i Suan Soo Foo 2/43, Durham Street St Lucia, QLD 4067 Australia October 2000 The Dean School of Engineering University of Queensland St Lucia QLD 4072 Dear Sir, In accordance and partial fulfillment of the requirements for the degree of Bachelor of Electrical Engineering (Honours) at the University of Queensland, I hereby submit for your consideration this thesis entitled: “Smart Antennas for Wireless Applications” This work was accomplished under the supervision of Associate Professor Marek E. Bialkowski. I declared that the work submitted in this thesis is my own, except as acknowledged in the text, and has not previous been submitted for a degree at the University of Queensland or any other institution. Yours faithfully, _____________ Suan Soo Foo
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i
Suan Soo Foo
2/43, Durham Street
St Lucia, QLD 4067
Australia
October 2000
The Dean
School of Engineering
University of Queensland
St Lucia QLD 4072
Dear Sir,
In accordance and partial fulfillment of the requirements for the degree of Bachelor of
Electrical Engineering (Honours) at the University of Queensland, I hereby submit for
your consideration this thesis entitled:
“Smart Antennas for Wireless Applications”
This work was accomplished under the supervision of Associate Professor Marek E.
Bialkowski.
I declared that the work submitted in this thesis is my own, except as acknowledged in
the text, and has not previous been submitted for a degree at the University of
Queensland or any other institution.
Yours faithfully, _____________ Suan Soo Foo
ii
Smart Antennas for
Wireless Applications
Suan Soo Foo
Approved by Assoc. Prof. Bialkowski
University of Queensland
School of Computer Science and Electrical Engineering
University of Queensland
Queensland 4072
Australia
iii
Acknowledgements
The author would like to express his appreciation to his supervisor, Associate Professor
M.E. Bialkowski, for providing the opportunity to research this interesting topic, for his
valuable advice and the direction he had shown throughout the year.
Many thanks to Danny Kai Pin Tan for giving the opportunity to work with him and the
tolerant he had shown while working together.
Thanks must also go to the laboratory supervisor, Damian Jones for his assistance and
the patient he had given while using the Microwave Laboratory throughout the thesis
project.
Last but no least, the author would like to extend his thanks to his girlfriend, Chai
Weichiun, and his family for their support and encouragement.
iv
ABSTRACT
The smart antenna is set to play a significant role in the development of next-generation
wireless communication system. The purpose of this thesis is to provide the concept on
smart antenna system by studying the performance of antenna array. A brief
introduction will be given before providing the overview of the thesis content.
Antenna theory and the description of different types of antennas will be discussed with
emphasis on array antennas. Two methods of antenna synthesis known as the
Woodward-Lawson and Dolph-Chebyshev will also be introduced before studying the
fundamental parameters of antenna.
With a basic understanding on antenna, this thesis will therefore discuss about the smart
antenna technology. The two types of smart antenna approaches known as the
Switching-Beam Array and Adaptive Array will be addressed after introducing the
benefits of the smart antenna technology. The smart antenna terminology together with
an adaptive algorithm called the Recursive Least Squares Algorithm will also be
presented.
After a brief introduction to the types multiple access schemes, array antennas
simulation and synthesis using the above mentioned methods and algorithm will be
carried out by varying different limiting parameters. Results will be tabulated and
antenna radiation patterns will also be plotted for discussion before wrapping up with a
conclusion and suggestion on future developments.
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CONTENTS
Page
Chapter 1 Introduction 1
1.1 Introduction 1 1.2 Aim of Thesis 2 1.3 Overview of Content 3
Chapter 6 Analysis of Array Antennas 49 6.1 Aim and Procedures 49 6.2 Microstrip Patch Antenna Design 49 6.3 Simulation on Linear Array Antenna 54 6.3.1 Effect of Varying Inter-element Spacing, d 54 6.3.2 Effect of Varying Number of Elements, N 56 6.3.3 Effect of Varying Amplitude Distribution 57 6.3.4 Effect of Varying Phase Excitation, β 59 6.4 Simulation on Planar Array Antenna 61 6.4.1 Effect of Varying Inter-element Spacing, d 61 6.4.2 Effect of Varying Number of Elements, N 64 6.4.3 Effect of Varying Amplitude Distribution 68 6.4.4 Effect of Varying Phase Excitation, βx and βy 68 6.5 Discussion 72
Chapter 7 Antenna Synthesis Investigation 74 7.1 Aim and Procedures 74 7.2 Woodward-Lawson Synthesis 75 7.2.1 Effect of Varying Number of Elements, N 75 7.2.2 Effect of Varying Inter-element Spacing, d 77 7.3 Dolph-Chebyshev Synthesis 80 7.3.1 Effect of Varying Number of Elements, N 80 7.3.2 Effect of Varying Sidelobe Level 82
vii
7.3.3 Effect of Varying Inter-element Spacing, d 84 7.4 Discussion 87
Chapter 8 Recursive Least Square Algorithm Analysis 88
Smart Antennas for Wireless Applications Chapter 2: Antennas
20
Assuming the elements of the array is placed along the z-axis, and thus, replacing cos u
with z in (2.20), will relate each of the expression to a Chebyshev polynomial Tm(z).
m = 0 cos(mu) = 1 = T0(z)
m = 1 cos(mu) = z = T1(z)
m = 2 cos(mu) = 2z2 –1 = T2(z)
m = 3 cos(mu) = 4z3 – 3z = T3(z)
m = 4 cos(mu) = 8z4 – 8z2 + 1 = T4(z) (2.22)
These relations between the cosine functions and the Chebyshev polynomials are valid
only in the range of –1 ≤ z ≤ +1. Because |cos(mu)| ≤ 1, each Chebyshev polynomial is
|Tm(z)| ≤ 1 for –1 ≤ z ≤ +1. For |z| > 1, the Chebyshev polynomials are related too the
hyperbolic cosine function [3].
The recursive formula can be used to determine the Chebyshev polynomial if the
polynomials of the previous two orders are known. This is given by
Tm(z) = 2zTm-1(z) – Tm-2(z) (2.23)
It can be seen that the array factor of an odd and even number of elements is a
summation of cosine terms whose form is similar with the Chebyshev polynomials.
Therefore, by equating the series representing the cosine terms of the array to the
appropriate Chebyshev polynomial, the unknown coefficients of the array factor can be
determine. Note that the order of the polynomial should be one less than the total
number of elements of the array.
Smart Antennas for Wireless Applications Chapter 3: Parameters of Antenna
21
Chapter 3 Parameters of Antenna
3.1 Introduction
Definitions of various parameters are necessary to describe the performance of an
antenna. Although the parameters may be interrelated, it is however, not a requirement
to specify all of the parameters for complete description of the antenna performance. An
antenna is chosen for operation in a particular application according to its physical and
electrical characteristics. Furthermore, the antenna must perform in a required mode for
the particular measurement system.
An antenna can be characterized by the following elements, not all of which apply to all
antenna types:
1. Radiation resistance;
2. Radiation pattern;
3. Beamwidth and gain of main lobe;
4. Position of magnitude of sidelobes;
5. Magnitude of back lobe;
6. Bandwidth;
7. Aperture;
8. Antenna correction factor;
9. Polarization of the electric field that it transmits or receive;
10. Power that it can handle in the case of a transmitting antenna.
Smart Antennas for Wireless Applications Chapter 3: Parameters of Antenna
22
Typically, antenna characteristics are measured in two principal planes and they are
known as the azimuth and elevation planes, which can also be considered as the
horizontal and vertical planes respectively, for land-based antennas. Conventionally, the
angle in the azimuth plane is denoted by the Greek letter phi, φ, while the Greek letter
theta, θ, represents the angle in the elevation plane.
