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School of Information Technology

and Electrical Engineering

The University of Queensland

Bachelor of Engineering (Hons) Thesis

Adaptive Equalisers and

Smart Antenna Systems

By Shannon Liew

Supervisor: Dr. John Homer

Submitted for the Degree of Bachelor of Engineering (Honours)

in the Division of Electrical Engineering

October 2002

11 Jefferson Pl

Stretton

Queensland

Australia 4116

18th October, 2002

Head of School

School of Information Technology and Electrical Engineering

The University of Queensland

St Lucia

Queensland

Australia 4067

Dear Professor Kaplan,

In accordance with the requirements of the degree of Bachelor of Engineering

(Honours) in the division of Electrical Engineering, I submit the following thesis

entitled “Adaptive Equalisers and Smart Antenna Systems”. This thesis was

performed under the guidance and supervision of Dr. John Homer.

I declare that the work submitted in this thesis is my own, except as acknowledged in

the text, or references, and has not been previously submitted for a degree at the

University of Queensland or any other institution.

Yours sincerely,

Shannon Liew

- i -

Acknowledgement

I would like to express great appreciation to my supervisor, Dr. John Homer, for his

complete patience, constructive guidance, and generous assistance throughout the year.

I would also like to thank him for giving me the opportunity to work on this topic as one

of his students.

My greatest thankfulness to my parents and brothers who have always been there to

support and encourage me and the decisions I have made, throughout my life. Their

constant love and support have shaped me into the person I am today.

Special thanks must also go to my girlfriend, Ngaire, who has had to endure a long

distance relationship over the last two years as I complete my degree. Her support, love

and encouragement have kept me going through the hard and stressful times.

- ii -

Abstract The massive expansion of mobile communications over the recent years has meant that

telecommunications companies continually need to increase the capacity and coverage

of their networks to keep up with demand. This has required more base station antennas

to be erected, resulting in increased costs and visual pollution. Smart antenna systems

have gained increased interest as they promise to provide significant increases in system

capacity and performance, and greater coverage, meaning less base stations are needed

to cover the same area compared to conventional antennas. The increase in demand for

fast data transmission rates in today’s society leads to greater inter-symbol interference

(ISI) in the received signal, due to multipath propagation. Adaptive equalisers can be

used to periodically estimate the communication channel and then perform equalisation

(inverse modelling) to reduce the effects of ISI.

This thesis revolves around the Least Mean Square (LMS) adaptive algorithm, chosen

for its computational simplicity and high stability. Using the Standard and Detection

Guided LMS algorithms, previous adaptive equalisation studies have modelled the

communication channel as a Finite Impulse Response (FIR) filter, and have found

success in negating the effects of inter-symbol interference caused by multipath

components. The first part of the thesis investigates whether these findings also stand

when modelling the channel as an Infinite Impulse Response (IIR) filter. The results

show that the FIR system findings do hold for the IIR system and we can therefore

confidently assume that further improvements to the LMS algorithms will also stand.

The second section implements the LMS algorithm into the MATLAB® simulation of

an adaptive array, smart antenna base station system to investigate its performance in

the presence of multipath components and multiple users. The simulations illustrate

that adaptive array antenna systems are able to adjust their antenna pattern to enhance

desired signals, and reduce interference. In theory, the implementation of appropriate

time delay filters suggests that time dispersed multipaths can be added constructively to

increase the signal to noise ratio (SNR), providing an increase in performance.

- iii -

Contents

Acknowledgement i

Abstract ii

List of Figures vi

List of Tables viii

1 Introduction 1

1.1 General Introduction 1

1.2 Motivation and Objectives of Thesis 1

1.3 Overview of Thesis 3

2 Cellular Telephone Systems 4

2.1 The Cellular Concept 4

2.2 Interference and System Capacity 5

2.2.1 Co-channel Interference 5

2.2.2 Adjacent Channel Interference 6

2.2.3 Power Control 6

2.3 Trunking 6

2.4 Increasing Capacity 7

2.4.1 Cell Splitting 7

2.4.2 Sectoring 8

2.5 Mobile Radio Propagation 9

2.5.1 Radio Wave Propagation 9

2.5.2 Propagation Mechanisms 9

2.5.3 Multipath Propagation 10

2.6 Multiple Access Techniques 11

2.6.1 Frequency Division Multiple Access 11

2.6.2 Time Division Multiple Access 12

2.6.3 Code Division Multiple Access 12

2.6.4 Space Division Multiple Access 13

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3 Adaptive Equalisation 15

3.1 Overview of LMS Adaptive FIR Filter 15

3.2 Standard LMS Algorithm 16

3.3 Stability of the LMS Algorithm 18

3.4 Detection Guided LMS 19

3.5 Normalised LMS Algorithm 21

4 Introduction to Smart Antennas 22

4.1 Key Benefits of Smart Antennas [7] 22

4.2 Switched Beam Antenna Systems 23

4.3 Introduction to Adaptive Antenna Technology 24

4.4 Adaptive Antenna Systems 27

4.5 Statistically Optimal Beamforming Techniques 28

4.6 Adaptive Algorithms 29

5 Analysis of Results 31

5.1 LMS Equalisation 31

5.1.1 Standard LMS with White Inputs 32

5.1.2 Detection Guided LMS with White Inputs 33

5.1.3 Standard LMS with Coloured Inputs 35

5.1.4 Detection Guided LMS with Coloured Inputs 36

5.1.5 LMS Summary 37

5.2 Normalised LMS Equalisation 38

5.2.1 Normalised LMS with White Inputs 39

5.2.2 Detection Guided NLMS with White Inputs 40

5.2.3 NLMS with Coloured Inputs 41

5.2.4 Detection Guided NLMS with Coloured Inputs 42

5.2.5 NLMS Summary 44

5.3 Smart Antenna Simulations 45

5.3.1 One White Signal with One DOA 45

5.3.2 One White Signal with Three DOAs 46

5.3.3 Two White Signals with One DOA Each 48

5.3.4 Two White Signals with Three DOAs Each 49

5.3.5 Smart Antenna Summary 51

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6 Conclusion and Future Work 52

6.1 Conclusion 52

6.2 Future Work 54

6.3 Final Thoughts 54

References 56

Appendix A LMS Code Listings 58

A.1 Standard LMS with White Inputs 58

A.2 Detection Guided LMS with White Inputs 60

A.3 Standard LMS with Coloured Inputs 62

A.4 Detection Guided LMS with Coloured Inputs 64

Appendix B NLMS Code Listings 66

B.1 Normalised LMS with White Inputs 66

B.2 Detection Guided NLMS with White Inputs 68

B.3 Normalised LMS with Coloured Inputs 70

B.4 Detection Guided NLMS with Coloured Inputs 72

Appendix C Smart Antenna Code Listings 74

C.1 Smart Antenna Receiving 1 White Signal with 1 DOA 74

C.2 Smart Antenna Receiving 2 White Signals with 1 DOA Each 76

C.3 Smart Antenna Receiving 1 White Signal with 3 DOAs 79

C.4 Smart Antenna Receiving 2 White Signals with 3 DOAs Each 82

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List of Figures Figure 2-1: An example of cell splitting [3]. 7

Figure 2-2: An example of sectoring [3]. 8

Figure 2-3: Multipaths resulting from reflectors [7]. 10

Figure 2-4: Frequency Division Multiple Access [3]. 11

Figure 2-5: Time Division Multiple Access [3]. 12

Figure 2-6: Code Division Multiple Access [3]. 13

Figure 2-7: Space Division Multiple Access [3]. 13

Figure 3-1: Block diagram of proposed adaptive equalisation system. 16

Figure 3-2: Example of an impulse response of a mobile communication channel. 19

Figure 4-1: Model of linear equally spaced array receiving a signal from angle θ

from perpendicular. 24

Figure 4-2: An adaptive array processor [7]. 28

Figure 5-1: Desired impulse response of F. 32

Figure 5-2: Received impulse response of F using Standard LMS with white

inputs. 32

Figure 5-3: Squared error of F and number of active taps detected using Standard

LMS with white inputs. 33

Figure 5-4: Received impulse response of F using Detection Guided LMS with

white inputs. 33

Figure 5-5: Squared error of F and number of active taps detected using

Detection Guided LMS with white inputs. 34

Figure 5-6: Received impulse response of F using Standard LMS with coloured

inputs. 35

Figure 5-7: Squared error of F and number of active taps detected using Standard

LMS with coloured inputs. 35

Figure 5-8: Received impulse response of F using Detection Guided LMS with

coloured inputs. 36


Detection Guided LMS with coloured inputs. 37

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Figure 5-10: Received impulse response of F using NLMS with white inputs. 39

Figure 5-11: Squared error of F and number of active taps detected using NLMS

with white inputs. 39

Figure 5-12: Received impulse response of F using Detection Guided NLMS with

white inputs. 40

Figure 5-13: Squared error of F and number of active taps detected Detection

Guided NLMS with white inputs. 41

Figure 5-14: Received impulse response of F using NLMS with coloured inputs. 41

Figure 5-15: Squared error of F and number of active taps detected using NLMS

with coloured inputs. 42

Figure 5-16: Received impulse response of F using Detection Guided NLMS with

coloured inputs. 43


Detection Guided NLMS with coloured inputs. 43

Figure 5-18: Smart antenna simulation received signal error for 1 white signal with

1 DOA. 46

Figure 5-19: Smart antenna simulation beam pattern for 1 white signal with 1

DOA. 46

Figure 5-20: Smart antenna simulation received signal error for 1 white signal with

3 DOAs. 47

Figure 5-21: Smart antenna simulation beam pattern for 1 white signal with 3

DOAs. 47

Figure 5-22: Smart antenna simulation received signal error for 2 white signals

with 1 DOA each. 48

Figure 5-23: Smart antenna simulation beam pattern for 2 white signals with 1

DOA each. 49

Figure 5-24: Smart antenna simulation received signal error for 2 white signals

with 3 DOAs each. 50

Figure 5-25: Smart antenna simulation beam pattern for 2 white signals with 3

DOAs each. 50

Figure 5-26: Block diagram of time delay summing system. 51

- viii -

List of Tables Table 5-1: Summary of results from LMS equalisation simulations. 38

Table 5-2: Summary of results from NLMS equalisation simulations. 44

Adaptive Equalisers and Smart Antenna Systems

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1 Introduction

1.1 General Introduction

Over the years, the expansion of mobile communications can be described as nothing

less than extraordinary. With the number of people using mobile phones increasing

continually, telecommunications companies are confronted with the problem of

increasing the capacity and coverage of their networks to keep up with demand. To

achieve this, more antenna base stations have needed to be erected, creating much visual

pollution and increased costs. The Optus Mobile Network in Australia has built an

average of one new base station everyday from January 1999 to June 2002, and has

invested $2.32 billion to provide the coverage that its subscribers enjoy today [1].

Smart antennas have promised to provide significant increases in system capacity and

performance in wireless communication systems [2]. In turn, this leads to increased

revenue for the telecommunications companies and also a reduction in dropped and

blocked calls. Other benefits include greater coverage, meaning less base stations are

needed to cover the same area compared to conventional antennas. For these reasons,

smart antennas have gained greater interest over the recent years.

1.2 Motivation and Objectives of Thesis

In most mobile channels, there is more than one propagation path between each

transmitter and receiver, and a received signal consists of two or more components,

each of which travelled a different path from the transmitter. Each multipath

component arrives with a delay depending on the path length. Delayed multipath

components can cause inter-symbol interference (ISI), and impose an upper limit on the


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data rate that the channel can support without the use of expensive equalisers. Fading is

another problem in a multipath channel. This "multipath fading" occurs because in

general multipath components arrive with different phases. At some points in space, the

components cancel each other, causing deep fades in the received signal level. Both ISI

and fading can be mitigated using adaptive antennas.

The increase in demand for fast data transmission rates in today’s society leads to

greater ISI in the received signal, due to multipath propagation. Adaptive equalisers can

be used to periodically estimate the communication channel, and then perform

equalisation to suppress ISI. Previous adaptive equalisation studies have been

successful in negating the effects of ISI caused by multipath components with the use of

the Standard and Detection Guided Least Mean Square (LMS) algorithms. However,

they have modelled the unknown communication channel as a Finite Impulse Response

(FIR) filter and the adaptive equaliser as an Infinite Impulse Response (IIR) filter.

The first part of this thesis aims to investigate whether the findings and algorithms used

in these studies also hold when we assume the unknown channel as an IIR filter, and

therefore the adaptive equaliser as an FIR filter. The advantage of this representation is

that an FIR equaliser provides better stability over an IIR equaliser. These

investigations are to be simulated using MATLAB®. The algorithms will need to be

modified slightly to cater for the changes in assumptions but it is expected that the

findings will still stand.

As described earlier, smart antennas have gained great interest over the recent years as

they have promised to increase capacity and performance. These benefits are a result of

the smart antenna system’s ability to direct beams in the direction of desired multipath

components and nulls in the direction of interference.

The second section of this thesis implements the LMS algorithm into the MATLAB®

simulation of an adaptive array, smart antenna base station system. The LMS adaptive

algorithm is chosen for its computational simplicity and high stability. The aim is to

investigate the antenna system’s performance in the presence of multipath components


- 3 -

and multiple users; in particular, whether the system can distinguish between multipath

components arriving from different directions at different time delays, as well as from

different directions arriving at the same time instant.

1.3 Overview of Thesis

The above serves as a brief introduction to why there is high interest in smart antenna

systems and adaptive equalisers, and their applications in telecommunications. The

motivation behind this research and the aims of this thesis are also noted.

Chapter 2 introduces cellular telephone systems in general and the methods of

increasing system capacity, as well as the propagation mechanisms and multiple access

techniques used in mobile communications.

Chapter 3 discusses the proposed adaptive equaliser system along with the Standard,

Detection Guided and Normalised variants of the Least Mean Square (LMS) algorithm,

which is used in this thesis. It also provides a block diagram of the system to aid in

understanding how the system will be implemented.

Chapter 4 provides a basic understanding of the two types of smart antenna systems

before concentrating on adaptive array antennas, which are investigated in this thesis.

The equations used in the MATLAB® simulation are also presented.

Chapter 5 presents and analyses the simulation results obtained using the program

MATLAB®. At the end of each section a summary of the results are discussed.

The thesis concludes with Chapter 6, in which a review of the findings, possible areas

for future research work, and final thoughts are given.


- 4 -

2 Cellular Telephone Systems

The objectives of early mobile radio systems were to cover the largest amount of area

possible with one high-powered antenna transmitter mounted on a tall tower [3]. This

method provided great coverage for the system. However, the system was unable to

reuse the same radio channels for another base station due to the amount of interference

experienced, thereby limiting the capacity of the system.

2.1 The Cellular Concept

A major innovation in reducing spectral congestion and increasing system capacity

while using a limited radio spectrum was the emergence of the cellular concept.

Dividing the large geographic area into many smaller cells meant moving from one

high-powered transmitter to many low-powered ones, each providing coverage to its

own cell.

