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EENG22000

Communications

Simon Armour

Version 2.1.1 (interim)


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This version of the EENG22000 unit was adapted for delivery during the Covid-19
pandemic. A number of adaptations have been made to accommodate the constraints
on learning, teaching and studying. These include:
 A reduction of the course syllabus of around 10% to reduce student workload
 Removal of the assessed laboratory assessment (commonly referred to as the
‘Modem lab’) to further reduce student workload
 Adjustments to the lecture delivery to create a suite of smaller modular ‘lectures’
for asynchronous online study
 Arrangements for Synchronous teaching activities
 Addition of online tests in preparation for the final summative assessment taking
place online as opposed to the normal written examination under controlled
conditions
Throughout the remainder of this course content directly addressing these Covid-19
adaptations will appear in this boxed format.
THIS VERSION IS AN INTERIM VERSION COVERING
PARTS 0-2. A REVISED VERSION COVERING ALL OF
SECTIONS 0-4 WILL BE PROVIDED AT A LATER DATE.




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0. Introduction
Throughout the last century electronic communications systems have made massive
progress in terms of the service that they are capable of providing. Analogue wireline
technology progressed from simple telegraph systems to telephony. Analogue radio has
facilitated television and radio broadcasting and even early mobile telephone technology.
However, with the advent of digital signal processing technology, communications has
evolved into a digital age where a previously unimaginable range of services can be
provided and the use of sophisticated communication technology is a ubiquitous feature
of human life.
Modern wireless communications systems are capable of supporting robust and reliable
access to data and the mobile phone and WiFi services are simply taken for granted. High
speed, low latency communications, video streaming, cloud computing access and more
or less any other application the individual user can dream of accessing can be facilitated.
Even highly demanding services such as mobile cloud gaming are now reaching the
mainstream user. Simultaneously, cable and fibre technologies have evolved to provide
the infrastructure capable of meeting our mass needs. Now, not all communication even
has a human user with machine to machine communications rapidly expanding for
application such as smart cities, smart grid, smart transport and digital health care.
Whilst wireless, cable and fibre links all pose significantly different challenges and
opportunities in the design of communications systems, many of the techniques employed
to address these challenge have a degree of commonality.
0.1. Who and what is this unit for?
This course is intended to provide the student with an introduction to the fundamental
concepts of communications systems as applied to a wide variety of different applications.
Emphasis is placed on the concepts and terminology used in communications systems
and not on complex mathematical proofs and analysis. The goals of the unit are for
students:
 To understand the language of communication engineers and be able to use it
competently themselves
 To understand the physical limits of communication systems and consequently
what is possible and what is not possible
 To understand the wide variety of significant challenges faced in the design of
communication systems
 To understand various options in system design (factors such as modulation
schemes, coding rates, filtering parameters)
 For to be able to design a good communication system for a given scenario

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For some students this unit will be the first in a sequence of communication focussed units
that you will study. This unit will provide the foundation for your communication
engineering education. A plethora of later units will enable you to specialise your expertise
further and study more advanced aspects of communications. These units include Mobile
Communication Systems, Advanced Mobile Radio Techniques, Broadband Wireless
Communications, Coding Theory, Networking Protocol Principles, Advanced Networking,
Optoelectronic Devices and Systems, Optical Communications Systems and Data
Networks, Data Centre Networking. You likely won’t be able to study all of these units but
many will be available, depending upon which programme you are studying.
Other students may choose to specialise in area other than communications (hey, no one
is perfect!) but many other areas of engineering and science make use of communications
and a basic knowledge of communications will still support your studies in those areas.
For example, take a look at a computer. Count the number of communications interfaces
you can see on the outside of a desktop PC or Laptop. If you are really keen, take it apart!
USB, HDMI, Ethernet, Wifi PCI, SATA, etc – these are all communication interfaces.
Over the course of this unit we will learn techniques used in 2G, 3G, 4G and 5G mobile,
WiFi, Bluetooth, ADSL, DOCSIS and Fibre (Broadband), USB, Ethernet, Television and
Broadcast Radio and many other technologies that people use in everyday life.
While you study this unit, think of yourself as a communication engineer. That’s the way
that I will address you in the lectures and in these notes. Whilst you may not pursue a
career in communications during study or beyond, try to retain that mindset whilst studying
this unit.
0.2. Unit Structure
This unit is structured into four parts:
 Part 1 provides an introduction to the fundamental concepts of communications,
including the basic building blocks of a communications system, various factors
engineers have to consider in their designs and then moves on to identify a number of
numerical metrics that apply to communication systems relates them together in a key
theorem; the Shannon capacity theorem.
 Part 2 covers the basic forms of digital bandpass modulation available to the
communications engineer. Part 2 keeps this simple by considering only binary
modulation and considering modulation of the Amplitude, Frequency and Phase of a
waveform in turn.
 Part 3 explores more advanced modulation schemes by considering what we can
achieve with non-binary modulation and by modulating more than one property of the
waveform.
 Throughout parts 2 and 3, filters are seen to be used in communications systems. The
details of these filters are important but aren’t discussed in detail in parts 2 and 3. Thus

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in part 4, we investigate these filters in detail. Part 4 also explores the concept of error
control coding which is another key tool in the design of communication systems.
0.3. Learning Materials
A suite of materials is provided to aid study. This suite consists of:
 This course handbook, which discusses all the topics in the syllabus in detail.
Sections 1-4 of this handbook correspond directly to parts 1-4 of the course.
 A set of lectures, recorded, so that you the student can watch them in your own
time (so called ‘asynchronous study’).
 A timetabled schedule of ‘synchronous’ classes. This is where we will meet as a
group (whether virtually or in person) to discuss the syllabus topics in detail.
 Problem sheets; 1 per part of the course, as outlined above, to be taken in your
own time. Questions will be released after the synchronous sessions for each part
of the course is complete.
 Online tests; 1 per part of the course to be taken within a scheduled window. These
tests will be indicative of the final unit assessment if it takes place online.
 Some software lab experiments which will enhance understanding through
experimentation. These software lab experiment activities will be led by Dr
Tommaso Capello.
All of this material will be provided on ‘Blackboard’, the Universities Online Learning
Environment.
0.4. Recommended Texts
Whilst the materials described in section 0.3 above provide the essential study material
the following texts are recommended as optional supporting reading material for this
course:
1. Digital Communications, Andy Bateman, Addison Wesley, 1998
2. Communication Systems (3rd/4th Edition), S Haykin, Wiley, 1994
3. Communication Systems, F G Stremler, Addison Wesley, 1990
4. Principles of Communication Systems (2nd Edition), Taub & Schilling, McGraw-Hill.
5. Communication Systems: An Introduction to Signals and Noise in Electrical
Communication, A B Carlson, McGraw Hill, 1968

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0.5. Mathematical Background
Students are expected to have good understanding of the following mathematical
concepts:
 Fourier series and analysis
 Fourier transforms
 Time/frequency representation of signals
 Trigonometric identities and substitutions.
Class 1 will provide an overview of the whole course and introduce the supporting
resources.


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1. Fundamentals of Communications
In this section key concepts of communications systems are introduced. These include
message types and quality measures, components of the communications system,
duplexing and multiple access strategies and the relationship between bandwidth and
capacity.
1.1. Components of the Communications System
The high level essential components of a communications system are illustrated in

Figure 1.
Transmitter
Channel
Receiver
Input
Message
Message
Source
Message
Destination
Transmitted
Signal
Received
Signal
Output
Message

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Figure 1: Outline of a Communications System
The message is whatever we are using the communication system to communicate. We
use communications for a plethora of applications and thus messages can be many, many
different things. For example:
 If you ‘chat’ to a friend on SMS or WhatsApp or similar then the message is the
text that you send them and that they send you.
 If you browse the web, the message is the url that you want to access and the
content of that webpage.
 If you stream music, the message is the audio content of the music you want to
hear.
 If you are watching a video, it’s the images you see, frame by frame, pixel by pixel,
RGB, chrominance and luminance (depending upon video format) and the
accompanying audio.
 If you play an online video game, an FPS shooter for example, then the message
consists of game state data, position, damage delivered and taken, loot drops etc.
This list is by no means exhaustive and the application of communications is only really
limited by the imaginations of those who design applications. The success of
communications engineers in making delivery of messages possible has led to an
explosion of applications and the challenges are ever evolving. As people seek to do more
and more things enabled by communications, we as engineers have to do more and more
to make it possible. Rarely these days are we able to design a communication system for
Transmitter
Channel
Receiver
Input
Message
Message
Source
Message
Destination
Transmitted
Signal
Received
Signal
Output
Message

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one specific type of message and often we have to design the communication system to
work for any application that it might be used for. This can be quite challenging!
The message has a source from which it originates and a destination. The whole purpose
of the communication system is to convey the message from the source to the destination
‘well’. What constitutes ‘well’ is another complex matter. We often refer to this as Quality
of Service (QoS) which is taught in detail in the unit EENGM4221 Broadband Wireless
Communications. Suffice to say here, QoS is typically about conveying the message
quickly and accurately from source to destination.
The transmitter takes the input message signal and processes it to produce the transmitted
signal. This processing may involve such steps as: source coding (compression), error
control coding, filtering and amplification. In all cases, it will involve modulation of the
message signal. All of these processes are designed with the purpose of adapting the
message signal into a form that will propagate successfully through the communication
channel.
The channel is the medium via which a signal is conveyed from the transmitter to the
receiver. Examples include ‘wire’ (typically copper cable either as twisted pair or coaxial
cable), optical fibres and radio waves. The channel is a crucial to communications. If you
don’t have a channel, there is no way to ‘connect’ your source and destination;
communication without a channel is impossible. However, the channel is typically also one
of the biggest challenges that we have to deal with as communication engineers. The
signal is conveyed from transmitter to receiver without being changed in any way by the
channel. These effects include loss (Attenuation and/or fading), limited bandwidth,
filtering, additive noise, interference, dispersion and many other forms of distortion. For
this reason we distinguish between the transmitted and received signals and as
communications engineers we must accept that the received signal will be very different
to the transmitted signal. The channel is also the component of the communication system
over which we have the least choice or control. It is always dictated to us by external
factors.
The receiver has the task of attempting to recover the input message signal from the
received signal. This involves reversing the effects of the transmitter and the channel. If
the receiver can perfectly cancel out whatever effects the transmitter and channel apply
to the original message then we should be able to reproduce the message at the
destination. Hence, filtering, amplification and coding often take place. Demodulation is
required in all cases. Due to the likely distortion introduced by the channel, it cannot be
guaranteed that the receiver will be entirely successful in this task. Hence it is necessary
to distinguish between the input message and the output message but our goal is usually
to make sure that the latter is very similar to the former.
Typically, the communications engineer has little control over the nature of the channel in
the communications system. This is usually dictated by the application for which the
system is intended. Thus, the task facing the communications engineer is to design the
modem to most effectively make use of the available channel. Many other factors must be
considered in this task: the size and cost constraints of the equipment to be used are often

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very important as is the speed of communication achieved. However, most important of
all, the output message must adequately resemble the input message. Otherwise there is
no point.
1.1.1. ‘Analogue’ and ‘digital’ in the context of communications
We are all familiar with the terms ‘analogue’ and ‘digital’ and it should come as no surprise
that these terms can be applied to the domain of communications. However there are
some subtleties to consider here.
Messages may be inherently analogue or digital. Many messages are derived from the
physical world (audio and video are good examples again) and are thus inherently
analogue but are often converted into digital form at source. Until the last couple of
decades, television was an all analogue process. An analogue property (visible light) was
captured, converted into an electrical domain (an analogue voltage), then to an
electrogmagnetic wave that would propagate through the channel and then the process
reversed to reproduce the image. Other messages originate in digital form. A word file, for
example, has no analogue property. It is inherently digital. If we want to send that word
file somewhere else, it is a digital message.
There are very few physical communications channels that are truly digital. There is no
digital light, no digital electricity, no digital radio waves. So more or less all channels, at a
physical level, are analogue.
Whilst we recognise and understand that messages come in many forms and may be
either analogue or digital and can classify a message as analogue or digital and we
recognise and understand that most, if not all channels are truly analogue, when we
classify a communication system as either analogue or digital, we are not referring the
message or the channel. Rather, we classify a communication system as analogue or
digital on the basis of the modulation that it uses.
Analogue communications systems transmit a continuously variant signal to represent
their message. Some property of the physical signal is varied in continuous fashion to
represent the message.
Digital communications systems transmit a discretely varying signal to represent their
message somehow. Some property of the physically continuous signal is varied in discrete
fashion between a finite alphabet of possible states. Whilst this sounds a little confusing,
just think about something like some basic electronic logic. We might use 0v and 5v to
represent logical 0 and logical 1 respectively. Its not that no other voltage can exist
(voltage is a continuous variable) but we constrain the possible voltages to a finite alphabet
(our two voltages).
Because the vast majority of communications systems are now digital communication
systems, its common to use digital communication even for analogue messages. For
example, if you take a picture with a digital camera on your phone and then want to send
it to a friend, whilst you are capturing an inherently analogue property (light) it is

