CHAPTER AN INTRODUCTION TO DIRECTION - OF - ARRIVAL

279

Handbook of Position Location: Theory, Practice, and Advances, First Edition.Edited by Seyed A. (Reza) Zekavat and R. Michael Buehrer.© 2012 the Institute of Electrical and Electronics Engineers, Inc.Published 2012 by John Wiley & Sons, Inc.

CHAPTER 9

AN INTRODUCTION TO DIRECTION - OF - ARRIVAL ESTIMATION TECHNIQUES VIA ANTENNA ARRAYS

Seyed A. (Reza) Zekavat Michigan Technological University, Houghton, MI

THIS CHAPTER introduces the fundamentals of direction - of - arrival ( DOA )

techniques for localization systems. The main goal is the introduction of DOA

methods via antenna arrays. In order to better introduce the principles of DOA,

fi rst, we briefl y introduce antenna systems and antenna arrays. The reader will

learn important parameters of an antenna and antenna arrays and the notion of

beamforming (BF). The chapter introduces different measures for comparing DOA

estimation methods and examples of DOA estimation techniques, and discusses

their pros and cons. Moreover, a new DOA estimation method that improves the

performance and complexity for implementation on software - defi ned radios is

introduced. Finally, we discuss a DOA estimation method that is suitable for

periodic sense transmissions in localization techniques such as wireless local

positioning systems (WLPSs) (see Chapter 1 ).

9.1 INTRODUCTION

DOA estimation techniques have applications in beamforming, detection, and local-ization. DOA estimation techniques can be divided in general into two categories: (1) online, those that possess lower complexity and can be easily implemented for online applications, and (2) offl ine, those that are very complex and can hardly be implemented for online applications. Beamforming and detection processes need

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280 CHAPTER 9 DIRECTION-OF-ARRIVAL ESTIMATION TECHNIQUES

online DOA estimation, while achieving high precision is not vital: Coarse DOA estimation is suffi cient for this process. Localization applications require high - performance DOA estimation; however, online DOA estimation is not essential in many localization applications.

Compared to time - of - arrival ( TOA ) estimation techniques, DOA estimation needs the implementation of antenna arrays. Each antenna element needs to get connected to a radio frequency (RF) component. RF components are one of expen-sive components of radio systems. In addition, the power consumption of these components is relatively high. Thus, it is expected that compared to TOA estimation techniques, DOA estimation needs higher complexity and power consumption. However, as discussed in Chapter 1 , in DOA estimation techniques, only two base nodes (equipped with antenna arrays) are suffi cient to maintain full localization of a target node. This adds a higher fl exibility to DOA estimation techniques compared with TOA or received signal strength indicator ( RSSI ) estimation methods.

In addition, TOA and RSSI estimation methods are suitable methods for homogeneous mediums with known specifi cations. Air and space are examples of those mediums. However, TOA and RSSI methods are not suitable for inhomoge-neous mediums such as water. The temperature and the percentage of salt in different layers of water are different. Accordingly, the speed of propagation of signals would be different. This creates error in TOA estimation. In addition, the loss of different layers of water is not the same. Accordingly, RSSI methods are involved with con-siderable error in inhomogeneous mediums. DOA estimation is also affected by the nonhomogeneity of environments; however, the impact is milder specifi cally if the DOA is getting closer to the antenna boresight. This makes DOA estimation tech-niques for localization good techniques for nonhomogeneous areas, such as the human body and the localization of sensors (e.g., endoscopy capsules) inside the body.

Section 9.2 offers an overview on antennas. This chapter reviews DOA estima-tion techniques that are based on antenna arrays; thus, Section 9.3 introduces antenna arrays. Driven by the demand, many DOA estimation techniques have been proposed [1, 2] ; examples are spatial spectral estimation methods, such as delay and sum (DAS) [3] , and eigenstructure methods, such as multiple signal classifi cation ( MUSIC ) [4] , root MUSIC [5] , and estimation of signal parameters via rotational invariance technique ( ESPRIT ) [6] . DOA estimation is possible via multiple anten-nas (separated by half wavelength) installed at the receiver. Section 9.4 discusses the details of many important DOA estimation techniques. It has been depicted that performance of DOA estimation in many systems such as WLPSs that use signals wherein periodic nature is low. Section 9.5 discusses this problem and introduces a solution to this problem. Section 9.6 concludes the chapter.