Cellular radio systems rely on intelligent allocation and reuse of channels throughout a

coverage area [4]. Each base station is assigned a set of radio channels to be used only

within its cell. Adjacent cells are allocated a completely different set of channels until

all available channels are used. This distribution of channels ensures interference to

both the base station and mobile are kept to a minimum. It also allows for frequency

reuse, where another base station, a calculated distance away, is able to use the same set

of channels so long as the interference between the co-channel base stations is kept

below acceptable levels.

When a mobile user is in conversation and moves into a new cell, the new base station

becomes active. The call is automatically transferred to a new set of channels belonging


- 5 -

to the new base station. This process is called a handoff and requires the mobile

switching centre to assign the new set of channels. Handoffs are an important task in

cellular radio systems and must occur as infrequently as possible and should be

transparent to the user.

As the number of users increase, a corresponding increase in the number of base

stations and decrease in transmitter power is needed in order to avoid interference. This

allows for the increase in user capacity without an increase in radio spectrum. The

drawback to the cellular concept is the visual pollution and costs for installation of new

antenna towers due to the increasing number of base stations needed [5].

2.2 Interference and System Capacity

The major limitation of cellular radio system performance is interference [3]. As a

result, much research into the reduction of interference in cellular systems has been

performed. Interference can be the result of a variety of sources, ranging from another

mobile in the same cell to another base station some distance away using the same set of

channels.

In highly developed, densely populated urban areas, the higher radio frequency (RF)

noise floor and greater number of base stations means the effects of interference are

much more significant. Interference is a major restriction in increasing capacity and is

usually responsible for dropped calls. The two major types of interference are co-

channel interference and adjacent channel interference. Interfering signals can be

generated from within the cellular system or from out-of-band users.

2.2.1 Co-channel Interference

Co-channel cells are those cells that share the same set of channels in the frequency

reuse concept described earlier. Co-channel interference is simply the interference

experienced between signals from co-channel cells. Increasing the signal-to-noise ratio

(SNR) does not reduce co-channel interference. Increasing the transmitter power of a


- 6 -

base station actually increases interference to co-channel cells. The only solution is to

increase the distance between the co-channel cells

2.2.2 Adjacent Channel Interference

Adjacent channel interference is interference resulting from signals that are adjacent in

frequency to the desired signal [3]. Adjacent channel interference is due to imperfect

receiver filters that allow frequencies outside of the desired range into the passband.

Careful filtering and channel assignments can minimise the effects of adjacent channel

interference. Smarter frequency allocation, by allocating adjacent channels to different

cells, can reduce adjacent channel interference considerably.

2.2.3 Power Control

In cellular communications, each mobile unit’s transmitter power levels are under

constant control of the serving base station [3]. This practice allows the base station to

ensure that the mobile transmits at the lowest power required to maintain a good quality

reverse channel link, thereby prolonging the battery life of the mobile unit and

minimising interference to other users. However, it also greatly reduces reverse channel

SNR in the system, increasing the probability of bit errors. There is therefore a trade-

off between prolonging battery life, minimising interference and probability of bit error.

Power control is especially important in systems using CDMA techniques as they allow

every user in the system to communicate sharing the same radio channel.

2.3 Trunking

Trunking is used to cater for large numbers of users in a limited radio spectrum of a

cellular radio system. The few channels in a cell are shared amongst the users, who are

provided access to the channels from a pool of available channels [3]. A user is given a

channel to make or receive a call, and when finished, the channel is returned to the pool.

If all channels are in use, the user is either blocked from the system or placed in a

queue.


- 7 -

2.4 Increasing Capacity

Increasing demands for mobile communication services means that eventually, the

number of channels in a cell will become insufficient to support the number of users.

Cellular design techniques are needed to provide more channels per unit coverage area.

Two such techniques are cell splitting and sectoring and are used in practice to increase

system capacity. Cell splitting allows for orderly growth of the system by increasing

the number of cells, and therefore base stations. Sectoring employs directional antennas

to control the interference and frequency reuse. Cell splitting does not suffer from the

trunking inefficiencies that sectoring does.

2.4.1 Cell Splitting

Cell splitting is the process of dividing cells that have the highest traffic congestion into

smaller microcells with their own base station. For the smaller sized microcells, there

must be a corresponding reduction in antenna height and transmitter power. Cell

splitting increases the capacity of a cellular system by increasing the number of times a

channel is reused and therefore increasing the number of channels per unit area.

Figure 2-1: An example of cell splitting [3].


- 8 -

In reality, different cells are split at different times, meaning different cell sizes exist

simultaneously. Care must be taken in such circumstances to ensure that the minimum

required distance is kept between co-channel cells; otherwise co-channel interference

may result [6].

2.4.2 Sectoring

Sectoring replaces a single omni-directional base station antenna with several

directional antennas, each radiating within its own sector. Sectoring results in reduced

co-channel interference as only a fraction of the available co-channel cells will be able

to receive interference and transmit within a particular cell. The amount of reduction of

co-channel interference depends on the amount of sectoring used, usually three 120º or

six 60º sectors.

Figure 2-2: An example of sectoring [3].

Figure 2-2 shows the centre cell, labelled 5, receiving co-channel interference from only

two other co-channel cells. This is much less than in the case for omni-directional

antennas where all six co-channel cells would be providing the centre cell with co-

channel interference.


- 9 -

The set of channels used in a particular cell is designated a sector and used only within

that sector. The increase in the number of antennas per base station is one drawback of

the improved signal-to-interference ratio (S/I) and resulting capacity improvement. An

increase in the frequency of handoffs results from the reduction in coverage area for

each group of channels. Dividing the set of channels into smaller ones for each sector

also splits the available pool of trunked channels, decreasing the trunking efficiency.

2.5 Mobile Radio Propagation

The communication medium of wireless systems is free space, as opposed to coaxial

cable or fibreoptic cable in wired communication systems. Transmission paths between

transmitters and receivers may vary from a direct line of sight (LOS) to a non line of

sight (NLOS) one that could be hindered by buildings or trees for example.

2.5.1 Radio Wave Propagation

Mechanisms driving electromagnetic wave propagation can generally be credited to

reflection, diffraction and scattering [3]. In general, a transmitted signal wave may

follow a number of different paths before reaching the receiver. Interaction between

these signal waves can cause multipath fading.

2.5.2 Propagation Mechanisms

In a mobile communication system, propagation is mainly influenced by reflection,

diffraction and scattering. The received power of a signal is generally the most

important factor predicted by large-scale propagation models [3]. It is used to predict

mean signal strength for a distance between a transmitter and receiver and is useful in

estimating the coverage area of a transmitter.

Reflection occurs when a propagating electromagnetic wave collides with an object that

has dimensions much larger than the wavelength of the wave. The reflecting objects

can vary from buildings, walls or the earth’s surface.


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Diffraction is defined as the bending of a propagating electromagnetic wave around

large objects, having sharp edges, obstructing the path between a transmitter and

receiver. Diffraction allows a receiver to receive a transmitted signal even when

positioned behind an object.

Scattering occurs when the path that the wave travels consists of objects with small

dimensions compared to the wavelength and where the density of these obstacles is

high. Rough surfaces, small objects and other irregularities such as foliage, street signs

and lampposts produce scattering.

2.5.3 Multipath Propagation

Multipath propagation in the radio channel causes small scale fading. Fading occurs in

built up areas because the heights of the mobile antennas are well below the height of

surrounding buildings, meaning there is no LOS path directly to the base station [3].

Even when a LOS path exists, multipath still occurs due to reflection from the ground

and surrounding objects.

Figure 2-3: Multipaths resulting from reflectors [7].

The incoming multipath waves arrive at the receiver from different directions with

different amplitudes, phases and time delays. These multipath components combine

vectorially at the receiver and can cause fading or time dispersion. Longer paths result

in delayed versions of the signal arriving at the receiver. When the difference in delays,


- 11 -

known as the time delay spread, is large, symbols spread into one another, causing inter-

symbol interference (ISI) at the receiver. This can result in poor signal reception.

2.6 Multiple Access Techniques

Multiple access schemes are used to allow a number of mobile users to share a finite

amount of radio spectrum [3]. This must be done without severe degradation in the

performance of the system for high quality communications.

Multiple access techniques require that the messages to the users be orthogonal in signal

space. There are four basic types of multiple access schemes: Frequency Division

Multiple Access (FDMA), Time Division Multiple Access (TDMA), Code Division

Multiple Access (CDMA), and Space Division Multiple Access (SDMA).

2.6.1 Frequency Division Multiple Access

Figure 2-4: Frequency Division Multiple Access [3].

FDMA assigns individual frequency channels to individual users [3]. Each user is

allocated a unique frequency band or channel, as can be seen in Figure 2-4. Channels

are assigned according to demand to users who request service. Only the assigned user

can use the specified frequency band. Therefore, signals assigned to different users are

clearly orthogonal. However in practice, out-of-band spectral components cannot be


- 12 -

completely suppressed, leaving the signal not quite orthogonal. This necessitates the

introduction of guard bands between frequency bands to reduce adjacent channel

interference.

2.6.2 Time Division Multiple Access

TDMA systems divide the radio spectrum into time slots, where each slot is allocated to

only one user. Each slot position within a frame is allocated to a different user and this

allocation reoccurs over the sequence of frames. This means that a particular user may

transmit during one particular slot in every frame. While TDMA users within a cell are

separated by their time slots, different cells use different frequency channels.

Figure 2-5: Time Division Multiple Access [3].

2.6.3 Code Division Multiple Access

All users in a CDMA system use the same carrier frequency and may transmit

simultaneously [3]. The channels are made orthogonal by using a different pseudo-

noise (PN) code sequence for each user that is approximately orthogonal to the PN

sequences of other users. A time correlation operation is implemented at the receiver in

order to detect only the desired PN sequence. All other PN sequences appear as noise

due to decorrelation; therefore the receiver needs to know the PN sequence used by the

transmitter. Each user operates independently with no knowledge of the other users.


- 13 -

Figure 2-6: Code Division Multiple Access [3].

2.6.4 Space Division Multiple Access

SDMA allows multiple users to operate in the same cell, on the same frequency or time

slot provided, using an array of antennas to spatially separate the signal [7]. This

technique is usually implemented with a secondary multiple access technique from the

above-mentioned techniques. This means that different areas covered by the antenna

beam may be served by the same timeslots in a TDMA system, frequency in an FDMA

system, or PN code in a CDMA system [3].

Figure 2-7: Space Division Multiple Access [3].


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Sectorised antennas can be thought of as a primitive form of SDMA. Adaptive smart

antennas make use of SDMA to steer their reception beams in the direction of many

users at once, and appear to be best suited for TDMA and CDMA base station

architectures.


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3 Adaptive Equalisation

Adaptive equalisation involves periodically training the equaliser to enable it to adapt

itself to any changes in the communication channel. This requires a known training

sequence of fixed length to be sent by the transmitter so that the receiver’s equaliser

may converge to the correct setting to maximise the signal by suppressing echoes

resulting from multipath propagation. Immediately following the training sequence, the

user data is sent.

Much time and effort has gone into the study of the LMS estimation of a channel that is

well modelled as a Finite Impulse Response (FIR) filter, and therefore using an Infinite

Impulse Response (IIR) filter as the equaliser. However, this thesis concentrates on the

study of the LMS algorithm while approximating the communication channel as an IIR

filter. The benefit of modelling the channel as an IIR filter is that an FIR filter can then

be used to equalise the system, providing better stability.

3.1 Overview of LMS Adaptive FIR Filter

The proposed LMS adaptive FIR filter consists of two basic processes. The first is a

filtering process that involves computing the output of the FIR filter, produced by a set

of tap inputs, and also generating an error estimate by comparing this output to a known

desired response. The second is an adaptive process involving the automatic adjustment

of the tap weights of the filter according to the error estimation computed in the first

process. These two processes combine to form a feedback loop around the LMS

algorithm described [8].


- 16 -

The system considered throughout this thesis is shown in Figure 3-1. At sampling

instant k, the input signal u(k) represents the signal sent by the transmitter. The

interference signal nn(k) represents the interference experienced by the receiver. The

signal r(k) is defined as the output from the unknown channel, 1/H, which is then added

with nn(k) to form the received signal y(k). The signal y(k) is then passed through the

equaliser, F, to give the signal x(k). Finally, x(k) is compared with the original

transmitted signal u(k), to give the error signal e(k).

Figure 3-1: Block diagram of proposed adaptive equalisation system.

The role of the equaliser is to resolve the distortion of the channel while minimising the

effect of additive noise at its output [9]. For an unknown channel 1/H, an equaliser with

the transfer function F = H produces an overall channel-equaliser transfer function of

F/H = 1. This implies that in the case of no interference being experienced, the output

from the equaliser, x(k), will be the original transmitted signal u(k). We can think of F

as being an equaliser of 1/H, or an estimator of H.

3.2 Standard LMS Algorithm

There are several assumptions that need to be made with regards to the system being

considered. Reference can be made to Figure 3-1 while reading the following

assumptions. We assume that the unknown channel 1/H is linear, time invariant and

able to be modelled as a discrete-time IIR filter with n taps.

H = [h0, h1, h2, …, hn-1] (3.1)

r(k) +

+

Error Signal e(k) x(k)

_

Transmitted Signal u(k) + y(k)

Adaptive EqualiserUnknown Channel

1 / H

F

Interference Signal nn(k)

+ +


- 17 -

The time invariant, n-tuple, IIR modelled, unknown channel 1/H is a sparse channel

with only m << n nonzero (active) taps, and h0 = 1.

The LMS adaptive FIR filter (equaliser) has a tap delay line structure and a length of n.

F(k) = [f0, f1, f2, …, fn-1] (3.2)

The tap coefficients of F(k) are initially set to zero.

Fi(0) = 0, for i = 0, 1, 2, …, n-1 (3.3)

The input signal u(k), and interference signal nn(k) are assumed to be zero mean,

bounded and wide-sense stationary processes. They are also assumed to be uncorrelated

with each other over time.

In order to calculate the output r(k) from the unknown channel 1/H, the system requires

knowledge of the last n –1 values of r. That is, the vector

R(k) = [r(k-1), r(k-2), r(k-3), …, r(k-n-1)] (3.4)

is used to calculate the output as

)1(

1)(H

kr = (u(k) – H(2: n) RT (k)) (3.5)

where H(1) = h0, and H(2:n) = [h1, h2, …, hn-1].

The output signal then has the interference signal nn(k) added to it to produce the

received signal

y(k) = r(k) + nn(k) (3.6)

The received signal y(k) is then added to an array of the last n-1 received signals to form

the received signal vector

Y(k) = [y(k), y(k-1), y(k-2), …, y(k-n-1)] (3.7)

This vector is then input into the adaptive equaliser F to produce the estimate x(k) of the

transmitted signal

x(k) = F(k-1)Y T(k) (3.8)


- 18 -

The estimate x(k) is then compared to the original transmitted signal u(k) to provide an

error signal

e(k) = u(k) – x(k) (3.9)

Ideally, the error signal e(k) should be equal to the interference signal nn(k). This

would indicate that the LMS adaptive equaliser has successfully estimated H. It can be

shown that the tap weights of the equaliser F are functions of the sampling instant k.