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immediately converted to a file, essentially a collection of binary data. That binary data is
then processed by the transmitter into a set of symbols from a finite digital alphabet. Those
signals are modulated onto radio waves which will propagate through a wireless channel
to the receiver. We classify this as a digital communication system, even though the
message was originally analogue and physical signal is analogue. It’s a digital
communication system because the transmitted signal was constrained to a finite set of
digital symbols for transmission.
1.1.2. Baseband and bandpass
Another classification that we apply to communication systems is that they are either
‘baseband’ or ‘bandpass’.
Messages will typically occupy a range of frequencies from dc upwards. We refer to this
frequency range as baseband. Where possible, this can be modulated and transmitted
directly into the channel without the need to change the frequency. This is known as
baseband signalling or baseband modulation and we classify such a communication
system as a baseband communication system. An example of this would be old (not
mobile) telephones. The sound of your voice, with frequency content in the range of a few
10s of Hz up to a few kHz, is converted to electricity by a microphone and that electrical
voltage is conveyed by cable without any need to change the frequency (it might be
filtered).
In some cases, however, it is not possible to transmit the baseband signal directly over
the channel. This results from the message signal being of unsuitable frequency for the
channel. Examples of this are radio and fibre optic communications systems. Radio bands
for communication are usually a finite range of frequencies in the range of MHz or GHz.
Its not usually legal to transmit at lower frequencies. The light which propagates along
optical fibres is inherently at very high frequencies. There’s no such thing as baseband
light! This finite range of frequencies that can be allowed in the channel effectively makes
it a filter of sorts; hence we describe it as a bandpass channel.
It thus becomes necessary for the communication system to change the frequencies for
transmission through the channel. The true frequency content of the message is shifted
up to a higher frequency, transmitted through the bandpass channel, and then shifted back
down to baseband again. If we employ this method to communicate through a bandpass
channel then we classify the communication system as a bandpass communication
system. An example of this would be a modern mobile phone where your voice, with
frequency content in the range of a few Hz up to a few kHz, is digitised and then shifted
up to a frequency range of ~900MHz, ~1.8GHz or ~3.5GHz (depending upon your mobile
system) for transmission as a radio wave.
The method for shifting frequencies up and down for the purposes of bandpass
communication is called carrier modulation is discussed in more detail in section 2.1.

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Lecture 1 will cover the components of the communication system, including analogue
and digital and baseband and bandpass.
1.2. Factors Affecting System Design
Many factors affect the design of the communication system.
Technological and logistical Issues often place limitations upon the design of
communications hardware. Often, what can be designed in theory cannot be implemented
in practice. Size and power consumption are often particularly relevant to mobile
technologies. Cost is also a big issue, particularly for mass market products. Hardware
and software availability and compatibility must also be considered. Further, issues to do
with installation of equipment and the associated time, cost and inconvenience must also
be considered, particularly for consumer electronic devices.
Government regulations are also crucial to the design of communications systems. The
worth of a communications device may be severely reduced if it is illegal to operate it!
Government regulations often restrict the use of different ‘bands’ in the radio spectrum to
particular applications and devices. Even when operating within regulations, further
limitations must be accounted for. Operating bandwidth will typically be limited to ensure
‘fair sharing’ of a particular band. Power limitations may be imposed for health reasons.
Also, in many bands, a specific standard for device design is imposed to ensure
interoperation of devices. The designer is restricted to design within the relevant standard
in such cases. A good example for all these cases is the mobile phone industry. Each
operator has been licensed a limited amount of radio spectrum in which to operate their
service – this ensures that all networks can operate successfully and fairly and (in theory)
provide competition between service providers for the benefit of the customer. Maximum
transmit power of the handset is limited to ensure the safety of the user and all phones
must conform to a particular standard. These different standards are more commonly
known by their colloquial generational number, 2G to 5G at the time of writing with 6G
inevitably on the horizon.
Of all the issues to consider, perhaps the two most important are the available bandwidth
and the sources of degradation. These are considered in more detail below.
1.2.1. Bandwidth Limitations
We will discuss in section 1.7 that the maximum symbol and data rate that can be achieved
by a communication system is directly related to the available bandwidth in which it can
operate. Since the range of applications that a communication system can support is
limited by data rate, the available bandwidth is a crucial factor in system design.
In metallic cables, there is a limited range of frequency of electrical signal that will
propagate effectively. In fibre optics, there is again a limited range of frequencies of
electromagnetic wave (visible light) that will propagate along the fibre. Sometimes it is
possible to increase the available bandwidth simply by increasing the number of cables or

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fibres (hence the bundles of fibres that you may see being installed when the roads are
dug up). At other times this is not practical for some reason.
In ‘wireless’ systems there is no physical ‘pipe’ into which the signal is constrained. Thus,
it is not possible to add another isolated medium for propagation (cable, fibre etc). There
is only one electromagnetic spectrum and there are many technologies wishing to make
use of it, not all of which are communications systems (weather radars and microwave
ovens for example). For these reasons, bandwidth is usually particularly limited in wireless
applications.
For much of the last few decades its was common ‘knowledge’ amongst communications
engineers that the radio spectrum was ‘congested’ or ‘full’. There was no more spectrum
to go around. Any new system had to share spectrum with existing systems or even take
away their rights to use the spectrum. This led to considerable problems allocating
spectrum to new systems. More recently though studies have shown that whilst spectrum
may be fully allocated ‘on paper’ its actual utilisation is quite low. In any given place and
at any given time much of the radio spectrum can be seen to be unused. This has led to
a fundamental rethink into how we use the radio spectrum. There has been much interest
in ‘Cognitive’ radio systems that are smart enough to work out when spectrum is not being
used and make use of it and then vacate (stop transmitting) when the systems that are
entitled to that spectrum do use it. There has also been an increased focus on spectrum
sharing to encourage better utilisation of spectrum.
Whatever the medium of signal propagation, it is inevitable that, due to one of the factors
described above, a bandwidth limit will be encountered. It is the job of the communications
design engineer to maximise the use of this bandwidth in terms of achieved data rate
within the practical limitations of the application. This is known as bandwidth efficiency –
a term to which we will often refer and which will be quantified in section 1.7.3.
A good example of the importance of bandwidth efficiency is given by the UK government’s
spectrum auction for ‘3G’. Error! Reference source not found. shows the price paid by
the five successful bidders for their licenses.
Table 1. Successful UK 3G License Bids
License Bidder Price
A (35MHz) TIW £4,384,700,000
B (30MHz) Vodaphone £5,964,000,000
C (25MHz) BT3G £4,030,100,000
D (25MHz) One2One £4,003,600,000
E (25MHz) Orange £4,095,000,000


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Given that operators paid billions of pounds for the right to operate their service it is clear
that they will need to make the very best use of their spectrum in order to recover their
investment.
Clearly, 3G is not the current generation and what we have seen, in the UK at least, is that
subsequent spectrum licensing processes for 4G and 5G have not been as spectacular in
their spectrum pricing as 3G was. This is a consequence of economic factors and a degree
of ‘pushback’ from the operators after paying these very high 3G costs. Nevertheless,
spectrum remains a precious and often expensive resource.
Not all bands are licensed. Some spectrum bands are made available on an ‘unlicensed’
or ‘license exempt’ basis. One example is the 2.4GHz Industrial Scientific and Medical
(ISM) band which has been very successfully used for communications by consumer
products such as WiFi and Bluetooth. Similar unlicensed bands have subsequently been
used by WiFi at 5GHz and 60GHz.
1.2.2. Signal/Message Degradation
The received signal can be degraded (distorted) with respect to the transmitted signal via
a variety of mechanisms. This signal degradation may be translated by the receiver into a
message degradation. Note that not all distortion is introduced into the communications
system by the channel – often it is introduced by the transmitter or receiver! Example
causes of degradation are:
 Gain/Attenuation of the signal. This may also be frequency selective (i.e. some
frequencies of the signal are subject to a different gain/attenuation than others) and
may also vary with time. We will look at this in a bit more detail in section 1.5.
 Frequency Offset may occur between the transmitter and receiver. This can be due to
error between the local oscillators (a hardware subsystem) in the transmitter and
receiver or via the mechanism of Doppler shifting in a mobile system.
 Distortion generated in the transmitter and/or receiver hardware. An example of this is
non-linear distortion in the transmit power amplifier (something you may study in the
Electronics 2 unit).
 Dispersion (also known as delay spread). This is the ‘smearing’ of the signal in time.
Also discussed in section 1.5.
 Interference. This is contamination of the communications channel by extraneous
sources, such as: power lines, ignition systems, machinery, other communication
channel users, etc.
 Additive Noise. This is characterised as random signals from natural sources. It may
occur for a number of reasons.

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 Thermal noise: This is due to the thermal agitation of electrons in the signal processing
circuits. In the case where the signal is attenuated severely between transmitter and
receiver, thermal noise in the receiver front end can be the dominant source of additive
noise in the system. This results from the fact that the signal input to the receiver is
very small and cannot be separated from the thermal noise. It is thus necessary to
amplify both the wanted signal and the noise before demodulation. Thermal noise in
the receiver is often the limiting constraint on the performance of many real
communications systems and so we will discuss it in more detail in section 1.2.3.
 Atmospheric noise: This is caused by random electrical processes in the atmosphere
at which an antenna is ‘pointing’.
1.2.3. Additive Noise
Additive Noise is commonly modelled as ‘Additive White Gaussian Noise’ (AWGN). This
is because the noise (N) typically has a randomly varying power conforming to a Gaussian
(G) statistical distribution, has equal power across all frequencies (hence, ‘white’ (W)) and
is added to the signal (A) – predominantly in the receiver.
The Gaussian probability density function is given in Equation 1 and is illustrated in Figure
2.
 



  2
2
2
exp
2
1)(
X
X
X
xxp



 x (Equation 1)
Where ௫ is the mean value of the variable , ௫ is its standard deviation and () is the
probability of the variable having a value of x . Very importantly, for AWGN, ௫ = 0. This
means that the average amplitude is 0 (it’s symmetric around 0) and the most likely
amplitudes of noise are smaller values. Also, the noise power corresponds to its variance,
௫ଶ.
Further note that AWGN has uniformly distributed phase.
The ratio of signal power to noise power or ‘Signal to Noise Ratio’ (SNR) is a key limiting
factor of communications systems. We will come back to it again and again in this course
– so get used to it!

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Figure 2: Gaussian Distribution
Lecture 2 will cover factors affecting system design with the focus on bandwidth, noise
and quality measures.

Class 2 will give us an opportunity to discuss the content of lectures 1 and 2, hence
components of the communication system and factors affecting system design. We will
consider the implications of bandwidth and noise in more detail.
1.3. Quality Measures and Metrics
1.3.1. Signal to Noise Ratio and related measures and metrics
It is intuitively obvious that if at the receiver we have to work with a combination of the
signal we deliberately transmitted and random signals that we didn’t want then our ability
to communicate successfully will be dependent upon how much of each we have. Lots of
good signal, not much bad noise? Its easy to communicate. Think of yourself having a
conversation in a quiet place (just the two of you in a room or a park); its easy to hear the
other person, to understand what they say and vice versa. Not much signal, lots of
background noise (a busy café, a rock gig or somewhere with lots of machinery in
operation) and things get much harder. It’s the same for any communication system.
One way which we can quantify this variable of how much wanted signal to unwanted
noise that we have is to directly calculate the ratio of Signal Power, to noise power .
This ratio is referred to as the Signal to Noise Ratio and often written as ‘SNR’ and is a
Value
Pr
ob
ab
ilit
y
of
g
iv
en
V
al
ue
µx
1/ Ö2x

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key piece of information for a communication engineer when designing a communication
system.
Note that the meaningful place in which we evaluate the SNR is at the receiver because
that is where the performance of the system will be determined. Its all very well having a
high SNR at the transmitter (this is easy to achieve) but after the signal experiences loss
in the channel, the SNR will be much lower.
We can calculate the theoretical thermal noise power in an electronic device from physical
parameters:
= (Equation 2)
Where is the Boltzmann constant, is Temperature (in Kelvin) and B is bandwidth. Note
that this is theoretical ideal and in practice imperfect electronic devices will typically
experience higher thermal noise that this minimum.
Other related measures that we can use to quantify the wanted signal and unwanted noise
are:
 The energy per symbol, ௦, which can be determined from the signal power and
the rate at which symbols are transmitted.
 The energy per bit, ௕, which can be determined from the signal power and the
rate at which bits are transmitted.
 The Noise Spectral Density, ଴, which can be determined from the noise power
and bandwidth.
1.3.2. Error measures and metrics
Digital messages vary in discrete fashion and take the form of a sequence of symbols from
an alphabet. We typically then measure the quality or accuracy of communication
according to the difference in the data at the receiver as compared to the transmitter. Any
differences are called errors. If we knew an error was an error it would be easy to fix it.
Typically, though, we do not. Errors occur randomly, due for example to AWGN. AWGN
is a random variable and will cause errors to occur at random. Thus we typically use
statistical measures of accuracy.
We can predict the probability of any given bit being in error, ௕. This will be a value
between 0 and 1.
We can evaluate the proportion of errors within a group of bits. For a large number of bits
this will be approximately equal to ௕. We term this the Bit Error Rate (BER).
We can use similar metrics for symbols rather than bits, hence ௦ and Symbol Error Rate
(SER).

