Lesson No. 01
AN OVERVIEW & NUMBER SYSTEMS
Analogue versus Digital
Most
of the quantities in nature that can be measured are continuous. Examples
include
• Intensity of light during the day: The intensity of light gradually
increases as the sun rises in the morning; it remains constant
throughout the day and then gradually decreases as the sun sets until it
becomes completely dark. The change in the light throughout the day is gradual
and continuous. Even with a sudden change in weather when the sun is obscured
by a cloud the fall in the light intensity although very sharp however is still
continuous and is not abrupt.
• Rise and fall in temperature during a
24-hour period: The
temperature also rises and falls with the passage of time during the day
and in the night. The change in temperature is never abrupt but gradual and
continuous.
• Velocity of a car travelling from A to
B: The velocity of a
car travelling from one city to another varies in a continuous manner.
Even if it abruptly accelerates or stops suddenly, the change in velocity
seemingly very sudden and abrupt is never abrupt in reality. This can be
confirmed by measuring the velocity in short time intervals of few
milliseconds.
The measurable values generally change over a continuous
range having a minimum and maximum value. The temperature values in a summer
month change between 23 0C to 45 0C. A car can travel at any velocity
between 0 to 120 mph.
Digital representing of quantities
Digital quantities unlike Analogue quantities are not
continuous but represent quantities measured at discrete intervals. Consider
the continuous signal as shown in the figure 1.1.
To represent this signal digitally the signal is sampled
at fixed and equal intervals. The continuous signal is sampled at 15 fixed and
equal intervals. Figure 1.2. The set of values (1, 2, 4, 7, 18, 34, 25, 23, 35,
37, 29, 42, 41, 25 and 22) measured at the sampling points represent the
continuous signal. The 15 samples do not exactly represent the original signal
but only approximate the original continuous signal. This can be confirmed by
plotting the 15 sample points. Figure 1.3. The reconstructed signal from the 15
samples has sharp corners and edges in contrast to the original signal that has
smooth curves.
If the number of samples that are collected is reduced by
half, the reconstructed signal will be very different from the original. The
reconstructed signal using 7 samples have missing peak and dip at 34 0C and 23 0C respectively. Figure 1.4. The reason
for the difference between the original and the reconstructed signal is due to
under-sampling. A more accurate representation of the continuous signal is
possible if the number of samples and sampling intervals are increased. If the
sampling is increased to infinity the number of values would still be discrete
but they would be very close and closely match the actual signal.
Electronic Processing of Continuous and Digital
Quantities
Electronic Processing of the continuous quantities or
their Digital representation requires that the continuous signals or the
discrete values be converted and represented in terms of voltages. There are
basically two types of Electronic Processing Systems.
• Analogue Electronic Systems: These systems accept and process
continuous signals represented in the form continuous voltage or current
signals. The continuous quantities are converted into continuous voltage or
current signals by transducers. The block diagram describes the processing by
an Analogue Electronic System. Figure 1.5.
• Digital Electronic Systems: These systems accept and process
discrete samples representing the actual continuous signal. Analogue to
Digital Converters are used to sample the continuous voltage signals
representing the original signal.
Do the Digital Electronic Systems use voltages to
represent the discrete samples of the continuous signal? This question can be
answered by considering a very simple example of a calculator which is a
Digital Electronic System. Assume that a calculator is calibrated to represents
the number 1 by 1 millivolt (mV). Thus the number 39 is represented by the
calculator in terms of voltage as 39 mV. Calculators can also represent large
numbers such as 6.25 x 1018 (as in 1 Coulomb = 6.25 x 1018 electrons). The value in terms of
volts is 6.25 x 1015
volts! This voltage value can not be practically represented by any electronic
circuit. Thus Digital Systems do not use discrete samples represented as
voltage values.
Digital Systems and Digital Values
Digital systems are designed to work with two voltage
values. A +5 volts represents a logic high state or logic 1 state and 0 volts
represents a logic low state or logic 0 state. The Digital Systems which are
based on two voltage values or two states can easily represent any two values.
For example,
• The numbers ‘0’ and ‘1’
• The state of a switch ‘on’ or ‘off’
• The colour ‘black’ and ‘white’
• The temperature ‘hot’ and ‘cold’
• An object ‘moving’ or ‘stationary’
Representing two values or two states is not very
practical, as many naturally occurring phenomenons have values or state that
are more than two. For example, numbers have widely varying ranges, a colour
palette might have 64 different shades of the colour red, the temperature of
boiling water at room temperature varies from 30 0C to 100 0C. Digital Systems are based on the
Binary Number system which allows more than two or multiple values to be
represented very conveniently.