Some characteristics such as beamwidth and sidelobes are the same in both planes for
symmetrical antennas such as circular waveguide horns and reflector. Other
characteristics such as the gain on boresight (i.e., where the azimuth and elevation
planes intersect) can only have a single value. In general, for unsymmetrical antennas
the characteristics are different in the two principal planes, with a gradual transition in
the intervening region between these two planes [6].
Not all of the antenna characteristic factors will be discussed here. The following
subsection will touch on some of the elements, which are essential for the understanding
of this thesis.
3.2 Radiation pattern
The antenna, which radiates or receives the electromagnetic energy in the same way, is
a reciprocal device. Radiation pattern is a very important characteristic of an antenna. It
facilitates a stronger understanding of the key features of an antenna that otherwise
cannot be achieved from the textual technical description of an antenna.
The radiation pattern is peculiar to class of antenna and its electrical characteristics as
well as its physical dimensions. It is gauged at a constant distance in the far field of the
antenna and its radiation pattern is usually plotted in terms of relative power. The power
at boresight, that is, at the position of maximum radiated power, is usually plotted at 0
dB; thus, the power at all other position appears as negative value. In other words, the
radiation power is normalized to the power at boresight. If the power were plotted in
linear units, the normalized power would be one at boreight [6].
Smart Antennas for Wireless Applications Chapter 3: Parameters of Antenna
23
The radiation is usually measured in the azimuth and the elevation planes and the
radiation power is plotted against the angle that is made with boresight direction. If the
antenna were not physically symmetrical about each of its principal planes, then it
would result in an unsymmetrical radiation patter in these planes.
The radiation pattern can be plotted using rectangular/cartesian or polar coordinates.
The rectangular plots can be read more precisely (since the angular scale can be
enlarged), but the polar plots offers a more pictorial representation and are thus easier to
visualize.
3.2.1 Rectangular/Cartesian Plots
Rectangular/Cartesian plots are standard x-y plots where the axes are plotted at right
angle to each other. The y-coordinate, which is called the ordinate, is used for the
dependent variable while the x-coordinate, known as abscissa, is used for the
independent variable.
In a radiation plot, the angle with respect to boresight is varied and the magnitude of the
power radiated is measured; thus, the angle is the independent variable and the power
radiated is the dependent variable. Thus, the magnitudes of the powers are the ordinate
while the angles are the abscissa. A typical rectangular plot of an antenna radiation
pattern is shown in Figure 3.2a.
Smart Antennas for Wireless Applications Chapter 3: Parameters of Antenna
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The y-axis can show two sets of scales: one graduated from 0 dB to 4 dB and another
from 0 dB to 8 dB. Scales of 40 dB and 80 dB are calculated by multiplying the scales
by ten. It should be noted that the numbers below should really by negative values of –4
dB and –8 dB because the zero is at the top.
On the hand, the x-axis can show three sets of angular scales of 5°, 30° and 180° on
either side of the zero, representing the angles measured clockwise and anti-clockwise
from the boresight position and in standard mathematical convention denoted by
positive and negative signs disregarded on radiation graph paper.
3.2.2 Polar Plots
In a polar plot the angles are plotted radially from boresight and the power or intensity is plotted along the radius as illustrated in Figure 3.2b.
Figure 3.2a Rectangular plot of an antenna radiation pattern
Smart Antennas for Wireless Applications Chapter 3: Parameters of Antenna
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This gives a pictorial representation of the radiation pattern of the antenna and is easier
to visualize than the rectangular/cartesian plots. Although the accuracy cannot be
increased as in the case of rectangular plot because the scale of the angular positions can
only be plotted from 0° to 360°, however, the scale of the intensity or power can be
varied.
Each circle on the polar plot represents a contour plot where the power has the same
magnitude and is shown relative to the power at boresight. These levels will always be
less than the power at boresight and values should be shown as negative because the
power is in generally a maximum value at boresight. However, they are normally
written without a sign and should be assumed to be negative, contrary to standard
arithmetic convention.
Figure 3.2b Polar plot of an antenna pattern
Smart Antennas for Wireless Applications Chapter 3: Parameters of Antenna
26
3.3 Main Lobe
The main lobe of the antenna is in the direction of maximum radiation. The
characteristics of an antenna such as beamwidth and gain are associated to the main lobe
alone. The peak/tip of the main beam is called the boresight of the antenna and the
radiation pattern is often positioned so that its boresight corresponds with the zero
angular position of the graph, even when the antenna is not physically symmetrical.
Figure 3.3a gives an idea of the main lobe, its maximum direction and beamwidth of a
typical power pattern polar plot.
3.3.1 Beamwidth – Half power and 10 dB
The beamwidth only relates to the main beam of the antenna and not the sidelobes and
in general, it is inversely proportional to its physical size. In other words, the larger the
antenna, the smaller is its beamwidth for the corresponding frequency. The plane
Figure 3.3a Typical power pattern polar plot
Smart Antennas for Wireless Applications Chapter 3: Parameters of Antenna
27
containing the largest dimension will have the narrowest beamwidth if the antenna does
not have the same dimensions in all planes.
The beam width of an antenna is usually defines in two ways. The most well known
definition is the 3-dB or half-power beamwidth. However, for antennas with very
narrow beams, the 10-dB beamwidth can also be applied. The 3-dB or half-power
beamwidth (HPBW) of an antenna is taken as the width at the points on either side of
the main beam where the radiated power is half the maximum value, and it is measured
in degrees or radians. Figure 3.3a shows the two points, half-power point (left) and half-
power (right), where the 3-dB beamwidth can be obtained.
3.3.2 Boresight Directivity/Gain
Although the terms directivity (or directive gain) and gain are frequently used
synonymously, but in fact they are not the same. The gain allows for efficiency of the
antenna, whereas directivity does not [6]. As a matter of fact, the gain of the antenna is
the product of the directivity and the efficiency. The IEEE definition of gain of an
antenna relates the power radiated by the antenna to that radiated by an isotropic
antenna (that radiates equally in all direction) and is quoted as a linear ratio or in
decibels [3].
The gain G as a linear ratio is defined as
antena isotropican by radiatedPower boresighton radiatedPower =G (3.1)
The gain GdB expressed in decibels is defined as
(G)10log 10=dBG (3.2)
Directivity of an antenna is defined as “the ratio of the radiation intensity in a given
direction from the antenna to the radiation intensity average over all direction. The
Smart Antennas for Wireless Applications Chapter 3: Parameters of Antenna
28
average radiation intensity is equal to the total power radiated by the antenna divided by
4π. If the direction is not specified, the direction of maximum radiation intensity is
implied” [3].
3.4 Sidelobes
The sidelobes are, strictly speaking, any of the maxima marked, for examples, as A, B,
C, D, E in Figure 3.4a. Nevertheless, in practice only the “near-in” lobes marked A are
referred to as sidelobes. Sometimes, due to the irregularities in the main beam of the
radiation pattern, it may result in small peaks such as those marked F in Figure 3.4a,
which could be mistaken for sidelobes.
Therefore, for this reason, the sidelobes are sometimes defined as the peaks, where the
difference between the peak and an adjacent trough is at least 3-dB. The sidelobes are
characterized by their level below the boresight gain and their angular position relative
to boresight. Although the sidelobe level (SLL) is usually cited as a positive quantity,
but it is a value in negative decibels since the radiation pattern is plotted with the
boresight gain at 0-dB.
Figure 3.4a A radiation pattern showing the sidelobe levels and positions
Smart Antennas for Wireless Applications Chapter 3: Parameters of Antenna
29
On top of sidelobes and main lobe, there are cases where multiple maxima occur, which
are referred to as grating lobes. Thus, one of the objectives in many designs is to avoid
grating lobes. Often it may be essential to select the largest spacing between elements
but with no grating lobes. However, the largest spacing between elements should be less
than one wavelength in order to avoid any grating lobes.