This indicates that the tap weights of the adaptive equaliser are time dependent, since

they are continuously being adapted.

The LMS algorithm adjusts the tap weights, or coefficients of the FIR equaliser in an

attempt to minimise the mean squared error (MSE) e2(k). However, the MSE requires

large amounts of memory, so the instantaneous error e(k) is used to estimate the

gradient of the MSE surface [8].

Eventually, the Standard LMS equation the FIR equaliser is given by

F(k) = F(k-1) + µ Y(k) e(k) (3.10)

where Y(k) is the received signal vector from (3.7) and µ is the step-size parameter.

3.3 Stability of the LMS Algorithm

A crucial parameter affecting the stability and convergence rate of the LMS algorithm is

the value of the step-size parameter, µ. There is a trade-off between the rate of

convergence and stability of the LMS adaptive equaliser. A large µ value results in a

faster convergence rate, but a reduction in the accuracy and stability of the equaliser.

On the other hand, a small µ value gives greater accuracy and stability but a slower

convergence rate [10]. In [9], [11], the value of µ is said to usually be chosen within the

range

0 < µ <∑

+−=

k

nki

iu1

2 )(

2 (3.11)


- 19 -

3.4 Detection Guided LMS

Mobile radio channels, and other communication channels, typically show impulse

responses having extended regions of inactivity in between active regions, as seen in

Figure 3-2. There are potential problems when using the Standard LMS approach while

using an FIR filter to represent the impulse response adequately because all the impulse

response taps are permitted to be active, or nonzero [12]. These problems are most

evident during LMS estimation of channels with a large number of adaptive taps as this

leads to high computational costs, and poor convergence rates and asymptotic

performance [13]. While this thesis considers modelling the unknown channel as an IIR

filter, it is expected that these problems will still exist.

Figure 3-2: Example of an impulse response of a mobile communication channel.

Homer et al. proposed an active tap detection scheme with the aim of detecting the

active taps and subsequently LMS estimating only these taps [10]. By reducing the

number of taps being estimated, faster convergence and accuracy occurs. The key to

this approach is to determine when a particular unknown channel tap is active or

inactive. Other researchers before them had proposed active tap thresholds based on

intuition whereas the theoretically based solution proposed in [10] focuses on the

minimisation of the least squares cost function. The activity measure and threshold

criterion developed are computationally efficient, which is important due to the fact that

it is being used in conjunction with the LMS algorithm, which is known for its

computational simplicity.


- 20 -

As this thesis models the unknown channel as an IIR filter, as opposed to an FIR filter

in [9] and [10], the formula for tap activity measure is derived as

X j(k) = ( )

∑

∑

+=

+=

−

−−+−−−

k

ji

k

jij

iky

ikykFikykxikyku

1

2

2

1

2

)(

)()1()()()()( (3.12)

where j is the tap number of the equaliser, k is the sampling instant, and Fj(k-1) the

estimate of hj at instant k -1.

One approach for active tap detection is to detect the m most active taps by finding the

m greatest values of Xj(k). This method requires prior knowledge of the unknown

channel in order to choose a correct value of m. A poor choice of m would lead to

either the inclusion of inactive taps or the exclusion of some active taps from the group

of taps being estimated [12].

A better method is to detect the taps that are active, rather than the m most active taps.

Homer et al. developed an activity threshold that is used to discriminate between the

active and inactive taps [12]. The formula for the activity threshold when modelling the

unknown channel as an IIR filter is

T(k) = ∑=

k

iiu

kk

1

2 )()log( (3.13)

where k is the sampling instant. That is, the jth tap at sampling instant k is deemed

active if

X j(k) > T(k) (3.14)

where Xj(k) is the result from (3.13). The accuracy of this criterion increases

proportionally with the number of sample intervals.

Each tap in the adaptive filter F is tested using (3.14) at each sample interval k. If the

tap is deemed to be active, the tap activity variable g(j) is set to 1, and 0 if the tap is

inactive. Using the following equation, only taps that are calculated as being active will

be adapted.

Fj(k) = g(j) Fj(k-1) + g(j)µ y(k-j+1) e(k) (3.15)


- 21 -

3.5 Normalised LMS Algorithm

The asymptotic performance of the LMS algorithms is dependent on the number of

adaptive taps. In the Normalised LMS (NLMS) approach µ can be viewed as a

normalised step-size parameter where the normalisation is with respect to total input

signal power within the filter.

µ → 2

unσµ (3.16)

This leads to the asymptotic performance being essentially independent of the number

of taps n. On the other hand, it leads to the convergence rate being strongly dependent

on n. This means that as n increases, the rate of convergence decreases.

The active tap detection algorithm can, consequently, be used to improve the

convergence rate of the NLMS estimator. In particular, this can be achieved by

modifying µ so it is proportional to the estimated number of active taps m̂

µnorm = 20

ˆˆ

umσµ

(3.17)

where 2ˆ uσ is an estimate of the input signal variance, and µ0 < 1 is a positive constant.

This normalised step-size parameter, µnorm, can now be substituted into the appropriate

equations given above for Standard and Detection Guided LMS algorithms to result in

NLMS and Detection Guided NLMS algorithms accordingly. The NLMS approach is

often preferred over LMS due to better stability.


- 22 -

4 Introduction to Smart Antennas

Smart antennas are basically an extension of sectoring, in which multiple beams replace

the coverage of sectors. This is achieved by the use of array structures, and the number

of beams in the sector is a function of the array extent [14]. The increase in beam

directionality can provide an increase in system capacity and expand cell coverage.

Smart antennas can be divided into two major types, being switched beam and adaptive

array systems. Both systems attempt to increase gain in the direction of the user [15].

This can be achieved by directing the main lobe, with increased gain, in the direction of

the user, and nulls in directions of the interference.

4.1 Key Benefits of Smart Antennas [7]

Smart antennas provide enhanced coverage through range extension, hole filling, and

better building penetration. Given the same transmitter power output at the base station

and subscriber, smart antennas can increase the gain of the base station antenna. The

uplink (mobile to base station) power received, in decibels (dB), from a mobile unit at a

base station is given by

Pr = Pt + Gs + Gb – PL (4.1)

where Pr is the received power at the base station, Pt is the transmitted power by the

subscriber, Gs is the gain of the subscriber unit antenna, and Gb is the gain of the base

station antenna. For a certain required base station received power, Pr, min, increasing

the base station gain, Gb, means that the link can tolerate greater path loss, PL.

Increasing the tolerable path loss means that the reception range of the base station can

be increased. As smart antennas can allow higher gain compared to conventional

antennas, smart antenna systems can provide range extension.


- 23 -

Initial deployment costs to install wireless systems can be reduced via range extension.

Smart antennas, through range extension, allow larger cell sizes than conventional

antennas when initially deploying wireless networks. The additional costs of using

smart antenna systems over the conventional systems must be taken into account when

calculating the benefits of smart antenna systems.

Link quality can be improved through the management of multipaths. As discussed

earlier, fading or time dispersion can be the results of multipath propagation in radio

channels. Smart antennas can manage multipaths by directing beams in the direction of

the desired signal and nulls towards interferers.

Smart antennas can improve system capacity. Smart antennas can be used to allow the

subscriber and base station to operate at the same range as a conventional system, but

with less power. This allows FDMA and TDMA systems to be rechannelised to

increase the amount of frequency reuse in the system. The multiple access interference

in CDMA systems is reduced if smart antennas are used to allow users to transmit less

power for each link, which then increases the simultaneous number of users that can be

supported by the system. Smart antennas can also be used to implement the SDMA

scheme presented earlier.

4.2 Switched Beam Antenna Systems

The switched beam technique further subdivides sectors into micro-sectors. Each

micro-sector contains a fixed beam pattern, with the greatest gain placed in the centre of

the beam. When a mobile user is near a micro-sector, the switched beam system selects

the beam with the strongest signal. During a call, the system monitors the signal

strength and switches to other fixed beams if necessary.

One of the major disadvantages of switched beam antenna systems is that the system is

unable to provide protection from multipath components received from directions close

to that of the desired signal.


- 24 -

4.3 Introduction to Adaptive Antenna Technology

Smart antennas use an array of low gain antenna elements that are connected by a

combining network [7]. To simplify the analysis of antenna arrays it is assumed that the

spacing between the array elements is small enough so that the signals received at

different elements do not vary in amplitude. It is also assumed that there is no mutual

coupling between antenna elements and that there are always a finite number of signals

arriving at the elements. Finally, the bandwidth of the arriving signal is small compared

to the carrier frequency.

The array of antennas is usually implemented as a linearly equally spaced (LES),

uniform circular, or uniformly spaced planar array of similar, co-polarised, low gain

elements, which are oriented in the same direction [7]. The capability to null out

interference depends on the number of interferers relative to the number of antenna

elements. An M element antenna array is capable of nulling M-1 interferers. If the

number of interferers is M-1 or less, the array is said to be underloaded and is able to

place a null in the direction of all the interferers.

Figure 4-1: Model of linear equally spaced array receiving a signal from angle θ from perpendicular.

dsinθ θ

θ

M 2d

x(t)

rM(t) rm(t) r2(t) r1 (t)

wM wm w2 w1

m 1

Σ

. . .

Phase front


- 25 -

We consider the axis of the M-element LES array, shown in Figure 4-1, to be the x-axis,

the axis perpendicular to the array to be the y-axis, and the 1st element to be lying at the

origin. We assume that all multipath components arrive at the base station in the

horizontal plane. The direction of arrival (DOA) θ is measured from the y-axis. Each

of the branches of the array in Figure 4-1 has a complex valued weighting element, wm,

that is, each weighting element has both a magnitude and a phase associated with it [7].

For the moment we assume that all antenna elements are noiseless isotropic antennas

having uniform gain in all directions. The element spacing d in an LES array must not

exceed λ/2 in order to avoid grating lobe problems, which produce undesired beams and

therefore amplify noise and interference [7]. Letting u(t) be the baseband complex

envelope of the incident signal, and the phase of u(t) at the 1st element be zero, the

signal received by antenna element m is

rm(t) = Au(t)e-j(2π(m-1/λ)dsinθ (4.2)

where A is the gain experienced by the signal.

The array output signal x(t) is then

x(t) = ∑=

M

mmm trw

1)( = Au(t)f(θ) (4.3)

where

f(θ) = ∑=

M

mmw

1e- j(2π(m-1)/λ) dsinθ (4.4)

is known as the array factor. The array factor determines the beam pattern of the array

by determining the gain in direction θ.

Adjusting the set of weights means it is possible to choose any desired direction as the

maximum gain direction [7]. To show this, let

wm = e j(2π(m-1)/λ)dsinψ (4.5)

then

f(θ) = ∑=

M

me

1

- j(2π(m-1)/λ)d(sinθ – sinψ) (4.6)

= DC e- j(2π(m-1)/λ)d(sinθ – sinψ)


- 26 -

where

C = sin [(2π M /λ)(d/2)(sinθ – sinψ)] (4.7)

D = sin [(2π/λ)(d/2)(sinθ – sinψ)]

This then means

| f(θ)|2 = 2

DC (4.8)

implying that the gain is at a maximum in the direction θ for which

sinθ = sinψ (4.9)

The ability of the LES array to reduce interference depends on the beam pattern. One

simple way of measuring this is the narrowness of the main lobe that is centred on ψ.

The first null relative to the centre of the main lobe is chosen as a measure of lobe

narrowness. If we let ψ = 0, then the first null occurs at

sinθ = Mdλ =

harraylengtwavelength (4.10)

so it can therefore be clearly seen that the ability of an LES array to reduce interference

improves with an increase in array length.

Now that a basic understanding of the adaptive antenna system process has been

grasped, we can introduce gain, interference and delay factors into the system.

It is convenient to make use of vector notation when working with array antennas. We

define the weight vector as

w = [w1 … wM]H (4.11)

where the superscript H represents the Hermitian transpose, which is a transposition

combined with complex conjugation.

The gain experienced by each multipath component differs from one another as they are

attenuated separately and arrive at the base station with different time delays depending

on the path taken to reach the receiver. The gain terms we introduce into the system

possess both amplitude and phase variation.


- 27 -

We introduce a further phase delay due to the propagation time of each multipath

component from the mobile transmitter to the base station. The propagation times vary

due to the different path lengths that each multipath travels.

The introduction of an interference signal nn(t) at the antenna elements means that all

samples of each received multipath component experiences the same interference. We

can then represent the “noisy” data vector as

y = [y1(t) … yM(t)]T (4.12)

where

ym(t) = rm(t) + nnm(t) (4.13)

and the signal received at the mth element is now

rm(t) = A u(t)e-j(2π(m-1/λ)dsinθe –j(2π f c T) (4.14)

where fc is the carrier frequency of the transmitted signal and T is the propagation time

of the particular multipath from transmitter to receiver.

Now the array output can be expressed by

x(t) = wHy(t) (4.15)

4.4 Adaptive Antenna Systems

Adaptive antenna arrays possess an ability to steer beams toward the desired signal, and

nulls toward interfering signals, as the user moves around a sector. This is the main

advantage that adaptive antenna systems have compared to switched beam systems [15],

and why this thesis is based on the investigation of adaptive antenna systems. In an

adaptive array, the phase and amplitude of each element output are controlled by

algorithms that iteratively adjust the weight vector of the signals at the array antennas.

The weight vectors are controlled depending on the signal and interference as well as

the system requirements. These weight vectors are complex in that they provide both

amplitude and phase information. As Figure 4-2 shows, the phase and/or amplitude of

the weights are continuously updated via feedback to minimise the interference and

optimise the signal.


- 28 -

Figure 4-2: An adaptive array processor [7].

4.5 Statistically Optimal Beamforming Techniques

There are many approaches that have been derived to form direct solutions for

statistically optimal beam patterns based on data received by the array [7]. In such a

beamformer, the pattern is optimised to minimise a cost function that is typically

inversely related to the quality of the signal at the array output. That is, the quality of

the signal at the array output is maximised as the cost function is minimised.

The Minimum Mean Square Error (MMSE) approach minimises the difference between

the output of the array and a desired response [16]. The advantage of MMSE is that no

knowledge of the DOA is needed while a disadvantage is that a training sequence is

required for the array to know where the signal is coming from.

The Max SNR technique maximises the signal-to-noise ratio (SNR) giving the

advantage of true maximisation of the SNR rather than improving via other aspects

[17]. Disadvantages of Max SNR are that the system must know the statistics of the

interfering noise and the DOA of the desired signal.


- 29 -

The Linearly Constrained Minimum Variance approach (LCMV) minimises the

variance at the output of the array subject to linear constraints [18]. Being a generalised

constraint technique is an advantage of LCMV. As with Max SNR, knowledge of the

DOA of the desired signal is required.