Binary Number System
The Binary Number System unlike the Decimal number system
is based on two values. Each digit or bit in Binary Number system can represent
only two values, a ‘0’ and a ‘1’. A single digit of the Decimal Number system
represents 10 values, 0, 1, 2 to 9. The Binary Number System can be used to
represent more than two values by combining binary digits or bits. In a Decimal
Number System a single digit can represent 10 different values (0 to 9),
representing more than 10 values requires a combination of two digits which
allows up to 100 values to be represented (0 to 99). A Combination of Binary
Numbers is used to represent different quantities.
• Represent Colours: A palette of four colours red, blue,
green and yellow can be represented by a combination of two digital
values 00, 01, 10 and 11 respectively.
• Representing Temperature: An analogue value such as 39oC can be represented in a digital
format by a combination of 0s and 1s. Thus 39 is 100111 in digital form.
Any quantity such as the intensity of light, temperature,
velocity, colour etc. can be represented through digital values. The number of
digits (0s and 1s) that represents a quantity is proportional to the range of
values that are to be represented. For example, to represent a palette of eight
colours a combination of three digits is used. Representing a temperature range
of 00
C to 1000
C requires a combination of up to seven digits.
Digital Systems uses the Binary Number System to
represent two or multiple values, stores and processes the binary values in
terms of 5 volts and 0 volts. Thus the number 39 represented in binary as
100111 is stored electronically in as +5 v, 0v, 0v, +5 v, +5 v and +5 v.
Advantages of working in the Digital Domain
Handling information digitally offers several advantages.
Some of the merits of a digital system are spelled out. Details of some these
aspects will be discussed and studied in the Digital Logic Design course. Other
aspects will be covered in several other courses.
• Storing and processing data in the
digital domain is more efficient: Computers
are very efficient in processing massive amounts of information and
data. Computers process information that is represented digitally in the form
of Binary Numbers. A Digital CD stores large number of video and audio clips.
Sam number of audio and video clips if stored in analogue form will require a
number of video and audio cassettes.
• Transmission of data in the digital
form is more efficient and reliable: Modern
information transmission techniques are relying more on digital transmission
due to its reliability as it is less prone to errors. Even if errors occur
during the transmission methods exist which allow for quick detection and
correction of errors.
• Detecting and Correcting errors in
digital data is easier: Coding
Theory is an area which deals with implementing digital codes that allow
for detection and correction of multi-bit errors. In the Digital Logic Design
course a simple method to detect single bit errors using the Parity bit will be
considered.
• Data can be easily and precisely
reproduced: The
picture quality and the sound quality of digital videos are far more
superior to those of analogue videos. The reason being that the digital video
stored as digital numbers can be exactly reproduced where as analogue video is
stored as a continuous signal can not be reproduced with exact precision.
• Digital systems are easy to design and
implement: Digital
Systems are based on two-state Binary Number System. Consequently the Digital
Circuitry is based on the two-voltage states, performing very simple
operations. Complex Microprocessors are implemented using simple digital
circuits. Several simple Digital Systems will be discussed in the Digital Logic
course.
• Digital circuits occupy small space: Digital circuits are based on two
logical states. Electronic circuitry that implements the two states is
very simple. Due to the simplicity of the circuitry it can be easily
implemented in a very small area. The PC motherboard having an area of
approximately 1 sq.ft has most of the circuitry of a powerful computer. A
memory chip small enough to be held in the palm of a hand is able to store an
entire collection of books.
Information Processing by a Digital System
A
Digital system such as a computer not only handles numbers but all kinds of
information.
•
Numbers: A
computer is able to store and process all types of numbers, integers, fractions
etc. and is able to perform different kinds of arithmetic operations on the
numbers.
• Text: A computer in a news reporting room is
used to write and edit news reports. A Mathematician uses a computer to
write mathematical equations explaining the dissipation of heat by a heat sink.
The computer is able to store and process text and symbols.
• Drawings, Diagrams and Pictures: A computer can store very conveniently
complex engineering drawings and diagrams. It allows real life still
pictures or videos to be processed and edited.
• Music and Sound: Musicians and Composers uses\ a
computer to work on a new compositions. Computers understand spoken
commands.