3.5 Front-to-back Ratio
The measure of the ability of a directional antenna to concentrate the beam in the
required forward direction is known as the front-to-back ratio (F/B). In linear terms, it is
determined as the ratio of the maximum power in the main beam to that in the back lobe
and it is usually expressed in decibels, as the different between the levels on boresight
and at 180° off boresight.
3.6 Aperture Size
The beamwidth is also influenced by the aperture size of an antenna. Generally, the
beamwidth gets narrower and the gain increases with an increasing aperture size at a
given frequency. The aperture size can be defined in two ways: either in terms of
wavelengths, or in terms of the actual physical size, in meters or feet.
3.7 Polarization
The polarization is another importance factor that would affect the radiation pattern.
The polarization of an antenna is defined as the polarization of the wave radiated by the
antenna in a given direction. However, the polarization is considered to be the
polarization in the direction of maximum gain when the direction is not stated.
Polarization may be classified as linear, circular, or elliptical. However, this thesis will
only touch on linear polarization.
Smart Antennas for Wireless Applications Chapter 3: Parameters of Antenna
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As shown in Figure 3.7a, the electric field varies sinusoidally in one plane for the case
of linear polarization. In this case of a vertical polarization, it is noted that the extremity
of the electric field vector at any fixed point in space is a straight line with maximum
value, which is equal to twice the amplitude of the sinc curve that depicts the variation
of the electric field with time.
While horizontal polarization is illustrated in Figure 3.7b, it is important to note that the
polarization of a receiving antenna must match that of the incident radiation in order to
detect the maximum field.
Figure 3.7a Variation of the electric field with time at a fixed point in space for vertical polarization
Figure 3.7b Variation of the electric field with time at a fixed point in space for horizontal polarization
Smart Antenna for Wireless Applications Chapter 4: Smart Antenna System
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Chapter 4
Smart Antenna System
4.1 Introduction
The field of wireless communication is growing at a dynamic rate, covering many
technical areas. Its sphere of influence is beyond imagination. An indication of its
importance is perhaps the immeasurable worldwide activities in this industry.
Since the early days of wireless communications, there have been simple antenna
designs that radiate signals omnidirectionally in a pattern resembling ripples in a pool of
water. Without the knowledge of the users’ locations, this unfocused technique
disseminates signals that reaches the intended user with a small percentage of overall
energy radiated out in the environment. Therefore, these strategies overcome the
problem by boosting the power level of the broadcasting signals. Moreover, there is also
additional problem of interference, which is likewise faced by directional antennas: a
system constructed to have certain fixed preferential transmission and reception
directions.
Therefore, the smart antenna systems, as shown in Figure 4.1a, have been introduced in
recent years to improve systems performance by increasing spectrum efficiency,
extending coverage area, tailoring beam shaping, steering multiple beams. Most
importantly, smart antenna system increases long-term channel capacity through Space
Division Multiple Access scheme (See Chapter 5 on Multiple Access Schemes). In
addition, it also reduces multipath fading, cochannel interferences, initial setup cost and
bit error rate (BER).
Smart Antenna for Wireless Applications Chapter 4: Smart Antenna System
32
In this chapter, the key benefits of the smart antenna technology are covered before
looking through the smart antenna systems and the types of approaches. This chapter
will wrap up with descriptions on the Recursive Least Squares Adaptive Algorithms
after introducing to Beam Forming and Steering Vector.
4.2 Key Benefits of Smart Antenna Technology
An understanding of signal propagation environment and channel characteristics is
significant to the efficient use of a transmission medium. In recent years, there have
been signal propagation problems associated with conventional antennas and
interference is the major limiting factor in the performance of wireless communication.
Thus, the introduction of smart antennas is considered to have the potential of leading to
a large increase in wireless communication systems performance.
A smart antenna system in the wireless communication contributes to the following
major benefits:
Figure 4.1a Concept of smart antenna systems: Able to form different beam for each user, extending coverage range, minimizing the impact of noise and interference for each subscriber.
Smart Antenna for Wireless Applications Chapter 4: Smart Antenna System
33
• Larger Range Coverage – 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 unit, smart antennas can increase
range by increasing the gain of the base station antenna [8].
• Reduced Initial Deployment Cost – Conventional wireless system networks are
initially often designed to satisfy coverage requirements, even though there are few
subscribers in the network. However, when the number of subscribers increases in
the network, system capacity can be increased at the expense of reducing the
coverage area and introducing additional cell sites. Nevertheless, smart antenna can
ease this problem by providing larger early cell sizes and thus, initial deployment
cost for the wireless system can be reduced through range extension.
• Reduced Multipath Fading – Multipath in radio channels can result in fading or time
dispersion. The effects of multipath fading in wireless communications
environments can be significantly reduced through smart antenna systems. This
reduction variation of the signal (i.e., fading) greatly enhances system performance
because the reliability and quality of a wireless communications system can strongly
depend on the depth and rate of fading [9].
• Better Security – The employment of smart antenna systems diminish the risk of
connection tapping. The intruder must be situated in the similar direction as the user
as seen from the transmitter base station.
• Better Services – Usage of the smart antenna system enables the network to have
access to spatial information about the users. This information can be used to assess
the positions of the users much more precisely than in existing network. This can be
applied in services such as emergency calls and location-specific billing.
• Increased Capacity – Smart antennas can also improve system capacity. They can
be used to allow the subscriber and base station to operate at the same range as a
conventional system, but a lower power. This may permit FDMA and TDMA
Smart Antenna for Wireless Applications Chapter 4: Smart Antenna System
34
systems, which will be discussed in the later section, to be rechannelized to reuse
frequency channels more frequently than conventional systems using fixed
antennas, since the carrier-to-interference ratio is much greater when smart antennas
are used. In CDMA systems, if smart antennas are used to allow subscribers to
transmit less power for each link, then the Multiple Access Interference is reduced,
which increases the number of simultaneous subscribers that can be supported in
each cell.
Although the smart antenna systems are favorable in many ways, there are also
drawbacks which include a more complex transceiver structure compared to traditional
base station transceiver and a growing need for development of efficient algorithm for
real-time optimizing and signal tracking. Thus, smart antenna base stations will no
doubt be much more expensive than conventional base stations and the advantages
should always be evaluated against the cost.
4.3 Smart Antenna System
A smart antenna system can be define as a system which uses an array of low gain
antenna elements with a signal-processing capability to optimize its radiation and/or
reception pattern automatically in response to the ever changing signal environment.
This can be visualized as the antenna focussing a beam towards the communication user
only.
Truly speaking, antennas are only mechanical construction transforming free
electromagnetic (EM) waves into radio frequency (RF) signals travelling on a shielded
cable or vice-versa. They are not smart but antenna systems are. The whole system
consists of the radiating antennas, a combining/dividing network and a control unit. The
control unit is usually realized using a digital signal processor (DSP), which controls
several input parameters of the antenna to optimize the communication link. This shows
that smart antennas are more than just the “antenna,” but rather a complete transceiver
concept.
Smart Antenna for Wireless Applications Chapter 4: Smart Antenna System
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Smart antenna systems are customarily classified as either Switching- Beam Array
(SBA) or Adaptive Array (also known as Tracking-Beam Array – TBA) systems, and
they are the two different approaches to realizing a smart antenna.
4.3.1 Switching-Beam Array (SBA)
In the smart antenna systems, the SBA approach forms multiple fixed beams with
enhanced sensitivity in specific area. These antenna systems will detect signal strength,
and select one of the best, predetermined, fixed beams for the subscribers as they move
throughout the coverage sector. Instead of modeling the directional antenna pattern with
the metallic properties and physical design of a single element, a SBA system couple
the outputs of multiple antennas in such a manner that it forms a finely sectorized
(directional) beams with spatial selectivity.
