Digital Logic Design:

In this course we will learn the following topics −

 Basics of digital signals and systems

 Different types of number systems and their conversions
 Binary codes and their conversions
 Boolean functions and their minimization
 Implementation of Boolean expressions
 Combinational logic circuits and their applications
 Sequential circuits and their applications
 Memory devices
 Logic Families

Digital Electronics:
Digital electronics deals with the study of digital signals and systems,
processing of digital signals and their applications. Under digital
electronics, several important concepts are covered such as logic gates,
Boolean operations, logic functions, combinational circuits, sequential
circuits, logic families, and more.

Importance of Digital Electronics in Computer Organization:

Here are the key points that highlight the importance of Digital
Electronics in the field of Computer Organization −
 The binary representation of digital electronics is used to design
different circuits of a computer system.
 Digital electronics provide logic gates and other digital circuits
which are used in designing different components of a computer
system like control units, arithmetic logic units (ALUs), memory
unit, and more.
 Digital electronics provide principles for design memory units and
data storage systems in computers.
  Digital electronics principles also empower computers to perform
various digital signal processing tasks such as modulation,
demodulation, filtering, etc.

Systems:
A system is defined as a group of various components interconnected
together to perform a specific task. For example, a digital computer
consists of several components such as monitor, CPU (Central
Processing Unit), memory, keyboard, mouse, printer, and more. All these
components are connected together to accomplish certain tasks. Hence, a
computer can be termed as a system.

We can broadly classify systems into the following two categories −

 Analog Systems
 Digital Systems
An analog system is a type of system that operates on continuous time
signals, while a digital system is one that can work on discrete time
signals.

Digital System
A Digital system is a system that processes or represents information
using discrete elements. Some common examples of digital systems
include smartphone, laptops, smart watch, tablet, desktop computers,
etc.

The working of a digital system is entirely based on digital signals or

binary signals. Where, a digital signal is a type of signal that is
represented as a discrete-elements. It can have two possible states
namely high or low. The high state is denoted by the logic 1 and the low
state is denoted by the logic 0.

In a digital system, if the state of the signal is logic 1, the system will be
on, and if the state of the signal is 0, the system will be off.
Characteristics of Digital Systems:
Today, digital systems are widely used in almost every aspect of life.
This is because of their high reliability and efficiency. The following are
some key characteristics of digital systems −

 Digital systems are relative less complex to implement as they use

binary number system having only two digits to represent the state
of a system.
 In digital systems, the information is represented in the form of a
group of 0s and 1s i.e., bits. This is called binary or digital
representation of information.
 Digital systems rely on digital signals having two well-defined
discrete states. This makes digital systems more reliable and
efficient in terms of processing, storage, and communication of
information.
 Digital systems use logical mathematics and operations to perform
computing tasks.
 Digital systems can be manufactured in the form of integrated
circuits (ICs) of very small sizes.
 Digital systems can be easily programmed to perform repeated
tasks that reduces human efforts and cost.
 Digital systems are highly immune to noise and distortions.

Types of Digital Systems:

Digital systems can be classified based on various parameters. Here are
some important types of digital systems that we commonly use in
practice:

Combinational Digital Systems

A combinational logic circuit or system is a type of digital circuit that
performs logical operations and produces output depending on the
present inputs. Hence, the output of a combinational digital circuit does
not depend on the past inputs and outputs of the system.
Example −The common examples of combinational digital systems are
binary adders, subtractors, logic gates, multiplexers, demultiplexers, etc.

Sequential Digital Systems

A type of digital system that has a memory element to store past history
of the system operation is called a sequential digital system. Therefore,
the output of a digital system depends on both present inputs and past
outputs of the system.
Example of sequential digital systems are flip-flops, registers, memory
devices, counters, etc.

Programmable Logic Devices (PLDs)

A programmable logic device is one that can be programmed to perform
a specific task automatically.
Example of programmable logic devices are microcontrollers, PLCs, etc.