4.6 Adaptive Algorithms

There are several reasons why it is not desirable to solve the normal equations from the

above techniques directly. The mobile environment is time dependent and therefore

weight vectors must be updated periodically [7]. The weight vector calculated at a

certain cycle is usually different to the one calculated in the previous cycle, even though

by only a small amount.

Adaptive algorithms are used to update the weight vector either in a block mode or

iterative mode. Block processing techniques calculate a new solution using estimates of

statistics obtained from the most recent block of data. In iterative algorithms, the

current weight vector is adjusted by an incremental amount to form a new weight vector

that approximates the optimal solution.

The LMS and Recursive Least Squares (RLS) algorithms require the desired signal to

be supplied using a training sequence or decision direction [7]. If using a training

sequence, a brief known sequence is sent to the receiver. The receiver uses an adaptive

algorithm to estimate the weight vector during the training period, and then holds the

weights constant while information is transmitted. The environment is required to be

stationary from one training period to the next and channel throughput is reduced

because of the use of channel symbols for training.

The simplicity and stability of the LMS algorithm, developed by Widrow and Hoff in

1960, have made it the standard against which other adaptive algorithms are measured

[14]. For these reasons, this thesis concentrates on the use of the LMS algorithm to

simulate an adaptive antenna system.


- 30 -

When using the LMS algorithm, the set of weighting elements are initially set to zero.

That is

w0 = 0 (4.16)

Using (4.15) an error signal is generated by comparing x(t) with the original known

signal u(t). This can be expressed as

e(t) = x(t) – u(t) (4.17)

The error signal is then used to update the weight vector using the following equation

w(t+1) = w(t) + µ y(t)e*(t) (4.18)

where µ is the step-size parameter.


- 31 -

5 Analysis of Results

The standard and normalised variants of the LMS algorithm were simulated using

MATLAB® and investigated to examine the differences in the convergence rate,

squared error performance and accuracy of active tap detection in terms of the number,

weight, and position of active taps in the unknown communication channel. Smart

antenna simulations were then conducted using the Standard LMS algorithm to

investigate the multipath and multi-user effects in the mobile environment. This chapter

presents and analyses the results of the MATLAB® simulations.

5.1 LMS Equalisation

All LMS equalisation simulations are assumed to have a time invariant unknown

communication channel 1/H, with input signal u(k) and interference signal nn(k) having

variances σu2 = 1 and σnn

2 = 0.1. The interference signal nn(k) is assumed to be a white

zero mean Gaussian signal. The number of sample intervals N is set at 5000 and the

fixed value of µ at 0.008. The unknown channel 1/H, was set according to H having a

tap length n = 25. Of these 25 taps, only 5 taps are nonzero being h0 = 1, h5 = 0.6, h10 =

0.36, h15 = 0.216 and h20 = 0.1296.

The two input signal models used for the equalisation simulations in this thesis are:

1. u(k) = w(k)

2. u(k) = w(k) / (1 - 0.8 z-1)

where w(k) is also a white zero mean Gaussian signal of unit variance and z-1 is the

sample delay operator. Note that the first model corresponds to a white input and the

second corresponds to a first order AR or coloured input.


- 32 -

As described in Chapter 4, the aim of the LMS equalisation of an IIR channel is to

estimate F to be equal to H. Therefore, the above-described channel is the desired

impulse response of F in the LMS equalisation simulations. Figure 5-1 shows a

graphical representation of the desired impulse response of F.

Figure 5-1: Desired impulse response of F.

5.1.1 Standard LMS with White Inputs

Figure 5-2, below, shows the received impulse response after the 5000th sample interval

using the Standard LMS algorithm with white inputs. The algorithm is able to estimate

the weights of the active taps quite well. However, the rest of the 20 inactive taps are

shown as being active, with small, but nonzero weights.

Figure 5-2: Received impulse response of F using Standard LMS with white inputs.

The squared error of F, (H – F)2, and the number of active taps detected by F are shown

in Figure 5-3. The first plot shows the asymptotic estimation error of F is roughly 0.004


- 33 -

and ranges from roughly 0.002 to 0.007 after converging. The second plot illustrates

that all 25 taps are being estimated even though 20 of them are inactive.

Figure 5-3: Squared error of F and number of active taps detected using Standard LMS with white

inputs.

5.1.2 Detection Guided LMS with White Inputs

Figure 5-4: Received impulse response of F using Detection Guided LMS with white inputs.

As can be seen from Figure 5-4, when using Detection Guided LMS the received

impulse response of F correctly contains only the 5 active taps. The other inactive taps

are considered inactive and therefore do not contribute to any erroneous estimation.


- 34 -

The first plot in Figure 5-5 shows the asymptotic estimation error is approximately

between 0.00007 and 0.004, with a mean of around 0.0004. The minimum value is

much lower compared to the results from the Standard LMS simulation due to the

removal of inactive taps during the Detection Guided LMS system.

Figure 5-5: Squared error of F and number of active taps detected using Detection Guided LMS with

white inputs.

An important point to note is that the activity threshold used to produce the above

results is 1/9th of the one proposed by (3.13). In the Standard LMS system, the

corresponding sample interval for the squared error value to converge to 0.01 was

approximately 1000, whereas the corresponding sample interval while using the

proposed threshold is 2800. This slow rate of convergence implied that the threshold

was too high as the correct number of active taps was not being estimated until about

the 2500th interval. By reducing the threshold, it takes fewer intervals to correctly

determine the number of active taps, as can be seen in the second plot of Figure 5-5,

leading to a faster rate of convergence. However, too small a threshold may result in

some inactive taps being calculated as being active. There is also a trade-off between

rate of convergence and stability. The chosen threshold was used as the resultant


- 35 -

convergence rate was similar to the Standard LMS algorithm, and therefore, the

estimation error could clearly be seen as being lower.

5.1.3 Standard LMS with Coloured Inputs Just like the simulation results of the Standard LMS Equalisation with white inputs, the

results of the simulation using coloured inputs, seen in Figure 5-6, shows good

estimation of the active tap weights, but also shows the inactive taps as being active.

Figure 5-6: Received impulse response of F using Standard LMS with coloured inputs.

Figure 5-7: Squared error of F and number of active taps detected using Standard LMS with coloured

inputs.


- 36 -

The first plot in Figure 5-7 shows that Standard LMS with coloured inputs converges to

0.01 after approximately 1400 sampling instants. This is slightly slower than with white

inputs, which was approximately 1000 intervals. The squared error mean using

coloured inputs is roughly 0.0019 with a range lying approximately between 0.0014 and

0.015, which is fractionally lower than the corresponding simulation using white inputs.

Once again, the second plot indicates that all 25 taps are active, which is of course

incorrect as only 5 of the 25 taps were set to active.

5.1.4 Detection Guided LMS with Coloured Inputs

Figure 5-8: Received impulse response of F using Detection Guided LMS with coloured inputs.

As expected, using Detection Guided LMS with coloured inputs estimates H more

accurately than the standard form of the LMS algorithm. Like the results using white

inputs, Figure 5-8 shows that the active tap detection scheme correctly assigns the 20

inactive taps a weight of zero and the 5 active taps weights close to the desired values

for coloured inputs as well.

Using the activity threshold proposed in (3.14) produces the results found in Figure 5-9.

The asymptotic error converges to 0.01 at approximately the 1000th sample interval

compared to the 1400th for Standard LMS. The asymptotic performance after

convergence ranges from 0.000013 to 0.0025, with a mean of approximately 0.00009,

which is a great deal lower than the Standard LMS simulation using coloured inputs.


- 37 -

The second plot in Figure 5-9 indicates that the correct number of active taps is detected

after approximately 2600 sample intervals, which contributes to the better asymptotic

error performance.

Figure 5-9: Squared error of F and number of active taps detected using Detection Guided LMS with

coloured inputs.

5.1.5 LMS Summary

A summary of the LMS equalisation results can be found above in Table 5-1. The table

displays the approximate mean values for the squared error of F after convergence, the

number of sample intervals taken to converge to a squared error value of 0.01, and the

number of active taps detected.

It can be clearly seen from the table that both Detection Guided schemes for white and

coloured inputs provide better channel equalisation, as their squared error values are

much lower compared to the Standard versions of the LMS algorithm. Both Detection

Guided schemes are also able to correctly detect five active taps whereas the Standard

schemes incorrectly detected all 25 taps as being active. These results are consistent

with the findings in [9], as expected.


- 38 -

Table 5-1: Summary of results from LMS equalisation simulations.

A very important point to note is that the proposed threshold in (3.13) was too large for

Detection Guided LMS with white inputs, most probably due to the IIR modelled

channel. This had the effect of a very slow rate of convergence. The results shown in

the table use a threshold that is 1/9th of the proposed one, which was chosen to provide a

similar rate of convergence, yet still be able to show the better performance of the

equaliser.

With the appropriate changes to the algorithms, we can now confidently assume that

other improvements of the LMS algorithm, such as those presented in [11], will also

hold when modelling the communication channel as an IIR filter.

5.2 Normalised LMS Equalisation

All NLMS equalisation simulations assume the same time invariant unknown

communication channel as the LMS simulations and therefore the desired impulse

response of F, seen in Figure 5-1. We also assume the same input signal u(k) for both

white and coloured inputs, and the interference signal nn(k). The number of sample

intervals N is still set at 5000. We set the fixed µ0 to 0.1 and are then able to calculate

µnorm using (3.17). This should see a slower rate of convergence when not using active

tap detection due to all 25 taps being assumed active, but also a higher stability of the

system.

Standard LMS with

white inputs

Detection Guided LMS

with white inputs

Standard LMS with coloured

inputs

Detection Guided LMS with coloured

inputs Mean squared

error

0.004

0.0004

0.0019

0.00009 Sample intervals

to 0.01

1000

1060

1400

1000 Active Taps

Detected

25 5

25

5


- 39 -

5.2.1 Normalised LMS with White Inputs

Figure 5-10: Received impulse response of F using NLMS with white inputs.

Just as the Standard LMS system for both white and coloured inputs showed, the

Normalised LMS system using white inputs does not distinguish between active and

inactive taps. Once again, as indicated in Figure 5-10, the estimation of the active taps

is quite good, but the other 20 inactive taps are shown to have a nonzero weight, and are

therefore estimated as being active.

Figure 5-11: Squared error of F and number of active taps detected using NLMS with white inputs.


- 40 -

The squared error, shown in the first plot of Figure 5-11, lies in the range of

approximately 0.0012 to 0.0027, with a mean of about 0.0019. As mentioned in

Chapter 4, NLMS is usually preferred due to its increased stability, which is confirmed

with the smaller range. However, the trade-off between stability and convergence rate

is clearly evident in this plot. As expected, the second plot in Figure 5-11 shows all taps

estimated as being active, even though only 5 of the taps are actually nonzero.

5.2.2 Detection Guided NLMS with White Inputs

As expected, and as shown in Figure 5-12, Detection Guided NLMS with white inputs

produces an impulse response of F that correctly estimates the positions and magnitudes

of the taps.

Figure 5-12: Received impulse response of F using Detection Guided NLMS with white inputs.

Using the activity threshold from (3.13), the simulation produced the squared error plot

seen in Figure 5-13. As can be seen, the rate of convergence over the first 2000

sampling instants is quite similar to the Standard LMS simulation results in Figure 5-11.

However, after converging and stabilising, the squared error mean is roughly 0.0004,

with values ranging from approximately 0.00005 to 0.0045, which is extremely low

compared to Standard LMS. This is due to the system correctly containing only 5

active taps, as can be seen in the second plot of Figure 5-13.


- 41 -

Figure 5-13: Squared error of F and number of active taps detected Detection Guided NLMS with white

inputs.

5.2.3 NLMS with Coloured Inputs

Figure 5-14: Received impulse response of F using NLMS with coloured inputs.

When using coloured inputs combined with the NLMS equaliser, the inactive taps are

still calculated as having small, nonzero weights and are therefore assumed as being

active, as shown above in Figure 5-14.


- 42 -

Figure 5-15: Squared error of F and number of active taps detected using NLMS with coloured inputs.

In comparison with the NLMS equaliser using white inputs, the results using coloured

inputs show a much slower rate of convergence. The approximate sample intervals for

each system to converge to 0.01 are 2000 and 3500 respectively. As can be seen in the

first plot of Figure 5-15, the system is extremely stable compared to the LMS version in

Figure 5-6, and remains about the 0.009 mark with a range from 0.0078 to 0.01.

5.2.4 Detection Guided NLMS with Coloured Inputs

The final equalisation simulation once again shows the benefit of using active tap

detection as the 20 inactive taps are given weight magnitudes of zero and the 5 active

taps are being estimated accurately. These results are consistent with the previous

findings and are illustrated in Figure 5-16.


- 43 -

Figure 5-16: Received impulse response of F using Detection Guided NLMS with coloured inputs.

Figure 5-17 below shows an erratic plot of the squared error of F over the 5000 sample

intervals. The initial stages, up to about 1000 intervals, can be contributed to the system

attempting to estimate the correct number of active taps, seen in the second plot below.

However, once the 5 active taps are correctly estimated there is a significant increase in

asymptotic performance. Also, because only 5 taps are deemed active, the step-size,

µnorm, is much larger than the NLMS system, and therefore the stability decreases. The

range of the squared error after convergence varies from approximately 0.00003 to

0.005, with a mean of about 0.00019.

Figure 5-17: Squared error of F and number of active taps detected using Detection Guided NLMS with

coloured inputs.


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5.2.5 NLMS Summary

Table 5-2 shows a summary of the NLMS equalisation results, displaying the

approximate mean values for the squared error of F after convergence, the number of

sample intervals taken to converge to a squared error value of 0.01, and the number of

active taps detected.

Table 5-2: Summary of results from NLMS equalisation simulations.

NLMS with white inputs

Detection Guided NMS

with white inputs

NLMS with coloured

inputs

Detection Guided NLMS with coloured

inputs Mean squared

error

0.0019

0.0004

0.0009

0.00019 Sample intervals

to 0.01

2090

2000

3600

1030 Active Taps

Detected

25 5

25

5

The table shows that as expected, the equaliser’s ability to estimate H improves when

using the Detection Guided LMS approach for both white and coloured inputs. This is

shown by the reduction of mean squared error of F and is due to the equaliser being able

to correctly identify the five active taps and the other twenty as being inactive.

As discussed earlier in the thesis and proven in the results, NLMS provides better

stability compared to LMS. However, this increased stability comes at a cost of

convergence rate, shown in the results above. The slower convergence rate is not

shown in the results of Detection Guided NLMS for white inputs as it uses the proposed

threshold, unlike its LMS counterpart.

The results do not show much improvement in convergence rate between NLMS and

Detection Guided NLMS for white inputs, but do show a dramatic increase in the rate of

convergence for coloured inputs. This is inconsistent with the findings of [10], in which

it was found that detection guided NLMS provided significantly better convergence

speed over standard NLMS for both coloured and white input signals.


- 45 -

5.3 Smart Antenna Simulations For all adaptive array smart antenna simulations, the 5000 input signals of the training

sequence have signed values of 1 or –1 to simulate a transmitter sending binary values.