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Consider that the worst case BER is effectively 0.5 as for any higher value we could
effectively improve the accuracy through a simple inversion. The same is not true for SER.
Another very important quality measure that may be associated with either analogue or
digital messages and signals relates the ratio of power in the wanted signal to the power
of the undesired noise. This is known as the Signal to Noise Ratio (SNR) and is usually
evaluated in dBs.
In digital systems a common method for comparing performance takes the form of a BER
vs. SNR graph. An example is shown in Figure 3.

Figure 3. Example BER vs. SNR Curves
This example diagram is just a sketch for illustrative purposes and isn’t based on any real
data. Elsewhere in the course we will use data from measurements, theory or simulations
to produce BER vs SNR for real communication systems.
Consider the BER versus SNR axes and think about how we would want a communication
system to perform. Ideally, a communication system would exhibit very low BER even with
a low SNR. Thus the more we can ‘move’ the BER vs SNR curve toward the bottom left
corner of the graph, the more we have succeeded. Its not that we want the SNR to be low
but if we can have less errors even when it is low then it is more reliable.
These diagrams enable both qualitative and quantitative comparisons of system
performance such as:
BE
R
SNR / dB
10-1
10-2
10-3
10-4
10-5
1 2
3
4
3dB

- 20 -




 The BERs of systems 1,2, and 3 reduce exponentially with SNR whilst system 4
reduces only linearly – this suggests that system 4 has more problems than just
AWGN (more on that in section 1.2.3. ).
 System 2 is better than System 1 for lower SNRs but worse for higher SNRs.
 System 2 outperforms System 3 by 3dB and is thus better at all SNRs and BERs.
Examples of quality measures associated with analogue messages are such quantities as
distortion and fidelity. However, since we focus on digital communications in this unit, we
will not discuss analogue quality measures any further.
Lecture 3 will cover quality measures and metrics.
1.4. Baseband Communications
In this section, we will consider some of the issues associated with baseband digital
signalling. Since, modulation is the process of varying a signal to represent the message,
we need a signal that we can modulate and transmit through the channel. An obvious
example here is an electrical signal, commonly used in many communication systems. An
electrical voltage can be varied via various electronic means and so we can electronically
modulate our message onto a voltage. Since this is such a convenient example and used
in so many real-world applications, we will assume electrical signals throughout this
section.
Much of the content of this section is also relevant to bandpass modulation as well, with
the additional of further processes. Thus, when we move on to bandpass modulation in
section 2, don’t leave this content behind.
1.4.1. On-Off vs Polar Signalling
Two schemes for representing binary data by different signal amplitudes (voltage for
example) are ‘On-Off’ and ‘Polar’ signalling. On-Off signalling uses a non-zero signal
amplitude to represent one binary symbol and a zero amplitude to represent the other.
Polar signalling uses a signal amplitude of one polarity to represent one binary symbol
and a signal of opposite polarity (and usually equal amplitude) to represent the other binary
symbol. This is illustrated in Figure 4 for the case where we transmit ‘one bit at a time’
(one bit per symbol).

- 21 -





Figure 4: On-Off and Polar Baseband Signalling.
The Symbol Error Rate (SER) of On-Off signalling for the AWGN channel is evaluated by:









022
1
N
E
erfcP ss (Equation 3)
The SER of Polar signalling for the AWGN channel is evaluated by:










02
1
N
E
erfcP ss (Equation 4)
Where ௌ is the probability of symbol error and ௌ is the average energy in each
transmitted symbol. [ ] is the complementary error function of its argument. The
complementary error function evaluates the area under part of the Gaussian pdf curve. It
is worth noting that this area is the ‘tail’ of the distribution and the output of the function
thus gets smaller as its input gets larger. This makes intuitive sense as we would expect
the probability of error to reduce as the ratio of signal to noise increases. Note:
 BER is the same as SER for binary signalling because each symbol represents one
bit.
 For the same reason, ௌ is equivalent to the energy per bit, ௕.
1 01 1 1 1 10 0 0 0
1 01 1 1 1 10 0 0 0
+V
0V
0V
+V
-V
Uni-Polar
1+V
0 0V
Also known as “On-Off”
Bipolar
1+V
0 -V
Also known as “Polar”

- 22 -




It can be seen that Polar signalling outperforms On-Off signalling since it requires only
half the average energy per symbol to noise power density ratio in order to achieve a given
BER.
Similar conclusions can be drawn by considering the BER versus Eb/N0 curves of the two
signalling schemes for an AWGN channel. These are shown in Figure 5.

Figure 5: BER versus Eb/N0 for On-Off and Polar Signalling.
1.4.2. Non-Binary Baseband Signalling Schemes
Baseband signalling schemes are not restricted to the binary case (one bit per symbol).
Larger alphabet/constellation sizes may be used to transmit multiple bits at a time. The
relationship between BER and Es/N0 for ‘M-ary’ baseband signalling is given by:






o
s
s
N
E
M
erfc
M
MP
1
31
2 (Equation 5)
Where is the number of different symbols in the alphabet. For electrical baseband
modulation, M is thus the number of discrete different voltages that can be transmitted to
represent the data. We will use ‘M-ary’ throughout this course to refer to modulation
schemes (both baseband and bandpass) which use a digital alphabet of size M.

- 23 -




The SER versus Es/N0 relationship for M-ary baseband signalling is plotted in Figure 6.
It can be seen from both the equation and the graph that as M is increased the Es/N0
requirement rises more rapidly than the bandwidth efficiency. This is to be expected, given
the shape of the Shannon Capacity curve.
For M-ary Signalling, the terms On-Off and Polar no longer make the appropriate
distinction. Instead, it is more accurate to define the signalling scheme as either Uni-Polar
or Bi-Polar, depending on whether or not the range of amplitudes used is between zero
and non-zero values or between two non-zero values of different polarity respectively.
Note that the SER relationship in equation 4 is for the case of bipolar signalling.
For M-ary signalling it is desirable to minimise the BER for a given SER. Thus, we want
each symbol error to result in as few bit errors as possible. The symbol error probability is
dominated by errors that occur between adjacent symbols. Thus, it is desirable for
adjacent symbols to differ by as few bits as possible – ie one. This can be achieved by a
Gray coding scheme. Gray codes are sequences of binary words for which adjacent words
only ever differ by one bit. Example Gray codes for different word lengths (and hence
constellation sizes) are shown in Error! Reference source not found., where letters A-P
represent adjacent symbols in the alphabet.

Figure 6: BER Performance of Bipolar M-ary Signalling


- 24 -




Table 2. Example Gray Codes.
M=2 M=4 M=8 M=16
A 0 00 000 0000
B 1 01 001 0001
C 11 011 0011
D 10 010 0010
E 110 0110
F 111 0111
G 101 0101
H 100 0100
I 1100
J 1101
K 1111
L 1110
M 1010
N 1011
O 1001
P 1000

If Gray Coding is employed:
b
P
P sb (Equation 6)
We will not explore in detail what happens if Gray coding in not employed. Suffice to say
that in the majority of cases ௕ increases and that is a bad thing. There are some
exceptions to this (related to coding schemes) but unless we know specifically to the
contrary, as communication engineers we use Gray coding by default.
Lecture 4 will present the basics of baseband modulation.

Class 3 will give us an opportunity to discuss the content of lecture 3 and 4, i.e quality
measures and baseband modulation. We will use this information to discuss how we

- 25 -




make decisions at the receiver and how the nature of AWGN affects this. We will design
some mapping rules for baseband modulation to reflect these factors.
1.5. Delay Spread and ISI
Delay Spread is the dispersion or ‘smearing’ in time of the energy of the signal. As a result,
the transmitted signal will be distorted by ‘echoes’ of earlier parts of the same signal. The
delay spread of the channel can be illustrated by its impulse response.
The concepts of dispersion and impulse responses should be familiar to you from your
Signals and Systems unit. Think of an FIR filter for example. This disperses the input signal
and the transfer function of the FIR filter is the delay spread.
The impulse response of a radio channel at 2GHz (the 3G UMTS band) as measured at
the University of Bristol is illustrated in Error! Reference source not found. for a single
instance of time. The effect of delay spread on the transmitted signal can be described
mathematically by:
     nhnxny  (2)
Where  nx ,  ny and  nh are the transmitted signal, the received signal and the impulse
response of the channel all in the discrete time domain.
Delay Spread is a temporal characteristic. It has an equivalent characteristic in the
frequency domain known as frequency selectivity – different frequencies experience
different gain or attenuation when propagating through the channel. Again, this should not
come as a surprise to you. An FIR filter has an impulse response and it has a frequency
response (that is why it is a filter). That is what it is for.
The frequency response corresponding to the impulse response in Figure 7a is shown in
Figure 7b. The effect of frequency selective channel on a signal can be described
mathematically by:
     kHkXkY  (3)
Where  kX ,  kY and  kH are the transmitted signal, the received signal and the
impulse response of the channel all in the discrete frequency domain.  kH and  nh are
related by the Fourier Transform.
The Delay Spread/Frequency Response is also often time variant. Delay Spread and
Frequency Responses for the 3G channel are shown in Figure 7c and Figure 7d for a
period of 10s.
Where the duration of each transmitted symbol is short relative to the duration of the delay
spread then the echoes will cause received symbols to be distorted by echoes of previous

- 26 -




symbols. This is known as Inter-Symbol Interference. In the frequency domain this is
equivalent to the signal experience frequency selective fading (filtering).
Systems with a symbol period that is long relative to the delay spread of the channel have
a bandwidth that is narrow enough that the frequency selectivity of the channel is
negligible. They are thus termed Narrowband. Narrowband systems experience little or no
ISI or frequency selective fading. Systems with a symbol period that is short relative to the
delay spread have a bandwidth that is wide enough that the frequency selectivity of the
channel is non-negligible. They are thus termed Wideband.


a: Impulse Response. b: Frequency Response.

c: Time Variant Impulse Response d: Time Variant Frequency Response
Figure 7: 3G Channel Measurements
If ISI occurs, then it has a major impact on the performance of communications systems.
It tends to introduce an error floor into the performance of a communications system. Error
floors are defined by the fact that BER does not reduce when SNR is increased. The error
floor tends to rise with increased amounts of ISI. This is illustrated in Figure 8. Note that
systems operating over channels which exhibit only AWGN do not experience ISI and
hence do not exhibit error floors.
10 12 14 16 18 20 22 24 26 28 30
-110
-105
-100
-95
-90

- 27 -





Figure 8. Illustration of Error Floors Introduced by ISI.
The issues of Narrowband/Wideband systems, ISI and frequency selectivity are
considered in much greater detail in the Masters level unit Advanced Mobile Radio
Techniques. For the remainder of this course we will concentrate primarily on Narrowband
communications systems – they are simpler!
Lecture 5 will present the topics of delay spread, frequency selectivity, ISI and
narrowband and wideband systems.
1.6. Duplexing and Multiple Access
Whilst not explicitly discussed earlier, Figure 1 shows a communications link that operates
in one direction only and involves just two terminals. In many cases, two way
communications are often required and many terminals may be involved in the
communications network. This requirement raises the issues of duplexing and multiple
access. Duplexing is the support provided for two way communications. Multiple Access
is the support provided for the sharing of the channel between multiple terminals.
1.6.1. Duplexing
A communications link may be classified in one of three ways depending on the degree of
support it provides for two way communications.
 ‘Simplex’ systems support communication in one direction only. TV and radio
broadcast systems are good examples of simplex systems.
B
E
R
10-1
10-2
10-3
10-4
10-5
SNR / dB
Increasing ISI
AWGN