Digital Communication Systems

A digital communication system is a type of digital system used for
transmission and reception of information in the form of digital signals.
Example of digital communication systems are internet, intranet, mobile
communication system, Wi-Fi, etc.

Digital Control Systems

A digital control system is a computerized control system used to
monitor and regulate the behavior of a dynamic system.
Example − Digital control systems are extensively used in robotics,
industrial automation, etc.
Signals:
Electromagnetic wave that is used for transmission of data or
information is called a signal.

Properties of Signal
In electronics, a signal is characterized by the following important
properties −
 Magnitude −The intensity or maximum value of a signal is termed
as its magnitude.
 Frequency −The number of oscillations per second is called
frequency of the signal.
 Time period −The time taken to complete one oscillation is called
the time period of the signal.

Types of Signals
In electronics, there are mainly two types of signals used, they are −
 Analog Signals
 Digital Signals

Analog Signal
A type of electronic signal that has continuous values within a given
range is called an analog signal. Analog signals are expressed as the
continuous functions of time. They are represented as the waveforms of
continuously varying current or voltage.
Example of analog signals are voice, speed, pressure, temperature, etc.

An important characteristic of analog signals is that they have a

definite value at every instant of time, known as instantaneous value
of the signal.
Analog signals have smooth waveforms as they are continuous in both
amplitude and time. That meant, there is no interruptions in their
representation over time.
Properties of Analog Signal
The following are main properties of analog signals −

 Analog signals are continuous signals in both amplitude and time.

 Analog signals have a certain value or magnitude at any given
instant of time.
 Analog signals have infinite resolution.
 Analog signals are best suited for representing the real-world
phenomena.
 Analog signals are represented by the continuously varying smooth
waveforms.

Digital Signal
A digital signal is a type of electronic signal that has a finite set of
discrete values representing information. It is also called binary signals,
as it uses binary 0 or 1 to represent the state of a signal. Where, the
binary 0 represents the off or low state of the signal, while the binary 1
represents the on or high state of the signal.

Properties of Digital Signal

The following are some key characteristics of digital signals −

 Digital signals have discrete or discontinuous values in terms of

both amplitude and time.
 Digital signals do not have values defined between any two distinct
instants of time.
 Digital signals are represented using binary system by sampling
the values of the signals at specific time instants.
 Digital signals represent information in the form of a sequence of
binary 0s and 1s.
 Digital signals have a finite resolution.
  Digital signals are capable to perform logical operations.
 Digital signals are more efficient and reliable when it comes to
storage and transmission.

Difference between Analog and Digital Signals

Let us now discuss the important differences between analog and digital
signals −

Key Analog Signals Digital Signals

Representation Analog signals are Digital signals are represented
represented as continuous as discrete functions of time.
functions or waveforms of
time.
Nature Analog signals are Digital signals are
continuous as they have discontinuous as they have
infinite values within a distinct values sampled at
specified range. specific time instants.
Resolution Analog signals have Digital signals have a finite
infinite resolution. resolution.
Accuracy Analog signals are more Digital signals are relatively
accurate. less accurate.
Storage Analog signals are difficult Digital signals are efficient to
to store. store.
Noise Analog signals are less Digital signals have high
immunity immune to noise. immunity against noise.
Examples Voice signals, temperature, Data transmitted over internet,
speed, etc. computer generated signals,
etc.

Applications of Signals:
Both analog and digital signals are widely used in the field of
electronics. The following are some key applications of signals −
 Signals are used for storage and transmission of information.
 Signals are used in control systems to regulate their behavior.
  Signals are also used in measurement of physical quantities like
temperature, pressure, speed, sound, light, and more.
 Signals are used in computing systems for data processing, etc.

Logic Levels and Pulse Waveforms:

Logic Level
In digital electronics, a voltage level that represents a specific binary
value either 0 or 1 is called a logic level. Here, the binary value 0
represents the low voltage level while the binary value 1 represents the
high value level.

