a coskun thesis revised

8/14/2019 A Coskun Thesis Revised

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Technische Universitt MnchenLehrschul fr Hochfrequenztechnik

Prof. Dr. Techn. Peter Russer

Master Thesis

Complexity-Reduced Wideband Beamforming

Ahmet Coskun

25.11.2004

Supervised by:

Prof. Dr. Techn. Peter Russer


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Dr. Tuan Do-Hong

This is a technical report of master thesis carried out at the Technische Universitt Mnchen in

Munich, Germany. This is in partial fulfillment of the degree leading to the Masters of Science in

Microwave Engineering. I would like to express my thanks to my supervisor Dr. Tuan Do-Hong

extensively for the wonderful help, guidance and support he provided throughout my thesis. I am also

grateful to Mr. Biscontini for his support.


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In memory of my grandfather


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Contents

List of Figures.......................................................................................................................6

List of Tables.........................................................................................................................7

Abstract.................................................................................................................................8

Chapter 1.............................................................................................................................10

1.1. Smart Antennas for Wireless Communication Systems.............................................10

1.1.1. Smart Antenna Classifications................................................................................11

1.1.2. Benefits of Smart Antennas....................................................................................14

1.1.3 Applications of Smart Antennas..............................................................................201.2. Antenna Arrays .........................................................................................................25

1.2.1. Basic Antenna Array Parameters............................................................................261.2.2. Linear Arrays..........................................................................................................30

1.2.3. Pattern Multiplication.............................................................................................32

1.2.4. Planar Arrays...........................................................................................................33Chapter 2.............................................................................................................................35

2.1. Introduction................................................................................................................35

2.2. Continuous Apertures.................................................................................................35

2.3. Discrete Apertures......................................................................................................38

2.4. Sparse Linear Arrays..................................................................................................40

2.4.1. Minimum Redundancy and Minimum Hole Arrays...............................................412.4.2. Simulation Results..................................................................................................43

Chapter 3.............................................................................................................................57

3.1.Introduction.................................................................................................................57

3.2. Beamforming..............................................................................................................57

3.2.1. Digital Beamforming vs. Analog Beamforming.....................................................57

3.2.2. Narrowband Beamforming vs. Wideband beamforming........................................593.3. Frequency-Invariant Wideband Beamforming............................................................63

3.4.Frequency-Domain Frequency-Invariant Beamformer (FDFIB)...................................65

3.4.1.Simulation Numerical Results.................................................................................66Conclusion..........................................................................................................................69

References...........................................................................................................................71


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List of Figures


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List of Tables


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Abstract

With the exponential growth of the demand for wireless communication systems, people are faced with

two main problems: Capacity and coverage area. Smart antennas offer many ways to improve

wireless system performance because of having the capability to overcome the major impairments

(multipath fading, delay spread, co-channel interference) of the wireless communication systems.

Smart antennas can provide potential improvements including capacity increase, range extension,

data rate increase, interference suppression, multipath diversity, and new services. Therefore, the

technology of smart antenna (SA) has received enormous interest worldwide in recent years.

There are many levels of smart antennas systems, ranging from those that require less signal

processing to extremely advanced signal processing systems requiring antenna arrays at both the

transmitter and receiver. Smart antennas offer a new form of multiple access, which is known as

space-division multiple access (SDMA). The latest form of SDMA uses adaptive antenna arrays, which

is mainly dependent on digital beamforming techniques. In this approach, the main beam of the

radiation pattern is directed towards the desired user directions, while the nulls or the side lobes with

very low levels are adjusted towards undesired users or interferers. Furthermore, the radiation pattern

can be adjusted to receive multipath signals that can be combined. This approach maximizes the

signal-to-interference and noise ratio (SINR) of a desired signal. However, digital beam forming

requires a large number of elements (including antenna elements, receiver modules, A/D converters

etc.) as well as more computational effort, resulting in high cost and system complexity.

Therefore, in this thesis, the first objective will be to reduce the number of elements, while retaining

same array size and same main beamwidth/side-lobe level (SLL). To reach to our aim, we will be

mainly concerned with the design of the array geometry. We will use non-uniform inter-element

spacing in the array instead of traditional uniform element spacing. These kinds of arrays are also

called sparse sampled (or thinned) arrays, which provide a large aperture with few antenna elements.

Sparse arrays are antenna arrays that originally were adequately sampled, but where several

elements have been removed. Sparsely sampled arrays have been used or proposed in several fields

such as radar, sonar, ultrasound imaging and seismics. We will show that this approach is also very

suitable for the smart antenna area.

On the other hand, in future wireless communication systems, wideband signals will be used to fulfil

the requirements of higher data services. Therefore, smart antennas for wideband signals (wideband


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smart antennas) will be the key solution for reliable high-data-rate wireless channels. However, when it

is desired to receive signals over a broad band of frequencies, the problem of wide-banding an

antenna array arises. Because the beamwidth of a linear array decreases as frequency increases.

Thus, as a second objective, we will investigate a frequency-invariant beam-forming scheme within atruly wide bandwidth. We first summarize traditional frequency-invariant beam-forming methods and

then propose a frequency-domain frequency-invariant beam former. Moreover, this method is suitable

to operate with arbitrary antenna arrays like uniformly/non-uniformly spaced arrays or one/multi

dimensional arrays. Therefore, we are able to apply the method to the array geometries, which have

been proposed in the first part.


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Chapter 1

1.1. Smart Antennas for Wireless Communication Systems

It is foreseen that in the future an enormous increase in traffic will be experienced for mobile and

personal communications systems. As the number of mobile subscribers increases rapidly, combined

with a demand for more sophisticated mobile services requiring higher data rates, the operators are

forced to investigate different methods to add more capacity into their networks [ 1]. An application of

antenna arrays has been suggested in recent years for mobile communication systems to overcome

the problem of limited channel bandwidth, thereby satisfying an ever growing demand for a large

number of mobiles on communication channels [2].

In the literature, adaptive antennas or intelligent antennas are sometimes preferred instead of the

expression smart antennas. However, we will always prefer to use smart antenna expression. Smart

antenna technology currently provides a viable solution to capacity-strained networks and lends itself

to the migration to third generation (3G) and fourth generation (4G) mobile communication systems.

This new method is to separate the users by their position, exploiting the fact that users normally are

positioned randomly in a cell. A smart antenna is an antenna system that is able to direct the beam ateach individual user, allowing the users to be separated in the spatial domain.

The technology of smart antennas for mobile communications has received huge interest worldwide in

recent years. The main motivation for the smart antennas is the capacity increase, however we should

take into account the other benefits which smart antennas provide like range increase, better signal

quality and new services.

Many base station antennas have up till now been omnidirectional or sectored. This leads to "waste"

of power owing to radiation in other directions than toward the user. Furthermore, other users will

experience the power radiated in other directions as interference. The idea of smart antennas is to use

antenna patterns that are not fixed, but adapt to the current conditions. This can be visualized as the

antenna directing a beam toward the communication partner only. Smart antennas will result in a

much more efficient use of the power and spectrum, increasing the useful received power as well as

reducing interference.


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The difference between a smart antenna and a dumb antenna is the property of having and adaptive

and fixed lobe pattern, respectively [3]. The main philosophy is that interferes seldom have the same

geographical location as the user. By maximizing the antenna gain in the desired direction and

simultaneously placing minimal radiation pattern in the directions of the interferers, the quality of thecommunication link can be significantly improved.

Not the antenna itself, but rather the complete antenna system including the signal processing is

adaptive or smart. As in Figure 1.1, a smart antenna system combines multiple antenna elements with

a digital signal processing capability to optimise its radiation and/or reception pattern automatically in

response to the signal environment. A smart antenna consists of antenna elements, whose signals are

processed adaptively in order to exploit the spatial dimension of the mobile radio channel. In the

simplest case, the signals received at the different antenna elements are multiplied with complex

weights, and then summed up; the weights are chosen adaptively. Such a system can automatically

change the directionality of its radiation patterns in response to its signal environment (current channel

and user characteristics). This can dramatically increase the performance characteristics (such as

capacity) of a wireless system.

Figure 1.1: Basic principle of smart antennas

1.1.1. Smart Antenna Classifications

Smart antenna systems are classified on the basis of their transmit strategy, into the following three

types. This kind of definition is also called levels of intelligence [3].

conventional


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Switched Beam Antennas

This is also called switch beam. Switched-beam systems consist of multiple narrow beams, the best of

which is used to serve the subscriber as it moves through the coverage of the cell. They have only abasic switching function between separate directive antennas or predefined beams of an array. The

setting that gives the best performance, usually in terms of received power, is chosen as in Figure 1.2.