A Digital System (computer) is capable of handling
different types of information, which is represented in the form of Binary
Numbers. The different types of information use different standards and binary
formats. For example, computers use the Binary number system to represent
numbers. Characters used in writing text are represented through yet another
standard known as ASCII which allows alphabets, punctuation marks and numbers
to be represented through a combination of 0s and 1s.
Digital Components and their internal working
Digital system process binary information electronically
through specialized circuits designed for handling digital information. These
circuits as mentioned earlier operate with two voltage values of +5 volts and 0
volts. These specialized electronic circuits are known as Logic Gates and are
considered to be the Basic Building Blocks of any Digital circuit.
The commonly used Logic Gates are the AND gate, the OR
gate and the Inverter or NOT Gate. Other gates that are frequently used include
NOR, NAND, XOR and XNOR. Each of these gates is designed to perform a unique
operation on the input information which is known as a logical or Boolean
operation.
Large and complex digital system such as a computer is
built using combinations of these basic Logic Gates. These basic building
blocks are available in the form of Integrated Circuit or ICs. These gates are
implemented using standard CMOS and TTL technologies that determine the
operational characteristics of the gates such as the power dissipation,
operational voltages (3.3v or 5 v), frequency response etc.
Combinational Logic Circuits and
Functional Devices
The logic
gates which form the basic building blocks of a digital system are designed to
perform simple logic operations. A single logic gate is not of much use unless
it is connected with other gates to collectively act upon the input data.
Different gates are combined to build a circuit that is capable of performing
some useful operation like adding three numbers. Such circuits are known as
Combinational Logic Circuits or Combinational Circuits. An Adder Combinational
Circuit that is able to add two single bit binary numbers and give a single bit
Sum and Carry output is shown. Figure 1.7.
Implementing
large digital system by connecting together logic gates is very tedious and
time consuming; the circuit implemented occupies large space, are power hungry,
slow and are difficult to troubleshoot.
Digital
circuits to perform specific functions are available as Integrated Circuits for
use. Implementing a Digital system in terms of these dedicated functional units
makes the system more economical and reliable. Thus an adder circuit does not
have to be implemented by connecting various gates, a standard Adder IC is
available that can be readily used. Other
commonly used combinational functional
devices are Comparators, Decoders, Encoders, Multiplexers and Demultiplexers.
Sequential logic and implementation
Digital
systems are used in vast variety of industrial applications and house hold
electronic gadgets. Many of these digital circuits generate an output that is
not only dependent on the current input but also some previously saved
information which is used by the digital circuit. Consider the example of a
digital counter which is used by many digital applications where a count value
or the time of the day has to be displayed. The digital counter which counts
downwards from 10 to 0 is initialized to the value 10. When the counter
receives an external signal in the form of a pulse the counter decrements the
count value to 9. On receiving successive pulses the counter decrements the
currently stored count value by one, until the counter has been decremented to
0. On reaching the count value zero, the counter could switch off a washing
machine, a microwave oven or switch on an air-conditioning system.
The counter
stores or remembers the previous count value. The new count value is determined
by the previously stored count value and the new input which it receives in the
form of a pulse (a binary 1). The diagram of the counter circuit is shown in
the figure. Figure 1.8.
Digital
circuits that generate a new output on the basis of some previously stored
information and the new input are known as Sequential circuits. Sequential
circuits are a combination of Combinational circuits and a memory element which
is able to store some previous information. Sequential circuits are a very
important part of digital systems. Most digital systems have sequential logic
in addition to the combinational logic. An example of sequential circuits is
counters such as the down -counter which generates a new decremented output
value based on the previous stored value and an external input. The storage
element or the memory element which is an essential part of a sequential
circuit is implemented a flip-flop using a very simple digital circuit known as
a flip-flop.
Programmable Logic Devices (PLDs)
The modern
trend in implementing specialized dedicated digital systems is through
configurable hardware; hardware which can be programmed by the end user. A
digital
Cost is
incurred on implementing a digital controller for a washing machine which
requires that an inventory of all its components such as its logic circuits,
functional devices and the counter circuits be maintained. The implementation
time is significantly high as all the circuit components have to be placed on a
circuit board and connected together. If there is a change in the controller
circuit the entire circuit board has to be redesigned. A PLD based washing
machine controller does not require a large inventory of components to be
maintained. Most of the functionality of the controller circuit is implemented
within a single PLD integrated circuit thereby considerably reducing the
circuit size. Changes in the controller design can be readily implemented by
programming the PLD.