Figure 4.3a shows the SBA patterns and Figure 4.3b illustrated the design network of a
typical SBA system. The SBA system network illustrated is relatively simple to
implement, requiring only a beamforming network, a RF switch, and control logic to
select a specific beam.
Figure 4.3b A Switch-Beam network uses a beamforming network to form M beams from M array elements
Figure 4.3a Switch-Beam Systems can select one of the several beams to enhance receive signals. Beam 2 is selected here for the desired signal
Smart Antenna for Wireless Applications Chapter 4: Smart Antenna System
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Switched beam systems offer numerous advantages of more elaborate smart antenna
systems at a fraction of the complexity and expense. Nevertheless, there are some
limitations to switched beam array, which comprise of the inability to provide any
protection from multipath components that arrive with Directions-of-Arrival (DOAs)
near that of the desire components, and also the inability to take advantage of path
diversity by combining coherent multipath components. Lastly, due to scalloping, the
received power from a user may fluctuate when he moves around the base station.
Scalloping is the roll-off of the antenna pattern as a function of angles as the DOA
varies from the boresight of each beam produced by the beamforming network [8].
In spite of the drawbacks, SBA systems are widespread for various reasons. They
provide some range extension benefits and offer reduction in delay spread in certain
propagation environments. In addition, the engineering costs to implement this low
technology approach are lesser than those associated with more complicated systems.
4.3.2 Adaptive Array
It is possible to achieve greater performance improvements than that obtained using the
SBA system. This can be accomplished by increasing the complexity of the array signal
processing to form the Adaptive Antenna Systems, which is considered to be the most
advance smart antenna approach to date.
The adaptive antenna systems approach communication between a user and the base
station in a different way, in effect adding a dimension in space. By adapting to the RF
environment as it changes, adaptive antenna technology can dynamically modify the
signal patterns to near infinity to optimize the performance of the wireless system.
Adaptive arrays continuously differentiate between the desired signals, multipath, and
interfering signals as well as calculate their directions of arrival by utilizing
sophisticated signal-processing algorithms. The technique constantly updates its
transmitting approach based on changes in both the desired and interfering signal
locations. It ensures that signal links are maximized by tracking and providing users
Smart Antenna for Wireless Applications Chapter 4: Smart Antenna System
37
with main lobes and interferers with nulls, because there are neither microsectors nor
predefined patterns.
Although both systems seek to increase gain with respect to the location of the users,
however, only the adaptive system is able to contribute optimal gain while
simultaneously identifying, tracking, and minimizing interfering signals. This can be
seen from Figure 4.3c that only the main lobe is directed towards the user while a null
being directed at a cochannel interferer. Illustrated in Figure 4.3d is the network
structure of an adaptive array.
Figure 4.3c An adaptive antenna can adjust its antenna pattern to enhance the desired signal, null or reduce interference, and collect correlated multipath power
Figure 4.3d Network structure of an adaptive array structure
Smart Antenna for Wireless Applications Chapter 4: Smart Antenna System
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4.4 Beam Forming
A single output of the array is formed when signals induced on different elements of the
array are combined. A plot of the array response as a function of angle is usually
specified as the array pattern or beam pattern. It can also be known as power pattern
when the power response is plotted.
This method of combining the signals from several elements is understood as beam
forming. The direction in which the array has maximum response is said to be the beam
pointing direction, and thus this is the bearing where the array has the utmost gain.
Conventional beam pointing or beam forming can be achieved by adjusting only the
phase of the signals from different elements. In other words, pointing a beam in the
desired direction. However, the shape of the antenna pattern in this case is fixed, that is,
the side lobes with respect to the main do not change when the main beam is pointed in
different directions by adjusting various phases. Nevertheless, this can be overcome by
adjusting the gain and phase of each signal to shape the pattern as required and the
degree of change will depend upon the number of elements in the array.
For example, signals can also be coupled together without any gain or phase shift in a
linear array, and it is known as broadside to the array, which is, perpendicular to the row
joining all the elements of the array. The array pattern formed thus falls to a low value
on either side of the beam pointing direction and the region of the low value is known as
a null. In this case, it must be noted that the null is actually a position where the array
response is zero and the term should not be misused to denote the low value of the
pattern.
Lastly, it is very convenient to make use of vector notation while working with array
antennas. Thus the term weight vector (w) is introduced. It is important because the
weight vector will have significant impact on the array output.
Smart Antenna for Wireless Applications Chapter 4: Smart Antenna System
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4.4.1 Null Beam Forming
The flexibility of array weighting to being adjusted to specify the array pattern is an
important property. This may be exploited to cancel directional sources operating at the
same frequency as that of the desired source, provided these are not in the direction of
the desired source [10].
In circumstances where the directions of these interferences are identified, cancellation
is feasible by positioning the nulls in the pattern corresponding to these directions and
concurrently steering the main beam in the direction of the desired signal. This approach
of beam forming by placing nulls in the directions of interferences is commonly referred
to as null beam forming or null steering.
4.4.2 Steering Vector
The steering vector contains the response of all elements of the array to a narrow-band
source of unit power. As the response of the array is different in different directions, a
steering vector is associated with each directional source. The uniqueness of this
association depends upon the array geometry [10].
Every component of this vector has unit magnitude for an array of identical elements.
The phase of its ith component is similar to the phase difference between signals
induced on the ith element and the reference element due to the source associated with
the steering vector.
This vector is also known as the space vector because each component of the vector
represents the phase delay that is resulted from the spatial position of the corresponding
element of the array. In addition, it can also be referred to as the array response vector
for it measures the response of the array due to the source under consideration.
Smart Antenna for Wireless Applications Chapter 4: Smart Antenna System
40
4.5 Recursive Least Squares Algorithm
For an adaptive array network as shown in Figure 4.3d, it is essential that the weight
vector to be updated or adapted periodically because the environment (e.g. mobile
environment) is time-variable. Generally, the weight vector computed differs by a small
but significant amount at different cycles.
In addition, because the necessary data to estimate the optimal solution is noisy, it is
beneficial to use an update technique, which uses previous solutions for the weight
vector to smooth the estimate of the optimal response. Thus, an adaptive algorithm is
exploited for updating the weight vector periodically.
There are many types of adaptive algorithms and the majorities are iterative. They
utilized the past information to minimize the computations required at each update
cycle. In iterative algorithms, the current weight vector, w(n), is modified by an
incremental value to form a new weight vector, w(n+1) at each iteration n.
In the later development of adaptive algorithm, the Least Mean Square (LMS) algorithm
and Recursive Least Squares (RLS) algorithm are viewed to be more efficient. However,
in this chapter, we will be only looking at the RLS algorithm as it is regarded to have a
faster convergence speed (the speed for the initial weight vector to reach the optimum
weight vector) compared to LMS. Nevertheless, it is a result of greater computation
complexity. Figure 4.5a illustrated the block diagram representation and signal flow
graph of the RLS algorithm.
Smart Antenna for Wireless Applications Chapter 4: Smart Antenna System
41
The RLS algorithm can be summarized as follow [14]:
Figure 4.5a Representation of RLS algorithm: (i) block diagram (ii) signal-flow graph
Smart Antenna for Wireless Applications Chapter 4: Smart Antenna System
42
P(n) = λ -1P(n-1) - λ -1k(n)uH(n)P(n-1) (4.6)
Convergence Coefficient
0 < λ <1
where,
δ is a small positive number,
I is the M X M identity matrix,
λ is the forgetting factor
k(n) is the gain vector,
αα (n) is the innovation,
w(n) is the weight vector,
P(n) is the inverse of the correlation matrix ΦΦ (n),
u(n) is the input vector and
d(n) is the desired response.