Because of the higher directivity compared to a conventional antenna, some gain is achieved. Such an

antenna is easier to implement in existing cell structures than the more sophisticated adaptive arrays,

but it gives a limited improvement.

Figure 1.2: Switched beam antennas

Dynamically Phased Arrays

The beams are predetermined and fixed in the case of a switched beam system. A user may be in the

range of one beam at a particular time but as he moves away from the centre of the beam, the

received signal becomes weaker and an intracell handover occurs. Once a signal becomes too weak,

the switching centre reassigns a new traffic channel closer to the phone and asks the phone to tune to

this new channel. This is known as handover, a process that is generally transparent to the mobile

user. But in dynamically phased arrays, by including a direction of arrival (DOA) algorithm for the

users signal as in Figure 1.3, continuous tracking can be achieved. So even when the intra-cell

handover occurs, the users signal is received with an optimal gain. DOA of user is first estimated, and

then the beam-forming weights are calculated in accordance with the DOA of user. In this case also,

the received power is maximized.


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Figure 1.3: Dynamically-phased arrays

Adaptive Antenna Arrays

Adaptive antenna arrays can be considered the most intelligent of all. In this case, DoA algorithm for

determining the direction toward interference sources is added as well. The radiation pattern can be

adjusted to null out the interferers. In addition, by using special algorithms and space diversity

techniques,

The radiation pattern can be adapted to receive multipath signals, which are combined.

These techniques will maximize the SINR (signal to interference and noise ratio) of a desired signal.

This procedure is also known as optimum combining, adaptive beam-forming or digital beam-

forming. In summary, adaptive antenna arrays can make three type of optimisation to the received

signal as in Figure 1.4.

1) DoA estimation for the desired user

2) DoA estimation for the interference (to null out)

3) Multipath user signal combination

DOA

DOA


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Figure 1.4: Adaptive antenna arrays

1.1.2. Benefits of Smart Antennas

We will explain the advantages of smart antennas briefly. More information can be found in [2], [3], [4],

[5], [6], [7].

Capacity and spectrum efficiency improvement

Smart antennas can increase the capacity of a wireless communication system significantly. Spectrum

efficiency implies the amount of traffic a system with certain spectrum allocation can handle. Channel

capacity refers to the maximum data rate a channel of given bandwidth could sustain. An increase in

the number of users of the communication system without a loss of performance causes the spectrum

efficiency and capacity increase.

The spectrum efficiency E, measured in channels/km2/MHz, is expressed as

1t ch

t c c ch c c

B BE

B N A B N A= = (1.1)

where tB is the total bandwidth of the system available for voice channels (transmit or receive), in

MHz, chB is the bandwidth per voice channel in MHz, cN is the number of cells per cluster, and cA is

the area per cell in square kilometres. The capacity of a system is measured in channels/km2

and is

given by


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t cht

ch c c c c

B NC EB

B N A N A= = = (1.2)

wheret

ch

ch

BN

B= is the total number of available transmit or receive voice channels in the system.

The actual number of users that can be supported can be calculated based on the traffic offered by

each user and the number of channels per cell. From (1.2), it is evident that capacity can be increased

in several ways. These include increasing the total bandwidth allocated to the system, reducing the

bandwidth of a channel through efficient modulation, decreasing the number of cells in a cluster, and

reducing the area of a cell through cell splitting. If somehow more than one user can be supported per

RF channel, this will also increase capacity.

Increased capacity can be achieved when the SNR level is improved through digital signal processing

techniques that place desired signals in or near the narrower main beam of the multibeam antenna

and place interferers into the pattern side lobes and/or nulls [8]. This can mainly accomplished in two

ways in smart antenna systems. First, the increased quality of service resulting from the reduced co-

channel interferences and reduced multipath fading may be traded to increase the number of users,

leading to increased spectrum efficiency and capacity improvement. Second, an array may be used in

order to create additional channels by forming multiple beams without any extra spectrum allocation,

which results in potentially extra users and thus increases the spectrum efficiency.

Range extension

The coverage, or coverage area, is simply the area in which communication between a mobile and thebase station is possible. In sparsely populated areas, extending coverage is often more important than

increasing capacity. In such areas, the gain provided by adaptive antennas can extend the range of a

cell to cover a larger area and more users than would be possible with omnidirectional or sector

antennas. In Fig. 1.5, it is easy to grasp the different approaches between a conventional

omnidirectional antenna and smart antenna. Hexagons in both cases show the coverage area of the

wireless system.

A coverage area varies with antenna gain as


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(2 )cA G (1.3)

where is the path loss exponent, which is typically between 3 and 4, G is either transmit or receive

antenna gain, and the gain of the other antenna is held constant.

(a) Omnidirectional antenna (b) Smart antenna

Figure 1.5: Range extension using a smart antenna

From (1.3), it can be seen that, the range increase can be achieved by improving the gain. Smart

antennas can improve the gain through more antenna directivity and interference reduction. Range

increase of the wireless network system means that base stations can be placed further apart,

potentially leading to a more cost-efficient deployment. The antenna gain compared to a single

element antenna can be increased by an amount equal to the number of array elements, e.g., an

eight-element array can provide a gain of eight (9 dB) [3].

Reduction in multipath fading and delay spread

Wireless communication systems are limited in performance and capacity by three major impairments

[9], as shown in Figure 1.6.


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Multipath fading is caused by the multipaths that the transmitted signal can take to the receive

antenna. The signals from these paths add with different phases, resulting in a received signal

amplitude and phase that vary with antenna location, direction and polarization, as well as with time

(with movement in the environment).

Figure 1.6: Wireless system impairments

The second impairment is delay spread, which is difference in propagation delays among the multiple

paths. A desired signal arriving from different directions is delayed due to the different travel distances

involved. Here, the main concern is that multiple reflections of the same signal may arrive at the

receiver at different times. This can result in intersymbol interference (or bits crashing into one

another) that the receiver cannot sort out. When this occurs, the bit error rate (BER) rises and

eventually causes noticeable degrading in signal quality.

A smart antenna with the capability to form beams in certain directions and nulls in others is able to

cancel some of these delayed arrivals in two ways. First, in the transmit mode, it focuses energy in the

required direction, which helps to reduce multipath reflections causing a reduction in the delay spread.

Second, in the receive mode, multipath fading is compensated by diversity combining technique, by

adding the signals belonging to different clusters after compensating for delays and by cancelling

delayed signals arriving from directions other than that of the main signal. A frequency-hopping system

might be used for correcting fading effects as well. More detailed information about multipath fading

and delay spread reduction techniques can be found in [2].


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Reduction in co-channel interference

Co-channel interference occurs when the same carrier frequency reaches the same receiver from two

separate transmitters. The signals that miss an intended user can become interference for users onthe same frequency in the same or adjoining cells. An antenna array allows the implementation of

spatial filtering, which may be exploited in transmitting as well as in receiving mode in order to reduce

co-channel interference. In the transmitting mode, the antenna is used to focus the radiated energy by

forming a directive beam in the area, where a receiver is likely to be. This, in turn means that there is

less interference in the other directions where the beam is not pointing. The co-channel interference

generated in transmit mode could be further reduced by forming specialized beams with nulls in the

directions of other receivers [2]. This scheme deliberately reduces the transmitted energy in the

direction of co-channel receivers and hence requires knowledge of their positions. In the receive

mode, it is not necessary to have a priori knowledge of the positions of the co-channel interferences,

however requires some information concerning the desired signal, such as the direction of its source,

a reference signal, such as a channel sounding sequence, or a signal that is correlated with the

desired signal.

Signal quality improvement or higher data rates

Reduced co-channel interference, multipath fading and delay spread also leads to an improvement

(reduction) in bit error rate (BER) and symbol error rate (SER) for a given signal-to-noise ratio (SNR).

This means that better signal quality and higher data rates can be achieved for a communication

system. In noise and interference limited environments, the gain that can be obtained with a smart

antenna can be exchanged for lower BER. Experimental results showed that, in a direct sequence

code division multiple access (DS-CDMA) system, if the smart antenna that is employed at the base

station of the central cell can achieve a radiation pattern with a beamwidth of 20 (ideal or effective),

then an improvement of 1-7 orders of magnitude for the BER can be accomplished with average side

lobe levels (ideal or effective) between 10 dB and 20 dB, respectively [7].