- 28 -




 ‘Half Duplex’ systems support communication in both directions but only one at a time.
A good example is CB radio.
 ‘Full Duplex’ systems support communication in both directions at the same time. The
obvious example of this is the telephone.
In the case of Half and Full Duplex systems then each device has a message source and
destination and each of those is the reciprocal of the other.
1.6.2. Duplexing and Multiple Access Strategies
Both duplexing and multiple access require the terminals to share the resources of the
channel in some way. Typically, this involves defining a protocol by which the different
message signals can be distinguished. Examples of multiple access/duplex schemes are:
 Frequency Division. Transmitters transmit their signals on different frequencies.
Receivers ‘tune in’ to the corresponding frequency to recover the desired signal from
the channel. Each transmitter may thus use all the available time but only some of the
available frequency. Analogue TV and radio broadcasting are examples of Frequency
Division Multiple Access systems. Different channels transmit signals on different
frequencies. When you select a channel, your TV tunes in to the appropriate
frequency. Note that this strategy can be employed in cable and fibre systems as well
as wireless. Because fibre systems operate at such high frequencies, the term
Wavelength Division is often used. However, this is just the same as Frequency
Division – frequency and wavelength are reciprocally related. Terrestrial Digital
Television Broadcast and 4G mobile use FDMA. The majority of mobile phone systems
use FDD.
 Time Division. Transmitters ‘take turns’ to transmit during ‘time slots’. Receivers simply
ignore the signal received in time slots that do not correspond to the desired message.
Similarly, transmitters may not transmit outside their assigned slots. If they do, their
transmitted signal will interfere with that of another transmitter. The 2G GSM mobile
standard is based on a Time Division Multiple Access. Most WiFi systems use
TDD/TDMA. Hence, many users all use all of a common frequency band but only
transmit one at a time time.
 Code Division. All transmitters may use all the available frequency and time. They are
required to code their signals with special orthogonal codes. Knowledge of these
codes enable the receivers to extract the desired signal from all the other signals. The
third generation (3G) mobile standard employs Code Division Multiple Access
(CDMA).
 ‘Space’ Division. This term can be used in both wired and wireless systems, although
it means different things in each. In wired (cable and fibre) systems, space division
simply involves providing a separate cable or fibre for different links. In wireless
systems, space division relies on signals propagating via unrelated paths through the
surrounding environment. The use of multiple antennas to transmit/receive signals via

- 29 -




different propagation paths can facilitate SDD or SDMA. As with code division, space
division allows all transmitted signals to operate over all available frequencies and all
available time. SDMA is one of the key defining features of the more recent WiFi
versions and of 5G New Radio.
Figure 9 illustrates these strategies. Note that in all cases, the overall system is bandwidth
limited but not time limited.
We will see in later sections that the rate of communications that a system can achieve is
dependent upon the range of frequencies that its signal is allowed to occupy. This is known
as the bandwidth of the signal.

Figure 9. Multiple Access Strategies. a) Frequency Division, b) Time Division c) Code or
Space Division
Lecture 6 will present the topics of Duplex and Multiple Access.

TimeFrequency
a
b
c
User 1 User 2 User 3 User 4All Users Other Systems
TimeFrequency
TimeFrequency

- 30 -




Class 4 will give us an opportunity to discuss the content of lecture 3 and we will classify
different channel responses as either narrowband or wideband. We will also discuss
some real world systems to determine whether they are wideband systems or
narrowband systems. Further we will discuss Duplex and Multiple Access using real
world examples; some prosaic and everyday, some related to cutting edge wireless
communications.
1.7. Rates, Capacities and Measures for Communications
Systems
In order to be able to consider communications systems in a quantitative manner, it is
necessary to understand and to be able to manipulate a vast range of variables. It is
beyond the scope of the course to consider all of these but several key fundamental
variables of digital communications will be considered here.
1.7.1. Symbol Rate
The rate at which the communications system changes the (discrete) symbol currently
being transmitted over the channel is known as the symbol rate, ௦. Unsurprisingly, this
has a major influence both on the speed of communication that can be achieved and the
complexity requirement of the system hardware and software used to implement the
system.
The symbol rate is usually evaluated in units of symbols/second. The unit of Bauds
(equivalent to symbols/second) is also commonly used. For the purposes of this unit these
terms can be used interchangeably.
The reciprocal of the symbol rate is the symbol period, Ts which has units of seconds.
1.7.2. Data Rate
Data Rate (or Information Transfer Rate) is usually a measure of the speed at which binary
information is transferred from transmitter to receiver. Clearly, it is related to the symbol
rate in some way.
Data rate is denoted as ௗ. The unit is bits/s. You may often see this written as bps (‘bits
per second’). These are simply different ways of writing the same unit.
1.7.3. Bandwidth Efficiency
Bandwidth efficiency is a measure of how effectively the communications system exploits
the available operating spectrum to achieve its data rate. If the bandwidth is denoted by
B, then the bandwidth efficiency B is given by:

- 31 -




஻ =
௙೏
஻
(Equation 7)
Hence the bandwidth efficiency evaluates the number of bits per second achieved per Hz
of bandwidth ‘occupied’, thus having units of bits/s/Hz.
1.7.4. Binary and M-ary signalling
In digital systems, symbols from a limited discrete alphabet are transmitted over the
channel to represent the message signal in some way. The size of the alphabet is defined
as . Thus we may describe a digital communications system as employing ‘M-ary’
signalling, thereby specifying the size of the signal alphabet which it employs.
For the case of 2M , we have an alphabet of two symbols. This defines a binary system.
One symbol transmitted represents a 0 (bit) and the other a 1 (bit). Typically, we
distinguish between binary systems (M=2) and M-ary systems (where M>2).
The relationship between the number of bits of data represented by one symbol, b, and
the size of the symbol alphabet is given by:
= ଶ (Equation 8)
Clearly then, we can also write:
= 2௕ (Equation 9)
Hence, the data rate achieved by a communications system transmitting symbols from an
alphabet of symbols is:
ௗ = ௦ = ௦ଶ (Equation 10)
The bandwidth efficiency of a communications system employing M-ary signalling is given
by:
஻ =
௙೏
஻
= ௙ೞ
஻
ଶ (Equation 11)
Review of these equations alone would suggest that there is no significant challenge
involved in achieving an arbitrarily large data rate or bandwidth efficiency. One merely
needs to choose an arbitrarily large value for symbol rate, alphabet size or both. However,
as engineers we understand that this is not practically feasible. For one thing, if it were
that easy, the world would not need highly skilled and intelligent people to become
communications engineers. The fact is that the world does need us and its because its not
so easy as these equations alone would suggest.

- 32 -




We will discuss the limitations of increasing symbol rate and alphabet size in the following
sections.
1.7.5. The Limit on Symbol Rate
The maximum symbol rate that can be achieved over a communications channel is directly
related to the available bandwidth. The physical reasons for this limitation will not be
explored here. They are covered in detail in some of the recommended textbooks. They
can be studied there for interest but are not an explicit learning outcome for this unit and
thus will not be covered by assessment.
For baseband signalling, the maximum symbol rate is given by:
௦,ெ௔௫ = 2 (Equation 12)
Equivalently, the minimum bandwidth required for a given symbol rate when using
baseband signalling is given by:
௠௜௡ =
௙ೞ
ଶ
(Equation 13)
The situation is different when it comes to bandpass signalling, but only slightly.
For bandpass signalling, the maximum symbol rate is given by:
௦,ெ௔௫ = (Equation 14)
Equivalently, the minimum bandwidth required for a given symbol rate when using
bandpass signalling is given by:
௠௜௡ = ௦ (Equation 15)
Clearly, the difference between the baseband and bandpass cases is a simple factor of 2.
The reasons for this will be discussed further in section 2.1.
In section 1.2.1. we established a number of practical reasons why we were constrained
to use limited bandwidths for communication systems. Given that we now know that
bandwidth is limited for practical reasons and symbol rate is limited by bandwidth (as in
equations 12 and 14), we know that symbol rate is effectively also limited by practical
constraints.
Given equations 10, 12 and 14, it can be seen that the maximum data rate of a baseband
communication system is:

- 33 -




ௗ,ெ௔௫ = 2ଶ (Equation 16)
The maximum data rate of a bandpass communication system is:
ௗ,ெ௔௫ = ଶ (Equation 17)
Thus we have ruled out choosing an arbitrarily large value of symbol rate as a means to
achieve an arbitrarily large data rate. In section 1.7.4. we will consider the implications of
large values of alphabet size.
1.7.5.1.A side note on bandwidth trends
It is worth observing that despite the point above that we cannot easily increase bandwidth
to increase data rate the communication industry has nevertheless, over time, consistently
moved towards developing higher and higher bandwidth systems. This has occurred as a
result of hardware developments, regulatory changes and demand. With the (so far) ever
evolving digital computational hardware comes the ability to sample and process symbols
faster and faster. Consumers demand higher and higher data rates for their applications.
One way to meet this demand is to exploit that faster digital hardware, allow signals to
occupy larger bandwidths and thereby achieve higher data rates. There is no doubt that
this works, where the hardware and regulations allow. But any barely competent
communications engineer can take this approach when allowed – effectively, this is just
shifting all the really difficult problems over to the digital electronics engineers to solve by
making smaller and smaller transistors (no doubt and oversimplification of the fine work
that they do but you get the idea). This has certainly happened in WiFi where a lot of the
increase in headline data rates from circa 54Mbit/s achieved by the ‘a’ version (this comes
from the standard version 802.11a) to close to 1Gbit/s achieved by the ‘ac’ version
(802.11ac) has come from simply allowing the user of larger bandwidths. This is not very
clever and in fact potentially counterproductive as the density of WiFi systems increases
and more and more of them interfere with each other because they all want to use all of
the bandwidth available to each of them. But there is a recurring trend in WiFi products to
want to market products with ‘big numbers’ and throwing bandwidth at the problem
achieves headline figures for sales purposes even if it isn’t good engineering. In WiFi and
other products we see increasing increase in ‘60GHz’ and ‘mmWave’ radio bands. Its not
that these are ‘good’ bands to use, they actually come with a whole plethora of different
additional challenges, but for now at least, there is plenty of bandwidth available in these
bands.
Despite the trend towards higher bandwidths being undeniably a factor over the last 20+
years and it almost certainly being a trend which will continue for the foreseeable future
(until demand or hardware developments cap off), there is a strong argument that the
really groundbreaking and clever work in communications comes not from taking
advantage of more bandwidth when it can be had but rather from making more efficient
use of whatever bandwidth is available – that takes really good engineering! To balance
the argument a little it is fair to say that some of this really clever work has found its way
into mobile phone systems and to WiFi and the high rates that we see in WiFi are partly

- 34 -




the result of good communication engineering and only partly the result of some fairly lazy
communication engineering.
1.7.6. The limit on alphabet size
It appeared in equations 10 and 11 that there was no effective limit on the data rate or
bandwidth efficiency that can be achieved – any rate can be achieved provided M is large
enough. This is in fact the case. But, is only the case if there is no distortion in the channel
and that is a big, bold ‘but’.
The advantage of M-ary signalling is that as M increases, more information can be
conveyed by a single symbol. The disadvantage is that as M increases, the system
becomes more vulnerable to the effects of signal degradation such as noise and
dispersion. To illustrate the susceptibility of larger alphabets to distortion, three signals
with different alphabet sizes are illustrated in Figure 10. In this example each of the three
signals above can be varied between 0V and 7V. However, it is clear that although all
three have the same maximum power, less and less distortion is required to produce a bit
error in the receiver. 3.5V of noise is required before the 2-ary (binary) signal will
experience an error. 1.15V of noise are required in the 4-ary case and 0.5V in the 8-ary
case. This simple example serves to illustrate an almost universal trend; larger alphabet
sizes are more susceptible to errors.
How many errors will this example system experience? We don’t know. It depends. It
depends upon the distortion it experiences. If that distortion is pure AWGN or if AWGN is
the dominant source of distortion then we can predict a probability of error. We know the
statistical distribution of the noise amplitude, so if we know its power, we can apply maths
to determine a probability equation. Equations such as equation 3 and Equation 5 may
apply here. Note that equation 5 is consistent with our conclusions about increasing M
increasing the probability of error. Could we reduce these errors? Yes, of course. One
thing we could do is reduce the noise power. But how do we do that? Looking at Equation
2, none of the terms looks very easy to manipulate. Another thing we could do is to
increase the maximum voltage. This will move the voltages used to represent different
data further apart. This works. But increasing voltage is increasing input power and there
are again costs involved and limits to what is safe, reliable, environmentally friendly and
economical.
Note also that whilst the susceptibility to noise increases linearly with M, the data rate only
increases logarithmically with M and this is thus not a particularly attractive engineering
tradeoff. It is nevertheless one we are often forced to contemplate.