9.2 ANTENNAS AND THEIR PARAMETERS

This section briefl y introduces antenna systems. Antenna is the key component in the transmission and reception of electromagnetic ( EM ) waves. Antennas radiate EM waves because of time - varying electric fi elds created by a time - varying signal

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9.2 ANTENNAS AND THEIR PARAMETERS 281

(e.g., sinusoidal waveform). Antennas come in all shapes and sizes but are essentially metallic structures used for the radiation and reception of radio waves. At high frequencies, even a short wire can act as an antenna. Antennas are divided into two main categories: (1) directional antennas and (2) omnidirectional antennas.

Directional antennas propagate energy only in some specifi c directions. Examples are dish antennas that are used for satellite communication or in radars. Omnidirectional antennas emit the EM energy in all directions. Practically, it is hard to build an omnidirectional antenna, however, some antennas such as dipole (or monopole) antennas are considered as omnidirectional. These antennas are used in applications such as broadcasting, for example, by TV or radio stations. In general, antennas possess many important parameters, which include the following:

1. antenna beam pattern,

2. antenna half - power beamwidth ( HPBW ),

3. main lobe power to the fi rst side lobe power ratio,

4. main lobe power to non - main lobe power (all side lobe power) ratio,

5. antenna impedance,

6. antenna return loss,

7. antenna bandwidth,

8. antenna gain, and

9. antenna polarization.

The antenna beam pattern represents the variation of the power or amplitude of the signal with angle. Here, angles are represented in terms of azimuth angle and eleva-tion angle. The azimuth angle is the angle that is defi ned by setting a reference direction (e.g., north) in the horizontal plane (e.g., a plane parallel to the earth), and the elevation angle is the angle that is defi ned with respect to the line perpendicular to this horizontal plane. These angles are shown in Figure 9.1 . The antenna beam pattern is generally a function of the antenna shape, dimension, and frequency.

As shown in Figure 9.2 , in general, the beam pattern consists of a main lobe, some side lobes, and some nulls. It is usually desired to transmit the signal in the

Figure 9.1 Azimuth and elevation angles.

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direction of the main lobe. Therefore, side lobes are not desirable and should be mitigated. Figure 9.2 shows the beam pattern of a directional antenna. The beam pattern of an ideal omnidirectional antenna is represented by a sphere.

9.2.1 Antenna HPBW

Considering that the y - axis in Figure 9.1 is the signal amplitude normalized to the main lobe maximum value, the HPBW can be found if we cross the 1 2 line and the beam pattern. The term “ half power ” comes from the fact that the output power has fallen to half of its midband level. This can be understood by examining the equations below:

Power ratio dB( ) = 100 2

1

log ,P

P

where P 2 is the output power and P 1 is the input power, and

Voltage ratio dB( ) = 20 2

1

log ,V

V

where V 2 is the output voltage and V 1 is the input voltage.

When P P2 11

2= ,

Power ratio dB dB( ) = = ( ) = −10

12 10 0 5 3 01

1

1

log log . .P

P

Voltage ratio dB dB( ) = = ( ) = −201

220 0 7071 3 01log log . . .

As a result, HPBW is also called a 3 - dB beamwidth. In other words, it is the angular distance (azimuth or elevation) between two antenna pattern points where the power

Figure 9.2 Antenna beam pattern. AF, array factor.

Azimuth or Elevation angle

2/1

Beam Pattern (AF): the Distribution of Amplitude

Main Lobe

Side LobesHPBW

φ

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becomes half of its maximum value. The elevation angle is measured with respect to its projection on the xy - plane.

The structure of a dipole antenna and its 3 - D beam pattern are shown in Figure 9.3 . As seen in this fi gure, its pattern is not fully isotropic or omnidirectional. Typically, the power at the top and bottom of the antenna is almost zero. This area is called the cone of silence.

9.2.2 First Side Lobe to the Main Lobe Power Ratio

This parameter refers to the maximum power of the largest (usually fi rst) side lobe to the maximum power of the main lobe. To avoid interference effects in many wireless communications systems, it is desirable to reduce this ratio as much as possible. For example, in cellular systems, to avoid interuser interference effects, it is desirable to use directional antennas and to direct the main lobe of these antennas toward the desired users. However, the side lobe of these antennas may create inter-ference. Adaptive antennas are those that direct their main lobe toward the desired users and adjust their nulls toward the interfering users.

9.2.3 Non - Main Lobe Power (All Side Lobe Power) to Main Lobe Power Ratio

This is another characteristic of a directional antenna and represents how well a directional antenna can suppress the side lobes. It could be considered as a better measure for an antenna compared with the fi rst side lobe to the main lobe power ratio , as it considers the power of all side lobes and, accordingly, the maximum possible interference level.

9.2.4 Antenna Impedance

This parameter represents the equivalent impedance of the antenna structure. Similar to other transmission lines (cable, waveguides, etc.), or any circuit element, an antenna can be well modeled by a combination of capacitors, inductors, and resistors.