In the RLS method, the desired signal must be supplied using either a training sequence
or decision direction. For the training sequence approach, a brief data sequence is
transmitted which is known by the receiver. The receiver uses the adaptive algorithm to
approximate the weight vector in the training duration, then retains the weights constant
while information is being transmitted. This technique requires that the environment be
stationary from one training period to the next, and it reduces channel throughput by
requiring the use of channel symbols for training. However, in the decision approach,
the receiver uses recreated modulated symbols based on symbol decisions, which are
used as the desired signal to adapt the weight vector [8].
Smart Antenna for Wireless Applications Chapter 5: Multiple Access Schemes
43
Chapter 5
Multiple Access Schemes
5.1 Introduction
Due to the recent development of wireless communication systems, the range of
frequencies available for wireless communication technologies can be utilized in
various ways/schemes, and this is referred to as multiple access schemes. These
techniques are adopted to allow numerous users to share simultaneously a finite amount
of signal spectrum.
The distribution of spectrum is required to achieve this high system capacity by
simultaneously allocating the available bandwidth (or available amount of channels) to
multiple users. This must be accomplished without severe degradation in the
performance of the system in order to achieve high quality communications.
Conventionally, there are three major access schemes used to share the available
bandwidth in a wireless communication. Nonetheless, they are known as the frequency
division multiple access (FDMA), time division multiple access (TDMA), and the code
division multiple access (CDMA).
As a result, there is a lot to debate about which schemes is better. However, the answer
to this depends on the combined techniques, such as the modulation scheme, anti-fading
techniques, forward error correction, and so on, as well as the requirements of services,
such as the coverage area, capacity, traffic, and types of information [11].
Smart Antenna for Wireless Applications Chapter 5: Multiple Access Schemes
44
5.2 Frequency Division Multiple Access (FDMA)
Frequency division multiple access (FDMA) is the most widespread multiple-access
scheme for land mobile communication system due to its ability to discriminate
channels effortlessly by filters in the frequency domain. In FDMA, every subscriber is
allocated to an individual unique frequency band or channel.
Figure 5.2a shows the spectrum of a FDMA system. The allocated system bandwidth is
divided into bands with bandwidth of Wch and guard space between adjacent channels to
prevent spectrum overlapping that may be resulted from carrier frequency instability.
When a user sends a call request, the system will assign one of the available channels to
the user, in which, the channel is used exclusively by that user during a call. However,
the system will reassign this channel to a different user when the previous call is
terminated.
One of the most important advantages in FDMA system is there isn’t any need for
synchronization or timing control and therefore, the hardware is simple. In addition,
there is only a need for flat fading consideration as for anti-fading technique because the
bandwidth of each channel in the FDMA is sufficiently narrow.
Figure 5.2a Spectrum of FDMA systems
Smart Antenna for Wireless Applications Chapter 5: Multiple Access Schemes
45
However, there are also various problems associated with FDMA systems and they are:
• Intermodulation interference increases with the number of carriers .
• Variable rate transmission is difficult because such a terminal has to prepare a lot of
modems. For the same reason, composite transmission of voice and non-voice data
is also difficult.
• High Q-value for the transmitter and receiver filters is required to guarantee high
channel selectivity [11].
5.3 Time Division Multiple Access (TDMA)
In the basic time division multiple access (TDMA) protocol, the transmission time axis
is divided into frames of equal duration, and each frame is divided into the same
number of time slots having equal duration. Each slot position within a frame is
allocated to a different user and this allocation stays the same over the sequence of
frames [12]. This means that a particular user may transmit during one particular slot in
every frame and thus, it has the entire channel bandwidth at its disposal during this slot.
Figure 5.3a illustrated the allocation in a basic TDMA frame with four time slots per
frame with the shaded areas representing the guard times in each slot in which
transmission is prohibited in this region. It is essential to have the guard times as it
prevents transmissions of different (spatially distributed) users from overlapping due to
transmission delay differences.
Figure 5.3a Frame and slot structure with basic TDMA
Smart Antenna for Wireless Applications Chapter 5: Multiple Access Schemes
46
5.4 Code Division Multiple Access (CDMA)
In code division multiple access (CDMA) systems, the signal is multiplied by a very
large bandwidth signal called the spreading signal. The spreading signal is a pseudo-
noise code sequence that as a chip rate which is in orders of magnitudes greater than the
data rate of message [8].
Having its own pseudorandom codeword, all subscribers in a CDMA system use the
same carrier frequency and may transmit simultaneously. Figure 5.4a(i) displays the
spectrum of a CDMA system. The most distinct feature of CDMA system is that all the
terminals share the whole bandwidth, and each terminal signal is discriminated by the
code.
When each user sends a call request to the base station, the base station assigns on of
the spreading codes to the user. When five users initial and hold the calls as shown in
Figure 5.4(ii), time and frequency are occupied as shown in Figure 5.4(iii) [13].
Therefore, CDMA requires a larger bandwidth as compared to FDMA and TDMA.
Furthermore, there is also a need for code synchronization in CDMA system.
Smart Antenna for Wireless Applications Chapter 5: Multiple Access Schemes
47
5.5 Space Division Multiple Access (SDMA)
In addition to these techniques, smart antennas provide a new method of multiple access
to the users, which is known as the space division multiple access (SDMA). The SDMA
(i)
(ii)
(iii)
Figure 5.4a Concept of a CDMA system: (i) spectrum of a CDMA system (ii) a call initiation and holding model for five-user case (iii) channel allocation to each user
Smart Antenna for Wireless Applications Chapter 5: Multiple Access Schemes
48
scheme, which is commonly referred to space diversity, uses smart antenna to provide
control of space by providing virtual channels in an angle domain. With the use of this
approach, simultaneous calls in various different cells can be established at the same
carrier frequency.
The SDMA scheme is based upon the fact that a signal arriving from a distant source
reaches different antennas in an array at different times due to their spatial distribution,
and this delay is utilized to differentiate one or more users in one area from those in
another area [10].
This technique enables an effective transmission to take place in one cell without
affecting the transmission in another cell. Without the use of an array, this can be
accomplished by having a separate base station for each cell and keeping cell size
permanent, while the use of space diversity enables dynamic changes of cell shapes to
reflect the user movement.
Thus, an array of antennas constitutes to an extra dimension in this system by providing
dynamic control in space and needless to say, it leads to improved capacity and better
system performance.
Smart Antenna for Wireless Applications Chapter 6: Analysis of Array Antennas
49
Chapter 6
Analysis of Array Antennas
6.1 Aim and Procedures
Previous chapters had provided basic concept on antennas and smart antenna systems.
Thus, this will greatly contribute to further understanding the operation of smart antenna
systems. As a result, it would be appropriate to study the basic of antenna arrays, its
radiation pattern and performance.
This chapter will be covering the analysis of linear and planar arrays of microstrip patch
antennas. The design of microstrip patch antennas was studied and implemented before
carrying out simulations using software programs known as the Personal Computer
Antenna Aided Design (PCAAD) and MATLAB.
Various parameters were altered to study the effects that would be reflected on antenna
arrays. The simulated results achieved were tabulated. In addition, polar plots were also
generated to cater for a better visualization and analysis. Last but not least, the chapter
will conclude with some discussions on the results achieved.
6.2 Microstrip Patch Antenna Design
The microstrip rectangular patch antenna is by far the most widely used configuration.
Therefore, we will be designing the rectangular patch for the linear and planar array
simulations. Several factors contribute to the design of a microstrip rectangular patch
antenna. Figure 2.2b(i) shows some of the parameters constrain for the design, which
Smart Antenna for Wireless Applications Chapter 6: Analysis of Array Antennas
50
include the length and width of the antenna patch, the type of substrate used and the
substrate thickness.