Reduced transmit power

Sometimes, the array gain cannot be used for range extension due to limitations on the maximum

EIRP (effective isotropic radiated power). Furthermore, recent public worries over health issues


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stemming from electromagnetic radiation will almost certainly force the governing/standardisation

bodies to change the current radiation standards in the future and adopt a lower emission policy, in

particular for cellular communication systems. In such cases, one can exploit the base station array

gain to reduce the power transmitted by the mobile. This reduction is also crucial, since it relaxes thebattery requirements and therefore the talk times are increased and the size/weight of the handsets is

reduced.

Moreover, if the received power requirement at the mobile remains same with an M element array at

the base station, then the output power from the base station power amplifiers can be reduced by M -2,

which will reduce the total transmitted power from the array by M -1. It is obvious that, it will reduce the

cost of a system, because high-power amplifiers are expensive hardware components of the system.

Reduction in handover rate

If the mobile phones movement causes the signal to become too weak, the switching centre, which

monitors the signal strength arriving from the phone at the base station, reassigns a new traffic

channel via a base station closer to the phone and asks the phone to tune this new channel. This

procedure is known as handover or handoff, a process that is generally transparent to the mobile user.

When the number of mobiles in a cell exceeds its capacity, cell splitting is used to create new cells,

each with its own base station and new frequency assignment. This result in an increased handover

due to reduced cell size. Smart antennas increase the capacity by creating independent beams using

more antenna elements instead of cell splitting. Each beam is adapted as the mobiles change their

positions. The beam follows a cluster of mobiles or a single mobile, and no handover is necessary as

long as the mobiles served by different beams using the same frequency do not cross each other.

Support of new services

An important use of adaptive antennas in future wireless systems will be direction finding. Smart

antennas can provide user location information which opens a road to the value added services like

enhanced emergency services, traffic congestion monitoring (by tracking the vehicles equipped with

cellular phones), location-sensitive billing, on-demand location specific services (roadside assistance,

tourist information, electronic yellow pages), vehicle and fleet management.


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1.1.3 Applications of Smart Antennas

In this section, we will describe two main applications of smart antenna systems. The first one space-

division multiple access (SDMA) is also known as frequency reuse in angle (or simply angle reuse).

SDMA uses beam-forming/directional antennas to support more than one user in the same frequency

channel. The second application, which is called multiple-input multiple-output (MIMO) system, is also

very popular nowadays. In MIMO systems, multiple antenna elements at both (reception and

transmission) end of the transmission link are used. This technique can dramatically increase the

quality of the communication system. The other applications of smart antenna systems are array gain,

diversity gain, and channel estimation. More information about these applications can be found in [ 10,

11].

Space-division multiple access (SDMA)

In wireless systems, there are several methods used for sharing the communication channel among

multiple users. The most popular methods are to separate the users in time Time Division Multiple

Access (TDMA), in frequency Frequency Division Multiple Access (FDMA) and by code Code

Division Multiple Access (CDMA).

Space is truly one of the final frontiers when it comes to new generation wireless communication

systems. Filtering in the space domain can separate spectrally and temporally overlapping signals

from multiple mobile units. Thus, the last stage in the development of multiple access forms is the full

space-division multiple access (SDMA). The spatial dimension can be exploited as a hybrid multiple

access technique complementing FDMA, TDMA and CDMA. This SDMA approach enables multiple

users within the same radio cell to be accommodated on the same frequency and time. The system

can allocate multiple users on the same cell, on the same frequency and on the same time slot, only

separated by angle (spatial domain).

There are various forms of SDMA approach, which provides improvement in the capacity and quality

over omnicells. These are sectorial cells, sectorial beams and adaptive beams as the latest form.

These forms are illustrated in Figure 1.7.


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(a) 7-cell system with 120 sectors (b) 4-cell system with 60 sectors

(c) 60 sectorial beams within a cell (d) Adaptive beam forming for SDMA

in a 7-cell system

Figure 1.7: Various SDMA approaches

In a frequency-reuse (channel-reuse) system [2], the term radio capacity is usually used to measure

the traffic capacity. The radio capacity rC is defined as [6]

.r

MC

K S= (1.4)

where M denotes the total number of frequency channels, K denotes the cell reuse factor, and S

denotes the number of sectors in a cell. In the case of omnicells ( 1S = and 7K= ) the radio

capacity is 7rC M= channels per cell.


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Sectorial cells can be exploited to reduce interference. Figure 1.7(a) and (b) show two kinds of

sectorial cell systems: the 7-cell with three 120 sectors ( 3S= and 7K= ) and 4-cell with six 60

sectors ( 6S= and 4K= ). In these systems, each sector has a set of unique designated channels.

The mobile user moving from one sector or one cell to another sector or cell requires an intracellhandover.

If directional antennas are used, the capacity can be further improved. In the case of 7K= , each cell

has a set of M K frequency channels. One can use six directional antennas to cover 360 in a cell

and divide the whole set of frequency channels that are assigned to the cell into two subsets, which

are alternating from sector to sector. In this arrangement, there are three co-channel sectors using

each subset in a cell, as shown in Figure 1.7(c).

The ultimate form of SDMA is to use independently steered high-gain beams at the same carrier

frequency to provide service to individual users within a cell, as shown in Figure 1.7(d). That is, a high

level of capacity can be achieved via frequency reuse within a cell. To carry out frequency reuse within

a cell, a certain level of spatial isolation of co-channel signals is required to maintain an acceptable

carrier-to-interference ratio. Adaptive beam forming can provide such a spatial isolation by pointing a

beam at the mobile user and at the same time nulling out the interference from co-channel users.

Therefore, spectrum efficiency can be improved.

A comparison of the capacity and SIR for various systems is presented in [6], as shown in Table 1.1.

Table 1.1: Radio capacity and carrier-to-interference ratio in SDMA

K S Capacity /C I

Omnicells 7 1 . /7M chs cell 18 dB

120 sectorial cells 7 3 ./sec21

Mchs tor 24.5 dB

60 sectorial cells 4 6 ./sec24

Mchs tor 26 dB

60 sectorial beams 7 63

. /7

Mchs cell 20 dB (worst case)

N adaptive beams 7 1 . /7

MNchs cell 18 dB (worst case)


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Multiple-input multiple-output (MIMO) systems

The MIMO technology figures prominently on the list of recent technical advances with a chance ofresolving the bottleneck of traffic capacity in future Internet-intensitive wireless networks. MIMO

communication systems can be defined simply [12], by considering a link for which the transmitting end

as well as at the receiving end is equipped with multiple elements. Such a setup is illustrated in Figure

1.8. Coding, modulation, and mapping of the signals onto the antenna may be realized jointly or

separately.

Figure 1.8: Diagram of a MIMO wireless transmission system

The core idea behind MIMO is that signals at both ends are "combined" in such a way that they either

create effective multiple parallel spatial data pipes (increasing therefore the data rate), and/or add

diversity to improve the quality (bit-error rate or BER) of the communication.

MIMO systems can be viewed as an extension of the smart antenna systems. A key feature of MIMO

systems is the ability to turn multipath propagation into a benefit a user. MIMO effectively takes

advantage of random fading and when available, multipath delay spread, for multiplying transfer rates.

The prospect of many orders of magnitude improvement in wireless communication performance at no

cost of extra spectrum (only hardware and complexity are added) is largely responsible for the

success of MIMO as a topic for new research.

Consider the multi-antenna system diagram in Fig. 1.8. A compressed digital source in the form of a

binary data stream is fed to a simplified transmitting block encompassing the functions of error control

coding and (possibly joined with) mapping to complex modulation symbols (quaternary phase-shift

keying (QPSK), M-QAM, etc.). The latter produces several separate symbol streams, which range

from independent to partially redundant to fully redundant. Each is then mapped onto one of the

multiple TX antennas. Mapping may include linear spatial weighting of the antenna elements or linear


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antenna spacetime precoding. After upward frequency conversion, filtering and amplification, the

signals are launched into the wireless channel. At the receiver, the signals are captured by possibly

multiple antennas and demodulation and demapping operations are performed to recover the

message. The level of intelligence, complexity, and a priori channel knowledge used in selecting thecoding and antenna mapping algorithms can vary a great deal depending on the application.

In the conventional smart antenna terminology, only the transmitter or the receiver is actually equipped

with more than one element, being typically the base station (BTS), where the extra cost and space

have so far been perceived as more easily affordable than on a small phone handset. Traditionally, the

intelligence of the multiantenna system is located in the weight selection algorithm rather than in the

coding side. In a MIMO link, the benefits of conventional smart antennas are retained since the

optimisation of the multiantenna signals is carried out in a larger space, thus providing additional

degrees of freedom. In particular, MIMO systems can provide a joint transmit-receive diversity gain, as

well as an array gain upon coherent combining of the antenna elements (assuming prior channel

estimation). Instead of demonstrating these gains rigorously, we will give an example of the

transmission algorithm over MIMO that is known as spatial multiplexing.

