers to use less logic gates two implement.

Complement of a f nction

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Complement of a function


DeMorgan's theorem can be extended to three or more

variables.

Examples on simplifications

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Examples on simplifications


Example 1:

AC + ABC + AC= AC + AC + ABC

= C(A+ A) + ABC

= C1 + ABC

= C + ABC

= (C+ AB)(C+C)

= AB + C

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Example 2:

(xy+z) + z + xy + wz = (xy+z) + z + wz + xy

= (xy+z) + z(1+ w) + xy

= (xy+z) + z + xy

= (x + y)z + z + xy => De Morgan

= (z + (x + y)) (z + z) + xy =>distributive

= (z + (x + y)) 1 + xy

= x + y + z + xy

= x + y + z [absorption]

lect2_blg231e_2013_02_28

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