Hence, the logic levels can be classified into the following two types −
 High Logic Level
 Low Logic Level

High Logic Level

In the case of a digital system, the voltage level closer to the maximum
voltage level that the system can handle without getting damaged is
called high logic level.

The high logic level is represented by the binary digit "1". The voltage
level for a high logic level depends on the technological standard used to
design the system. Typically, the voltage value between 2 V and 5 V
represents the high logic level or 1.

Low Logic Level

In a digital system, the low logic level is defined as the maximum
voltage level for which the system will remain in the OFF state.

The low logic level is represented by the binary digit "0". Similar to the
high logic level, the voltage level for a low logic level depends on the
technology standard used to design the system. In actual practice, the
voltage value between 0 V and 0.8 V represents the low logic level or
logic 0.
Pulse
A pulse is a type of an electronic signal that can change suddenly
between two possible states i.e., high state and low state.
Depending on the switching characteristics, the pulses can be classified
into the following two types −

Positive Pulse − When a signal normally goes from low logic level to
the high logic level and then returns to its normal low logic level, then it
is called a positive pulse.
Negative Pulse − When a signal normally goes from high logic level to
the low logic level and then returns to its normal high logic level, then it
is known as a negative pulse.

The pulse waveforms for positive and negative pulses are depicted in the
following figure.

A pulse has two edges namely, a leading edge and a trailing edge.
In the case of a positive pulse, the edge going from low logic level to
high logic level is called the leading edge, and the edge going from high
logic level to low logic level is called the trailing edge.

In the case of a negative pulse, the edge going from high logic level to
low logic level is called the leading edge, whereas the edge going from
low logic level to high logic level is called the trailing edge.

Components of a Digital System

A typical digital system consists of the following main components −

 Central Processing Unit (CPU)

 Memory
 Input Devices
 Output Devices
 Logic Gates
 Power Supply
 Communication Channels

Number Systems
The following number systems are the most commonly used.

 Decimal Number system

 Binary Number system
 Octal Number system
 Hexadecimal Number system

Decimal Number System

The base or radix of Decimal number system is 10. So, the numbers
ranging from 0 to 9 are used in this number system. The part of the
number that lies to the left of the decimal point is known as integer
part. Similarly, the part of the number that lies to the right of the
decimal point is known as fractional part.
Example
Consider the decimal number 1358.246. Integer part of this number is
1358 and fractional part of this number is 0.246.
Mathematically, we can write it as
1358.246 = (1 × 103) + (3 × 102) + (5 × 101) + (8 × 100) + (2 × 10-1) +
(4 × 10-2) + (6 × 10-3)
After simplifying the right hand side terms, we will get the decimal
number, which is on left hand side.

Binary Number System

All digital circuits and systems use this binary number system.
The base or radix of this number system is 2. So, the numbers 0 and 1
are used in this number system.

Example
Consider the binary number 1101.011. Integer part of this number is
1101 and fractional part of this number is 0.011.
Mathematically, we can write it as
1101.011 = (1 × 23) + (1 × 22) + (0 × 21) + (1 × 20) + (0 × 2-1) +
(1 × 2-2) + (1 × 2-3)= 13.375
After simplifying the right hand side terms, we will get a decimal
number, which is an equivalent of binary number on left hand side.

Octal Number System

The base or radix of octal number system is 8. So, the numbers ranging
from 0 to 7 are used in this number system. The part of the number that
lies to the left of the octal point is known as integer part. Similarly, the
part of the number that lies to the right of the octal point is known as
fractional part.
Example
Consider the octal number 1457.236. Integer part of this number is
1457 and fractional part of this number is 0.236.
Mathematically, we can write it as
1457.236 = (1 × 83) + (4 × 82) + (5 × 81) + (7 × 80) + (2 × 8-1) +
(3 × 8-2) + (6 × 8-3)= 815.30859375
After simplifying the right hand side terms, we will get a decimal
number, which is an equivalent of octal number on left hand side.