In addition, the center/resonant frequency must also be determined. A resonant
frequency of 2GHz is chosen because in 1992, the World Administrative Radio
Commission (WARC) of the International Telecommunications Union (ITU)
formulated a plan to implement a global frequency band in the 2000 MHz range that
would be common to all countries for the universal wireless communication systems
[8].
The dimensions of a rectangular patch antenna can be determined using the following
equations:
Width, W = 2λ
2/1
2
)1( −
+rε (6.1)
Length, L = le
∆−
2
)*2( ελ
(6.2)
where the effective dielectric constant, εe and l∆ are given by:
Effective dielectric constant, εe = 2/112
12
12
1 −
+
−++
Wtrr εε
(6.3)
where t is the thickness of the substrate.
l∆ = (0.412t)( )
( )
+−
++
8.0258.0
264.03.0
tW
e
tW
e
ε
ε (6.4)
Patch size calculation:
Assuming a rectangular line-fed configuration.
Smart Antenna for Wireless Applications Chapter 6: Analysis of Array Antennas
51
Assuming resonant frequency, f = 2GHz
Assuming typical substrate of dielectric constant, εr = 2.2
Assuming substrate thickness, t = 0.5 cm
Wavelength, λ = C/f
= (3*108)/2GHz
= 0.15m
where C is the free space velocity of light.
Width, W = 2λ
2/1
2
)1( −
+rε
= 2/1
2)12.2(
215.0 −
+
= 0.0593 m
= 5.93 cm
Effective dielectric constant, εe = 2/112
12
12
1 −
+
−++
Wtrr εε
= 2/1
93.55.0*12
12
12.22
12.2 −
+
−++
= 2.02300444384
l∆ = (0.412t)( )
( )
+−
++
8.0258.0
264.03.0
tW
e
tW
e
ε
ε
= (0.412*0.5)
+−
++
8.05.093.5
)258.002.2(
264.05.093.5
)3.002.2(
= 0.2597
Smart Antenna for Wireless Applications Chapter 6: Analysis of Array Antennas
52
Length, L = le
∆−
2
)*2( ελ
= )2597.0*2(02.2*2(
100*15.0 −
= 4.75 cm
Nevertheless, after much testing, it was observed from PCAAD simulations that a
length of 4.75 cm (Figure 6.2a) does not produce a minimum voltage standing wave
ratio (VSWR)* compared to a length of 4.706 cm (Figure 6.2b) at 2GHz. Hence, the
new dimension of Length = 4.706 cm and Width = 5.93 cm is selected for applications
to the simulations.
*Note: The reflected waves from the interface between the source and the antenna
create, along with the travelling waves from the source towards the antenna,
constructive and destructive patterns, referred to as standing waves. Thus, when the
impedance of the antenna (load) to the characteristic impedance of the transmission line
matched, a desired minimum VSWR is achieved.
Figure 6.2a VSWR plot for length of 4.75 cm
VSWR ≈ 4.6
Smart Antenna for Wireless Applications Chapter 6: Analysis of Array Antennas
53
Figure 6.2c illustrated the radiation pattern of the single microstrip patch antenna with
simulated results of:
Bandwidth = 3.9%
Efficiency = 97.6%
Directivity = 7.2
Figure 6.2b VSWR plot for length of 4.706 cm
VSWR ≈ 4.4
Figure 6.2c Radiation pattern for single microstrip patch antenna
-30 -20 - 10 0
Note that the values here will be the same for all polar plots unless otherwise stated
Smart Antenna for Wireless Applications Chapter 6: Analysis of Array Antennas
54
6.3 Simulation on Linear Array Antenna
Linear array is the simplest and commonly used configuration. Therefore, it is essential
to investigate its performance. There are four basic factors influencing the performance
of the linear array antenna and this section will be examining a linear array of microstrip
patch antennas. The four influencing factors, which consists of the inter-element
spacing, number of elements in an array, the amplitude distribution and the phase
excitation, will be varied, and all observation will be monitored. The subsequent
simulations on linear array will be performed using the PCAAD program with the
following predefined parameters:
Microstrip antenna patch length, L = 4.706 cm
Microstrip antenna patch width, W = 5.93 cm
Substrate thickness, t = 0.5 cm
Dielectric constant = 2.2
Center frequency, f = 2GHz
Wavelength, λ = 15 cm
Assuming the element polarization is in the X-direction.
6.3.1 Effect of Varying Inter-element Spacing, d
The following assumptions are made:
• Phase shift = zero degree
• Amplitude distribution = uniform
• Number of elements in the array = 8
PCAAD simulations were carried out and the results were tabulated in Table 6.1. Figure
6.3a illustrated the radiation pattern for an inter-element spacing of ¼ wavelength and
Figure 6.3b displayed the pattern for an inter-element spacing of one wavelength.
Smart Antenna for Wireless Applications Chapter 6: Analysis of Array Antennas
8. J.C. Liberti Jr. and T.S. Rappaport: “Smart Antennas for Wireless
Communications: IS-95 and Third Generation CDMA Applications,” Prentice
Hall, Upper Saddle River, New Jersey, 1999.
9. I.E. Sutherland et al., “Experimental Evaluation of Smart Antenna System
Performance for Wireless Communications,” IEEE Transactions on Antennas
and Propagation, Vol. 46, No. 6, Jun. 1998, pp. 794-757.
10. L.C. Godara, “Applications of Antenna Arrays to Mobile Communication, Part
I: Performance, Improvement, Feasibility and Systems Considerations,”
Proceedings of the IEEE, Vol. 85, No. 7, Jul. 1997, pp. 1029-1060.
11. S. Sampei: “Applications of Digital Wireless Technologies to Global Wireless
Communications,” Prentice Hall, Upper Saddle River, New Jersey, 1997.
12. R. Prasad: “CDMA for Wireless Personal Communications,” Artech House,
Boston, 1996.
13. T.S. Rappaport: “Wireless Communications: Principles & Practice,” Prentice
Hall, Upper Saddle River, New Jersey, 1996.