- 35 -






Figure 10. An example of M-ary baseband signalling with M=2,4 and 8.
1.8. The Shannon Capacity Theorem
If our job is to design a communication system and maximisation of data rate and/or
bandwidth efficiency is one of our key objectives then it would be extremely useful to be
able to determine the upper limit of what is practically achievable. Fortunately for us, a
theoretical upper bound on data rate and bandwidth efficiency has already been
determined. This is the result of work undertaken by Claude Shannon in the 1940s,
building upon earlier work by Ralph Hartley. This work identifies the maximum rate that
can be communicated through a channel and defines that as the channel’s capacity. Note
that this is in no way related to the electronic parameter capacitance. Because Shannon’s
work introduced the concept of capacity building upon Hartley’s work, the terms ‘Shannon-
Hartley theorem’ and ‘Shannon Capacity’ are variously used to refer to essentially the
same concept.
Shannon Capacity considers the capacity of a channel with distortion in the form of AWGN.
It does not consider any other source of impairment or distortion. As stated earlier,
although not the only form of distortion that can occur in a communications system, AWGN
is perhaps the most important. It occurs in all systems and is often the dominant form of
0V
7V
1s
0V
7V
1s
0V
7V
1s
0
1
00
11
01
10
000
001
010
011
100
101
110
111

- 36 -




distortion. Hence the Shannon Capacity is very useful despite this limitation. The theorem
states:
= ଶ ቀ1 +
ௌ
ே
ቁ (Equation 18)
Where denotes the channel capacity. As stated earlier denotes the received signal
power, denotes the received noise power (and ‘received’ is an important distinction as
explained in section 1.3) and denotes the bandwidth. As stated earlier, this ratio of ⁄
is often written (less mathematically accurately) as .
It is often convenient to rearrange the Shannon-Hartley Equation into one of a number of
other forms.
We can note that:
௕ =
ௌ
௙೏
(Equation 19)
If we operate the communication system at the limit of the channel capacity (ௗ = ):
௕ =
ௌ
஼
(Equation 20)
And:
଴ =
ே
஻
(Equation 21)
Given equations, 18-21 we may also state:
= ଶ ቀ1 +
ா್஼
ேబ஻
ቁ (Equation 22)
஼
஻
= ଶ ቀ1 +
ா್஼
ேబ஻
ቁ (Equation 23)
1.8.1. The Shannon Capacity Curve
It is often useful to present the Shannon Theorem graphically using two axes to represent
‘received signal quality’ and bandwidth efficiency. Two common forms of the Shannon
Curve, using different axes are shown in Figure 11.

- 37 -




In sections 2 and 4 we will investigate different bandpass modulation schemes that we
can use in communications systems. Once we have done that we will plot them all on the
Shannon capacity graphs to see how close they get to the limit of what is theoretically
possible.

(a) (b)
Figure 11: Plots of Shannon Capacity
1.9. Summary – Challenges for the Communications
Engineer
Throughout section 1 we have discussed the fundamentals of communications and the
challenges that we face as engineers designing good solutions to communications
problems. We identified a wide range of practical factors to consider but focussed on two
in particular: the limitations of bandwidth and AWGN. We identified the key numerical
metrics that we can apply to communications and we were able to relate key performance
metrics, data rate and bandwidth efficiency, to the primary limiting factors of bandwidth
and noise through the Shannon-Hartley Capacity theorem. We also identified important
distinctions between binary and M-ary signalling and between baseband and bandpass
systems. We briefly considered baseband systems in section 1.4. Bandpass systems
create more challenges. In section 2 we will investigate binary bandpass modulation
schemes because these are simpler than M-ary ones. In section 3 we will build upon our
understanding of binary bandpass modulation to investigate M-ary bandpass modulation.
Lecture 7 will present Quantitative metrics and the Shannon Theorem.
12
10
8
6
4
2
2
2 13 1 32
Power
Efficient
Bandwidth
Efficient
Error Free
Region
Boundary
log2(C/B)
log10(Eb/N0)
10
1
Error Free
Region,
fdBoundary,
fd=C
0.1
20
30
0 6 12 18 24 30 36
fd > C
Eb/N0 dB-1.6
fd/B, Bandwidth Efficiency

- 38 -





Class 5 will give us an opportunity to discuss quantitative metrics and to practice some
numeric computations and to discuss our challenges as communications engineers. In
class 5 we will also begin to consider the material in section 2 of these notes.


- 39 -




2. Binary Bandpass Modulation
As discussed in section 1.1.2. bandpass modulation is employed when the channel
available to the communications system is not suitable for baseband communications.
Obvious examples are radio and optical communications. Electromagnetic waves only
propagate effectively at higher frequencies than those of our typical baseband messages.
In such situations it is necessary to modulate the baseband, message-bearing signal onto
a higher frequency carrier wave.
2.1. Carrier Modulation
As with many electronic applications, a continuous wave, sinusoidal signal is convenient
to use as our carrier. One key parameter of a sinusoid is that its frequency content is a
single, pure tone. Whilst some communications systems do not use sinusoidal carrier
waves (some forms of Ultra Wideband Communications, for example), the vast majority
do and so in this unit we will focus exclusively upon the case where the carrier wave is a
single, simple sinusoid. We will describe this carrier mathematically by:
() = ( + ) (Equation 24)
Where is the amplitude, is the angular frequency, is the phase and is time.
Fundamental to the concept of carrier modulation for bandpass communication is the idea
that we must vary some property of this carrier wave to represent this message. Time is
not an easily manipulable thing, not just within the domain of electronics. So, realistically
we must focus our attention on the other three variables, all of which we can in fact readily
manipulate in electronics for the purpose of modulating this carrier wave with our
message. This leads us to three options, then; we can modulate the amplitude, frequency
or phase of the carrier to represent our data and we term these methods Amplitude Shift
Keying (ASK), Frequency Shift Keying (FSK) and Phase Shift Keying (PSK) respectively.
Note that we typically don’t use the term ‘modulation’ in the names of these modulation
schemes as this was first used in reference to analogue modulation schemes (e.g. AM
and FM for Amplitude Modulation and Frequency Modulation). We use ‘Shift Keying’
instead of ‘modulation’ to emphasise that these are digital schemes.
Throughout section 2 we will only consider methods for modulating binary symbols onto
the carrier (i.e. M=2) Non-binary bandpass modulation (i.e. for M>2) will be considered in
section 3.
Lecture 8 will present an overview of digital modulation schemes and the types of ‘Shift
Keying’ we employ.


- 40 -




2.1.1. Carrier Modulation through Multiplication
If we wish to modulate our carrier wave with a message we require some means of
achieving that modulation. One very familiar mathematical function which is electronically
viable via various methods and which lends itself conveniently to the purposes of
modulation is multiplication. Let us consider what happens when we multiply two signals
together. Figure 12 shows a simple scenario where two sinusoids are multiplied together.
One is the carrier wave. The other, nominally, is the message.
In this simple block diagram, multiplication is achieved through a device called a mixer. A
mixer, quite simply, is an electronic device designed to achieve multiplication. For the
purposes of this unit, we will keep it simple; the mixer is a multiplier. This somewhat
trivialises the potentially complex challenges involved in making that mixer work. But this
is very often the way that engineers work. When designing a system, we are not always
familiar in detail with the design of each individual sub-system; we rely upon our
colleagues’ good work to design them for us. A mixer is a good example of this. For the
mixer to exist, a very intelligent and talented electronic engineer has to design it. For us
as communications engineers, we don’t take that for granted but we do use it without
needing to understand its design in detail. We do need to understand the parameters and
limitations of the device when integrating it into our larger design. Throughout the
remainder of this unit we will view the mixer simply as a device which achieves electronic
multiplication of two signals. In reality, if we were designing a real world communication
system we can do so in the knowledge that such devices do exist and do perform well in
other communication systems already in existence. Nevertheless we would need to
carefully select a mixer device or design to integrate into our wider communication system
to make sure that it is suitable for our purpose (for example that it works well at the
frequencies of signal that we will be mulitiplying).
The diagram also shows a signal source for the carrier wave, (௖). This can be
achieved in electronics through another sub-system called a synthesiser and similar points
can be made here. We know these exist, they work and we may use them in our design.
No synthesizer is perfect and we would need to take care to select a suitable one in our
hardware design. But for the purposes of this unit, a synthesizer outputs the carrier wave
of our choice.

Figure 12: Carrier Modulation by multiplication
The message is not obtained from a synthesizer. The message is obtained from the
message source in the transmitter as discussed in section 1.1. The message will also not
Cos(mt)
Cos(ct)

- 41 -




be a sinusoid in reality because a sinusoid contains no information. Real messages will
be much more complicated signals which vary over time in a quasi-random fashion
(because it is a sequence of information). Considering the message as a sinusoid is
mathematically convenient for this discussion in the first instance, though, so that is what
we will do first.
We know from mathematics what happens when we multiply two sinusoids together. We
can make use of the well-known mathematical identify:
()() = ଵ
ଶ
[( + ) + ( − )] (Equation 25)
Substituting ௖ for and ௠ for , we can equivalently write:
(௖)(௠) =
ଵ
ଶ
[(௖ + ௠) + (௖ − ௠)] (Equation 26)
Now we can see that multiplying the two input signals together produces an output signal
which has two frequency components: one the sum of the two input frequencies and one
the difference of the two input frequencies. This is generally mathematically true. In
communications, we use specific names for these two frequency components: the upper
sideband (sum) and the lower sideband (difference).
If we now reflect back on our original motivation for carrier modulation we can see that the
carrier frequency is a much higher frequency than the message frequency (that’s why we
had to shift the message up to get it to propagate through the channel). Thus in our carrier
modulation scenario, the difference between the upper and lower sidebands is typically
much smaller than the absolute value of the carrier frequency. For example, the 4G and
5G mobile and WiFi systems that we all use modulate message frequencies of the order
of 10s of MHz onto carrier waves of the order of GHz.
We now need to consider what happens in the realistic case that our message is not a
sinusoid. Our message is some time variant, quasi-random data sequence, the frequency
content of which is essentially unknown. However, we could in principle describe that
message in the frequency domain as the combination of many different sinusoids of
different amplitudes, frequencies and phases. If we multiply that message by a carrier
wave we can still apply the trigonometric identify in equation 25. Every frequency
component in the message will produce two frequency components in the carrier
modulated signal, one at the sum of the two frequencies, one at the difference. We have
successfully shifted our message up to a higher frequency by modulating it onto a carrier
wave. In the process of doing so though, we have doubled its frequency content because
every frequency in the message produces both an upper and lower sideband. This is the
reason for the difference in maximum symbol rate/minimum bandwidth identified in section
1.7.5. The 2 in equations 12 and 13 is cancelled out by the 2 resulting from carrier
modulation (the two sidebands) to yield equations 14 and 15.

- 42 -




2.1.2. Carrier Demodulation
If we can successfully carrier modulate our message to convey it through the channel, we
need a method to reverse this process at the receiver and shift the signal back down to
baseband so that we can recover the message.
Various methods for carrier demodulation exist. We will classify them into two types:
coherent and non-coherent.
The defining characteristic of a coherent demodulator is that it uses the carrier wave (or
an approximation of it) in the demodulator. How this is used is discussed in section 2.1.3.
Non-coherent demodulators do not use a local replica of the carrier signal.
Various coherent and non-coherent demodulators for different binary modulation schemes
are presented throughout section 2. These methods could be readily extended to M-ary
modulation schemes in section 3 but this is not presented explicitly as such extensions
are relatively trivial.
2.1.3. Carrier Demodulation through multiplication (for coherent
demodulation)
Whilst it may not be intuitively obvious, the means by which the carrier wave can be used
for carrier demodulation lies in repeating the process of multiplying with the carrier wave,
just as we did in the modulator.
(௖)(௠)(௖) =
1
2
[(௖ + ௠) + (௖ − ௠)](௖)
= ଵ
ସ
[(2௖ + ௠) + (௠) + (2௖ − ௠) + (−௠)] (Equation 27)
Naturally, the multiplication process doubles the number of frequency components again
so two in yields four out. Because both the terms being multiplied include ௖ terms these
components either cancel to 0 or double to 2௖ depending upon whether they are sum or
difference terms. Again, we should reflect on the fact that ௖ is much bigger than ௠ and
recall our goal of recovering the message back to its original frequency. This means that
the terms we want are the ones at frequency ௠ (positive and negative don’t really matter).
We don’t want the terms at 2௖. Because ௖ is much larger than ௠, 2௖ must be
substantially higher still. It is now easy to remove these unwanted, very high frequency
components from the signal with a low pass filter. This filter doesn’t need to be very
complex or expensive because we don’t need a steep roll off in the transition band
because the signal components that we want to pass and the signal components we want
to attenuate are so far apart in frequency.
Note that we are making some important assumptions here. The first is that we have a
signal at the receiver which is the same as that in the transmitter, i.e. we have access to