Hexadecimal Number System

The base or radix of Hexa-decimal number system is 16. So, the
numbers ranging from 0 to 9 and the letters from A to F are used in this
number system. The decimal equivalent of Hexa-decimal digits from A
to F are 10 to 15.

Example
Consider the Hexa-decimal number 1A05.2C4. Integer part of this
number is 1A05 and fractional part of this number is 0.2C4.
Mathematically, we can write it as
1A05.2C4 = (1 × 163) + (10 × 162) + (0 × 161) + (5 × 160) + (2 × 16-1) +
(12 × 16-2) + (4 × 16-3)
After simplifying the right hand side terms, we will get a decimal
number, which is an equivalent of Hexa-decimal number on left hand
side.

Digital Circuits - Base Conversions

Decimal Number to other Bases Conversion
If the decimal number contains both integer part and fractional part,
then convert both the parts of decimal number into other base
individually. Follow these steps for converting the decimal number into
its equivalent number of any base ‘r’.
 Do division of integer part of decimal number and successive
quotients with base ‘r’ and note down the remainders till the
quotient is zero. Consider the remainders in reverse order to get
the integer part of equivalent number of base ‘r’. That means, first
and last remainders denote the least significant digit and most
significant digit respectively.
 Do multiplication of fractional part of decimal number
and successive fractions with base ‘r’ and note down the carry till
the result is zero or the desired number of equivalent digits is
obtained. Consider the normal sequence of carry in order to get
the fractional part of equivalent number of base ‘r’.

Decimal to Binary Conversion

The following two types of operations take place, while converting
decimal number into its equivalent binary number.

 Division of integer part and successive quotients with base 2.

 Multiplication of fractional part and successive fractions with base
2.

Example
Consider the decimal number 58.25. Here, the integer part is 58 and
fractional part is 0.25.
Step 1 − Division of 58 and successive quotients with base 2.

Operation Quotient Remainder

58/2 29 0 (LSB)

29/2 14 1

14/2 7 0

7/2 3 1

3/2 1 1

1/2 0 1(MSB)

⇒(58)10 = (111010)2
Therefore, the integer part of equivalent binary number is 111010.
Step 2 − Multiplication of 0.25 and successive fractions with base 2.

Operation Result Carry

0.25 x 2 0.5 0

0.5 x 2 1.0 1
- 0.0 -

⇒(.25)10 = (.01)2
Therefore, the fractional part of equivalent binary number is .01
⇒(58.25)10 = (111010.01)2
Therefore, the binary equivalent of decimal number 58.25 is
111010.01.

Decimal to Octal Conversion

The following two types of operations take place, while converting
decimal number into its equivalent octal number.
 Division of integer part and successive quotients with base 8.
 Multiplication of fractional part and successive fractions with base
8.
Example
Consider the decimal number 58.25. Here, the integer part is 58 and
fractional part is 0.25.
Step 1 − Division of 58 and successive quotients with base 8.

Operation Quotient Remainder

58/8 7 2

7/8 0 7
⇒(58)10 = (72)8
Therefore, the integer part of equivalent octal number is 72.
Step 2 − Multiplication of 0.25 and successive fractions with base 8.

Operation Result Carry

0.25 x 8 2.00 2

- 0.00 -

⇒ (.25)10 = (.2)8
Therefore, the fractional part of equivalent octal number is .2
⇒ (58.25)10 = (72.2)8
Therefore, the octal equivalent of decimal number 58.25 is 72.2.

Decimal to Hexa-Decimal Conversion

The following two types of operations take place, while converting
decimal number into its equivalent hexa-decimal number.

 Division of integer part and successive quotients with base 16.