In Fig. 1.9, a high-rate bit stream (left) is decomposed into three independent -rate bit sequences

which are then transmitted simultaneously using multiple antennas, therefore consuming one third of

the nominal spectrum. The signals are launched and naturally mix together in the wireless channel as

they use the same frequency spectrum. At the receiver, after having identified the mixing channel

matrix through training symbols, the individual bit streams are separated and estimated. This occurs in

the same way as three unknowns are resolved from a linear system of three equations. This assumes

that each pair of transmit receive antennas yields a single scalar channel coefficient, hence flat fading

conditions. However, extensions to frequency selective cases are indeed possible using either a

straightforward multiple-carrier approach (e.g., in orthogonal frequency division multiplexing (OFDM),

the detection is performed over each flat subcarrier) or in the time domain by combining the MIMO

spacetime detector with an equalizer. The separation is possible only if the equations are

independent which can be interpreted by each antenna seeing a sufficiently different channel in

which case the bit streams can be detected and merged together to yield the original high rate signal.

Iterative versions of this detection algorithm can be used to enhance performance.


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Figure 1.9: Basic spatial multiplexing (SM) scheme with three TX and three RX antennas yielding

three-fold improvement in spectral efficiency

A strong analogy can be made with code-division multiple-access (CDMA) transmission in which

multiple users sharing the same time/frequency channel are mixed upon transmission and recovered

through their unique codes. Here, however, the advantage of MIMO is that the unique signatures of

input streams (virtual users) are provided by nature in a close-to- orthogonal manner (depending

however on the fading correlation) without frequency spreading, hence at no cost of spectrum

efficiency. Another advantage of MIMO is the ability to jointly code and decode the multiple streams

since those are intended to the same user. However, the isomorphism between MIMO and CDMA can

extend quite far into the domain of receiver algorithm design.

1.2. Antenna Arrays

In many applications, it is necessary to design antennas with directive characteristics (very high gains)

to meet the demands of long-distance wireless communications. This can be accomplished by forming

an assembly of radiating elements in an electrical and geometrical configuration. This antenna, formed

by multielements, is referred to as an array. There are five-control mechanisms in an antenna array in

order to shape the overall pattern of the antenna. These are:


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1. The geometrical configuration of the overall array (linear, circular, rectangular, etc)

2. The relative displacement between the elements

3. The excitation amplitude of the individual elements4. The excitation phase of the individual elements

5. The relative pattern of the individual elements

Before we start to introduce antenna arrays, some antenna array parameters, which will be used in the

second and third chapters to describe the performance of an antenna array, should be provided. More

information about terms and definitions that are commonly used in the study of antennas and arrays

can be found in [13,14]. Afterwards we will explain linear arrays (one dimensional) and planar arrays

(two dimensional). Meanwhile we will introduce pattern multiplication, which let us pass to two-

dimensional arrays from one-dimensional arrays easily.

1.2.1. Basic Antenna Array Parameters

Radiation pattern

An antenna radiation pattern or beam pattern is defined as a mathematical function or a graphical

representation of the radiation properties of the antenna as a function of space coordinates.

Main lobe

The main lobe (also called major lobe or main beam) of an antenna radiation pattern is the lobe

containing the direction of maximum radiation power.

Side lobes

Side lobes are lobes in any direction other than that of the main lobe (intended lobe). For an equally

weighted linear array, the first side lobe (i.e., the one nearest the main lobe) in the radiation pattern isabout 13 dB below the peak of the main lobe.


27/74

Beamwidth

The beamwidth of an antenna is the angular width of the main lobe in its far-field radiation pattern.

Half-power beamwidth (HPBW), or 3-dB beamwidth, is the angular with measured between the pointson the main lobe that are 3 dB below the peak of the main lobe.

An example of radiation (beam) pattern is depicted in Figure 1.10. HPBW is usually expressed in

angle (azimuth or elevation angle).

Figure 1.10: A radiation pattern and its associated side lobes and beamwidths

Grating Lobe

Only the lobe centered at the center angle 0 = (or at the beam-steering angle 0 if it is not equal to

zero) is the desired lobe. All additional lobes that fall into the real or visible region are called grating

lobes [15]. In an antenna array, if the element spacing is too large, extra main lobes (grating lobes) will

be formed on each side of the array plane. To prevent grating lobes the spacing between antenna

elements should be properly designed. The spacing between the antenna elements should be

maximum half wavelength. That is,

2d

(1.6)


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Since lowering the array spacing below this upper limit only provides redundant information and

directly conflicts with the desire to have as much as aperture as possible for a fixed number of antenna

elements, for uniform linear arrays (ULA) 2d = is used.

In Figure 1.11, we compare the beam patterns with different element spacing while keeping the

aperture size constant. The element spacings are 4 , 2 , , 2 for an equal-sized apertures of

10 with 40, 20, 10, and 5 elements, respectively.

(a) 4, 40d M= = (b) 2, 20d M= =

(c) , 10d M= = (d) 2 , 5d M= =

Figure 1.11: Beampatterns of uniform linear arrays for different element spacing with an equal-sized

aperture

The beam patterns for 4d = and 2d = spacings are identical with equal 3-dB beamwidths

around 7 and equal first side lobes having a height of 13 dB. The oversampling for the array with an

element spacing of 4 does not provide any extra information and therefore does not improve the


29/74

beamformer response in terms of resolution. Actually, if the sensors are too close together

(oversampling case), spatial discrimination suffers (worse angular resolution), because of the smaller

than necessary aperture. In this example, in order to keep aperture size constant, we made the

number of elements twice. Thus, resolution remained same. In the case of the undersampled arrays (d = and 2), we see similar main beamwidth response, however the additional peaks in the beam

pattern appear at 90o

for d = and in even closer for 2d = . As we described before, these extra

lobes are grating lobes. Grating lobes create spatial ambiguities; that is, signals incident on the array

from the angle associated with a grating lobe will look just like signals from the direction of interest.

The beamformer has o means of distinguishing signals from these various directions. In some

applications, grating lobes may be acceptable if it is determined that it is either impossible or very

improbable to receive returns from these angles.

Array aperture and beamforming resolution

The aperture is the finite area over which an antenna element collects spatial energy. In general, the

designer of an array yearns for as much aperture as possible. The greater the aperture, the finer the

resolution of the array. The resolution can be defined as the ability to distinguish between closely

spaced sources. Improved resolution results in better angle estimation [16

]. The angular resolution of

an antenna array is measured in beamwidth. Usually 3-dB beamwidth is used for the comparison of

resolution capabilities of different array structures. The narrower the 3-dB beamwidth, the better the

resolution of the array.

In order to illustrate the effect of aperture on resolution, we compare the beam patterns for M=

4,8,16, and 32 with interelement spacing fixed at 2d = (nonaliasing condition). Therefore, the

corresponding apertures in wavelengths are 2 , 4 , 8 ,16D = . It can be observed in Figure 1.12

that, increasing the aperture yields narrower main lobe width and thus provides a better resolution.


30/74

(a) 4M= (b) 8M=

(c) 16M = (d) 32M =

Figure 1.12: Beampatterns of uniform linear arrays for different aperture sizes with equal element

spacing

1.2.2. Linear Arrays

In Figure 1.13, a uniformly spaced linear array is depicted with M identical isotropic elements. This is

the most common structure due to its low complexity. It can perform beam forming in azimuth angle

within an angular sector. Each element is weighted with a complex weight mW with 0,1, , -1m M= ,

and the interelement spacing is denoted by d . If a plane wave impinges upon the array at angle

where is the angle between broadside of the array and the direction from the wavefield (usually

called azimuth angle), the wavefront arrives at element m , since travel-distance between two

neighbor elements is sind . By setting the phase of the signal at the origin arbitrarily to zero, the


31/74

phase lead of the signal at element m relative to that at element 0 is sinmd , where 2=

and = wavelength.

Adding all the element outputs together gives array factor AF. The array factor represents the far-

field radiation pattern of an array of isotropically radiating elements.

1sin 2 sin sin

0 1 2

0

M j d j d jk d

m

m

AF W W e W e W e

=

= + + + = (1.7)

Figure 1.13: A uniformly spaced linear array

The array factor in (1.7) can also be expressed in terms of vector inner product as

( ) TAF = W w (1.8)

where

0 1 1[ ]T

MW W W = W (1.9)


32/74

is the weighting vector and

sin ( 1) sin[1, , , ] j d j M d T e e = w (1.10)

is the array propagation vector that contains the information on the angle of arrival of the signal. If the

complex weight is

jm

m mW A e= (1.11)

where the phase of thethm element leads that of the

th( 1)m element by , the array factor

becomes

1( sin )

0

( )M

j md m

m

m

AF A e

+

=

= (1.12)

If 0sind = , a maximum response of ( )AF will result at the angle 0 . That is, the antenna

beam has been steered towards the wave source in 0 direction.