Although there are 5000 sampling instants, the results only show up to 150 intervals due

to the extremely high rate of convergence of the system. The step-size parameter µ for

the LMS algorithm is set to 0.008. To keep simulations as realistic as possible, for

those simulations with more than one multipath, each multipath experiences a different

gain, which contains both amplitude and phase components. It was found that the

amplitude of the gain had the most effect on the system, with the phase having little to

no effect.

The carrier frequency fc of transmitted training sequences is set to 400MHz, which

means the value of the wavelength λ is set to 0.75m. To satisfy an element spacing d of

λ/2 then means that d is set to 0.375m. For simulations with only one transmitted

signal, the propagation delay from transmission to reaching the first antenna element is

set to 100µs, and for those with a second transmitted signal, the second propagation

delay is set at 150µs.

Even though only four simulation results are being presented, there were many other

simulations that were used to progress to the final simulations. For example, the gain

and noise terms were initially left out of the system to ensure that the simulations were

achieving the correct results in the ideal environment. Also, in order to reach a

simulation of signals with three multipaths, a simulation with two multipaths was first

examined. Such simulations have not been included to avoid repetition.

5.3.1 One White Signal with One DOA

To ensure that the system worked correctly, the first simulation investigated was the

reception of one signal with the one path that arrives at the base station at angle of 60˚.

A gain with amplitude of 0.5 was introduced to the input signal as it was propagated to

the antenna. Figure 5-18 illustrates that the received signal error converges at

approximately 54 sample intervals and reaches 0.01 after 43 intervals. The mean


- 46 -

received signal error after convergence lies approximately at 0.0006. Figure 5-19

shows that the beam pattern of the system correctly steers the main beam in the

direction of 60 with beam strength of two. This is due to the signal experiencing a gain

of amplitude 0.5, which reduces the power of the signal by half. To counter this, the

beam adjusts its gain to the inverse of the signal power in order to receive a signal

similar to the original signal.

Figure 5-18: Smart antenna simulation received signal error for 1 white signal with 1 DOA.

Figure 5-19: Smart antenna simulation beam pattern for 1 white signal with 1 DOA.

5.3.2 One White Signal with Three DOAs

The next simulation is again for the transmission of one training sequence, but this time

with three multipaths that have directions of arrival of 60˚, 30˚ and -20˚. Each

multipath arrives at the antenna system with a difference of one sampling period 1/fc, so


- 47 -

we can denote the signals arriving at time instant t as u(t), u(t-1), and u(t-2). The

corresponding gains introduced to each of the multipath components have amplitudes of

0.5, 0.66, and 1.0. In this instance, three different weight vectors are used in the

adaptive antenna system, one for each multipath. This means that three LMS equations

are running simultaneously to each produce a main lobe in the direction of a multipath.

Figure 5-20: Smart antenna simulation received signal error for 1 white signal with 3 DOAs.

The received signal error plot shown in Figure 5-20 illustrates the effects of having

different gain terms. The smaller the gain amplitude, the longer it takes for the antenna

array to adapt and correctly estimate the transmitted signal. The number of intervals for

the received signal error of each multipath to converge to 0.01 is 38, 22 and 9

respectively. The mean values are approximately 0.005, 0.0015 and 0.00034. The

mean of the 1st multipath is approximately 10 times more than the mean when only one

path exists.

Figure 5-21: Smart antenna simulation beam pattern for 1 white signal with 3 DOAs.


- 48 -

Figure 5-21 shows the antenna systems beam pattern. It also demonstrates its ability to

steer separate beams in multiple directions and nulls in the directions of interferers.

Once again, the gain of each main beam is the inverse of the gain introduced to each

corresponding multipath component. To prove that the weights are able to place nulls

in the interference directions, the gain of DOA1 in the directions of DOA2 and DOA3

are 0.0133 and 0.0151 respectively. This means that when receiving the 1st multipath

signal, the other two multipath signals are also received but with multiples of 0.0133

and 0.0151, which we can deem as negligible. This is also the case for the reception of

the other two multipaths.

5.3.3 Two White Signals with One DOA Each

Figure 5-22: Smart antenna simulation received signal error for 2 white signals with 1 DOA each.

The simulation of transmitting two different signals with one DOA each is in effect the

same as sending one signal with two multipaths separated by at least one sample period.

This is because in both situations the two signals are uncorrelated with each other. The

1st signal is exposed to a gain with amplitude 0.5 and the 2nd signal 1.0. Figure 5-22

once again shows that it takes longer for the system to converge when the gain term is

smaller. In this case two LMS equations are running simultaneously to determine the

weight vectors to produce the two beams in each desired direction.

The beam pattern in Figure 5-23 shows the two beams from each set of weights is able

to correctly identify the DOAs of each signal as being 60˚ and –25˚. The gain of the 1st


- 49 -

set of weights in the direction of the 2nd signal is 0.0537 and the gain of the 2nd set of

weights in the direction of the 1st signal is 0.0105, demonstrating the smart antenna’s

ability to distinguish between desired signals and interfering ones.

Figure 5-23: Smart antenna simulation beam pattern for 2 white signals with 1 DOA each.

5.3.4 Two White Signals with Three DOAs Each The final smart antenna simulation is the most complex and provided the most

unexpected results. In this simulation we transmit two training sequences, each with

three multipath components. However, the 2nd and 3rd multipath components of each

signal are both set to arrive at the antenna array one sample period behind the 1st

multipath. Essentially, this means that the 2nd and 3rd multipaths are arriving at the base

station at the same time but from different directions.

As can be seen from Figure 5-24, although there were three multipaths for each signal

in the system, only two sets of received signal errors are being displayed. That is, only

four unique weight vectors exist. This is because the weight vectors for the 2nd and 3rd

multipaths are exactly the same due to these signals arriving at the same time. This

means that for multipath components of the same signal that arrive at the same time,

only one weight vector is needed. Also, the mean received signal error of the 1st

multipath of the 1st signal is roughly the same as for one signal with three multipaths,

lying at approximately 0.003.


- 50 -

Figure 5-24: Smart antenna simulation received signal error for 2 white signals with 3 DOAs each.

Figure 5-25: Smart antenna simulation beam pattern for 2 white signals with 3 DOAs each.

After this finding, it was expected that the main beam would either be directed in the

direction of the closest multipath or the one with the greatest gain. However, the beam

pattern shown in Figure 5-25 displays the four different beam patterns but the patterns


- 51 -

for the 3rd multipath of both signals have two main lobes in the correct directions of the

2nd and 3rd multipaths. The gains of these beams are half what they would normally be

and swapped between the multipath components. This was a major result, as neither Dr.

Homer nor I knew of the ability for weight vectors to steer multiple beams in multiple

directions. We were always under the impression that each set of weight vectors could

only steer one beam in one specified direction.

5.3.5 Smart Antenna Summary

The smart antenna simulations confirmed that smart antenna systems have an ability to

distinguish between signals of interest and interferers by directing beams in the

directions of the desired signals and nulls in the directions of interferers. These

interferers can either be other transmitted signals from other mobile or multipath

components of the same signal. The major finding of the smart antenna simulations,

and perhaps the major finding of the whole thesis, is that adaptive array smart antenna

systems are able to deploy multiple main beams in multiple directions if multipaths of

the same desired signal arrive at the base station at the same time.

Given that we know that the antenna system can direct beams in the direction of a

desired signal and nulls in the direction of interferers, let us consider three multipaths of

one signal as in 5.3.2. The signal output from the 1st antenna array weight vector is

essentially u(t). Similarly, the outputs from the 2nd and 3rd antenna array weight vectors

are essentially u(t-1) and u(t-2). We can then apply time delay filters of 2Ts and Ts,

where Ts is the sample period, to the signals u(t) and u(t-1) respectively. These signals

can then be summed together constructively to increase the received signal power,

therefore increasing SNR and providing better performance, as shown below.

Figure 5-26: Block diagram of time delay summing system.

u(t)

u(t-2)

u(t-1)

2Ts

Ts + 3u(t)


- 52 -

6 Conclusion and Future Work

6.1 Conclusion

The first part of this thesis presents an alternative model that can be used for an adaptive

equalisation system. Instead of modelling the unknown communication channel as an

FIR filter and therefore the adaptive equaliser as an IIR filter, the channel is modelled as

an IIR filter and the equaliser as an FIR filter.

The aim was not to investigate the model in depth, rather to examine whether the

system is successful when equalising using the Standard and Detection Guided LMS

and NLMS algorithms. The results found some inconsistencies with the findings of

previous studies. For example, the proposed activity threshold for Detection Guided

LMS using white inputs was far too large for the proposed system. To show the

increased convergence rate that is expected, a threshold smaller by 1/9th of the proposed

threshold would be needed. This value was chosen to show the better asymptotic

performance for roughly the same convergence rate as the Standard LMS.

Using the proposed threshold for Detection Guided NLMS with white inputs showed

only a slight increase in convergence rate. This is in contrast to the significant increase

found in [10]. On the other hand the results using coloured inputs were consistent with

the findings of [10]. That is, the Detection Guided NLMS showed a dramatic increase

in convergence rate over the Standard NLMS.

Overall, the results show that the proposed adaptive equalisation system needs to be

investigated in more detail. The system does show promise at being a better alternative

in adaptive equalisation. At a glance, the performance of the system seems to be better


- 53 -

than the system proposed in [9] but there are a number of factors that need to be

examined. These include the difference in the number of active taps and the total

number of taps, along with the difference in step-size parameter µ.

The second part of the thesis examined adaptive array smart antenna systems and the

effects that multipath components had on their performance. The results confirmed the

great interest in smart antenna systems as they proved that smart antenna systems could

steer beams for reception in the direction of desired incoming signals. Furthermore,

they can also place nulls in the direction of interfering signals.

It was also found that signals from multiple users might as well be multipath

components from the one signal arriving at different times. This is because both are

uncorrelated with the desired transmitted signal.

Both multipaths arriving at the smart antenna at different times and at the same time

were investigated. When investigating multipaths arriving at the same time, it was

found that only one set of weights is needed no matter how many multipaths are

arriving at the same time. The major finding of the thesis was that a set of weights is

not only able to steer a beam in a desired direction, but also able to steer multiple beams

in multiple desired directions.

In theory, the inclusion of appropriate time delay filters at the output of the smart

antenna system would facilitate the constructive summation of the output signal,

therefore resulting in an increased signal power, meaning an increase in SNR and

therefore performance is achieved.

Once again, these results confirm why smart antennas have gained such popularity and

increased attention and that they will be the future of mobile communications.


- 54 -

6.2 Future Work

The investigation of adaptive equalisation when modelling the communication channel

as an IIR filter is far from complete. We can assume with some confidence that the

proposed improvements to the LMS algorithm performed in previous studies will still

hold for this system. However, there may need to be some subtle changes such as

adjusting the activity threshold in this thesis for Detection Guided LMS with white

inputs.

It would be suggested that a thorough investigation be conducted into the performance

of the system proposed in this thesis compared to the one proposed in [9]. This could

be achieved by using common filter tap lengths, number and weights of active taps and

step-size. Only then would it be possible to determine which system is able to equalise

the channel to a better degree.

The analysis of the adaptive array smart antenna system can be expanded in many ways.

The theory of the time delay filters at the output of the smart antenna system could be

investigated to prove that this can in fact be done. Only the reception of signals has

been investigated, which logically leads to an investigation of transmission and then

combining the two together.

Perhaps other future work may look at applying smart antennas into a CDMA system.

Particularly, incorporating Rake receivers into the system may enhance the system by

receiving only the strongest multipaths instead of all the multipaths arriving at the

antenna.

6.3 Final Thoughts

This thesis has been quite successful in terms of achieving the objectives that were

agreed upon at the beginning of the year. However, more thorough results and more in

depth investigations may have been able to be performed if more detailed research had

been carried out at the beginning of the year. This was not due to a lack of effort but


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more an improper understanding of the subject matter and the processes involved in

smart antenna systems. This also meant that the plan and schedule that was produced

for the Progress Report was not adhered to as the tasks were inappropriate.

The experience in research and development has been a thoroughly enjoyable journey

and the interest and knowledge gained on smart antenna systems and mobile

communications has been tremendous and extremely valuable.


- 56 -

References

[1] Optus Administration Pty Ltd Website, No date, [Online]. Available at

http://www.optus.com.au [Accessed 30/09/02].

[2] Wong, K.K., Murch, R.D. & Letaief, K.B. 2001. Optimizing Time and Space

MIMO Antenna System for Frequency Selective Fading Channels. IEEE

Journal on Selected Areas in Communications, July, pp.1395-1406.

[3] Rappaport, T.S. 1996 Wireless Communications: Principles & Practice,

Prentice Hall Communications Engineering and Emerging Technology Series.

[4] Oeting, J. 1983. Cellular Mobile Radio – An Emerging Technology. IEEE

Communications Magazine, November, pp. 10-15.

[5] Rosol, G. 1995. Base Station Antennas: Part 1, Part 2, Part 3. Microwaves &

RF, August, pp. 117-123, September, pp. 127-131, October, pp. 116-124.

[6] Brickhouse, R.A., and Rappaport, T.S. 1997. A Simulation Study of Urban In-

Building Frequency Reuse. IEEE Personal Communications Magazine,

February, pp. 19-23.

[7] Liberti, J. C. & Rappaport, T.S. 1999. Smart Antennas for Wireless

Communications: IS-95 and Third Generation CDMA Applications, Prentice

Hall Communications Engineering and Emerging Technology Series, NJ.

[8] Haykin, S. 1996. Adaptive Filter Theory, Third Edition, Prentice Hall Inc., pp.

365 – 405.

[9] Beng, C. L. 2001. Detection Guided LMS Based Channel Equalization.

Undergraduate Thesis, School of Information Technology and Electrical

Engineering, University of Queensland, Brisbane.


- 57 -

[10] Homer, J. 2000. Detection Guided NLMS Estimation of Sparsely Paramtrized

Channels. IEEE Transactions on Circuits and Systems II: Analog and Digital

Signal Processing, Vol. 47, No. 12, pp. 1437-1442.

[11] How, L. 2001. Signed LMS Adaptive Filtering with Detection.

Undergraduate Thesis, School of Information Technology and Electrical

Engineering, University of Queensland, Brisbane.

[12] Homer, J., Mareels, I., Bitmead, R., Wahlberg, B., & Gustafsson, F. 1998.

LMS Estimation via Structural Detection. IEEE Transactions on Signal

Processing, Vol. 46, No. 10, pp. 2651-2663.

[13] Homer, J. 1994. Adaptive Echo Cancellation in Telecommunications. Ph.D.

Dissertation, The Australian National University, Canberra.

[14] Pattan, B. 2000. Robust Modulation methods and Smart Antennas in Wireless

Communications, Prentice Hall PTR, NJ.

[15] Zooghby, A. 2001. Potentials of Smart Antennas in CDMA Systems and

Uplink Improvements. IEEE Antennas and Propagation Magazine, pp. 172-

177.

