2
Digital Electronics
Digital techniques and systems have the advantages of being relatively much easier to design and
having higher accuracy, programmability, noise immunity, easier storage of data and ease of fabrication
in integrated circuit form, leading to availability of more complex functions in a smaller size. The
real world, however, is analogue. Most physical quantities – position, velocity, acceleration, force,
pressure, temperature and flowrate, for example – are analogue in nature. That is why analogue
variables representing these quantities need to be digitized or discretized at the input if we want to
benefit from the features and facilities that come with the use of digital techniques. In a typical system
dealing with analogue inputs and outputs, analogue variables are digitized at the input with the help
of an analogue-to-digital converter block and reconverted back to analogue form at the output using a
digital-to-analogue converter block. Analogue-to-digital and digital-to-analogue converter circuits are
discussed at length in the latter part of the book. In the following sections we will discuss various
number systems commonly used for digital representation of data.
1.2 Introduction to Number Systems
We will begin our discussion on various number systems by briefly describing the parameters that are
common to all number systems. An understanding of these parameters and their relevance to number
systems is fundamental to the understanding of how various systems operate. Different characteristics
that define a number system include the number of independent digits used in the number system,
the place values of the different digits constituting the number and the maximum numbers that can
be written with the given number of digits. Among the three characteristic parameters, the most
fundamental is the number of independent digits or symbols used in the number system. It is known as
the radix or base of the number system. The decimal number system with which we are all so familiar
can be said to have a radix of 10 as it has 10 independent digits, i.e. 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9.
Similarly, the binary number system with only two independent digits, 0 and 1, is a radix-2 number
system. The octal and hexadecimal number systems have a radix (or base) of 8 and 16 respectively.
We will see in the following sections that the radix of the number system also determines the other
two characteristics. The place values of different digits in the integer part of the number are given by
r
0 , r
1 , r
2 , r
3 and so on, starting with the digit adjacent to the radix point. For the fractional part, these
are r
−1 , r
−2 , r
−3 and so on, again starting with the digit next to the radix point. Here, r is the radix
of the number system. Also, maximum numbers that can be written with n digits in a given number
system are equal to r
n .
1.3 Decimal Number System
The decimal number system is a radix-10 number system and therefore has 10 different digits or
symbols. These are 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. All higher numbers after ‘9’ are represented in terms
of these 10 digits only. The process of writing higher-order numbers after ‘9’ consists in writing the
second digit (i.e. ‘1’) first, followed by the other digits, one by one, to obtain the next 10 numbers
from ‘10’ to ‘19’. The next 10 numbers from ‘20’ to ‘29’ are obtained by writing the third digit (i.e.
‘2’) first, followed by digits ‘0’ to ‘9’, one by one. The process continues until we have exhausted all
possible two-digit combinations and reached ‘99’. Then we begin with three-digit combinations. The
first three-digit number consists of the lowest two-digit number followed by ‘0’ (i.e. 100), and the
process goes on endlessly.
The place values of different digits in a mixed decimal number, starting from the decimal point, are
10
0 , 10
1 , 10
2 and so on (for the integer part) and 10
−1 , 10
−2 , 10
−3 and so on (for the fractional part).
Digital Electronics
Digital techniques and systems have the advantages of being relatively much easier to design and
having higher accuracy, programmability, noise immunity, easier storage of data and ease of fabrication
in integrated circuit form, leading to availability of more complex functions in a smaller size. The
real world, however, is analogue. Most physical quantities – position, velocity, acceleration, force,
pressure, temperature and flowrate, for example – are analogue in nature. That is why analogue
variables representing these quantities need to be digitized or discretized at the input if we want to
benefit from the features and facilities that come with the use of digital techniques. In a typical system
dealing with analogue inputs and outputs, analogue variables are digitized at the input with the help
of an analogue-to-digital converter block and reconverted back to analogue form at the output using a
digital-to-analogue converter block. Analogue-to-digital and digital-to-analogue converter circuits are
discussed at length in the latter part of the book. In the following sections we will discuss various
number systems commonly used for digital representation of data.
1.2 Introduction to Number Systems
We will begin our discussion on various number systems by briefly describing the parameters that are
common to all number systems. An understanding of these parameters and their relevance to number
systems is fundamental to the understanding of how various systems operate. Different characteristics
that define a number system include the number of independent digits used in the number system,
the place values of the different digits constituting the number and the maximum numbers that can
be written with the given number of digits. Among the three characteristic parameters, the most
fundamental is the number of independent digits or symbols used in the number system. It is known as
the radix or base of the number system. The decimal number system with which we are all so familiar
can be said to have a radix of 10 as it has 10 independent digits, i.e. 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9.
Similarly, the binary number system with only two independent digits, 0 and 1, is a radix-2 number
system. The octal and hexadecimal number systems have a radix (or base) of 8 and 16 respectively.
We will see in the following sections that the radix of the number system also determines the other
two characteristics. The place values of different digits in the integer part of the number are given by
r
0 , r
1 , r
2 , r
3 and so on, starting with the digit adjacent to the radix point. For the fractional part, these
are r
−1 , r
−2 , r
−3 and so on, again starting with the digit next to the radix point. Here, r is the radix
of the number system. Also, maximum numbers that can be written with n digits in a given number
system are equal to r
n .
1.3 Decimal Number System
The decimal number system is a radix-10 number system and therefore has 10 different digits or
symbols. These are 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. All higher numbers after ‘9’ are represented in terms
of these 10 digits only. The process of writing higher-order numbers after ‘9’ consists in writing the
second digit (i.e. ‘1’) first, followed by the other digits, one by one, to obtain the next 10 numbers
from ‘10’ to ‘19’. The next 10 numbers from ‘20’ to ‘29’ are obtained by writing the third digit (i.e.
‘2’) first, followed by digits ‘0’ to ‘9’, one by one. The process continues until we have exhausted all
possible two-digit combinations and reached ‘99’. Then we begin with three-digit combinations. The
first three-digit number consists of the lowest two-digit number followed by ‘0’ (i.e. 100), and the
process goes on endlessly.
The place values of different digits in a mixed decimal number, starting from the decimal point, are
10
0 , 10
1 , 10
2 and so on (for the integer part) and 10
−1 , 10
−2 , 10
−3 and so on (for the fractional part).