1.2.3. Pattern Multiplication

We have only considered arrays of isotropic elements until now. An isotropic element can transmit or

receive energy uniformly in all directions. However, the isotropic antenna is just a mathematical fiction.

In practice, all antenna elements have nonuniform radiation patterns. Let us consider an array

consisting of identical antenna elements that have radiation patterns decided by ( , )e . The principle

of pattern multiplication states that the beampattern of an array is the product of the element pattern

and the array factor [6]. That is, the array beampattern ( , )G is given by

( , ) ( , ) ( , )G e AF = (1.13)


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where ( , )AF is the array factor. It contains the geometric information of the array, that is, the

position coordinates of the elements. The first term ( , )e is usually called element pattern. Its effect

on the array is determined by the current excitation across the element.

The principle of pattern multiplication (1.13) is an important result. It decomposes the array properties

into those associated with the excitation of the element and those resulting from the geometric

positioning of the elements. It shows how theorems relating to array design are independent of the

particular antenna element used to form the array. Furthermore, this principle is also very useful to

determine the array factor of a complicated array that is composed of simple subarrays.

1.2.4. Planar Arrays

In addition to placing elements along a line to form a linear array, one can position them on a plane to

form a planar array. Planar arrays can be designed two-dimensional or three-dimensional according to

the spatial requirement of the beam forming. We will only concentrate in two-dimensional planar arrays

that perform beam forming in both elevation and azimuth angles.

A two-dimensional rectangular grid array is one of the common configurations of planar arrays, as

shown in Figure 1.14.

Figure 1.14: A rectangular planar array geometry


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A rectangular planar array can be seen as a composition of two linear arrays consisting of M-element

in one plane and N-element in another plane. The array factor for the M-element linear array is given

by

1( sin )

1

0

( ) xM

j md u m

m

m

AF u A e

+

=

= (1.14)

where sin sin cosu = and { }1

0

Mjm

m mA e

=are the complex weights. The array factor for the N-element

linear array is given by

1( sin )

2

0

( ) yN

j nd v n

n

n

AF v A e

+

=

= (1.15)

where sin sin sinv = and { }1

0

Njn

n nA e

=are the complex weights. According to the principle of pattern

multiplication, the overall array factor for the rectangular array is the given by

1 2( ) ( ) AF AFuAF v=

(1.16)


35/74

Chapter 2

2.1. Introduction

In this chapter, we will start with a brief description of finite continuous apertures, proceed with discrete

apertures. We will describe some necessary concepts like aperture smoothing function and co-array.

We will introduce two important array geometry approaches, which are known as minimum

redundancy arrays and minimum hole arrays. We will show that the minimum redundancy and

minimum hole arrays provide a narrowed main beamwidth and optimal or close to optimal peak side

lobe levels (SLL). For six active elements, we will search minimum redundancy and minimum hole

array geometries, and all the possible array geometries between minimum redundancy and minimum

hole array geometries. Finally, the chapter ends with the application of the approach to the two

dimensional case.

2.2. Continuous Apertures

Apertures are finite areas where antenna array elements gather signal energy. The aperture function

( )w x embodies two kinds of information about the aperture. The spatial extent of ( )w x reflects the

size and shape of the aperture. Actually, the aperture acts like a window through which we observe

the wavefield. Moreover, aperture functions can take on any real value between 0 and 1 inside the

aperture. This second aspect of aperture functions allows us to represent the relative weighting of the

field within the aperture. Aperture weighting is sometimes referred to as shading, tapering, or

apodization as well.

Aperture Smoothing Function

Let us assume a space-time signal ( , )f x t . When we observe a field through the finite aperture, the

output of the sensor

( , ) ( ) ( , )z x t w x f x t = (2.1)


36/74

where ( )w x is the aperture or weighting function.

After calculating the space-time Fourier Transform of this relationship, we obtain

3

1( , ) ( ) ( , )

(2 )Z W l F l dl

= r r rr r

(2.2)

where represents the temporal frequency variable. ( , )Z is a convolution over wavenumber

between the Fourier Transform of the field.

The aperture smoothing function is defined as

{ }31

( ) ( )exp .(2 )

W w x j x dx

= r rr r r

(2.3)

This convolution means that the wavefields spectrum becomes smoothed by the kernel ( )W once

we observe it through an aperture.

Co-array Function

The coarray is defined as the autocorrelation of the weighting function ( )w x [17] and for the continuous

aperture is given by

( ) ( ) ( )c w x w x dx +r rr r r

(2.4)

The variable is called a lag, and we term its domain lag space. The coarray becomes important

when array-processing algorithms employ the waves spatiotemporal correlation function to

characterize the waves energy content. The Fourier transform of ( )c equals ( )W k2

, the squared

magnitude of the aperture smoothing function.


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To clarify the above discussion, let us assume a basic linear aperture. A linear aperture has an

aperture function that is nonzero only along a finite-length line segment in three-dimensional space.

For example, if we let

1, / 2( )

0,

x Db x

otherwise

=

(2.5)

the three-dimensional aperture function ( , , )w x y z can be written as ( , , ) ( ) ( ) ( )w x y z b x y z = .

The aperture smoothing function for the linear aperture can be found from Eq. 2.3 as

sin / 2( )

/ 2

x

x

k DW k

k=

r

(2.6)

Fig. 2.1 illustrates the geometry associated with a typical linear aperture.

Figure 2.1: The linear aperture smoothing function

Because the aperture function is nonzero only along a small segment of the x axis, the aperture

smoothing function W depends only on the x component of the wavenumber vector k .

The co-array function for the linear aperture can be found from Eq. 2.4 as

( )0,

D x D Dc

otherwise

=

, -

(2.7)


38/74

2.3. Discrete Apertures

In many practical applications, arrays composed of individual antenna elements sample the wavefield

at discrete spatial locations. We will give the aperture smoothing function and co-array function like in

the continuous case.

Aperture Smoothing Function

Let us assume M antenna elements be placed anywhere in a three-dimensional space characterized

by the variable x . Let the wavenumber vector be3 with norm 2 = , and for three-

dimensional space, it can be characterized by ,x y and z , where 2 sin cos= x ,

2 sin sin= y and 2 cos= z . (The minus signs are needed because a wave

propagating into the origin from the first octant (where x , y and z are all positive) would have

negative wavenumber vector characteristics). These angles are usually called azimuth angle for

and elevation angle for . Let the thm antenna element be located at the position mx , where the

array element locations are3( , , )= m m m md x y z and yield the element signal ( )my t taken here to be

( , )mf x t , which in turn can be represented by the Fourier transform

{ }41

( ) ( , ) ( . )(2 )

= r rr r

m my t F exp j t d dkd

(2.8)

The wavenumber-frequency spectrum ( , )Z of the array output is given by

3

1( , ) ( ) ( , )

(2 )Z W l F l dl

= r r rr r

(2.9)

where ( )W is the aperture smoothing function, which is given by

( 2 )(sin cos sin sin cos )1 1

0 0

( ) + +

= =

= = rrr j x y z m m m

m

M M j d

m m

m m

W w e w e (2.10)


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For a one-dimensional array that lies in the x axis like in Figure 1.15, this equation can be rewritten

as

1 2 (sin )

0

( )

=

= mM j x

m

m

W w e (2.11)

since elevation angle is equal to zero, =0.

One can notice the similarity between the array factor formula for a one-dimensional uniformly spaced

array in (1.21) and aperture smoothing function formula for one-dimensional array in (2.11). Actually,

(2.11) can be thought a general beampattern formula that can be used for both equally-spaced (filled)

and non-equally spaced (sparse) linear arrays. Also, in (2.11) no steering has been taken into account,

that is 0 0 = . However, beampattern of sparse arrays can also be computed with (1.21), considering

the weighting of the removed element to be zero. That is, if the third, forth and sixth elements are

removed from a seven-element aperture in order to make a sparse array, with taking 3 4 6 0w w w= = =

beampattern can be computed with (1.21).

The aperture smoothing function is the output after weighting and summing all elements in the array

for a wave from infinite distance hitting the array. The aperture smoothing function determines how the

wavefield Fourier transform is smoothed by observation through a finite aperture [18].

Co-array Function

As said before, the co-array is defined as the autocorrelation of the element weights and for the

discrete case is given by

1

0

( )

+=

= M l

m m lm

c l w w (2.12)

where l is the spatial lag between two elements.