14. S.Haykin: “Adaptive Filter Theory,” Prentice Hall, New Jersey, 1991.
Appendix A
% Planar Array teta = -90:1:90; theta = teta*pi/180; ph = 0; % E-plane, phi = 90 for H-plane phi = ph*pi/180; % Convert to radian bx = 0; % Phase shift in x-direction by = 0; % Phase shift in y-direction beta_x = (bx/180)*pi; % Convert to radian beta_y = (by/180)*pi; % Convert to radian x = 8.3; % Inter-element spacing in x-direction y = 9; % Inter-element spacing in y-direction k = (2*pi)/(3e10/2e9); % Wave number % Progreesive phase value in x-direction phix = k*x*sin(theta).*cos(phi) + beta_x; % Progreesive phase value in y-direction phiy = k*y*sin(theta).*sin(phi) + beta_y; ix1 = 1; % Excitation of each element ix2 = 1; iy1 = 1; iy2 = 1; % Array factor of array in the x-direction Sx = ix1+ix2*exp(j*phix); % Array factor of array in the y-direction Sy = iy1+iy2*exp(j*phiy); load A:\eplane.dat % load data of single element Edb = eplane(:,2); E = 10.^(Edb/20); % Covert to ratio subplot(2,2,1); % Define plot area % Plot radiation pattern of a Single microstrip element h = polar(theta',abs(E)); set(h,'color','red'); h = ylabel('Single Element'); set(h,'color','red');
subplot(2,2,2); % Define plot area AF = abs(Sx.*Sy); % Array factor of planar array % Plot radiation pattern of array factor of planar array h1 = polar(theta,AF); set(h1,'color','magenta'); h1 = ylabel('Array Factor'); set(h1,'color','magenta'); subplot(2,2,3); % Define plot area overall = abs(AF'.*E); % Pattern multiplication % Plot overall radiation pattern of planar array h2 = polar(theta',overall); set(h2,'color','blue'); h2 = ylabel('Overall pattern'); set(h2,'color','blue'); subplot(2,2,4); % Define plot area range_x = pi*(90/pi); range_x1 = -pi*(90/pi); theta = linspace(range_x1,range_x,181); h3 = plot(theta',overall); % Plot in rectangular pattern % Initialize y-axis for linear plot x_axis = pi*(90/pi); % Initialize y-axis for linear plot x_axis1 = -pi*(90/pi); axis([x_axis1 x_axis exp(-4) 5]); % Plot linear pattern set(h3,'color','green'); h3 = ylabel('Linear Plot'); set(h3,'color','green'); grid;
Appendix B
%This program uses the Woodward-Lawson synthesis, to design a %radiation pattern for a 10 elements uniform linear %array with an element spacing of one half the wavelength. t = 0:1:180; theta = t*pi/180; p5m = 1; n5m = -1; %cos(theta-m) p4m = 0.8; n4m = -0.8; p3m = 0.6; n3m = -0.6; p2m = 0.4; n2m = -0.4; p1m = 0.2; n1m = -0.2; p0m = 0; b5m = 0; nb5m = 0; %Excitation at the sample points b4m = 0; nb4m = 0; b3m = 1; nb3m = 1; b2m = 1; nb2m = 1; b1m = 1; nb1m = 1; b0m = 1; a5 = cos(theta) - p5m; %Pattern of each composing function AF5 = ((sin(5.*pi.*a5))./(sin((pi.*a5)./2)).*b5m)./10; a4 = cos(theta) - p4m; AF4 = ((sin(5.*pi.*a4))./(sin((pi.*a4)./2)).*b4m)./10; a3 = cos(theta) - p3m; AF3 = ((sin(5.*pi.*a3))./(sin((pi.*a3)./2)).*b3m)./10; a2 = cos(theta) - p2m; AF1 = ((sin(5.*pi.*a2))./(sin((pi.*a2)./2)).*b2m)./10; a1 = cos(theta) - p1m; AF2 = ((sin(5.*pi.*a1))./(sin((pi.*a1)./2)).*b1m)./10; a0 = cos(theta) - p0m; AF0 = ((sin(5.*pi.*a0))./(sin((pi.*a0)./2)).*b0m)./10; an1 = cos(theta) - n1m; AFn1 = ((sin(5.*pi.*an1))./(sin((pi.*an1)./2)).*nb1m)./10; an2 = cos(theta) - n2m; AFn2 = ((sin(5.*pi.*an2))./(sin((pi.*an2)./2)).*nb2m)./10;
clc; % Clear screen clear; % Clear all variables % Prompt user for number of elements in an array. disp(' ') disp('Please enter the number of elements in the array.') disp(' ') disp('Assuming that the array has at least 2 elements') disp('but not more than 10 elements.') disp(' ') N = input(['Number of elements in the array = ']); % Prompt user for the required Side Lobe Level. clc; disp(' ') disp('Please enter the required side lobe level in decibels.') disp(' ') SLL = input(['Side lobe level(dB) = ']); R = 10^(SLL/20); % Convert to ratio Zo = cosh((1/(N-1))*acosh(R)); % Determine Zo % Prompt user for the Normalised Inter-element Spacing. clc; disp(' ') disp('Please enter one of the following inter-element spacing (Normalised).') disp(' ') disp('Press "a" for 1/4 wavelength.') disp('Press "b" for 1/2 wavelength.') disp('Press "c" for 3/4 wavelength.') disp('Press "d" for full wavelength.') disp(' ') a=0.25; b=0.5; c=0.75; d=1; spacing = input(['Inter-element spacing (Normalised) = ']); t = 0:1:179; theta = t*pi/180; % Convert to radian u = pi*spacing*cos(theta); clc; disp(' ')
disp(['Number of elements = ' num2str(N)]) disp(['Side lobe level = ' num2str(SLL) ' dB']) disp(['Inter-element spacing (Normalised) = ' num2str(spacing)]) if N<=10 if N == 2; AFp = [1]; % Polynomial of excitation coefficient AFc = [1*Zo]; % Chebyshev polynomial X = AFp\AFc; % Determine the excitation coefficient Xo = X/X(1,1); % Normalized with respect to the amplitude % of the elements at the edge AF = abs(Xo(1,1)*cos(u)); % Determine the array factor subplot(2,2,1); polar(theta,AF); % Generate polar plot AF1=20*log10(AF); % Convert to decibels max=max(AF1); % Setting maximum value of the % array factor to "max" AF2=AF1-max; % Set values of array factor % with respect to maximum value theta1=(180/pi)*theta; subplot(2,2,2); plot(theta1,AF2); % Generate linear plot axis([0 180 -40 0]); % Set maximum and minimum %values for X and Y scales grid % Turn grid on elseif N == 3; AFp = [0,2; 1,-1]; % Polynomial of excitation coefficient AFc = [2*Zo^2; % Chebyshev polynomial -1]; X = AFp\AFc; % Determine the excitation coefficient Xo = X/X(2,1); % Normalized with respect to the amplitude % of the elements at the edge % Determine the array factor AF = abs(Xo(1,1)+Xo(2,1)*cos(2*u));
subplot(2,2,1); polar(theta,AF); % Generate polar plot AF1=20*log10(AF); % Convert to decibels max=max(AF1); % Setting maximum value of the % array factor to "max" AF2=AF1-max; % Set values of array factor % with respect to maximum value theta1=(180/pi)*theta; subplot(2,2,2); plot(theta1,AF2); % Generate linear plot axis([0 180 -40 0]); % Set maximum and minimum % values for X and Y scales grid % Turn grid on elseif N==4; AFp = [0,4; 1,-3]; % Polynomial of excitation coefficient AFc = [4*Zo^3; -3*Zo]; % Chebyshev polynomial X = AFp\AFc; % Determine the excitation coefficient Xo = X/X(2,1); % Normalized with respect to the % amplitude of the elements at the edge % Determine the array factor AF = abs(Xo(1,1)*cos(u)+Xo(2,1)*cos(3*u)); subplot(2,2,1); polar(theta,AF); % Generate polar plot AF1=20*log10(AF); % Convert to decibels max=max(AF1); % Setting maximum value of the % array factor to "max" AF2=AF1-max; % Set values of array factor % with respect to maximum value theta1=(180/pi)*theta; subplot(2,2,2); plot(theta1,AF2); % Generate linear plot axis([0 180 -40 0]); % Set maximum and minimum % values for X and Y scales grid % Turn grid on elseif N == 5; AFp = [0,0,8; 0,2,-8; 1,-1,1]; % Polynomial of excitation coefficient