[16] Widrow, B., Mantey, P. E., Griffiths, L. J., and Goode, B. B. 1967. Adaptive

Antenna Systems. Proc. of the IEEE, December.

[17] Monzingo, R. & Miller, T. 1980. Introduction to Adaptive Arrays, Wiley and

Sons, NY.

[18] Frost, O. L., III. 1972. An Algorithm for Linearly Constrained Adaptive Array

Processing. Proc. of the IEEE, August.


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Appendix A LMS Code Listings

A.1 Standard LMS with White Inputs

% Standard LMS Adaptive Equalisation with White Inputs % File name: LMS_White.m u = randn (1, 5024); % Input signals N = length(u); % Number of input signals H = [1, zeros(1, 4), 0.6, zeros(1, 4), 0.36, zeros(1, 4), 0.216, zeros(1, 4), 0.1296, zeros(1, 4)];

% Channel impulse response n = length(H); % Number of taps in the channel nn = randn (1, N) * 0.1; % Noise at receiver mu = 0.008; % Step-size F = zeros(1, n); % Weight vectors initialised to zero He = zeros(1, N-n+1); % Error between H and F r = zeros (1, N); % Output from channel y = zeros (1, N); % Received signal x = zeros (1, N); % Estimate of input signals e = zeros (1, N); % Error between u and x % for each sample interval % calculate received signal y(k), equalised signal x(k), error signal e(k) % update weights according to e(k) and calculate squared error He for k = n: N

R = [r(k-1), r(k-2), r(k-3), r(k-4), r(k-5), r(k-6), r(k-7), r(k-8), r(k-9), r(k-10), r(k-11), r(k-12), r(k-13), r(k-14), r(k-15), r(k-16), r(k-17), r(k-18), r(k-19), r(k-20), r(k-21), r(k-22), r(k-23), r(k-24)];

r (k) = 1 / H(1) * (u(k) - H(2: n) * R'); y (k) = r(k) + nn(k);

Y = [y(k), y(k-1), y(k-2), y(k-3), y(k-4), y(k-5), y(k-6), y(k-7), y(k-8), y(k-9), y(k-10), y(k-11), y(k-12), y(k-13), y(k-14), y(k-15), y(k-16), y(k-17), y(k-18), y(k-19), y(k-20), y(k-21), y(k-22), y(k-23), y(k-24)];

x (k) = F * Y'; e(k) = u(k) - x(k); F = F + mu * Y * e(k); He(k-n+1) = (H - F) * (H - F)'; end; figure(1); clf; subplot(2, 1, 1); impz(H); grid on; zoom on;


- 59 -

title ('Desired Impulse Response of F'); xlabel ('Tap Number'); ylabel ('Tap Weight'); subplot(2, 1, 2); impz(F); grid on; zoom on; title ('Received Impulse Response of F'); xlabel ('Tap Number'); ylabel ('Tap Weight'); figure(2); clf; subplot (2, 1, 1); semilogy(He); YLIM([10^-5, 10^1]); zoom on; grid on; title ('Squared Error of F'); ylabel ('[H - F]^2 Error'); xlabel ('Sample Interval'); xcons = [0: 10: N-n]; ycons = n; subplot(2, 1, 2); plot(xcons, ycons, 'b-'); title ('Number of Active Taps Detected'); ylabel ('Active Taps Detected'); xlabel ('Sample Interval'); grid on; zoom on;


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A.2 Detection Guided LMS with White Inputs

% LMS Active Tap Detection Adaptive Filter Equalisation with White Inputs % Filename: LMS_Active_Tap_White.m u = randn (1, 5024); % Input signals N = length(u); % Number of input signals H = [1, zeros(1, 4), 0.6, zeros(1, 4), 0.36, zeros(1, 4), 0.216, zeros(1, 4), 0.1296, zeros(1, 4)];

% Channel impulse response n = length(H); % Number of taps in the channel nn = randn (1, N) * 0.1; % Noise at receiver mu = 0.008; % Step-size F = zeros(1, n); % Weight vectors initialised to zero He = zeros(1, N-n+1); % Error between H and F r = zeros (1, N); % Output from channel y = zeros (1, N); % Received signal x = zeros (1, N); % Estimate of input signals a = zeros(1, n); % Numerator of tap activity measure b = zeros(1, n); % Denominator of tap activity measure c = zeros(1, n); % Tap activity measure g = zeros(1, n); % Tap activity d = 0; % Last activity threshold plus u(k)^2 T = 0; % Activity threshold sg = zeros(1, N-n+1); % Number of active taps e = zeros (1, N); % Error between u and x % for each sample interval % calculate received signal y(k), equalised signal x(k), % activity threshold T, and calculate squared error He for k = n: N,




x (k) = F * Y'; d = d + u(k)^2; T = (d * log(k-n+1)) / (k-n+1);

% for each tap % calculate tap activity measure c(i) for i = 1: n, if (i < (k-n+1)) & (k <= N) a(i) = a(i) + (u(k) * y(k-i+1) - x(k) * y(k-i+1) + F(i) * y(k-i+1)^2); b(i) = b(i) + (y(k-i+1)^2); c(i) = a(i)^2 / b(i);

% if tap activity is greater than threshold


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% set tap to active, otherwise inactive if c(i) > T / 1, g(i) = 1; else g(i) = 0; end;

% calculate number of active taps, error signal e(k) % and update weight of tap sg(k-n+1) = sum(g); e(k) = u(k) - x(k); F(i) = g(i) * F(i) + g(i) * mu * y(k-i+1) * e(k); end; end; He(k-n+1) = (H - F) * (H - F)'; end; figure(1); clf; subplot(2, 1, 1); impz(H); grid on; zoom on; title ('Desired Impulse Response of F'); xlabel ('Tap Number'); ylabel ('Tap Weight'); subplot(2, 1, 2); impz(F); grid on; zoom on; title ('Received Impulse Response of F'); xlabel ('Tap Number'); ylabel ('Tap Weight'); figure(2); clf; subplot(2, 1, 1); semilogy(He); YLIM([10^-5, 10^1]); grid on; zoom on; title ('Squared Error of F'); xlabel ('Sample Interval'); ylabel ('[H -F]^2 Error'); subplot(2, 1, 2); plot(sg); grid on; zoom on; title ('Number of Active Taps Detected'); xlabel ('Sample Interval'); ylabel ('Active Taps Detected');


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A.3 Standard LMS with Coloured Inputs

% Standard LMS Adaptive Filter Equalisation with Coloured Inputs % File name: LMS_Col.m u = randn (1, 5024); C = 0.8; % Correlation of input signals W = u; u = filter(1, [1,-C], W); % Coloured input signals N = length(u); % Number of input signals H = [1, zeros(1, 4), 0.6, zeros(1, 4), 0.36, zeros(1, 4), 0.216, zeros(1, 4), 0.1296, zeros(1, 4)];





x (k) = F * Y'; e(k) = u(k) - x(k); F = F + mu * Y * e(k); He(k-n+1) = (H - F) * (H - F)'; end; figure(1); clf; subplot(2, 1, 1); impz(H); grid on; zoom on; title ('Desired Impulse Response of F'); xlabel ('Tap Number'); ylabel ('Tap Weight'); subplot(2, 1, 2); impz(F); grid on; zoom on; title ('Received Impulse Response of F');


- 63 -

xlabel ('Tap Number'); ylabel ('Tap Weight'); figure(2); clf; subplot (2, 1, 1); semilogy(He); YLIM([10^-5, 10^1]); grid on; zoom on; title ('Squared Error of F'); ylabel ('[H - F]^2 Error'); xlabel ('Sample Interval'); xcons = [0: 10: N-n]; ycons = n; subplot(2, 1, 2), plot(xcons, ycons, 'b-'); title ('Number of Active Taps Detected in F'); ylabel ('Active Taps Detected'); xlabel ('Sample Interval'); grid on; zoom on;


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A.4 Detection Guided LMS with Coloured Inputs % LMS Active Tap Detection Adaptive Filter Equalisation with Coloured Inputs % Filename: LMS_Active_Tap_Col.m u = randn (1, 5024); C = 0.8; % Correlation of input signals W = u; u = filter(1, [1,-C], W); % Coloured input signals N = length(u); % Number of input signals H = [1, zeros(1, 4), 0.6, zeros(1, 4), 0.36, zeros(1, 4), 0.216, zeros(1, 4), 0.1296, zeros(1, 4)];






% for each tap % calculate tap activity measure c(i) for i = 1: n, if (i < (k-n+1)) & (k <= N) a(i) = a(i) + (u(k) * y(k-i+1) - x(k) * y(k-i+1) + F(i) * y(k-i+1)^2); b(i) = b(i) + (y(k-i+1)^2);


- 65 -

c(i) = a(i)^2 / b(i);

% if tap activity is greater than threshold % set tap to active, otherwise inactive if c(i) > T / 1, g(i) = 1; else g(i) = 0; end;

% calculate number of active taps, error signal e(k) % and update weight of tap sg(k-n+1) = sum(g); e(k) = u(k) - x(k); F(i) = g(i) * F(i) + g(i) * mu * y(k-i+1) * e(k); end;

end; He(k-n+1) = (H - F) * (H - F)';

end; figure(1); clf; subplot(2, 1, 1); impz(H); grid on; zoom on; title ('Desired Impulse Response of F'); xlabel ('Tap Number'); ylabel ('Tap Weight'); subplot(2, 1, 2); impz(F); grid on; zoom on; title ('Received Impulse Response of F'); xlabel ('Tap Number'); ylabel ('Tap Weight'); figure(2); clf; subplot(2, 1, 1); semilogy(He); YLIM([10^-5, 10^1]); grid on; zoom on; title ('Squared Error of F'); xlabel ('Sample Interval'); ylabel ('[H -F]^2 Error'); subplot(2, 1, 2); plot(sg); grid on; zoom on; title ('Number of Active Taps Detected'); xlabel ('Sample Interval'); ylabel ('Active Taps Detected');


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Appendix B NLMS Code Listings

B.1 Normalised LMS with White Inputs

% Normalised LMS Adaptive Filter with White Inputs % File name: NLMS_White.m u = randn (1, 5024); % Input signals N = length(u); % Number of input signals H = [1, zeros(1, 4), 0.6, zeros(1, 4), 0.36, zeros(1, 4), 0.216, zeros(1, 4), 0.1296, zeros(1, 4)];





x (k) = F * Y'; e(k) = u(k) - x(k); F = F + (mu / 25) * Y * (e(k)); He(k-n+1) = (H - F) * (H - F)'; end; figure(1); clf; subplot(2, 1, 1); impz(H); grid on; zoom on;


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title ('Desired Impulse Response of F'); xlabel ('Tap Number'); ylabel ('Tap Weight'); subplot(2, 1, 2); impz(F); grid on; zoom on; title ('Received Impulse Response of F'); xlabel ('Tap Number'); ylabel ('Tap Weight'); figure(2); clf; subplot (2, 1, 1); semilogy(He); YLIM([10^-5, 10^1]); grid on; zoom on; title ('Squared Error of F'); ylabel ('[H - F]^2 Error'); xlabel ('Sample Interval'); xcons = [0: 10: N-n]; ycons = n; subplot(2, 1, 2); plot(xcons, ycons, 'b-'); title ('Number of Active Taps Detected'); ylabel ('Active Taps Detected'); xlabel ('Sample Interval'); grid on; zoom on;


- 68 -

B.2 Detection Guided NLMS with White Inputs

% NLMS Active Tap Detection Adaptive Filter Equalisation with White Inputs % Filename: NLMS_Active_Tap_White.m u = randn (1, 5024); % Input signals N = length(u); % Number of input signals H = [1, zeros(1, 4), 0.6, zeros(1, 4), 0.36, zeros(1, 4), 0.216, zeros(1, 4), 0.1296, zeros(1, 4)];






% for each tap % calculate tap activity measure c(i) for i = 1: n, if (i < (k-n+1)) & (k <= N) a(i) = a(i) + (u(k) * y(k-i+1) - x(k) * y(k-i+1) + F(i) * y(k-i+1)^2); b(i) = b(i) + (y(k-i+1)^2); c(i) = a(i)^2 / b(i);

% if tap activity is greater than threshold


- 69 -

% set tap to active, otherwise inactive if c(i) > T / 1, g(i) = 1; else g(i) = 0; end;

% calculate number of active taps, error signal e(k) % and update weight of tap sg(k-n+1) = sum(g); e(k) = u(k) - x(k); F(i) = g(i) * F(i) + g(i) * (mu / (sg(k-n+1) + 1)) * y(k-i+1) * e(k); end; end; He(k-n+1) = (H - F) * (H - F)'; end; figure(1); clf; subplot(2, 1, 1); impz(H); grid on; zoom on; title ('Desired Impulse Response of F'); xlabel ('Tap Number'); ylabel ('Tap Weight'); subplot(2, 1, 2); impz(F); grid on; zoom on; title ('Received Impulse Response of F'); xlabel ('Tap Number'); ylabel ('Tap Weight'); figure(2); clf; subplot(2, 1, 1); semilogy(He); YLIM([10^-5, 10^1]); grid on; zoom on; title ('Squared Error of F'); xlabel ('Sample Interval'); ylabel ('[H -F] ^2 Error'); subplot(2, 1, 2); plot(sg); grid on; zoom on; title ('Number of Active Taps Detected'); xlabel ('Sample Interval'); ylabel ('Active Taps Detected');


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B.3 Normalised LMS with Coloured Inputs

% Normalised LMS Adaptive Filter with Coloured Inputs % File name: NLMS_Col.m u = randn (1, 5024); C = 0.8; % Correlation of input signals W = u; u = filter(1, [1,-C], W); % Coloured input signals N = length(u); % Number of input signals H = [1, zeros(1, 4), 0.6, zeros(1, 4), 0.36, zeros(1, 4), 0.216, zeros(1, 4), 0.1296, zeros(1, 4)]; n = length(H); % Number of taps in the channel nn = randn (1, N) * 0.1; % Noise at receiver mu = 0.1; % Step-size F = zeros(1, n); % Weight vectors initialised to zero He = zeros(1, N-n+1); % Error between H and F r = zeros (1, N); % Output from channel y = zeros (1, N); % Received signal x = zeros (1, N); % Estimate of input signals e = zeros (1, N); % Error between u and x % for each sample interval % calculate received signal y(k), equalised signal x(k), error signal e(k) % update weights according to e(k) and calculate squared error He for k = n: N




x (k) = F * Y'; e(k) = u(k) - x(k); F = F + (mu / 25) * Y * (e(k)); He(k-n+1) = (H - F) * (H - F)'; end; figure(1); clf; subplot(2, 1, 1); impz(H); grid on; zoom on; title ('Desired Impulse Response of F'); xlabel ('Tap Number'); ylabel ('Tap Weight'); subplot(2, 1, 2); impz(F); grid on; zoom on; title ('Received Impulse Response of F'); xlabel ('Tap Number');