- 43 -




a (௖) signal at both transmitter and receiver. This is not trivial, as these two devices
are physically separate and we cannot just share the signal (from the synthesizer) between
them. We need one synthesizer in transmitter and one in the receiver. The maths above
is applicable on the assumption that they are identical. If they are not, then phase and
frequency terms will be introduced into the demodulated signal which relate to the
difference between the signals produced by the two synthesizers and this will degrade the
performance of the communication system. Ensuring that the two synthesizer outputs are
the same is referred to as frequency synchronisation. This is beyond the scope of this
course but can be achieved by various means, one of which is a sub-system called a
Phase Locked Loop.
Because this method of carrier modulation and demodulation through multiplication is
electronically viable, in the following sections, where modulator and demodulator block
diagrams are presented for different modulation schemes, we will frequently see mixers
used. Similarly, in the demodulator block diagrams the mixer is followed by a low pass
filter to remove the signal components at twice the carrier frequency.
2.1.4. Square Wave Messages
Since we are focussed on digital communications systems, it is reasonable to assume that
our message will be some sort of square wave. For a binary system it will be a square
wave with two voltage levels to represent binary data. For an M-ary system it will be a
square wave with different voltages to represent combinations of bits. In either case
it is a square wave and we know about the frequency content of square waves.
A square wave signal consists of frequency components at odd multiples of the
fundamental frequency with diminishing amplitude. This forms an infinite series. , e.g
(௦) +
ଵ
ଷ
(3௦) +
ଵ
ହ
(5௦) +
ଵ
଻
(7௦) … where the fundamental frequency is
determined by the symbol rate, ௦ = 2௦. Following on from the discussion above in
section 2.1.1. we can see that if we modulate this square wave baseband message onto
a carrier wave we will create an infinite series of both upper and lower sidebands, e.g.
(௖ + ௦) +
ଵ
ଷ
(௖ + 3௦) +
ଵ
ହ
(௖ + 5௦) +
ଵ
଻
(௖ + 7௦) … and (௖ −
௦) +
ଵ
ଷ
(௖ − 3௦) +
ଵ
ହ
(௖ − 5௦) +
ଵ
଻
(௖ − 7௦) … . This outcome is
unacceptable because we now have an infinite bandwidth signal centred around our
carrier frequency. This is a consequence of the infinite bandwidth squarewave. Whilst it is
convenient to think of squarewave messages at baseband we must bandlimit the
bandpass signal. Fundamentally, we can do that in two ways. We can either low pass filter
the squarewave message at baseband before carrier modulating it (to remove the higher
frequency components) or we can apply a bandpass filter at the higher frequency,
probably centred at the carrier frequency, to remove the higher and lower frequency
sidebands. Consequently in the following sections where we review different bandpass
modulation schemes we will see either low pass or bandpass filters used in the modulator
circuits for bandlimiting purposes. During sections 2 and 3 it is sufficient to recognise the
purpose of these filters. In section 4 we will consider the design of bandlimiting filters in

- 44 -




more detail and we will see that there are some very specific criteria that we need these
filters to meet to ensure that they are fit for purpose.
Figure 13 illustrates a baseband square wave spectrum in its unfiltered form (dashed line)
and low pass filtered form (solid line). Figure 14 illustrates a bandpass signal modulated
with a square wave message in its unfiltered form (dashed line) and filtered form (solid
line).

Figure 13. Baseband Square Wave Signal Spectrum.

Figure 14. Bandpass Signal Spectrum.
2.1.5. Constellation Diagrams
Constellation diagrams are often a useful way to represent the different symbols in the
alphabet of a communications system. We will refer to them frequently throughout sections
2 and 3.
Constellation diagrams show the different symbols of the alphabet in Euclidean space.
Thus, the horizontal axis of the diagram shows the amplitude of the real or ‘in-phase’
component of the signal as modulated onto the carrier (௖). The vertical axis of the
diagram shows the amplitude of the imaginary or ‘quadrature’ component of the signal as
E
ne
rg
y
0 fm 3fm 5fm 7fm
Frequency
En
er
gy
fc fc+fm fc+3fm fc+5fmfc-fm fc+7fm
Frequency

- 45 -




modulated onto a carrier rotated 90 relative to the carrier, (௖). Thus, the diagram
shows symbols in a constellation either in terms of their amplitude and phase or –
equivalently – in terms of a complex number (recall that () = () = ( + 2⁄ )).
The constellation diagram is a relatively simple concept but best explained by example.
We will see many examples of these diagrams in the following sections as each
modulation scheme is discussed.
Lecture 9 will present the concepts of carrier modulation, square wave spectral content,
carrier demodulation and constellation diagrams.

As well as covering Shannon examples, in Class 5 we will review our understanding of
Carrier Modulation and the types of Shift Keying available to us.

2.2. Amplitude Shift Keying
Amplitude Shift Keying represents binary data by two discrete amplitudes of the carrier
signal. The simplest form of ASK is On-Off Keying (OOK). A mapping table for OOK ASK
is given in Table 3 where A is peak amplitude of the transmitted signal.
Table 3: OOK ASK Mapping
Data Amplitude (V)
0 0
1 A

Thus, the carrier is ‘turned on’ to represent a binary 1 and ‘turned off’ to represent a binary
0.
Now, at this point, you might reflect back to section 1.4.1. and reasonably ask the question:
‘if there is OOK baseband and Polar baseband and we can do OOK ASK, can we not do
Polar ASK?’. The answer is ‘yes, we can.’ But we are not going to consider that here. We
will come to it later and when we do, we will understand why. For now, we will focus our
discussion of binary ASK on the case of OOK ASK.

- 46 -




2.2.1. The OOK ASK Constellation
The constellation diagram of OOK ASK is as illustrated in Figure 15.

Figure 15: OOK-ASK Constellation.
2.2.2. Modulation of ASK
Two methods for implementing OOK ASK modulation are shown in Figure 16.
In case a, the data signal can be used to control a switch connected to a signal source
generating the carrier wave. Bandlimiting is applied subsequently using a bandpass filter
centred on the carrier frequency. Using a switch in this way is actually a form of
multiplication since the message is either ‘on’ or ‘off’ and switching the carrier on or off is
analogous to multiplying it with the message. In this case the switch is sufficient for
multiplication and we don’t need a mixer. This switching method serves to illustrate the
OOK ASK concept but it encounters problems when the carrier frequency is high.
Implementation of a quality bandpass filter at high frequencies is highly problematic.
In case b, bandlimiting (pulse-shaping) is implemented at baseband. The bandlimited data
signal is then multiplied with the carrier signal. Multiplication must now be implemented
using a mixer because neither of the two signals being multiplied together is binary. The
message was binary but once we bandlimit it, it can no longer be a square wave, cannot
vary instantaneously between two states and has become a continuously variant signal.
Remember, we will look at the topic of bandlimiting filters in detail in section 4. Compared
to switching method we have circumvented the need for the high frequency bandpass filter
by doing our bandlimiting at baseband. The cost of this is we can no longer use a simple
switch and must use a mixer. Ultimately, in most practical implementations the mixer is
easier to implement than the bandpass filter and so case b is the preferred option in
practice. Nevertheless, case a helps to understand the concept.

- 47 -





a: OOK Modulation via a switch

b: ASK Modulation via a mixer
Figure 16: ASK Modulation
2.2.3. Demodulation of ASK
As stated in section 2.1.2. , demodulation can be either coherent or non-coherent. Both
options exist for ASK and have difference advantages and disadvantages so we will
consider both options here.
A circuit capable of non-coherent detection of OOK ASK is shown in Figure 17. This is
called an envelope detector. The received signal is first half wave rectified by a diode and
then low pass filtered. The purpose of the filter is to retain the message frequency ௠
whilst removing the carrier frequency ௖. As discussed above, since ௠ and ௖ are far
apart, this can potentially be a very simple filter, (hence it is illustrated here as a simple
RC combination). The output of the filter should thus approximate the output of the
bandlimiter in the modulator. The final block of the demodulator is called a slicer or
hardlimiter. The purpose of this is to ‘square up’ the output of the filter, effectively reversing
the bandlimiting effect applied in the modulator. This process is commonly used
throughout communication systems. If you think about what you have learned about
comparators in other units, it should be easy to see how something like an op-amp can be
used to achieve this purpose. Clearly, this is a very simple, low cost circuit to implement:
a diode, a few capacitors and resistors (maybe just one of each) and an op-amp in open
loop configuration. However simple it may be, it is a viable method for detection of OOK
ASK (as you may find out in the laboratory exercise!).
).cos( tc
switch
Bandpass filter
cos(ct
Bandlimiting Filter

- 48 -





Figure 17: Non-Coherent ASK Demodulator
A coherent ASK demodulator is shown in Figure 18. This demodulator follows the method
described in section 2.1.3. and thus consists of a synthesiser to provide the local replica
of the carrier wave, a mixer and a low pass filter. The output of the filter should thus be
approximate the bandlimited message signal in the modulator and because we don’t want
it to be bandlimited anymore, we square it up using a hard limiter, just as in the non-
coherent demodulator. At a block level, this does not look significantly more complex than
the non-coherent demodulator. However, there are two things that we must reflect on here.
Firstly, the mixer is a much more complex device than a diode. Mixers exist, we can buy
them and use them but they will add significantly more cost and complexity to our receiver
than a mere diode. Secondly, we require a local replica of the carrier signal. We need a
synthesizer in our receiver. This is again a more complex component but also viable. The
non-obvious cost here is that we must ensure that our synthesizer in the receiver is
synchronised with the one in the transmitter. That can be done via a phase-locked loop
but that task is not simple and the design effort and component cost are not captured by
this block diagram. Thus, this coherent receiver is significantly more complex and
expensive than the non-coherent one.
Comparator

- 49 -





Figure 18: Coherent ASK Demodulator

2.2.4. Spectrum and Bandwidth Efficiency of the ASK Signal
The spectrum of the ASK signal is exactly as illustrated in Figure 14. The minimum
bandwidth required for ASK transmission is equal to the symbol rate and the number of
bits transmitted per symbol is 1. Thus, the maximum bandwidth efficiency of binary ASK
is 1bit/s/Hz.
2.2.5. BER Performance of OOK ASK
The probability of symbol error (and hence probability of bit error, because its binary) of
non-coherently demodulated OOK ASK is given by:
௘ =
ଵ
ଶ
൬ට ா್ଶேబ൰ +
ଵ
ଶ
ቀିா್
ேబ
ቁ (Equation 28)
The SER and BER of coherently demodulated OOK ASK is given by:
௘ =
ଵ
ଶ
൬ට ா್ଶேబ൰ (Equation 29)
The obvious feature of these two equations is that the error performance of non-coherent
demodulation is worse. The equation for non-coherent error probability consists of two
terms, one being the same as the equation for coherent demodulation and the other being
‘more errors’.
To aid in comparison this performance is presented graphically in Figure 19. The
difference in performance can be seen to be around 4dB. Thus, although non-coherent
cos(ct)

- 50 -




detection offers low receiver complexity, its performance is often unacceptable because
in most communication scenarios we deem 4dB of performance loss to be quite large and
thus the additional hardware cost of a coherent receiver is preferable. Note that this is not
a black and white decision. There is no right and wrong choice here. It is an engineering
trade-off and it will be you, the communication engineer, who has to make these kinds of
choices for the systems which you design.

Figure 19: BER Performance of OOK ASK.
Lecture 9 will present ASK.
2.3. Frequency Shift Keying
Frequency Shift Keying (FSK) modulates the frequency of the carrier wave according to a
discrete message signal. Binary Thus binary FSK represents digital data by transmitting
one of two distinct carrier frequencies. One or the other frequency is transmitted
depending on the bit being represented. The mapping table for binary FSK can thus as
shown in Table 4. Note that it doesn’t matter which of the two frequencies is used to
represent a binary 0 and which represents a binary 1. We could easily flip them and it
would make no difference.

- 51 -




Table 4: Binary FSK Mapping
Data Frequency
0 ଵ
1 ଶ

Note that since constellation diagrams show amplitude and phase information and FSK
varies only in frequency then the FSK constellation is trivial – a single point – and thus
somewhat uniformative.
2.3.1. FSK Spectrum
Mathematical analysis of the FSK spectrum is not so easy. However, it is fairly easy to
consider FSK spectrum graphically. We can think of the FSK signal as being a bit like two
OOK signals on different frequencies (ଵ and ଶ) with opposite bit to state mapping –
think about it! If we turn one on, we turn the other off and vice versa. We already know
from the discussion in section 2.1.4. and 2.2.4. what those individual OOK ASK spectra
look like. Figure 20 illustrates this with the two red dashed lines being the individual OOK
ASK spectra and the solid blue line being the effective FSK spectrum. Note that the shape
of this compound spectrum will depend upon the spacing between the two frequencies
relative to the symbol rate.