40/74

The co-array describes the morphology of the antenna array, rather than describing the angular

response [19]. This means that, it describes the weight with which the array samples the different lags

of the incoming fields correlation function. Usually the co-array has been used to design arrays with

as high resolution as possible. This is equivalent to having a co-array, which is as uniform as possible,and which spans the maximum number of lags undersampled [20]. For the zero lag, 0l= , the co-array

is always equal to the number of antenna elements.

For an M element linear array with element distance d , the co-array is related to the beampattern

with:

21

( 1)( ) ( ) exp( )

M

l MW c l j ld

= =

r r

(2.13)

where2

( )W is squared magnitude of the aperture smoothing function. That is, the discrete co-array

function is equal to the inverse Fourier transform of the squared magnitude of the aperture smoothing

function.

Owing to the symmetry of the co-array, this implies

21

1

( ) (0) 2 ( )cos( )M

l

W c c l j ld

=

= + r r

(2.14)

2.4. Sparse Linear Arrays

Sparse arrays are antenna arrays that originally were adequately sampled, but where several

elements have been removed. This is called thinning, and it results in the array being undersampled

[19]. Such undersampling, in traditional sampling theory, creates aliasing. In the context of spatial

sampling, and if the aliasing is discrete, it is usually referred to as grating lobes. In any case, this is

unwanted energy in the side lobe region. (Please state that in sparse arrays, if the positions of

antenna elements are selected appropriately, no grating lobes appear !)


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What is the motivation for using sparse arrays rather than full arrays? The main reason for their use is

economy. Each of the elements needs to be connected to a transmitter and a preamplifier for

reception, in addition to receive and transmit beam formers. Therefore, the increase in the number of

antenna elements means the increase in cost. For example, medical ultrasound imaging is a fieldwhere most of the sparse arrays work was done, illustrates this: Conventional 2-D scans is done with

1-D arrays with between 32 and 192 elements. 3-D ultrasound imaging is now in development and this

requires 2-D arrays in order to perform a volumetric scan without mechanical movement. Such arrays

require thousands of elements in order to cover the desired aperture. Another reason is the system

complexity. The antenna array should always have low number of elements to avoid unnecessarily

high complexity in signal processing.

2.4.1. Minimum Redundancy and Minimum Hole Arrays

In situations where one must obtain maximum spatial resolution from a limited number of antenna

elements, array configurations known as minimum redundancy arrays [21, 22, 23] and minimum hole

arrays [25,24] are often employed.

To describe the minimum redundancy arrays and minimum hole arrays, we refer to co-array equation

in (2.11). If the array has more than one pair of antenna elements separated by the same distance,

these pairs produce redundant estimates of the correlation function at that lag. In this case, the co-

array of that array is said to have redundancies. Mathematically, this means, in the co-array equation,

if the co-array of a lag is greater than unity, ( ) 1>c l , then a redundancy occurs in that lag. If there is

no pair of antenna elements separated by some distance (lag) that is smaller than the aperture of the

array, the array is said to have a hole in its co-array at that location. This means, if the co-array of a

lag is zero, ( ) 0=c l , then a hole occurs in that lag. In order to have an even sampling of the incoming

wave field, it is required a co-array with the same weight for all lags. A perfect array is such an array. It

is defined as an array with a coarray with no holes or redundancies except for zero lag. Thus, each lag

(excluding the zero lag) of the spatial correlation up to the lag corresponding to the array aperture is

sampled exactly once. Unfortunately, perfect arrays only exist for four or fewer elements in the array.

Therefore, we study arrays that approximate perfect arrays: the Minimum Redundancy (MR) and the

Minimum Hole (MH) arrays. They are defined by the number of redundancies R , and holes H .

Redundancy arrays are defined as an array with redundancies but no holes in its co-array. Minimum

redundancy arrays are those element configurations that have no holes and minimize the number ofredundancies. They are sometimes called redundant arrays [21] as well. Such an array has the largest

A figu

descriarray

holes?


42/74

possible aperture for a redundancy array with a given number of antenna elements. Similarly, hole

arrays are defined as an array with holes but no redundancies in its co-array. Minimum hole arrays

minimize the number of holes in the co-array without any redundancies. These arrays are also known

as non-redundant arrays [21] or Golomb rulers [

25

]. Such an array has the minimum aperture possiblefor a hole array with a given number of antenna elements.

The number of total elements M in the aperture as shown in [26] given by

( 1)1

2

= + +

n nM H R (2.15)

where n is the number of active elements, H is the number of holes and R is the number of

redundancies.

For n antenna elements, there are ( 1) 2M M pairwise element separations. If each pair were

separated with a different distance (no redundancies and holes were allowed), the number of total

elements M in the aperture would be

( 1)1

2

n nM

= + (2.16)

As one can notice easily, this is the case for perfect arrays where 0R H= = .

(2.15) implies that an array with n active elements and M total elements bounded by

<


43/74

(a) A filled array (b) A thinned array

Figure 2.2: Array geometries for filled and thinned (sparse) arrays

In Figure 2.2(a), a six-element equally spaced filled antenna array geometry is given. All the element

spacing between antenna elements is equal to d . The aperture size can be calculated easily using

( 1)D M d = (2.18)

which gives (6 1) 5 D d d = = . However, in Figure 2.2(b) two elements have been removed and a

sparse array was obtained. In the array geometry, symbol shows the positions of active elements,

whereas symbol shows the removed elements. In this case, we have four active elements and six

total elements (active plus removed elements). Therefore, in sparse arrays, both the number of total

elements and the number of active elements are given. With the knowledge of the number of total and

active elements, one can know the aperture size and the degree of thinning.

2.4.2. Simulation Results

1D Sparse Arrays (Sparse Linear Arrays)

We search minimum redundancy arrays and minimum hole arrays for six active antenna elements.

However, the set of possible apertures where minimum hole and minimum redundancy solutions are

restricted, and therefore we also study arrays with both holes and redundancies. For 6n = , minimum

redundancy arrays are observed for 14M = . There are 3 different minimum redundancy arrays (+3

mirrored). For 6n = and 14M = , when the two end elements are fixed, we obtain a total of

( )12 4954 = possible thinning patterns. In Figure 2.3(a), we show the maximum side lobe levels vs. 3


44/74

dB beamwidths of these 495 possible array geometries. As one can see, most of the arrays reside in

one part of the figure. We focus on this part in Figure 2.3(b).

(a) All array possibilities (b) Focus on some array possibilities

Figure 2.3: Maximum SLL vs. 3 dB beamwidth for a minimum redundancy array with 6n = and

14M =

From Figure 2.3(b), one can notice that there are only a few arrays, which satisfy low peak SLL and

narrow beamwidth at the same time. This kind of arrays lies on the lower and left boundary of this

figure.

We will first calculate co-array by using (2.11) where { }0,1mw in this case and we obtain

beampattern (squared magnitude of the aperture smoothing function) with the information of co-array

as in Figure 2.4.

The one with the smallest peak sidelobe and narrowest 3 dB beamwidth from the three minimum

redundancy array possibilities is choosen as an example in Figure 2.4. In Figure 2.4(a), the element

positions of the aperture are given. We prefer to describe the element positions by giving distances

between them, that is, 15322 is the geometry for this array. However, the array geometry can also be

described using the exact positions of the elements, that is, 11000010010101. In Figure 2.4(b), the

resulting co-array design is shown. Only positive lags for the co-array are shown; the negative lags

mirror the positive ones due to the symmetry. Except for 0l= , 2l= and 4l= l=5 (?), the co-array

has a uniform shape, that is ( ) 1c l = . As said before, for 0l= , ( )c l is always equal to the number of

antenna elements, here ( ) 6c l = . Therefore, there are two redundancies (at and 4l= 5(?)) for this

Explathe me

of 15

array.state t

indicathe po

of act

elemewhile

for reeleme


45/74

array, 2R = . In Figure 2.4(c), the beampattern, which is calculated using by using (2.13) is shown. It

has a 3 dB beamwidth of 6.0 and maximum side lobe of -6.3212 dB.

(a) Array geometry

(b) Co-array (positive side) (c) Beampattern

Figure 2.4: Array geometry, co-array and beampattern for a minimum redundancy array with 6n =

and 14M =

For 6n = , minimum hole arrays are observed for 18M= . There are 4 different minimum redundancy

arrays (+4 mirrored). For 6n = and 18M = , when the two end elements are fixed, we obtain a total of

( )16 18204 = possible thinning patterns.