Figure 9.3 Dipole antenna structure and its beam pattern. See color insert.

dB (Gain Total)

Y

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Then, an antenna can be represented by its impedance. In general, the impedance of any element is not constant and varies with frequency. In addition, in circuit theory, we have learned that a given Thevenin circuit has the maximum energy transfer when the load impedance Z L matches the circuit impedance Z , as shown in Figure 9.4 . Because Z and Z L are both functions of frequency, we can match the antenna load only for certain frequencies.

At the transmitter end, the antenna is connected to the source as shown in Figure 9.5 . It is important to match the antenna impedance to the source to allow for maximum power transfer from the source to the antenna. The impedance is normally seen through the antenna terminals ( Z a ). In Figure 9.5 , Z 0 refers to the transmission line (e.g., connecting cable) characteristic impedance.

If the impedance is not matched properly, the maximum transfer of energy does not occur. Under these circumstances, the refl ections of energy can occur, which leads to interference and unwanted standing waves in the transmission line. This reduces the power transform effi ciency. As we mentioned, the impedance of the antenna varies with frequency. Therefore, the antenna would be properly matched only at certain frequencies.

9.2.5 Antenna Return Loss

Antenna return loss refers to the amount of power refl ected by the antenna divided by the amount of power transmitted to the antenna due to the mismatch between the antenna and the course and transmission line, as shown in Figure 9.5 ; that is,

Figure 9.4 Thevenin circuit.

AC

Z

Thevenin Equivalent

Z = ZL*

ZL

Figure 9.5 Antenna impedance.

Source *0 aZZ =

aZ

Transmission Line

0Z

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Return loss Reflected

Transmitted

= 10 log .P

P (9.1)

Here, P Refl ected refers to the refl ected power by the antenna, and P Transmitted refers to the transmitted power to the antenna. As P Refl ected reduces to zero, the return loss reduces. The return loss is measured with respect to the frequency. As frequency changes, the antenna impedance would vary, and accordingly, the mismatch increases. As shown in Figure 9.6 , in some frequencies, the return loss is minimum.

9.2.6 Antenna Bandwidth

Antenna bandwidth is defi ned based on the antenna return loss. Typically, the fre-quencies that the return loss is less than − 10 ° dB determine the antenna bandwidth. Higher antenna bandwidth allows higher transmission bandwidth. Higher transmis-sion bandwidth allows higher throughput, transmission rate, or capacity. Later, we will observe that the transmission rate of a transmitter is not only a function of the antenna but also of many other transmitter components.

9.2.7 Antenna Gain

Antenna gain measures the change of the antenna radiation intensity with respect to the azimuth and elevation direction. The following expressions illustrate how the antenna gain is determined:

Antenna gain = ( ) = ( )G Dθ φ η θ φ, , . (9.2)

Here,

η = antenna effi ciency = R

R Rr

r d+,

R r = radiation resistance, and

R d = antenna resistance.

Figure 9.6 Antenna return loss.

0 dB

−10 dB

−30 dB

Bandwidth2.2 2.25 2.3 2.35 2.4 2.45 2.5 2.55 2.6

−30

−25

−20

−15

−10

−5

0

2.469 GHz2.41 GHz

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In addition,

D θ φ,( ) = =Antenna directivityRadiation power in space angle uunit

Average radiation power in space angle unit

and

Space angle unit = = ⋅ ⋅d d dΩ sin .θ θ φ

It is clear that if the antenna loss is zero, R d would be equal to zero. In this case, antenna effi ciency would be unity; however, in general, R d > 0. In addition, the radiation resistance R r is a function of antenna length and/or dimensions. If the antenna dimension ( d ) increases with respect to the antenna wavelength λ , for example, if d approaches infi nity, the radiation resistance R r increases. As a result, as the antenna length becomes longer, its effi ciency η tends toward its maximum value of 1. Hence, theoretically, the length of an antenna should be infi nitely large to achieve full effi ciency. In other words,

If then and thend Rr→ ∞ → ∞ →, .η 1

However, in practice, it is not acceptable to infi nitely increase the antenna dimen-sion. The length of λ /2 leads to a reasonably high antenna effi ciency for dipole antennas. As shown in Figure 9.7 , monopole antenna has a length that is half of a dipole antenna. Monopole antennas are usually installed on (perfect) conductors, such as the ground surface or the top of vehicles. The monopole antenna along with its projection with respect to the ground surface creates a beam pattern similar to a dipole antenna. An example of monopole antennas is your car radio antenna. The length of a monopole antenna is usually selected λ /4.