Figure 20: FSK Spectrum – graphical illustration
It is common to space the two signalling frequencies at a multiple of one half of the symbol
rate. This ensures that the two carriers are orthogonal i.e. that one has zero spectral
energy at the centre frequency of the other and vice versa. This ensures that when the
receiver attempts to demodulate at one frequency, then the energy of one carrier is
E
ne
rg
y
fc fc+fmfc-fm
Frequency

- 52 -




maximised whilst the other is minimised. This ensures optimal performance under additive
noise conditions.
The smallest frequency spacing at which the two carriers are orthogonal is one half of the
symbol rate. When this spacing is employed, the central ‘lobe’ of the signal spectrum
narrows considerably and the FSK modulation is known as Minimum Shift Keying (MSK).
2.3.2. FSK Modulation
An example of the FSK modulation concept is given in Figure 21.

Figure 21: FSK Modulation Concept
his method of generating FSK serves to illustrate the concept but it is not practical. The
switching between the two independent frequency sources tends to result in phase
discontinuities. An alternative method, based on the use of a Voltage Controlled Oscillator,
avoids such discontinuities and achieves Continuous Phase (CP) FSK. It also allows
filtering of the baseband data by a bandlimiting filter. Just as discussed in section 2.2.2.
use of a low pass filter is preferable to the use of a bandpass filter.
A common bandlimiting filter employed in FSK communications systems is a Gaussian
filter. The result is known as Gaussian FSK (GFSK). GFSK is specified in the Bluetooth
communications standard for Wireless Personal Area Networks (WPAN). GMSK is
specified in the GSM standard for cellular mobile (2G). It is worth noting that these are
both highly successful communication technologies.
2.3.3. FSK Demodulation
As with ASK, FSK may be demodulated either coherently or non-coherently.
A circuit that can be employed for non-coherent detection of FSK is shown in Figure 22.
Two bandpass filters centred on the two signalling frequencies are used to ‘separate’ the
two symbols into two branches. An envelope detector and low pass filter are then
employed in a similar fashion to that used in a non-coherent ASK detector. A comparator
is then used to make a decision on the output symbol on the basis of the branch with the
largest energy. This is called a discriminator detector.
0 01 1
ASK-2
ASK-1
Switch

- 53 -





Figure 22: Non-Coherent FSK Detection.
A Coherent FSK receiver is shown in Figure 23. Two local carrier replicas locked to the
two signalling frequencies are mixed with the received signal in two parallel processes. If
the two frequencies are orthogonal then one – the ‘incorrect’ one – should produce a zero
output. The maths of orthogonality tells us that. The other should produce a non-zero
output. Low pass filtering of the two results enables a decision to be made on which
frequency (and hence what binary data) is currently being received. This illustrates the
importance of using orthogonal frequencies to ensure that the output of one branch tends
toward zero to maximise the difference between the two branches. Keep in mind that in
the presence of additive noise, perfect orthogonality will not be possible because there will
be some energy at all frequencies (because thermal noise is white).

Figure 23: Coherent FSK Detection

Bandpass filter
f 1
Bandpass filter
f 2
Peak detectors
Comparator
FSK
Input
Data
output
FSK Input
Comparatorcos(1t)
LPF
LPF
Data
output
cos(2t)

- 54 -




2.3.4. BER Performance of FSK
The SER (and hence BER, in the binary case) of binary, non-coherently demodulated FSK
is given by:



 
02
exp
N
E
P bs (Equation 30)
The SER and BER of binary, coherently demodulated FSK is given by:









02N
E
erfcP bs (Equation 31)
This BER performance is plotted in Figure 24, which shows that the difference in
performance is only around 1dB. Thus, non-coherent detection of FSK offers a lower
complexity option for implementation of the receiver at a relatively small penalty in terms
of performance (compared with the 4dB difference for ASK). For this reason non-coherent
detection of FSK is often employed in practical systems (though it should be noted that
the discriminator detector doesn’t perform as well as theory suggests).
The MSK version of FSK also has the advantage of achieving good bandwidth efficiency
(non MSK FSK is inferior to ASK and PSK).
Because FSK modulates the frequency of the carrier, rather than the amplitude or phase,
the resultant modulated waveform is constant envelope. We know (think about your
Electronics 2 unit) that this property makes it easier to achieve efficient power amplification
since we can use a non-linear amplifier without distorting the message. This makes FSK
an attractive modulation scheme where power efficiency is an important design limitation.
It is undoubtedly because of these qualities that FSK was adopted in 2G mobile and
Bluetooth. The greater bandwidth efficiency of MSK made it the preference for 2G. For
Bluetooth, the choice was to sacrifice bandwidth efficiency to reduce hardware costs and
so non-minimum spacing was chosen there.

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Figure 24: BER Performance of FSK.
Lecture 10 will present FSK.

In class 6 we will review our understanding of ASK and FSK and discuss some of the
key similarities in the modulator and demodulator circuits.
2.4. Phase Shift Keying
Phase Shift Keying (PSK) modulates the phase of the carrier signal to represent the
message. Binary PSK (BPSK) uses two phases to represent the binary data. Performance
is maximised by achieving the maximum phase spacing between the two symbols in the
alphabet, i.e. 180. The absolute phases are not that important but it is usually convenient
to have one of them be 0. Hence the example mapping shown in table Table 5.
Table 5: Binary PSK Mapping
Data Phase (radians) Phase (degrees)
0 0 0
1 180


- 56 -




It is worthwhile to note that (௖ + ) = −(௖). In section 0 when first
considering ASK we decided to focus our attention on OOK ASK and not consider Polar
ASK for reasons that would be explained ‘later’. This simple fact that phase shifting a
cosine by is equivalent to an inversion is the explanation for that. There is in fact no
difference between PSK and Polar ASK in the binary case so, conventionally, we use
BPSK to describe this modulation scheme rather than Polar ASK.
2.4.1. Spectrum of a BPSK Signal
The spectral analysis presented in section 2.1.4. is equally applicable to PSK as it was to
ASK in 2.2.4. (to justify this, consider that the only difference between OOK ASK and
BPSK is the average voltage of the message (OOK ASK has a dc offset, BPSK does not).
2.4.2. Constellation of BPSK
The constellation diagram for PSK is shown in Figure 25. PSK is said to have an antipodal
(equal and opposite) constellation. Comparison with Figure 15 should also help to illustrate
how BPSK and Bipolar Binary ASK are equivalent.

Figure 25: Binary PSK Constellation Diagram
2.4.3. PSK Modulation
An example of a system capable of implementing BPSK modulation is shown in Figure
26. A signal source and inverter are used to generate the carrier signal in inverted and
non-inverted forms. The output of the modulator is switched between these two forms
according to the input data. Again, just as we discussed in section 2.2.2. for ASK, this is

- 57 -




not practical for high carrier frequencies, because we do not want to implement a
bandlimiting filter at the carrier frequency. Instead, we would prefer to bandlimit at
baseband and then modulate with a mixer. We can do that for BPSK. The practical
modulator for BPSK looks just like the one for OOK ASK in Figure 16b. The only difference
between the two cases is that for OOK ASK the message is an On-Off message. For
BPSK the message is a Bipolar message.

Figure 26: BPSK Modulation Concept
2.4.4. SK Detection
There is no non-coherent detection of PSK. The same coherent detection process as
applied to ASK (as illustrated in Figure 18) may be applied to PSK. Again, the only
difference is that the baseband message output is a Bipolar one and so the decision
threshold is at 0 (remember, we said this could be implemented with a comparator so we
just make the threshold voltage 0).
However, in PSK there is ambiguity in the phase of the signal. Whatever method we
employ to frequency synchronise the synthesizer in the demodulator to the one in the
modulator, it might generate the right frequency but the opposite phase. If this happens,
all 1s will be demodulated as 0s and all 0s as 1s! This needs to be fixed somehow. There
are two options for this: Training Sequences and Differential Encoding.
2.4.5. Training Sequences in PSK
Training Sequences are used to address many challenges in communications systems. In
essence, they are a black box test. You put a known input into an unknown system, you
measure the output and you compare the output to the input to characterise the system.
cos(ct)
Data Input
Switch
Inverter
x(t)

- 58 -




A training sequence is a predefined sequence of bits or symbols which are known by both
the transmitter and the receiver in the communication system and which are transmitted
at pre-agreed times. Since the receiver already knows what the training sequence was
when it was transmitted, it can compare what it receives with that known transmitted
sequence to characterise the unknown function. You will see in later units how this might
be used to estimate the response (the transfer functions in equations 2 and 3 for example).
Here though, what we are trying to identify is: has the data been inverted or not. Provided
that the received sequence can be demodulated without too many errors (although it may
possibly be inverted) then the receiver can identify the presence of a phase inversion by
comparing the received sequence with known transmitted sequence. If most of the bits
are the same, then no phase inversion exists. If most bits are inverted then a phase
inversion is detected. Once detected, a phase inversion may be corrected by simply
inverting all bits output from the demodulator.
Training sequences have two disadvantages. Firstly, since they are already known at the
receiver they convey no information and thus they reduce the bandwidth efficiency of a
communications system. Every bit of training sequence transmitted takes the place of a
bit of data. Further, if the receiver fails for any reason, it must wait for the next training
sequence before it can resume demodulation. Thus, training sequences are required to
be transmitted regularly, thereby further reducing bandwidth efficiency. The longer the
training sequence, the less likely it is that mistakes will be made when trying to identify the
inversions. So there are design tradeoffs to be made in terms of how often the training
sequences are sent and how long they are.
Lecture 11 will present PSK in its coherent form.

2.4.6. Differentially Encoded PSK
Differential encoding of PSK can be employed in combination with conventional PSK
modulation and coherent demodulation. However, the data is pre-coded before the
modulation process and decoded after the demodulation process. The benefit of this will
be illustrated.
The result of the encoding process is to change the bit to phase mapping into a bit to
phase change mapping.
The differential encoding and decoding processes are shown in Figure 27.

- 59 -





a: Encoder b: Decoder
Figure 27: Differential Codec
Using this codec a binary 1 is represented by a phase inversion relative to the previous
symbol whilst a binary 0 is represented by no phase change relative to the previous
symbol. Thus it is the change of phase which represents the message, not the absolute
phase.
This can be confirmed by considering the encoder circuit. This circuit will output a zero if
the current input is the same as the previous output and a one if the current input differs
from the previous output. The decoder reverses this process. This is illustrated in Table 6:
Differentially Encoded PSK Example for an arbitrary sequence of data.
Table 6: Differentially Encoded PSK Example
Transmitter
Input Bit 1 1 1 0 0 1 0 1 0 1
Delayed Bit 1* 0 1 0 0 0 1 1 0 0
XOR 0 1 0 0 0 1 1 0 0 1
Receiver
Input Bit 0 1 0 0 0 1 1 0 0 1
Delayed Bit 1* 0 1 0 0 0 1 1 0 0
XOR 1 1 1 0 0 1 0 1 0 1
* = Arbitrary Value.

2.4.6.1.Differentially Coherent Detection of PSK
If Differential Encoding of PSK is employed, it makes is possible to undertake Differentially
Coherent Demodulation using the receiver shown in Figure 28. A PSK system using a
differentially coherent demodulation process is often referred to as a DPSK system.
Ex -Or
Ts
In Out
Ts
Ex -Or
OutIn

- 60 -





Figure 28: DPSK Demodulator

Note that this receiver is broadly similar to that used for Coherent Demodulation. However,
instead of generating the local carrier replica by means of a PLL or similar, each received
PSK symbol is mixed with its predecessor. This eliminates the need for a synthesizer in
the receiver and hence any requirement to frequency synchronise the receiver with the
transmitter.
The function of this system is illustrated in Table 7 using the same arbitrary data sequence
as used in Table 6.
Note that for the Differentially Coherent Detector to work, the message must still be
explicitly differentially encoded in the modulator using the encoder in Figure 27a. The
decoder in Figure 27b is not required though – its function is implicit in the demodulator in
Figure 28.








Data output
Comparator
Received
Signal
T
s

- 61 -




Table 7: Differential PSK Example
Transmitter
Input Bit 1 1 1 0 0 1 0 1 0 1
Delayed Bit 1* 0 1 0 0 0 1 1 0 0
XOR 0 1 0 0 0 1 1 0 0 1
Tx Phase
Change
0 180 0 0 0 180 180 0 0 180
Receiver
Rx Phase
Change
0 180 0 0 0 180 180 0 0 180
Delayed
Phase
180 0 180 0 0 0 180 180 0 0
Output
Phase
180 180 180 0 0 180 0 180 0 180
Data Out 1 1 1 0 0 1 0 1 0 1
* = Arbitrary Value.
The BER performance of DPSK is inferior to that of coherent PSK. This is due to the fact
that the phase reference for differential demodulation is noisy (it’s the received signal delay
by one symbol period).
2.4.7. BER Performance of BPSK
The SER (and hence BER) performance of Coherent BPSK is given by:





02
1
N
E
erfcP bs (Equation 32)
Note that this equation is the same as equation 4 for Polar Baseband.
Differential Encoded PSK performs identically to non-differentially Encoded PSK if a
Coherent Demdulator (with decoder) is used.
The SER and BER performance of Differentially demodulated PSK is given by:

- 62 -







 
0
exp
2
1
N
E
Ps b (Equation 33)
The BER performances of coherent and Differential PSK are plotted in Figure 29.