For the minimum hole array, the one with the lowest peak sidelobe but largest 3 dB beamwidth fromthe three possibilities is chosen as an example in Figure 2.5. For a 6-element minimum hole array 3

dB beamwidth of 4.8 and maximum side lobe level of -5.8592 dB is observed.

(a) Array geometry


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(b) Co-array (positive side) (c) Beampattern

Figure 2.5: Array geometry, co-array and beampattern for a minimum hole array with 6n = and18M=

Regular uniform linear arrays and sparse arrays can be compared in two ways. The first is to keep the

aperture fixed while reducing the number of antenna elements to obtain a sparse array. The second

keeps the number of antenna elements fixed and extends the aperture in order to create a sparse

array. The latter is fairer for comparison, because the number of antenna elements decides the cost

and system complexity, as emphasized before. We compare uniform linear array (ULA), minimum

redundancy (MRA) and minimum hole array (MHA) for 6n = as shown in Figure 2.6.

(a) Co-array for ULA (b) Beampatterns for ULA, MRA and MHA

Figure 2.6: Co-array for ULA and beampattern comparison for ULA, MRA and MHA with 6n =


47/74

In Figure 2.6(a), co-array of a uniform linear array is shown. A uniform linear array gives a discrete

triangular shaped co-array pattern. Clearly there is a very high degree of redundancy present.

Therefore, it is impossible to obtain a narrow beamwidth (good resolution) with a uniform linear array

as one can observe in Figure 2.6(b). The 3 dB bandwith beamwidth of a this six-element uniform lineararray is found of17.2. This is naturally far from being a good result. But However it gives a max. SLL

of -12.426 dB, which is almost twice better of the minimum redundancy and minimum hole solutions.

However, we have to remind you that there are only a few arrays, which satisfy low peak SLL and

narrow beamwidth at the same time. Minimum redundancy and minimum hole arrays are those with

optimal or close to optimal peak side lobe level. In Table 2.1, a comparison of 3 dB beamwidth and

peak side lobe levels is given for 6, 14, and 18-element ULAs, 6-element MRA, and 6-element MHA.

Table 2.1: Comparison of 6,14, and 18-element ULA and 6-element MRA and 6-element MHA

n M R H -3 dBBeamwidth

Max.SLL[dB]

Array Geometry ArrayType

6 6 15 0 17.2 -12.426 11111 ULA14 14 91 0 7.3 -13.112 1111111111111 ULA18 18 153 0 5.7 -13.171 11111111111111111 ULA

6 14 2 06.16.0

6.1

-6.0606-6.3212

-6.0606

1316215322

11443

MRA

6 18 0 2

4.84.44.84.6

-5.8592-5.3444-5.8592-5.7285

13625136521732417423

MHA

14-, and 18-element ULAs are given for the comparison while keeping the aperture fixed for minimum

redundancy and minimum hole arrays, respectively. If the aperture is fixed, MRAs and MHAs have

beampatterns with slightly narrower mainlobe beamwidth then ULAs. If one increases the number of

elements for ULAs, it naturally implies an increase in the aperture size as well. Thus, which parameter

(aperture size or number of elements) determines the main beamwidth or max. SLL? For the answer,

we can compare our minimum redundancy ( 6n = , 14M = ) and minimum hole ( 6n = , 18M = )

arrays, because only aperture size changes, number of elements is fixed. The increased aperture

leads to a narrower mainlobe. There is a slight increase in side lobe level, however this is not related

to the increased aperture, side lobe changes are due to the changes in the arrays geometry array

geometry.


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Table 2.2 and Table 2.3 show a comparison of 3 dB beamwidth and maximum side lobe levels of

minimum redundancy (from 3n = to 17n = ) and minimum hole arrays (from 3n = to 19n = ),

respectively. These tables include all known minimum redundancy and minimum hole arrays and can

partially be found in [

27

,

28

].

Table 2.2: Comparison of minimum redundancy arrays for 3,4,...,17n =

n M R -3 dBBeamwidth

[deg]

Max. SLL[dB]

Array Geometry

3 4 0 23.5 -4.6112 12 (perfect)4 7 0 12.2 -5.2547 132 (perfect)

5 10 18.58.7

-5.4566-5.0110

13323411

6 14 26.16.06.1

-6.0606-6.3212-6.0606

131621532211443

7 18 4

4.64.74.94.64.8

-6.3353-5.6402-5.3530-5.7666-6.2271

136232114443111554116423173222

8 24 53.43.4

-5.9243-5.6572

13662321194332

9 30 7

2.7

2.72.7

-6.2428

-5.4375-5.7076

13666232

1237744111(12)43332

10 37 9 2.2 -5.8592 12377744111 44 12 1.8 -6.3603 1237777441

12 51 161.61.6

-6.5199-5.8598

12377777441111(20)5444433

13 59 201.41.41.4

-5.8919-5.8936-5.4177

111(24)5444443311671(10)(10)(10)3423143499995122

14 69 231.21.2

-5.9051-5.2866

11671(10) (10)(10)(10)342311355(11)(11)(11)66611

15 80 26 1.0 -5.9791 11355(11)(11)(11)(11)6661116 91 30 0.9 -5.8312 11355(11)(11)(11)(11)(11)66611

17 102 35 0.8 -6.1546 11355(11)(11)(11)(11)(11)(11)66611

We can observe some interesting results from Table 2.2 and Table 2.3. First, only one MRA array

geometry (15322) shows better side lobe level compared to MHA, which includes same number of

antenna elements. For all the others, MHA shows lower peak side lobe level and narrower beamwidth.

Therefore, it is a better solution to use MHA instead of MRA for a given number of antenna elements,

if there is no requirement that antenna array size should be limited. We also see that for large number

of elements, the peak side lobe level of a MHA is approaching to the peak side lobe level of a ULA.

For instance, for 18-element MHA provides peak SLL of -9.1451 dB 9.2018 dB (?), whereas same


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number of element ULA gives -13.171 dB peak SLL. Moreover, while the increase in the number of

elements for a ULA does not result in a noteworthy decrease in peak SLL, the increase in the number

of elements for a MHA leads to a considerable improvement in peak SLL. Thus, side lobe level may

not be a problem for sparse arrays in case of the usage of large number of elements. If we increasethe number of elements both for MRA and MHA, the mainlobe becomes narrower due to bigger

aperture size, as shown in Fig 2.8(a) and (c) Fig 2.7(a) and (c) (?). However, there is not always

decrease in the peak side lobe level while increasing the number of elements, as shown in Fig 2.7(a)

and (c) Fig 2.7(b) and (d) (?).

.

Table 2.3: Comparison of minimum hole arrays for 3,4,...,19n =

n M H 3 dBBeamwidth

[deg]

Max.SLL

[dB]

Array Geometry

3 4 0 23.5 -4.6112 12 (perfect)4 7 0 12.2 -5.2547 132 (perfect)

5 12 16.77.3

-4.7564-5.5632

13522513

6 18 2

4.84.44.84.6

-5.8592-5.3444-5.8592-5.7285

13625136521732417423

7 26 4

3.03.13.23.23.1

-5.2997-5.8501-6.3911-6.1730-5.9691

1368521649322176541(10)5342256813

8 35 6 2.4 -6.2299 13567(10)29 45 8 1.8 -6.7747 147(13)286310 56 10 1.5 -7.0929 154(13)387(12)2

11 73 171.11.2

-7.3838-7.3485

139(15)5(14)7(10)6218(10)57(21)42(11)3

12 86 19 1.1 -7.6831 24(18)5(11)3(12)(13)71913 107 28 0.8 -8.1373 23(20)(12)6(16)(11)(15)491714 128 36 0.8 -7.7829 5(23)(10)381(18)7(17)(15)(14)24

15 152 46 0.6 -7.0393 618(13)(12)(11)(24)(14)32(27)(10)(16)416 178 57 0.5 -7.5132 137(15)6(24)(12)8(39)2(17)(16)(13)5917 200 63 0.5 -8.7866 8(23)36(21)(16)(22)(19)1(13)(11)4(35)(10)2518 217 63 0.5 -9.2018 (11)(13)4(21)(14)5(18)(32)961(26)3(31)(12)8219 247 75 0.4 -9.1451 492(27)(14)3(18)(16)(23)(10)(12)8(28)(40)7(19)51

We have two important results that can guide the selection of particular array geometry, aperture size

and number of elements.

Aperture size fundamentally determines the width of the beampatterns mainlobe, which in

turn determines the spatial resolution of the array. A greater aperture leads to a narrower

beamwidth.


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Both the number of antenna elements and the geometry of an array determine the level of the

beampatterns side lobes.