Antenna Gain Is Usually Measured in d B i : dBi refers to the gain of antenna, when it is measured with respect to an omnidirectional antenna. An omnidirectional antenna (also called isotropic antenna) is an antenna that emits EM energy uniformly in all directions. The term “ i ” in dBi refers to isotropic . As the antenna gain increases, the energy transferred to certain points would increase compared to an isotropic antenna. For example, if two antennas both transmit 1 W of power, one is an omni-

Figure 9.7 Monopole antenna.

Reflection of the monopole with

respect to ground

Monopoleantenna

Perfectconductor

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9.3 ANTENNA ARRAYS 287

directional antenna and the other is directional, a 5 - dBi gain of the directional antenna represents that the gain of that antenna is 5 dB more than the omnidirectional antenna that transmits the same amount of power. In other words, there are points in the space in the direction of maximum power that receive 10 0.5 = 3 times more power with the directional antenna compared with an omnidirectional one.

9.2.8 Antenna Polarization

Antenna polarization refers to the orientation of an EM wave ’ s electric fi eld. The three most common antenna polarizations are

1. horizontal polarization,

2. vertical polarization, and

3. circular polarization.

A horizontal polarization antenna ’ s electric fi eld is parallel to the ground (Fig. 9.8 ). “ Fishbone ” - type television antennas are examples of horizontal polarized antennas. A vertical polarization antenna ’ s electric fi eld is perpendicular to the ground (Fig. 9.9 ). Examples of vertical polarized antennas include AM radio towers and car radio antennas.

Horizontal and vertical polarized antennas radiate EM waves in a linear plane only. A circular polarized antenna radiates waves in the horizontal, vertical, and all planes in between both linear planes. Circular polarization is commonly used for satellite communications. Helix antenna is an example of circular polarized antennas.

9.3 ANTENNA ARRAYS

An antenna array is an array of antenna elements located at a distance, d , from each other. Figure 9.10 represents a patch antenna element, and Figure 9.11 is an antenna array made of patch antenna elements. Antenna arrays are also called

Figure 9.8 Horizontal polarization.

E-Field

Figure 9.9 Vertical polarization.

E-Field

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multi - input – multioutput ( MIMO ) systems. MIMO systems that experience a coher-ent channel across all antenna elements can be used for BF and DOA estimation. If the channel is not coherent across antenna elements, MIMO systems are mainly used for space diversity in order to improve the performance of detection.

Antenna arrays that are used for BF and DOA estimation are examples of directional antennas. As shown in Figure 9.12 , these antennas can direct the main part of the produced energy toward a specifi c direction in the main lobe. Antenna arrays that are used for BF and DOA estimation are also called smart antennas .

9.3.1 Smart Antennas

These antennas are called smart not because of the antenna, as it is usually a metallic structure, but because of signal processing algorithms that enable them to fi nd the direction of the desired users and to form a beam optimally toward that user. Two important categories of smart antennas are switched beam smart antennas and adap-tive antennas . Figure 9.12 represents these categories of smart antennas.

Switched beam antennas are formed by fi xed beam patterns. As the source moves from one direction to another, the antenna is switched from one beam to another one. In this category of smart antennas, the interference effects of other

Figure 9.10 Patch antenna element.

Figure 9.11 Patch antenna array.

Figure 9.12 Switched beam smart antennas (left) and adaptive smart antennas (optimal beamformers) (right).

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sources are not optimally removed. In switched beam smart antennas, a simple BF technique such as conventional BF is applied to the antennas.

Adaptive smart antennas are the second category of antennas in which, based on the direction of the desired and interfering sources, an adaptive beam is formed. The goal is to optimally receive the signal of desired source and to remove the signal of interfering sources. Adaptive smart antennas are equipped with optimal BF tech-niques. The goal of an optimal beamformer is to direct the signal toward a specifi c user or source and to create nulls toward undesired users or sources.

An optimal beamformer is indeed a fi lter that incorporates the signals observed over each antenna element (space domain) and each time (time domain) to extract the desired signal and null interfering signals. As shown in Figure 9.13 , the input to the beamformer x is a matrix that consists of all observed signals in (in time, space, or frequency domains). The optimal beamformer fi nds the weight matrix w such that the output y represents the best estimate of the desired signal.

As shown in Figure 9.12 , an optimal beamformer forms a beam toward desired users and steers nulls toward interfering users. Different BF techniques have been introduced, which include capon and linear constrained minimum variance ( LCMV ) [7] , and maximum noise fraction ( MNF ) [8] .