Figure 29: BER Performance of PSK
Lecture 12 will present the DESPK and DSPK variants of PSK

2.5. Summary and Comparison of Binary Bandpass
Modulation Options
We have now assembled a basic ‘tool box’ of binary bandpass modulation options. We
can modulate the amplitude, frequency or phase of a carrier with our message data and
we can use a variety of non-coherent, coherent or differentially-coherent methods to
demodulate the message at the receiver. It is useful now to compare these different
options. However, engineering is complex and ‘which is best’ is rarely a one dimensional
decision. We should reflect back on all the factors affecting design (quantitative and
qualitative) that we reviewed in section 1.2 as well as the BER performance, spectral
content and modulator and demodulator block diagrams presented throughout section 2.

- 63 -




To aid in comparison, Figure 30 combines the six BER vs SNR curves presented earlier
in section 2 (note that only 5 curves are distinctly visible since coherent ASK and coherent
FSK have identical performance, consistent with equations 29 and 31). There are obvious
conclusions to draw here. Strictly in terms of BER vs SNR performance, coherent PSK
performs the best and non-coherent ASK performs the worst and we can quantify the
difference in performance between any two modulation/demodulation options. BER versus
SNR performance is important. But it is not everything. As engineers, we must look beyond
the maths and take a more holistic view. It is our job to design communications systems
which best fit our design objectives taking all of these factors into account.
Taking into account the quantitative information in Figure 30 and other design factors,
some key conclusions can be drawn when we apply our knowledge to the design of real
world systems:
 Of all the binary modulation schemes, Coherent BPSK is the most robust to additive
noise. For this reason it is a commonly used modulation scheme (sometimes
combined with additional advanced techniques) in things like digital television, WiFi
and most generations of cellular mobile (excluding 1G which was analog).
 FSK performs less well in BER versus SNR terms than PSK but has an advantage of
producing a constant envelope signal. This ameliorates hardware design challenges
and typically makes for more power efficient transmission. When implemented as MSK
it also retains good spectral efficiency. For this reason it was used to great success in
2G cellular mobile. Whilst 2G might be considered old news today it was really 2G
which made mobile phones the phenomenal success that they are today. Without the
use of FSK then, we might not all have mobile phones now!
 Non-coherent FSK performs less well still in BER versus SNR terms but places even
lower demands on hardware. It has the benefits of constant envelope discussed above
and allows a cheaper, lower complexity receiver design. For this reason it is still used
today in Bluetooth.
 ASK offers a poor compromise between SNR requirements and spectral efficiency and
is thus rarely used in practical applications.
So, based on this review, it would seem that for binary modulation schemes PSK and FSK
provide good options and design tradeoffs, whilst ASK looks like a poor performer. Its hard
to think of a reason to use binary ASK or an example of a real world system which does
use it.
It is not enough though to consider only binary modulation schemes. We chose to do that
first in section 2 because they are simpler and because they still have real world relevance.
Now, however, we must look beyond the limits of binary modulation and consider what
can be achieved by M-ary modulation schemes.

- 64 -





Figure 30: BER Performance of Binary Bandpass Modulation Schemes
Class 7 will first review PSK. Also in class 6 we will work through a set of simplified
design briefs for different communication systems and select which modulation scheme
we deem best for the task. We will finish with some conclusions about our options and
set some challenges for M-ary modulation.

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3. M-Ary Bandpass Modulation
In section 2 we explicitly limited ourselves to binary modulation (every transmitted symbol
represented just one bit) and identified three methods for bandpass signalling – ASK, FSK
and PSK. In this section we will consider possible extensions to Mary signalling. We will
discuss M-Ary ASK, M-ary FSK and M-ary PSK in the next three subsections. After that,
we will also consider bandpass modulation schemes that modulate data onto more than
one property of the carrier signal. There was no point doing the latter for binary modulation
because with only one bit there was no point modulating more than one property of the
carrier wave but for M-ary, that is an option that should be considered and in some cases
does yield some useful modulation schemes.
At this point it is worthwhile to reflect upon the fact that we are repeating ourselves
somewhat. We have studied ASK, FSK and PSK already. Now we are doing so again.
Arguably, this could be inefficient and perhaps we should have just presented all of ASK
(binary and M-ary) at once. However, there is method in this approach and indeed in the
order in which ASK, FSK and PSK were covered. We are gradually building up the
complexity of the modulators and demodulators that we consider. In fact, in the following
sections, you should see that very little is actually new content. It is just an increment in
detail on things we have already studies and hopefully understood.
3.1. M-ary ASK
M-ary ASK represents b bits of data by allowing the carrier signal to take one of 2b
amplitudes. As an example, the constellation diagram for 8-state unipoplar ASK is shown
in Figure 31.

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Figure 31. Unipolar 8-ASK Constellation

3.1.1. M-ary ASK Modulation
An M-ary ASK modulator is shown in Figure 32.

Cos(ct)
Sin(ct)
Data
Input
M-ary ASK
output
Cos(ct)
2M

- 67 -




Figure 32. M-ASK Modulator.

The input binary signal is grouped into groups of bits that map to one of amplitudes.
The resultant time variant amplitude is then bandlimited before being mixed with the carrier
signal.
This is a good example of the point made above, that M-ary ASK is only incrementally
more complex to understand and implement than binary ASK. We know why we bandlimit
the baseband M-ary waveform. We know why we multiply it by a carrier wave. The only
additional factor to consider is the issue of mapping bits to one of possible amplitudes.
Table 8 presents a suitable mapping table for 8-ary uni-polar ASK which would yield the
example constellation given in Figure 31, where ெ௔௫ is the maximum signal amplitude
that our communication system is able to transmit. We don’t need to concern ourselves
here with how we would implement this process. It could be implemented in principle using
basic electronic components (shift registers, logic, op amps, a few resistors) but more
likely in practice it would be implemented in programmable hardware (a microprocessor,
FPGA, DSP or whatever) or within an communications ASIC. The key point for us as
communications engineers is that it is neither hard nor expensive to do this.
Table 8: Uni-polar 8-ASK Mapping
Data Amplitude (V)
000 0
001 ௠௔௫
7

011 2௠௔௫
7

010 3௠௔௫
7

110 4௠௔௫
7

111 5௠௔௫
7

101 6௠௔௫
7


- 68 -




100 A

3.1.2. M-ary ASK Detection
A coherent M-ary ASK demodulator is shown in Figure 33.

Figure 33: Coherent M-ary ASK Demodulation
The received signal is mixed and filtered to return the signal to baseband. A multilevel
comparator or hard limiter is then use to ‘square up’ the signal to force it to one of the M
amplitude states. This can then be demapped to recover the corresponding b bits of output
data.
There is almost nothing new in this diagram! If we understand how coherent demodulation
of binary ASK works then we understand why we use a mixer and a filter here. If we
understood the mapping process in the modulator (see Table 8 for example) then we
understand the demapping (just flip the two columns in Table 8 for example). The only
slight extension of our knowledge here is that the comparator is not binary. It cannot set a
single voltage threshold and say ‘anything above the threshold is high, anything below is
low’. Rather it must set − 1 thresholds and hard limit the output whichever one of the
possible outputs is nearest to the input.
Note that non-coherent detection of uni-polar ASK is also possible using a combination of
the amplitude detector circuit in Figure 17 and the multilevel comparator/hard limiter in
Figure 33. Go ahead and design one yourself!
Data
outputM-ary ASK
input
Multi-level comparator
M2
Cos(ct)

- 69 -




3.1.3. BER Performance of ASK
With M-ary ASK, there is no opportunity to exploit orthogonality (we will see that there is
with FSK and PSK) and so the SNR requirement increases as soon as we increase the
number of bits per symbol. The SER of (Bipolar, Coherent) M-ASK is given by:
௘ =
ெିଵ
ெ
ቂ ଷா್
ேబ(ெమିଵ)
ቃ (Equation 34)
This performance is shown in Error! Reference source not found. for M=2,4,8 and 16.

Figure 34: M-ary ASK BER Performance
3.2. M-ary FSK
M-ary FSK maps unique combinations of bits onto one of frequencies. Just as with
ASK, we can build upon our understanding of the binary modulator and demodulators and
extend that to the M-ary case.
3.2.1. M-ary FSK Modulation
An M-ary FSK Modulator is shown in Figure 35.

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Figure 35: M-FSK Modulator
A group of b bits is mapped into one of 2b amplitudes, bandlimited and input to a VCO.
There is nothing new here and you could have designed this yourself based on what you
already knew about FSK and M-ary ASK.
Its worthwhile to note that the despite the difference between ASK and FSK, the mapping
of bits to symbols is identical for both, because for both cases we wish to map unique bit
combinations to unique voltages. Although voltage is amplitude, for FSK it is the function
of the VCO to convert that input amplitude into a frequency.
3.2.2. M-ary FSK Demodulation
A coherent M-ary FSK Demodulator is shown in Figure 36.

FSK
Output
Data
Input
VCO
2M
Data output
Multi-way comparator
M2
M-ary FSK
input
Cos(c+mt)
Cos(c+2mt)
Cos(c+Mmt)

- 71 -




Figure 36: M-ary FSK Demodulator.
This demodulator introduces a couple of slightly different concepts to those that we already
understood earlier. Firstly, the M-ary FSK detector uses a similar parallel coherent
demodulation process as for binary FSK but with parallel coherent demodulators rather
than 2. If orthogonal signalling frequencies are used then only one of these demodulators
should produce a significant output. Secondly, a multi-way (as opposed to multi-level)
comparator can be used to choose the correct symbol and thus the corresponding correct
set of 2b bits. This comparator isn’t comparing one input against − 1 thresholds as in M-
ary ASK, it is comparing inputs against each other and deciding, which of them is
biggest.
3.2.3. BER Performance of M-ary FSK and Bandwidth efficiency
The BER performance of M-ary FSK is complex and it is beyond the scope of this course
to present it mathematically. However it is illustrated graphically in Figure 37 for a range
of values of M.

Figure 37: BER Performance of M-ary FSK
It can be seen that increased M results in an improving BER and ‘steepening’ of the curve.
In the limit of M , an Eb/N0 of –1.6dB achieves an infinitely small BER.

P
0
M=2
M=4
M=8
M=2048
M10-1
10-2
10-3
10-4
10-5
10-6
1
10 20Eb/N0

- 72 -




This may initially seem quite exciting (remember you are supposed to be thinking like a
communications engineer) because it suggests something unepected. The symbols we
have in our alphabet, the more robust our system will be to AWGN. And if we can transmit
more bits per symbol if there are more symbols in the alphabet, it may seem that we can
‘have our cake and eat it’ here. Larger means more robust to noise and more bandwidth
efficiency. We have a modulation scheme which avoids the fundamental tradeoff we
identified in section 1.7.6. Well, regrettably, this is not true. The explanation for this lies
back in Figure 20. Each of our possible frequencies in our modulation scheme is being
turned on and off (we transmit one at a time). Each will have an approximate sinx/x
spectrum (not allowing for bandlimiting effects). The width of each of these is determined
by the symbol rate and we must space them accordingly. So if we have frequencies
spaced at ௦ then to total bandwidth occupied is approximately ௦, i.e. the bandwidth
occupied increases approximately linearly with . Meanwhile we know that the number of
bits per symbol increases logarithmically with . As a consequence we find that the
bandwidth efficiency of M-ary FSK is proportional to (ଶ) ⁄ which of course
decreases as M increases.
Whilst its not the ‘best of both’ trend that we might hope for, this remains interesting
because what it means is that M-ary FSK does the opposite of what we expect (and any
other modulation scheme that we know). By increasing we can be more robust to noise
at the cost of being less bandwidth efficient. For many applications, low bandwidth
efficiency isn’t acceptable. But for some niche applications where very low SNRs are
unavoidable, M-ary FSK with large values of may be used.
There is also the specific case of 4-ary FSK. It is noteworthy here that the bandwidth
efficiency of 4-ary FSK is the same as binary FSK but the BER performance is better. The
cost of this is more hardware. 4-ary FSK is believed to be used today in some proprietary
video game controllers.