(a) HPBW vs. number of elements for MRA (b) Peak SLL vs. number of elements for MRA

c) HPBW vs. number of elements for MHA (b) Peak SLL vs. number of elements for MHA

Figure 2.7: -3 dB Beamwidth/Peak side lobe level vs. number of elements for MRA (a), (b) and for

MHA (c), (d)

In Figure 2.7, the lowest sidelobe level and the lowest 3 dB beamwidth values is chosen for the

numbers where there is more then one MRA or MHA geometry. In Figure 2.7, the values of SLL and 3

dB beamwidth are chosen for smallest values. For instance, for 5n = and 12M = (MHA), 6.7 and

-5.5632 dB values 3 dB beamwidth of 6.7 and SLL of -5.5632 dB are used in order to create the

graphs, but they belong to different geometries.


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For a given number of antenna elements n , an array that has a larger aperture than the MRA with n

elements has holes. Similarly, an array with aperture smaller than that of the n -element MHA has

redundancies. Therefore, any array whose aperture is between that of the MRA and MHA with n

elements must have both redundancies and holes. Aperture sizes of the known minimum redundancyand minimum hole arrays are presented in Figure 2.8. The region between the two curves represents

arrays that must have both holes and redundancies.

Figure 2.8: Aperture size vs. number of elements for known MRA and MHA

We investigate the arrays for 6n = , which are near to perfect arrays like minimum redundancy and

minimum hole arrays, but allow redundancies and holes. The arrays with 6n = active elements for

each of the apertures 14,15,...,18M = are presented in Table 2.4.

Table 2.4: Properties of the arrays with minimum number of redundancies (R ) and minimum number

of holes (H) for 14,15,...,18n =

M R H 3 dBBeamwidth

Max. SLL[dB]

ArrayGeometry

14 2 06.16.06.1

-6.0606-6.3212-6.0606

131621532211443

15 2 15.25.95.6

-4.1007-6.3635-5.8481

146214161224341

16 2 2

4.8

5.65.1

-3.5487

-5.7223-5.6498

14721

241531552217 1 2 4.8 -5.6512 25531


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5.45.2

-6.2830-5.8432

4117354421

18 0 2

4.84.44.84.6

-5.8592-5.3444-5.8592-5.7285

13625136521732417423

We see that, near perfect arrays that allow redundancies and holes may give lower side lobe levels.

The best maximum side lobe level has been observed for 41612 array geometry ( 15M= ) which has

2R = and 1H= . Another this kind of array, 41173, might also be an optimal solution with 5.4 3 dB

beamwidth of 5.4 and -6.2830 peak side lobe level of -6.2830 dB.

Side lobe suppression

As we have showed before, minimum redundancy and minimum hole arrays have relatively high side

lobes in comparison with uniform linear arrays. The side lobe suppression can be achieved by using

non-uniform weighting instead of uniform array weights. However, it is difficult or sometimes

impossible to achieve uniform side lobe levels for MRA and MHA over all the angles. Such

suppression can be achieved only over a limited range of angles near the main beam. The number of

side lobe peaks that can be suppressed does not excess 1n .

Several researchers have investigated ways to control the side lobe levels for sparse arrays. In [29], an

elegant technique is proposed to find appropriate weighting coefficients, which suppress a maximum

number of side lobes in minimum redundancy arrays to an arbitrarily specified level.

The employed technique is based on iteratively aligning the side lobe peaks to a prespecified level.

The weighting coefficients were obtained in [29], for the MRA that produce a beampattern with the side

lobe level specified as 30 dB. These weights were found to be 0.1401, -0.0966, 0.0329, 0.1303,

0.2096, 0.2308, 0.1914, 0.1135, 0.0115, 0.0337, 0.0027, for an 11-element MRA. The resulting

beampattern and 11-element MRA with uniform weighting is given for the visual comparison in Figure

2.9.

The MRA with non-uniform weightings has uniform 30 dB side lobe level of 30 dB extending to

about 13 from both sides of the main lobe. Compared to the uniform weighting pattern, a

substantial side lobe improvement was achieved over the region [-13, 13] in which 10 side lobe

peaks were suppressed to be 30 dB. Since there are only 11 elements in the array, the side peaks in

other regions cannot be controlled. However, the 3 dB beamwidth is now 2.7, which is wider than of


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the uniform weighting case. The 3 dB beamwidth for 11-element MRA with uniform weighting had a

value of1.8 -3 dB beamwidth. This is a result of the trade-off between the mainlobe width and the

side lobe levels.

Figure 2.9: The beampatterns for 11-element MRA with uniform and non-uniform weightings

2D Sparse Arrays

There are different antenna geometries, which can be used in (smart) antenna systems. These

antenna geometries can be one-, two-, or three-dimensional, depending on the dimension of the space

one wants to access. Although increasing the dimension increases the complexity in the signal-

processing unit, it provides additional scanning (so information) in space, which might be necessary

for some applications. For example, linear arrays and circular arrays are both examples of one-

dimensional arrays and they are used for beam forming in the horizontal plane (azimuth) only. This will

normally be sufficient for outdoor environments, at least in large cells in mobile communication

systems [3]. However, for indoor or dense urban environments, two-dimensional arrays may be

necessary due to their two-dimensional beam-forming capability, in both azimuth and elevation angles.

Like one-dimensional case, we search minimum redundancy arrays and minimum hole arrays for six

active antenna elements. We will find the beampattern using (1.25), after calculating (1.23) and (1.24)

and for the sake of consistency, we will take the square of (1.25) to obtain the squared aperture

smoothing function.


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A 6-element 1D minimum redundancy array (6-element linear minimum redundancy array) structure

corresponds to a 36-element 2D minimum redundancy array (36-element rectangular minimum

redundancy array) structure. A 36-element 2D minimum redundancy array structure and the

corresponding beampattern for this structure are given in Figure 2.10.

(a) Array structure (b) Beampattern

Figure 2.10: Array structure and beampattern for a 36-element rectangularMRA

The array stucture in Figure 2.10(a) consists of 6 6 36 = active antenna elements. Active antenna

elements are showed with symbol, whereas symbol shows the removed elements like in linear

arrays. The unit spacing between antenna elements (including non-active elements) is d , which is

equal to 2 . The aperture size is equal to 13 13d d or 6.5 6.5 . The beampattern in Figure

2.10(b) is same for both azimuth and elevation angles because there is no steering applied. For a 36-

element 2D rectangular minimum redundancy array 3.0 3 dB beamwidth of 3.0 and -7.8923 dB

maximum side lobe level of -7.8923 dB is obtained.

Similar to MRA, 36-element 2D rectangular minimum hole array structure and the corresponding

beampattern for this structure are given in Figure 2.11.


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(a) Array structure (b) Beampattern

Figure 2.11: Array structure and beampattern for a 36-element rectangularMHA

This array structure is the extension of 13625 geometry of the 1D linear MHA structure to 2D. For a

36-element 2D rectangularminimum hole array, 3 dB beamwidth of2.4 and peak SLL of-8.7106 dB

are found. An interesting observation is that, extending the 1D geometry to 2D geometry, that is

doubling the number of elements, results with an improvement in the resolution twice. For 6-element

MRA and 6-element MHA, 3 dB values beamwidths changed from 6.0 and 4.8 to 3.0 and 2.4,

respectively.

Finally, we compare two-dimensional 36-element rectangularminimum redundancy array, 36-element

rectangular minimum hole array and 36-element uniform rectangular linear array (URA) in Figure

2.12(b).

(a) Array structure for 2D ULA URA (b) Beampattern

Figure 2.12: (a) Array structure for 2D ULA URA and (b) beampattern for 36-element rectangular

MRA, 36-elementrectangularMHA and 36-element URA


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The array geometry for a two-dimensional uniform linear array is given in Figure 2.12(a). Different

types of arrays are compared in Table 2.5.

Table 2.5: Comparison of 36-element 2D ULA, MRA and MHA 36-element rectangularMRA, 36-

elementrectangularMHA and 36-element URA

n M Max. SLL[dB]

3 dBBeamwidth

ArrayGeometry

Array Type

36 36 -21.361 8.6 11111

11111 ULA URA36 196 -7.8923 3.0 15322 15322 MRA

36 324 -8.7106 2.4 13625 13625 MHA


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Chapter 3

3.1. Introduction

In this chapter, we will begin with a very brief description of beamforming. Then, we will make a

comparison of classical analog beamforming and digital beamforming and continue with a discussion

of narrowband beamforming. After a quick view to narrowband beamforming, we will introduce

broadband beamforming techniques. Our aim is to investigate a broadband frequency-invariant

beamforming method. Therefore, in section 3.2, we address ourselve

a coskun thesis revised

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