Programmable
Logic Devices can be used to implement Combinational and Sequential Digital
Circuits.
Memory
Memory
plays a very important role in Digital systems. A research article being edited
by a scientist on a computer is stored electronically in the digital memory
whilst changes are being made to the document. Once the document has be
finalized and stored on some media for subsequent printing the memory can be
reused to work on some other document. Memory also needs to store information
permanently even when the electrical power is turned off. Permanent memories
usually contain essential information required for operating the digital
system. This important information is provided by the manufacturer of a digital
system.
Memory is
organized to allow large amounts of data storage and quick access. Memory (ROM)
which permanently stores data allows data to be read only. The Memory does not
allow writing of data. Volatile memory (RAM) does not store information
permanently. If the power supplied to the RAM circuitry is turned off, the
contents of the RAM are permanently lost and can not be recovered when power is
restored. RAM allows reading and writing of data. Both RAM and ROM are an
essential part of a digital system.
Analogue to Digital and Digital to
Analogue conversion and Interfacing
Real-world
quantities as mention earlier are continuous in nature and have widely varying
ranges. Processing of real-world information can be efficiently and reliably
done in the digital domain. This requires real-world quantities to be read and
converted into equivalent digital values which can be processed by a digital
system. In most cases the processed output needs to be converted back into
real-world quantities. Thus two conversions are required, one from the
real-world to the digital domain and then back from the digital domain to the
real-world.
Modern
digitally controlled industrial units extensively use Analogue to Digital (A/D)
and Digital to Analogue (D/A) converters to covert quantities represented as an
analogue voltage into an equivalent digital representation and vice versa.
Consider the example of an industrial controller that controls a chemical
reaction vessel which is being heated to expedite the chemical reaction. Figure
1.9. Temperature of the vessel is monitored to control the chemical reaction.
As the temperature of the vessel rises the heat has to be reduced by a
proportional
A/D and D/A converters are an important aspect of digital
systems. These devices serve as a bridge between the real and digital world
allow the two to communicate and interact together.
Number Systems and Codes
Decimal Number System
The decimal
number system has ten unique digits 0, 1, 2, 3… 9. Using these single digits,
ten different values can be represented. Values greater than ten can be
represented by using the same digits in different combinations. Thus ten is
represented by the number 10, two hundred seventy five is represented by 275
etc. Thus same set of numbers 0,1 2… 9 are repeated in a specific order to
represent larger numbers.
The decimal number system is a positional number system
as the position of a digit represents its true magnitude. For example, 2 is
less than 7, however 2 in 275 represents 200, whereas 7 represents 70. The left
most digit has the highest weight and the right most digit has the lowest
weight. 275 can be written in the form of an expression in terms of the base
value of the number system and weights.
2 x 102
+ 7 x 101 + 5 x 100 = 200 + 70 + 5 = 275
where, 10 represents the base or radix
102,
101, 100 represent the weights 100, 10 and 1
of the numbers 2, 7 and 5
Fractions in Decimal Number System
In a Decimal Number System the fraction part is separated
from the Integer part by a decimal point. The Integer part of a number is
written on the left hand side of the decimal point. The Fraction part is
written on the right side of the decimal point. The digits of the Integer part
on the left hand side of the decimal point have weights 100, 101, 10 2 etc. respectively starting from the
digit to the immediate left of the decimal point and moving away from the
decimal point towards the most significant digit on the left hand side.
Fractions in decimal number system are also represented in terms of the base
value of the number system and weights. The weights of the fraction part are represented
by 10-1,
10-2, 10-3 etc. The weights decrease by a factor
of 10 moving right of the decimal point. The number 382.91 in terms of the base
number and weights is represented as
3 x 102 + 8 x 101 + 2 x 100 + 9 x 10-1 + 1 x 10-2 = 300 + 80 + 2 + 0.9 + 0.01 = 382.91
Caveman number system
A number system discovered by
archaeologists in a prehistoric cave indicates that the caveman used a number
system that has 5 distinct shapes ∑, ∆, >, Ω and ↑.
Furthermore it has been determined that the symbols ∑ to ↑represents the
decimal equivalents 0 to 5 respectively.