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ylabel ('Tap Weight'); figure(2); clf; subplot (2, 1, 1); semilogy(He); YLIM([10^-5, 10^1]); grid on; zoom on; title ('Squared Error of F'); ylabel ('[H - F]^2 Error'); xlabel ('Sample Interval'); xcons = [0:10: N-n]; ycons = n; subplot(2, 1, 2); plot(xcons, ycons, 'b-'); title ('Number of Active Taps Detected'); ylabel ('Active Taps Detected'); xlabel ('Sample Interval'); grid on; zoom on;


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B.4 Detection Guided NLMS with Coloured Inputs

% NLMS Active Tap Detection Adaptive Filter Equalisation with Coloured Inputs % Filename: NLMS_Active_Tap_Col.m u = randn (1, 5024); C = 0.8; % Correlation of input signals W = u; u = filter(1, [1,-C], W); % Coloured input signals N = length(u); % Number of input signals H = [1, zeros(1, 4), 0.6, zeros(1, 4), 0.36, zeros(1, 4), 0.216, zeros(1, 4), 0.1296, zeros(1, 4)];






% for each tap % calculate tap activity measure c(i) for i = 1: n, if (i < (k-n+1)) & (k <= N) a(i) = a(i) + (u(k) * y(k-i+1) - x(k) * y(k-i+1) + F(i) * y(k-i+1)^2); b(i) = b(i) + (y(k-i+1)^2);


- 73 -

c(i) = a(i)^2 / b(i);

% if tap activity is greater than threshold % set tap to active, otherwise inactive if c(i) > T / 1, g(i) = 1; else g(i) = 0; end;

% calculate number of active taps, error signal e(k) % and update weight of tap sg(k-n+1) = sum(g); e(k) = u(k) - x(k); F(i) = g(i) * F(i) + g(i) * (mu / (sg(k-n+1) + 1)) * y(k-i+1) * e(k); end;

end; He(k-n+1) = (H - F) * (H - F)';

end; figure(1); clf; subplot(2, 1, 1); impz(H); grid on; zoom on; title ('Desired Impulse Response of F'); xlabel ('Tap Number'); ylabel ('Tap Weight'); subplot(2, 1, 2); impz(F); grid on; zoom on; title ('Received Impulse Response of F'); xlabel ('Tap Number'); ylabel ('Tap Weight'); figure(2); clf; subplot(2, 1, 1); semilogy(He); YLIM([10^-5, 10^1]); grid on; zoom on; title ('Squared Error of F'); xlabel ('Sample Interval'); ylabel ('[H -F] ^2 Error'); subplot(2, 1, 2); plot(sg); grid on; zoom on; title ('Number of Active Taps Detected'); xlabel ('Sample Interval'); ylabel ('Active Taps Detected');


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Appendix C Smart Antenna Code Listings

C.1 Smart Antenna Receiving 1 White Signal with 1 DOA

% LMS Smart Antenna Simulation % File name: LMS_1DOA_1Sig_White_Gain.m % This program simulates a Smart Antenna systems receiving a white input signal % from one source. u = sign(randn(1, 5000)); % Inputs from source N = length(u); % Number of input signals n = 25; % Number of antenna elements nn = (randn(N, n) + j * randn(N, n)) * 0.1; % Noise inputs for SA mu = 0.008; % Step-size F = zeros(1, n); % Initialise SA weight vectors to zero T1 = 100 * 10^(-6); % Time for signal to arrive at first element fc = 4 * 10^8; % Carrier frequency c = 3 * 10^8; % Speed of light lambda = c / fc; % Wavelength d = lambda / 2; % Element spacing DOA = 60; % Direction of Arrival of u DOA_rad = DOA * pi / 180; % DOA in radians sin_DOA = sin(DOA_rad); % Sine of DOA_rad r = zeros (1, n); % Received signal at each element y = zeros (1, n); % r + noise x = zeros (1, N); % Estimate of transmitted signal e = zeros (1, N); % Error between u and x B = zeros (1, n); % Mean squared error expA = 0; % Phase delay due to propagation time expB = zeros(1, n); % Additional delay at each element Gain = 0.5 * (exp(j * pi / 3)); % Gain experienced by u for k = 1: N

U = Gain * [u(k), u(k), u(k), u(k), u(k), u(k), u(k), u(k), u(k), u(k), u(k), u(k), u(k), u(k), u(k), u(k), u(k), u(k), u(k), u(k), u(k), u(k), u(k), u(k), u(k)];

for m = 1: n expA = exp(-j * 2 * pi * fc * T1); expB(m) = exp(-j * 2 * pi * (m-1) * d * sin_DOA / lambda); r(m) = U(m) * expA * expB(m); y(m) = r(m) + nn(k, m); end; x(k) = y * F'; e(k) = u(k) - x(k);


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F = F + mu * y * conj(e(k)); B(k) = e(k) * e(k)'; end; figure (1); clf; subplot(2, 1, 1); semilogy (abs(B)); XLIM([0, 150]); YLIM([10^-7, 10^1]); grid on; zoom on; title ('Received Signal Error: 1 White Signal'); xlabel ('Sample Interval'); ylabel ('[U - X] ^2 Error'); angle_min = -90 * pi / 180; angle_max = 90 * pi / 180; angle_incr = 1 * pi / 180; q = 0; F = conj(F); for angle1 = angle_min : angle_incr : angle_max q = q + 1; angle2(q) = 2 * pi * d * sin(angle1) / lambda; for t = 1: n G(t) = exp(j * angle2(q) * (t-1)); end; beam(q) = abs(F * G'); end; figure (2); clf; polar (angle_min : angle_incr : angle_max, beam); view(90, -90); zoom on;


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C.2 Smart Antenna Receiving 2 White Signals with 1 DOA Each

% LMS Smart Antenna Simulation % File name: LMS_2DOA_2_Sig_White_Gain.m % This program simulates a Smart Antenna system receiving 2 white input % signals and distinguishing each signal. u1 = sign(randn(1, 5000)); % Inputs from 1st source v1 = sign(randn(1, 5000)); % Inputs from 2nd source N = length(u1); % Number of input signals n = 25; % Number of antenna elements nn = (randn(N, n) + j * randn(N, n)) * 0.1; % Noise inputs for SA mu = 0.008; % Step-size F1 = zeros(1, n); % Initialise 1st DOA weight vectors to zero F2 = zeros(1, n); % Initialise 2nd DOA weight vectors to zero T1 = 100 * 10^(-6); % Time for 1st signal to arrive at first element T2 = 150 * 10^(-6); % Time for 2nd signal to arrive at first element fc = 4 * 10^8; % Carrier frequency c = 3 * 10^8; % Speed of light lambda = c / fc; % Wavelength d = lambda / 2; % Element spacing DOA1 = 60; % Direction of Arrival of u1 DOA2 = -25; % Direction of Arrival of u2 DOA_rad1 = DOA1 * pi / 180; % DOA1 in radians DOA_rad2 = DOA2 * pi / 180; % DOA2 in radians sin1 = sin(DOA_rad1); % Sine of DOA1 sin2 = sin(DOA_rad2); % Sine of DOA2 r = zeros (2, n); % Received signals at each element y = zeros (2, n); % r + noise x = zeros (2, N); % Estimate of transmitted signal e = zeros (2, N); % Error between u and x for each signal B = zeros (2, n); % Squared error of each signal F = zeros(2, n); % Initialise SA weight vectors to zero exp1A = 0; % Phase delay due to propagation time of u1 exp2A = 0; % Phase delay due to propagation time of v1 exp1B = zeros(1, n); % Additional delay at each element exp2B = zeros(1, n); % Additional delay at each element Gain1 = 0.5 * (exp(j * pi / 3)); % Gain experienced by u1 Gain2 = 1 * (exp(j * pi / 4)); % Gain experienced by u2 for k = 1: N

U1 = Gain1 * [u1(k), u1(k), u1(k), u1(k), u1(k), u1(k), u1(k), u1(k), u1(k), u1(k), u1(k), u1(k), u1(k), u1(k), u1(k), u1(k), u1(k), u1(k), u1(k), u1(k), u1(k), u1(k), u1(k), u1(k), u1(k)];

V1 = Gain2 * [v1(k), v1(k), v1(k), v1(k), v1(k), v1(k), v1(k), v1(k), v1(k), v1(k), v1(k), v1(k), v1(k), v1(k), v1(k), v1(k), v1(k), v1(k), v1(k), v1(k), v1(k), v1(k), v1(k), v1(k), v1(k)];

for m = 1: n exp1A = exp(-j * 2 * pi * fc * T1); exp1B(m) = exp(-j * 2 * pi * (m-1) * d * sin1 / lambda); exp2A = exp(-j * 2 * pi * fc * T2);


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exp2B(m) = exp(-j * 2 * pi * (m-1) * d * sin2 / lambda); r(1, m) = U1(m) * exp1A * exp1B(m) + V1(m) * exp2A * exp2B(m); y(1, m) = r(1, m) + nn(k, m); r(2, m) = V1(m) * exp2A * exp2B(m) + U1(m) * exp1A * exp1B(m); y(2, m) = r(2, m) + nn(k, m); end; x(1, k) = y(1, :) * F(1, :)'; e(1, k) = u1(k) - x(1, k); F(1, :) = F(1, :) + mu * y(1, :) * conj(e(1, k)); B(1, k) = e(1, k) * e(1, k)'; x(2, k) = y(2, :) * F(2, :)'; e(2, k) = v1(k) - x(2, k); F(2, :) = F(2, :) + mu * y(2, :) * conj(e(2, k)); B(2, k) = e(2, k) * e(2, k)'; end; figure (1); clf; subplot(2, 1, 1); semilogy (abs(B(1, :)), 'b'); hold on; XLIM([0, 100]); YLIM([10^-7, 10^1]); semilogy (abs(B(2, :)), 'r'); XLIM([0, 150]); YLIM([10^-7, 10^1]); hold off; grid on; zoom on; title ('Received Signal Error: 2 White Signals'); xlabel ('Sample Interval'); ylabel ('[U,V - X] ^2 Error'); legend ('1st Sig', '2nd Sig', 4); angle_min = -90 * pi / 180; angle_max = 90 * pi / 180; angle_incr = 1 * pi / 180; q = 0; F(1, :) = conj(F(1, :)); F(2, :) = conj(F(2, :)); for angle1 = angle_min : angle_incr : angle_max q = q + 1;

angle2(q) = 2 * pi * d * sin(angle1) / lambda; for t = 1: n G(t) = exp(j * angle2(q) * (t-1)); end; beam1(q) = abs(F(1, :) * G'); beam2(q) = abs(F(2, :) * G'); end; angle_range = angle_min : angle_incr : angle_max;


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figure (2); clf; polar (angle_range, beam1, 'b'); hold on; polar (angle_range, beam2, 'r'); hold off; view(90, -90); legend ('1st Sig', '2nd Sig', 4); zoom on;


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C.3 Smart Antenna Receiving 1 White Signal with 3 DOAs

% LMS Smart Antenna Simulation % File name: LMS_3DOA_1Sig_White_Gain.m % This program simulates a Smart Antenna system receiving three white input multipath % signals from one source and distinguishing each multipath. u1 = sign(randn(1, 5000)); % Inputs from 1st multipath u2 = [0, u1]; % Inputs from 2nd multipath u3 = [0, u2]; % Inputs from 3rd multipath N = length(u1); % Number of input signals n = 25; % Number of antenna elements nn = (randn(N, n) + j * randn(N, n)) * 0.1; % Noise inputs for SA mu = 0.008; % Step-size fc = 4 * 10^8; % Carrier frequency T1 = 100 * 10^(-6); % Time for 1st signal to arrive at first element T2 = T1 + 1 / fc; % Time for 1st signal to arrive at first element T3 = T2 + 1 / fc; % Time for 1st signal to arrive at first element c = 3 * 10^8; % Speed of light lambda = c / fc; % Wavelength d = lambda / 2; % Element spacing DOA1 = 60; % Direction of Arrival of u1 DOA2 = 30; % Direction of Arrival of u2 DOA3 = -20; % Direction of Arrival of u3 DOA_rad1 = DOA1 * pi / 180; % DOA1 in radians DOA_rad2 = DOA2 * pi / 180; % DOA2 in radians DOA_rad3 = DOA3 * pi / 180; % DOA2 in radians sin1 = sin(DOA_rad1); % Sine of DOA_rad1 sin2 = sin(DOA_rad2); % Sine of DOA_rad2 sin3 = sin(DOA_rad3); % Sine of DOA_rad3 r = zeros (3, n); % Received signals at each element y = zeros (3, n); % r + noise x = zeros (3, N); % Estimate of transmitted signal e = zeros (3, N); % Error between u and x for each multipath B = zeros (3, N); % Squared error of each SA F = zeros (3, n); % Initialise SA weight vectors to zero exp1A = 0; % Phase delay due to propagation time of u1 exp2A = 0; % Phase delay due to propagation time of u2 exp3A = 0; % Phase delay due to propagation time of u3 exp1B = zeros(1, n); % Additional delay at each element exp2B = zeros(1, n); % Additional delay at each element exp3B = zeros(1, n); % Additional delay at each element Gain1 = 0.5 * (exp(j * pi / 3)); % Gain experienced by u1 Gain2 = 0.66 * (exp(j * pi / 6)); % Gain experienced by u2 Gain3 = 1.0 * (exp(j * pi / 4)); % Gain experienced by u3 for k = 1: N


U2 = Gain2 * [u2(k), u2(k), u2(k), u2(k), u2(k), u2(k), u2(k), u2(k), u2(k), u2(k), u2(k),


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u2(k), u2(k), u2(k), u2(k), u2(k), u2(k), u2(k), u2(k), u2(k), u2(k), u2(k), u2(k), u2(k), u2(k)];


for m = 1: n exp1A = exp(-j * 2 * pi * fc * T1); exp2A = exp(-j * 2 * pi * fc * T2); exp3A = exp(-j * 2 * pi * fc * T3); exp1B(m) = exp(-j * 2 * pi * (m-1) * d * sin1 / lambda); exp2B(m) = exp(-j * 2 * pi * (m-1) * d * sin2 / lambda); exp3B(m) = exp(-j * 2 * pi * (m-1) * d * sin3 / lambda);

r(1, m) = U1(m) * exp1A * exp1B(m) + U2(m) * exp2A * exp2B(m) + U3(m) * exp3A * exp3B(m);

y(1, m) = r(1, m) + nn(k, m); r(2, m) = U2(m) * exp2A * exp2B(m) + U3(m) * exp3A * exp3B(m) + U1(m)

* exp1A * exp1B(m); y(2, m) = r(2, m) + nn(k, m); r(3, m) = U3(m) * exp3A * exp3B(m) + U1(m) * exp1A * exp1B(m) + U2(m)