AFc = [8*Zo^4; -8*Zo^2; 1]; % Chebyshev polynomial X = AFp\AFc; % Determine the excitation coefficient Xo = X/X(3,1); % Normalized with respect to the % amplitude of the elements at the edge % Determine the array factor AF = abs(Xo(1,1)+Xo(2,1)*cos(2*u)+Xo(3,1)*cos(4*u)); subplot(2,2,1); polar(theta,AF); % Generate polar plot AF1=20*log10(AF); % Convert to decibels max=max(AF1); % Setting maximum value of the % array factor to "max" AF2=AF1-max; % Set values of array factor % with respect to maximum value theta1=(180/pi)*theta; subplot(2,2,2); plot(theta1,AF2); % Generate linear plot axis([0 180 -40 0]); % Set maximum and minimum % values for X and Y scales grid % Turn grid on elseif N==6; AFp = [0,0,16; 0,4,-20; 1,-3,5,]; % Polynomial of excitation coefficient AFc = [16*Zo^5; -20*Zo^3; 5*Zo]; % Chebyshev polynomial X = AFp\AFc; % Determine the excitation coefficient Xo = X/X(3,1); % Normalized with respect to the % amplitude of the elements at the edge % Determine the array factor AF = abs(Xo(1,1)*cos(u)+Xo(2,1)*cos(3*u)+Xo(3,1)*cos(5*u)); subplot(2,2,1); polar(theta,AF); % Generate polar plot AF1=20*log10(AF); % Convert to decibels max=max(AF1); % Setting maximum value of the % array factor to "max" AF2=AF1-max; % Set values of array factor % with respect to maximum value theta1=(180/pi)*theta;
subplot(2,2,2); plot(theta1,AF2); % Generate linear plot axis([0 180 -40 0]); % Set maximum and minimum % values for X and Y scales grid % Turn grid on elseif N == 7; AFp = [0,0,0,32; 0,0,8,-48; 0,2,-8,18 1,-1,1,-1]; % Polynomial of excitation coefficient AFc = [32*Zo^6; -48*Zo^4; 18*Zo^2; -1]; % Chebyshev polynomial X = AFp\AFc; % Determine the excitation coefficient Xo = X/X(4,1); % Normalized with respect to the % amplitude of the elements at the edge % Determine the array factor AF = abs(Xo(1,1)+Xo(2,1)*cos(2*u)+Xo(3,1)*cos(4*u)+Xo(4,1)*cos(6*u)); subplot(2,2,1); polar(theta,AF); % Generate polar plot AF1=20*log10(AF); % Convert to decibels max=max(AF1); %Setting maximum value of the % array factor to "max" AF2=AF1-max; % Set values of array factor % with respect to maximum value theta1=(180/pi)*theta; subplot(2,2,2); plot(theta1,AF2); % Generate linear plot axis([0 180 -40 0]); % Set maximum and minimum % values for X and Y scales grid % Turn grid on elseif N==8; AFp = [0,0,0,64; 0,0,16,-112; 0,4,-20,56; 1,-3,5,-7,]; % Polynomial of excitation coefficient
AFc = [64*Zo^7; -112*Zo^5; 56*Zo^3; -7*Zo]; % Chebyshev polynomial X = AFp\AFc; % Determine the excitation coefficient Xo = X/X(4,1); % Normalized with respect to the % amplitude of the elements at the edge % Determine the array factor AF = abs(Xo(1,1)*cos(u)+Xo(2,1)*cos(3*u)+Xo(3,1)*cos(5*u)+Xo(4,1)*cos(7*u)); subplot(2,2,1); polar(theta,AF); % Generate polar plot AF1=20*log10(AF); % Convert to decibels max=max(AF1); % Setting maximum value of the % array factor to "max" AF2=AF1-max; % Set values of array factor % with respect to maximum value theta1=(180/pi)*theta; subplot(2,2,2); plot(theta1,AF2); % Generate linear plot axis([0 180 -40 0]); % Set maximum and minimum % values for X and Y scales grid % Turn grid on elseif N == 9; AFp = [0,0,0,0,128; 0,0,0,32,-256; 0,0,8,-48,160; 0,2,-8,18,-32; 1,-1,1,-1,1]; % Polynomial of excitation coefficient AFc = [128*Zo^8; -256*Zo^6; 160*Zo^4; -32*Zo^2; 1]; % Chebyshev polynomial X = AFp\AFc; % Determine the excitation coefficient Xo = X/X(5,1); % Normalized with respect to the % amplitude of the elements at the edge
% Determine the array factor AF = abs(Xo(1,1)+Xo(2,1)*cos(2*u)+Xo(3,1)*cos(4*u)+Xo(4,1)*cos(6*u)+Xo(5,1)*cos(8*u)); subplot(2,2,1); polar(theta,AF); % Generate polar plot AF1=20*log10(AF); % Convert to decibels max=max(AF1); % Setting maximum value of the % array factor to "max" AF2=AF1-max; % Set values of array factor % with respect to maximum value theta1=(180/pi)*theta; subplot(2,2,2); plot(theta1,AF2); % Generate linear plot axis([0 180 -40 0]); % Set maximum and minimum % values for X and Y scales grid % Turn grid on elseif N==10; AFp = [0,0,0,0,256; 0,0,0,64,-576; 0,0,16,-112,432; 0,4,-20,56,-120; 1,-3,5,-7,9]; % Polynomial of excitation coefficient AFc = [256*Zo^9; -576*Zo^7; 432*Zo^5; -120*Zo^3; 9*Zo]; % Chebyshev polynomial X = AFp\AFc; % Determine the excitation coefficient Xo = X/X(5,1); % Normalized with respect to the % amplitude of the elements at the edge % Determine the array factor AF = abs(Xo(1,1)*cos(u)+Xo(2,1)*cos(3*u)+Xo(3,1)*cos(5*u)+Xo(4,1)*cos(7*u)+Xo(5,1)*cos(9*u)); subplot(2,2,1); polar(theta,AF); % Generate polar plot AF1=20*log10(AF); % Convert to decibels max=max(AF1); % Setting maximum value of the % array factor to "max"
AF2=AF1-max; % Set values of array factor % with respect to maximum value theta1=(180/pi)*theta; subplot(2,2,2); plot(theta1,AF2); % Generate linear plot axis([0 180 -40 0]); % Set maximum and minimum values %for X and Y scales grid % Turn grid on end else disp(' ') disp(' ') disp(' ') disp(' ') disp(' ') disp(' ') disp(' ') disp(' ') disp(' ') disp(' ') disp('Invalid value !!!') disp(' ') disp('Press any key to exit........') pause end
Appendix D
%Number of element M = input(['Number of elements in array : ']); %Direction of desired signal a = input(['Steering angle in degrees : ']); x = a*pi/180; %Convert to radian % initializing the algorithm I = eye(M); %M X M identity matrix delta = 1e-6; %Small positive constant P0 = inv(delta)*I; %Initialize the algorithm w0 = (linspace(0,0,M))'; %Initial weight vector for n = 1:100 %Number of iterations B = steeringv(M,x); %Steering vector X = B'*n; wq = B'; dn = conj(wq)'*X; %Desired response vector un = X; %Input data vector forget = 0.95; %Forgetting factor pin = un'*P0; %Calculate pi(n) kn = forget + (pin*un); %Calculate k(n) Kn = (P0*un)/(kn); %Calculate K(n) an = dn - (conj(w0)'*un); %Calculate alpha(n) wn = w0 + (Kn*conj(an)); %Calculate w(n) Pn = (1/forget)*[P0 - [P0*un*un'*P0]/kn];%Calculate P(n) P0 = Pn; w0 = wn; end for ang = -90:1:90 %linear plot n = 91 + ang; angl(n) = ang; x = ang*pi/180; A = steeringv(M,x); out = (w0'*A'); %Multiply with steering vector output(n) = abs(out); output1 = output/max(output); %Normalize to unity xlabel('Angle in degree'); ylabel('Normalize array gain') plot(angl,output1); %Plot linear pattern axis tight;
xlabel('Angle (Degrees)'); ylabel('Normalized Array Gain (Ratio)'); grid; end pause ang = -90:1:90; y = ang*pi/180; polar(y,output1); %polar plot function S = steeringv(M,x); %define steering vector %free space wavelength of 15cm at resonant freq of 2GHz lamda = 0.15; d = lamda/2; %inter-element spacing K = 1:M; %x is DOA of the received signal S = exp((-2*pi*j*(K-1)*d*sin(x))/lamda);