 Multiplication of fractional part and successive fractions with base
16.
Example
Consider the decimal number 58.25. Here, the integer part is 58 and
decimal part is 0.25.
Step 1 − Division of 58 and successive quotients with base 16.
Operation Quotient Remainder

58/16 3 10=A

3/16 0 3

⇒ (58)10 = (3A)16
Therefore, the integer part of equivalent Hexa-decimal number is 3A.
Step 2 − Multiplication of 0.25 and successive fractions with base 16.

Operation Result Carry

0.25 x 16 4.00 4

- 0.00 -

⇒ (.25)10 = (.4)16
Therefore, the fractional part of equivalent Hexa-decimal number
is .4.
⇒ (58.25)10 = (3A.4)16
Therefore, the Hexa-decimal equivalent of decimal number 58.25 is
3A.4.

Binary Number to other Bases Conversion

The process of converting a number from binary to decimal is different
to the process of converting a binary number to other bases. Now, let us
discuss about the conversion of a binary number to decimal, octal and
Hexa-decimal number systems one by one.
Binary to Decimal Conversion
For converting a binary number into its equivalent decimal number,
first multiply the bits of binary number with the respective positional
weights and then add all those products.
Example
Consider the binary number 1101.11.
Mathematically, we can write it as
(1101.11)2 = (1 × 23) + (1 × 22) + (0 × 21) + (1 × 20) + (1 × 2-1) +
(1 × 2-2)
⇒ (1101.11)2 = 8 + 4 + 0 + 1 + 0.5 + 0.25 = 13.75
⇒ (1101.11)2 = (13.75)10
Therefore, the decimal equivalent of binary number 1101.11 is 13.75.

Binary to Octal Conversion

We know that the bases of binary and octal number systems are 2 and 8
respectively. Three bits of binary number is equivalent to one octal
digit, since 23 = 8.
Follow these two steps for converting a binary number into its
equivalent octal number.
 Start from the binary point and make the groups of 3 bits on both
sides of binary point. If one or two bits are less while making the
group of 3 bits, then include required number of zeros on extreme
sides.
 Write the octal digits corresponding to each group of 3 bits.
Example
Consider the binary number 101110.01101.
Step 1 − Make the groups of 3 bits on both sides of binary point.
101 110.011 01
Here, on right side of binary point, the last group is having only 2 bits.
So, include one zero on extreme side in order to make it as group of 3
bits.
⇒ 101 110.011 010
Step 2 − Write the octal digits corresponding to each group of 3 bits.
⇒ (101 110.011 010)2 = (56.32)8
Therefore, the octal equivalent of binary number 101110.01101 is
56.32.

Binary to Hexa-Decimal Conversion

We know that the bases of binary and Hexa-decimal number systems
are 2 and 16 respectively. Four bits of binary number is equivalent to
one Hexa-decimal digit, since 24 = 16.
Follow these two steps for converting a binary number into its
equivalent Hexa-decimal number.
 Start from the binary point and make the groups of 4 bits on both
sides of binary point. If some bits are less while making the group
of 4 bits, then include required number of zeros on extreme sides.
 Write the Hexa-decimal digits corresponding to each group of 4
bits.

Example
Consider the binary number 101110.01101
Step 1 − Make the groups of 4 bits on both sides of binary point.
10 1110.0110 1
Here, the first group is having only 2 bits. So, include two zeros on
extreme side in order to make it as group of 4 bits. Similarly, include
three zeros on extreme side in order to make the last group also as
group of 4 bits.
⇒ 0010 1110.0110 1000
Step 2 − Write the Hexa-decimal digits corresponding to each group of
4 bits.
⇒ (0010 1110.0110 1000)2 = (2E.68)16
Therefore, the Hexa-decimal equivalent of binary number
101110.01101 is (2E.68).

Octal Number to other Bases Conversion

The process of converting a number from octal to decimal is different to
the process of converting an octal number to other bases. Now, let us
discuss about the conversion of an octal number to decimal, binary and
Hexa-decimal number systems one by one.