* exp2A * exp2B(m) ; y(3, m) = r(3, m) + nn(k, m);

end; x(1, k) = y(1, :) * F(1, :)'; e(1, k) = u1(k) - x(1, k); F(1, :) = F(1, :) + mu * y(1, :) * conj(e(1, k)); B(1, k) = e(1, k) * e(1, k)'; x(2, k) = y(2, :) * F(2, :)'; e(2, k) = u2(k) - x(2, k); F(2, :) = F(2, :) + mu * y(2, :) * conj(e(2, k)); B(2, k) = e(2, k) * e(2, k)'; x(3, k) = y(3, :) * F(3, :)'; e(3, k) = u3(k) - x(3, k); F(3, :) = F(3, :) + mu * y(3, :) * conj(e(3, k)); B(3, k) = e(3, k) * e(3, k)'; end; figure (1); clf; subplot(2, 1, 1); semilogy (abs(B(1, :)), 'b'); XLIM([0, 150]); YLIM([10^-7, 10^1]); hold on; semilogy (abs(B(2, :)), 'r'); XLIM([0, 150]); YLIM([10^-7, 10^1]); hold on; semilogy (abs(B(3, :)), 'g'); hold off;


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XLIM([0, 150]); YLIM([10^-7, 10^1]); grid on; zoom on; title ('Received Signal Error: 1 White Signal, 3 DOAs'); xlabel ('Sample Interval'); ylabel ('[U - X] ^2 Error'); legend ('DOA1', 'DOA2', 'DOA3', 4); angle_min = -90 * pi / 180; angle_max = 90 * pi / 180; angle_incr = 1 * pi / 180; q = 0; F(1, :) = conj(F(1, :)); F(2, :) = conj(F(2, :)); F(3, :) = conj(F(3, :)); for angle1 = angle_min : angle_incr : angle_max q = q + 1; angle2(q) = 2 * pi * d * sin(angle1) / lambda; for t = 1: n G(t) = exp(j * angle2(q) * (t-1)); end; beam1(q) = abs(F(1, :) * G'); beam2(q) = abs(F(2, :) * G'); beam3(q) = abs(F(3, :) * G'); end; angle_range = angle_min : angle_incr : angle_max; figure (2); clf; polar (angle_range, beam1, 'b'); hold on; polar (angle_range, beam2, 'r'); hold on; polar (angle_range, beam3, 'g'); hold off; view(90, -90); legend ('DOA1', 'DOA2', 'DOA3', 4); zoom on;


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C.4 Smart Antenna Receiving 2 White Signals with 3 DOAs Each

% LMS Smart Antenna Simulation % File name: LMS_6DOA_2Sig_White_Gain.m % This program simulates a Smart Antenna system receiving three white input multipath % signals from two sources (3 each) and distinguishing each multipath. u1 = sign(randn(1, 5000)); % Inputs from 1st multipath, 1st signal u2 = [0, u1]; % Inputs from 2nd multipath, 1st signal u3 = u2; % Inputs from 3rd multipath, 1st signal v1 = sign(randn(1, 5000)); % Inputs from 1st multipath, 2nd signal v2 = [0, v1]; % Inputs from 2nd multipath, 2nd signal v3 = v2; % Inputs from 3rd multipath, 2nd signal N = length(u1); % Number of input signals n = 25; % Number of antenna elements nn = (randn(N, n) + j * randn(N, n)) * 0.1; % Noise inputs for SA mu = 0.008; % Step-size fc = 4 * 10^8; % Carrier frequency Tu1 = 100 * 10^(-6); % Time for 1st signal to arrive at first element Tu2 = Tu1 + 1 / fc; % Time for 1st signal to arrive at first element Tu3 = Tu2; % Time for 1st signal to arrive at first element Tv1 = 150 * 10^(-6); % Time for 1st signal to arrive at first element Tv2 = Tv1 + 1 / fc; % Time for 1st signal to arrive at first element Tv3 = Tv2; % Time for 1st signal to arrive at first element c = 3 * 10^8; % Speed of light lambda = c / fc; % Wavelength d = lambda / 2; % Element spacing DOAu1 = 60; % Direction of Arrival of u1 DOAu2 = 30; % Direction of Arrival of u2 DOAu3 = -20; % Direction of Arrival of u3 DOA_radu1 = DOAu1 * pi / 180; % DOAu1 in radians DOA_radu2 = DOAu2 * pi / 180; % DOAu2 in radians DOA_radu3 = DOAu3 * pi / 180; % DOAu2 in radians sin_u1 = sin(DOA_radu1); % Sine of DOA_radu1 sin_u2 = sin(DOA_radu2); % Sine of DOA_radu2 sin_u3 = sin(DOA_radu3); % Sine of DOA_radu3 DOAv1 = -50; % Direction of Arrival of v1 DOAv2 = 0; % Direction of Arrival of v2 DOAv3 = 45; % Direction of Arrival of v3 DOA_radv1 = DOAv1 * pi / 180; % DOAv1 in radians DOA_radv2 = DOAv2 * pi / 180; % DOAv2 in radians DOA_radv3 = DOAv3 * pi / 180; % DOAv2 in radians sin_v1 = sin(DOA_radv1); % Sine of DOA_radv1 sin_v2 = sin(DOA_radv2); % Sine of DOA_radv2 sin_v3 = sin(DOA_radv3); % Sine of DOA_radv3 ru = zeros (3, n); % Received 1st signal at each element yu = zeros (3, n); % ru + noise xu = zeros (3, N); % Estimate of 1st signal eu = zeros (3, N); % Error between u and x for each multipath of

1st signal Bu = zeros (3, N); % Squared error of multipath of 1st signal


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Fu = zeros (3, n); % Initialise SA weight vectors to zero rv = zeros (3, n); % Received 2nd signal at each element yv = zeros (3, n); % rv + noise xv = zeros (3, N); % Estimate of 2nd signal ev = zeros (3, N); % Error between v and x for each multipath of

2nd signal Bv = zeros (3, N); % Squared error of multipath of 2nd signal Fv = zeros (3, n); % Initialise SA weight vectors to zero Gainu1 = 0.5 * (exp(j * pi / 2)); % Gain experienced by u1 Gainu2 = 0.66 * (exp(j * pi / 6)); % Gain experienced by u2 Gainu3 = 0.75 * (exp(j * pi / 4)); % Gain experienced by u3 Gainv1 = 0.6 * (exp(j * pi / 7)); % Gain experienced by v1 Gainv2 = 0.8 * (exp(j * pi / 5)); % Gain experienced by v2 Gainv3 = 0.7 * (exp(j * pi / 3)); % Gain experienced by v3 for k = 1: N

U1 = Gainu1 * [u1(k), u1(k), u1(k), u1(k), u1(k), u1(k), u1(k), u1(k), u1(k), u1(k), u1(k), u1(k), u1(k), u1(k), u1(k), u1(k), u1(k), u1(k), u1(k), u1(k), u1(k), u1(k), u1(k), u1(k), u1(k)];



V1 = Gainv1 * [v1(k), v1(k), v1(k), v1(k), v1(k), v1(k), v1(k), v1(k), v1(k), v1(k), v1(k), v1(k), v1(k), v1(k), v1(k), v1(k), v1(k), v1(k), v1(k), v1(k), v1(k), v1(k), v1(k), v1(k), v1(k)];



for m = 1: n expu1A = exp(-j * 2 * pi * fc * Tu1); expu2A = exp(-j * 2 * pi * fc * Tu2); expu3A = exp(-j * 2 * pi * fc * Tu3); expu1B(m) = exp(-j * 2 * pi * (m-1) * d * sin_u1 / lambda); expu2B(m) = exp(-j * 2 * pi * (m-1) * d * sin_u2 / lambda); expu3B(m) = exp(-j * 2 * pi * (m-1) * d * sin_u3 / lambda); expv1A = exp(-j * 2 * pi * fc * Tv1); expv2A = exp(-j * 2 * pi * fc * Tv2); expv3A = exp(-j * 2 * pi * fc * Tv3); expv1B(m) = exp(-j * 2 * pi * (m-1) * d * sin_v1 / lambda); expv2B(m) = exp(-j * 2 * pi * (m-1) * d * sin_v2 / lambda); expv3B(m) = exp(-j * 2 * pi * (m-1) * d * sin_v3 / lambda);

ru(1, m) = U1(m) * expu1A * expu1B(m) + U2(m) * expu2A * expu2B(m) + U3(m) * expu3A * expu3B(m) + V1(m) * expv1A * expv1B(m) + V2(m) * expv2A * expv2B(m) + V3(m) * expv3A * expv3B(m);


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yu(1, m) = ru(1, m) + nn(k, m); ru(2, m) = U2(m) * expu2A * expu2B(m) + U3(m) * expu3A * expu3B(m) +

V1(m) * expv1A * expv1B(m) + V2(m) * expv2A * expv2B(m) + V3(m) * expv3A * expv3B(m) + U1(m) * expu1A * expu1B(m);

yu(2, m) = ru(2, m) + nn(k, m); ru(3, m) = U3(m) * expu3A * expu3B(m) + V1(m) * expv1A * expv1B(m) +

V2(m) * expv2A * expv2B(m) + V3(m) * expv3A * expv3B(m) + U1(m) * expu1A * expu1B(m) + U2(m) * expu2A * expu2B(m) ;

yu(3, m) = ru(3, m) + nn(k, m);

rv(1, m) = V1(m) * expv1A * expv1B(m) + V2(m) * expv2A * expv2B(m) + V3(m) * expv3A * expv3B(m) + U1(m) * expu1A * expu1B(m) + U2(m) * expu2A * expu2B(m) + U3(m) * expu3A * expu3B(m);

yv(1, m) = rv(1, m) + nn(k, m); rv(2, m) = V2(m) * expv2A * expv2B(m) + V3(m) * expv3A * expv3B(m) +

U1(m) * expu1A * expu1B(m) + U2(m) * expu2A * expu2B(m) + U3(m) * expu3A * expu3B(m) + V1(m) * expv1A * expv1B(m);

yv(2, m) = rv(2, m) + nn(k, m); rv(3, m) = V3(m) * expv3A * expv3B(m) + U1(m) * expu1A * expu1B(m) +

U2(m) * expu2A * expu2B(m) + U3(m) * expu3A * expu3B(m) + V1(m) * expv1A * expv1B(m) + V2(m) * expv2A * expv2B(m) ;

yv(3, m) = rv(3, m) + nn(k, m); end; xu(1, k) = yu(1, :) * Fu(1, :)';

eu(1, k) = u1(k) - xu(1, k); Fu(1, :) = Fu(1, :) + mu * yu(1, :) * conj(eu(1, k)); Bu(1, k) = eu(1, k) * eu(1, k)';

xu(2, k) = yu(2, :) * Fu(2, :)'; eu(2, k) = u2(k) - xu(2, k); Fu(2, :) = Fu(2, :) + mu * yu(2, :) * conj(eu(2, k)); Bu(2, k) = eu(2, k) * eu(2, k)'; xu(3, k) = yu(3, :) * Fu(3, :)'; eu(3, k) = u3(k) - xu(3, k); Fu(3, :) = Fu(3, :) + mu * yu(3, :) * conj(eu(3, k)); Bu(3, k) = eu(3, k) * eu(3, k)'; xv(1, k) = yv(1, :) * Fv(1, :)'; ev(1, k) = v1(k) - xv(1, k); Fv(1, :) = Fv(1, :) + mu * yv(1, :) * conj(ev(1, k)); Bv(1, k) = ev(1, k) * ev(1, k)'; xv(2, k) = yv(2, :) * Fv(2, :)'; ev(2, k) = v2(k) - xv(2, k); Fv(2, :) = Fv(2, :) + mu * yv(2, :) * conj(ev(2, k)); Bv(2, k) = ev(2, k) * ev(2, k)'; xv(3, k) = yv(3, :) * Fv(3, :)'; ev(3, k) = v3(k) - xv(3, k); Fv(3, :) = Fv(3, :) + mu * yv(3, :) * conj(ev(3, k)); Bv(3, k) = ev(3, k) * ev(3, k)';


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end; figure (1); clf; subplot(2, 1, 1); semilogy (abs(Bu(1, :)), 'b'); XLIM([0, 150]); YLIM([10^-7, 10^1]); hold on; semilogy (abs(Bu(2, :)), 'r'); XLIM([0, 150]); YLIM([10^-7, 10^1]); hold on; semilogy (abs(Bu(3, :)), 'g'); XLIM([0, 150]); YLIM([10^-7, 10^1]); hold off; grid on; zoom on; title ('Received Signal Error: 2 White Signals, 3 DOAs Each (1st Sig)'); xlabel ('Sample Interval'); ylabel ('[U - XU] ^2 Error'); legend ('DOA1, Sig1', 'DOA2, Sig1', 'DOA3, Sig1', 4); subplot(2, 1, 2); semilogy (abs(Bv(1, :)), 'm'); XLIM([0, 150]); YLIM([10^-7, 10^1]); hold on; semilogy (abs(Bv(2, :)), 'c'); XLIM([0, 150]); YLIM([10^-7, 10^1]); hold on; semilogy (abs(Bv(3, :)), 'k'); XLIM([0, 150]); YLIM([10^-7, 10^1]); hold off; grid on; zoom on; title ('Received Signal Error: 2 White Signals, 3 DOAs Each (2nd Sig)'); xlabel ('Sample Interval'); ylabel ('[V - XV] ^2 Error'); legend ('DOA1, Sig2', 'DOA2, Sig2', 'DOA3, Sig2', 4); angle_min = -90 * pi / 180; angle_max = 90 * pi / 180; angle_incr = 1 * pi / 180; q = 0; Fu(1, :) = conj(Fu(1, :)); Fu(2, :) = conj(Fu(2, :)); Fu(3, :) = conj(Fu(3, :)); Fv(1, :) = conj(Fv(1, :)); Fv(2, :) = conj(Fv(2, :)); Fv(3, :) = conj(Fv(3, :)); for angle1 = angle_min : angle_incr : angle_max


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q = q + 1; angle2(q) = 2 * pi * d * sin(angle1) / lambda; for t = 1: n G(t) = exp(j * angle2(q) * (t-1)); end; beamu1(q) = abs(Fu(1, :) * G'); beamu2(q) = abs(Fu(2, :) * G'); beamu3(q) = abs(Fu(3, :) * G'); beamv1(q) = abs(Fv(1, :) * G'); beamv2(q) = abs(Fv(2, :) * G'); beamv3(q) = abs(Fv(3, :) * G'); end; angle_range = angle_min : angle_incr : angle_max; figure (2); clf; polar (angle_range, beamu1, 'b'); hold on; polar (angle_range, beamu2, 'r'); hold on; polar (angle_range, beamu3, 'g'); hold on; polar (angle_range, beamv1, 'm'); hold on; polar (angle_range, beamv2, 'c'); hold on; polar (angle_range, beamv3, 'k'); hold off; view(90, -90); legend ('DOA1, Sig1', 'DOA2, Sig1', 'DOA3, Sig1', 'DOA1, Sig2', 'DOA2, Sig2', 'DOA3, Sig2',

4); zoom on;

adaptive equalisers and smart antenna systemsread.pudn.com/downloads67/doc/comm/240733/adaptive...

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