Centuries
ago a caveman returning after a successful hunting expedition records his
successful hunt on the cave wall by carving out the numbers ∆↑.What does the
number ∆↑ represent? The table 1.1 indicates that the Caveman numbers ∆↑represents
decimal number 9.
The Caveman is using a Base-5 number system. A Base-5
number system has five unique symbols representing numbers 0 to 4. To represent
numbers larger than 4, a combination of 2, 3, 4 or more combinations of Caveman
numbers are used. Therefore, to represent the decimal number 5, a two number
combination of the Caveman number system is used. The most significant digit is
∆which is equivalent to decimal 1. The least significant digit is ∑which is
equivalent to decimal 0. The five combinations of Caveman numbers having the
most significant digit ∆, represent decimal values 5 to 9 respectively. This is
similar to the Decimal Number system, where a 2-digit combination of numbers is
used to represent values
greater
than 9. The most significant digit is set to 1 and the least significant digit
varies from 0 to 9 to represent the next 10 values after the largest single
decimal number digit 9.
The Caveman number ∆↑can be written in expression form
based on the Base value 5 and weights 50, 51, 52 etc.
= ∆x 51
+ ↑x 50 = ∆x 5 + ↑x 1
Replacing
the Caveman numbers ∆and ↑with equivalent decimal values in the expression
yields
= ∆x 51
+ ↑x 50 = 1 x 5 + 4 x 1 = 9
The
number ∆Ω↑∑indecimal
is represented in expression form as
∆x 53 + Ω x 52 + ↑x 51 + ∑x 50 = ∆x 125 + Ω
x 25 + ↑x 5 + ∑x 1
Replacing the Caveman numbers with
equivalent decimal values in the expression yields
= (1) x 125 + (3) x 25 + (4) x 5 + (0) x 1 = 125 + 75 +
20 + 0 = 220
Binary Number System
The Caveman Number system is a hypothetical number system
introduced to explain that number system other than the Decimal Number system
can exist and can be used to represent and count numbers. Digital systems use a
Binary number system. Binary as the name indicates is a Base -2 number system
having only two numbers 0 and 1. The Binary digit 0 or 1 is known as a ‘Bit’.
Table 1.2
Counting in Binary Number system is similar to
counting in Decimal or Caveman Number systems. In a decimal Number system a
value larger than 9 has to be represented by 2, 3, 4 or more digits. In the
Caveman Number System a value larger than 4 has to be represented by 2, 3, 4 or
more digits of the Caveman Number System. Similarly, in the Binary Number
System a Binary number larger than 1 has to be represented by 2, 3, 4 or more
binary digits.Any binary
number comprising of Binary 0 and 1 can be easily represented in terms of its
decimal equivalent by writing the Binary Number in the form of an expression
using the Base value 2 and weights 20, 21, 22 etc.
The number
100112
(the subscript 2 indicates that the number is a binary number and not a decimal
number ten thousand and eleven) can be rewritten in terms of the expression
100112
= (1 x 24) + (0 x 23) + (0 x 22) + (1 x 21) + (1 x 20)
= (1 x 16) + (0 x 8) + (0 x 4) + (1 x 2)
+ (1 x 1)
= 16 + 0 + 0 + 2 + 1
= 19
Fractions in Binary Number System
In a Decimal number system the Integer part and the
Fraction part of a number are separated by a decimal point. In a Binary Number
System the Integer part and the Fraction part of a Binary Number can be
similarly represented separated by a decimal point. The Binary number 1011.1012 has
an Integer part represented by 1011 and a fraction part 101 separated by a
decimal point. The subscript 2 indicates that the number is a binary number and
not a
decimal number. The Binary number
1011.101 2
can be written in terms of an expression using the Base value 2 and weights 23, 22, 21, 20, 2-1, 2-2 and 2-3.
1011.1012
= (1 x 23) + (0 x 22) + (1 x 21) + (1 x 20) + (1 x 2-1) + (0 x 2-2) + (1 x 2-3)
= (1 x 8) + (0 x 4) + (1 x 2) + (1 x 1)
+ (1 x 1/2) + (0 x 1/4) + (1 x 1/8)
= 8 + 0 + 2 + 1 + 0.5 + 0 + 0.125
= 11.625
Computers
do handle numbers such as 11.625 that have an integer part and a fraction part.
However, it does not use the binary representation 1011.101. Such numbers are
represented and used in Floating-Point Numbers notation which will be discussed
latter.
No comments:
Post a Comment