Octal to Decimal Conversion

For converting an octal number into its equivalent decimal number, first
multiply the digits of octal number with the respective positional
weights and then add all those products.
Example
Consider the octal number 145.23.
Mathematically, we can write it as
(145.23)8 = (1 × 82) + (4 × 81) + (5 × 80) + (2 × 8-1) + (3 × 8-2)
⇒ (145.23)8 = 64 + 32 + 5 + 0.25 + 0.05 = 101.3
⇒ (145.23)8 = (101.3)10
Therefore, the decimal equivalent of octal number 145.23 is 101.3.
Octal to Binary Conversion
The process of converting an octal number to an equivalent binary
number is just opposite to that of binary to octal conversion. By
representing each octal digit with 3 bits, we will get the equivalent
binary number.
Example
Consider the octal number 145.23.
Represent each octal digit with 3 bits.
(145.23)8 = (001 100 101.010 011)2
The value doesn’t change by removing the zeros, which are on the
extreme side.
⇒ (145.23)8 = (1100101.010011)2
Therefore, the binary equivalent of octal number 145.23 is
1100101.010011.

Octal to Hexa-Decimal Conversion

Follow these two steps for converting an octal number into its
equivalent Hexa-decimal number.

 Convert octal number into its equivalent binary number.

 Convert the above binary number into its equivalent Hexa-decimal
number.
Example
Consider the octal number 145.23
In previous example, we got the binary equivalent of octal number
145.23 as 1100101.010011.
By following the procedure of binary to Hexa-decimal conversion, we
will get
(1100101.010011)2 = (65.4C)16
⇒(145.23)8 = (65.4C)16
Therefore, the Hexa-decimal equivalent of octal number 145.23 is
65.4C.

Hexa-Decimal Number to other Bases Conversion

The process of converting a number from Hexa-decimal to decimal is
different to the process of converting Hexa-decimal number into other
bases. Now, let us discuss about the conversion of Hexa-decimal
number to decimal, binary and octal number systems one by one.

Hexa-Decimal to Decimal Conversion

For converting Hexa-decimal number into its equivalent decimal
number, first multiply the digits of Hexa-decimal number with the
respective positional weights and then add all those products.
Example
Consider the Hexa-decimal number 1A5.2
Mathematically, we can write it as
(1A5.2)16 = (1 × 162) + (10 × 161) + (5 × 160) + (2 × 16-1)
⇒ (1A5.2)16 = 256 + 160 + 5 + 0.125 = 421.125
⇒ (1A5.2)16 = (421.125)10
Therefore, the decimal equivalent of Hexa-decimal number 1A5.2 is
421.125.

Hexa-Decimal to Binary Conversion

The process of converting Hexa-decimal number into its equivalent
binary number is just opposite to that of binary to Hexa-decimal
conversion. By representing each Hexa-decimal digit with 4 bits, we
will get the equivalent binary number.
Example
Consider the Hexa-decimal number 65.4C
Represent each Hexa-decimal digit with 4 bits.
(65.4C)6 = (0110 0101.0100 1100)2
The value doesn’t change by removing the zeros, which are at two
extreme sides.
⇒ (65.4C)16 = (1100101.010011)2
Therefore, the binary equivalent of Hexa-decimal number 65.4C is
1100101.010011.

Hexa-Decimal to Octal Conversion

Follow these two steps for converting Hexa-decimal number into its
equivalent octal number.

 Convert Hexa-decimal number into its equivalent binary number.

 Convert the above binary number into its equivalent octal number.
Example
Consider the Hexa-decimal number 65.4C
In previous example, we got the binary equivalent of Hexa-decimal
number 65.4C as 1100101.010011.
By following the procedure of binary to octal conversion, we will get
(1100101.010011)2 = (145.23)8
⇒(65.4C)16 = (145.23)𝟖
Therefore, the octal equivalent of Hexa-decimal number 65.4C is
145.23.