The conventional or Fourier BF technique is the simplest BF method in which a fi xed phase consistent (based on known DOA) is applied to each antenna element to form a beam toward the desired users. In conventional BF, the DOA of a desired user should be known in priori. In addition, this technique does not optimally remove the impact of interfering users.

An important factor in all DOA estimation techniques is that the phase and frequency across all antenna elements should be properly synchronized. In other words, the same carrier signal reference should be applied to all antenna elements.

9.3.2 Important Parameters of Antenna Arrays

Besides the parameters defi ned for antenna elements, antenna arrays possess other parameters, which include the following:

1. array vector,

2. array factor, and

3. mutual impedance.

Array Vector: This parameter of an antenna array corresponds to

v e ej

dj

M d

= …⎡

⎣⎢

⎤

⎦⎥

−⋅

−−( ) ⋅

12 2

1

, , , .sin sin

πφ

λπ

φλ (9.3)

Figure 9.13 Optimal beamforming is equivalent to optimal fi ltering.

x y = wHx

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In this equation, M is the number of antenna elements, ϕ corresponds to the DOA, and λ is the wavelength. The array vector indeed represents the relative signal current (or voltage) of each antenna element with respect to the other antenna ele-ments. Based on Figure 9.14 , it is observed that if the signal arrived in the fi rst antenna element can be represented by the phasor v , the signal of the second antenna

with respect to the fi rst one would have the phase difference of ej

d−

⋅2π

φλsin

, and the third antenna with respect to the fi rst one corresponds to

ej

d−

⋅2

2π

φλsin

,

and the M th antenna with respect to the fi rst one,

ej

M d−

−( ) ⋅2

1π

φλ

sin

.

Array Factor: Considering Figure 9.14 and the array vector, it is observed that the signal received at the antenna receiver after application of the weights w = [ w 1 , w 2 , … , w M ], and the addition process corresponds to w · v T ; that is,

AF w em

jm d

m

M

=−

−( ) ⋅

=∑ 2

1

1

πφ

λsin

. (9.4)

The normalized array vector assuming w m = 1, ∀ m corresponds to

AFM

eM

e

en

jm d

m

M jMd

jd

= = ⋅−

−

−−( ) ⋅

=

−⋅

−⋅∑1 1 1

1

21

1

2

2

πφ

λ

πφ

λ

π

sinsin

sinn.φ

λ

The magnitude of the array factor represents the antenna beam pattern and corre-sponds to

AFM

M d

dn = ⋅⋅ ⋅⎛

⎝⎜⎞⎠⎟

⋅ ⋅⎛⎝⎜

⎞⎠⎟

1 22

12

2

sinsin

sinsin

.

πλ

πλ

φ

φ (9.5)

Figure 9.14 Computing the array vector of an antenna array.

Element #0

Element #1

Element #2

Element #M-1

φs(t)

w0

w1

w2

wN–1

d

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Example 9.1 Antenna Array Pattern

Use the magnitude of the normalized array factor in Equation 9.5 , assume d = λ /2, and write a MATLAB code and sketch the beam pattern of an antenna array com-posed of four and six antenna elements. Compare their HPBW.

In addition, sketch the beam pattern assuming higher antenna element distanc-ing, for example, d = 2 λ for six element antenna arrays. What happens if the number of antenna elements increases beyond six elements?

Solution

The beam pattern MATLAB code is in the fi le Chapter_9_Example1.m. MATLAB codes can be found online at ftp://ftp.wiley.com/public/sci_tech_med/matlab_codes . As it is observed, the code does not need any specifi c MATLAB toolbox. The cor-responding beam pattern for four and six element antenna arrays and d = λ /2 is sketched in Figure 9.15 . Based on this fi gure and the MATLAB code results, the HPBW of four element antenna arrays corresponds to 26.3 ° and that for six element antenna arrays corresponds to 17.18 ° . As the number of antenna elements increases from 4 to 6 elements, the HPBW decreases. Therefore, it is expected that as we increase the number of antenna elements, using any DOA algorithm, we will be able to better resolve sources. In other words, we expect that the resolution of the DOA estimation technique increases.

It is observed that as the number of antenna elements increases, HPBW decreases. In addition, the side lobe level of six element antenna arrays is lower than the four element antennas. Figure 9.16 sketches the beam pattern for d = 2 λ and six element antenna arrays. It is shown that as the number of antenna elements increases, a higher number of grating lobes are generated. As a result, for DOA estimation techniques to avoid any confusion, the spacing of antenna elements should be selected to be d = λ /2.

Mutual Coupling: The signal incident on one antenna element is usually refl ected from that antenna element and impacts the signal received by neighboring antenna elements. In other words, each antenna acts as a parasitic element for neighboring

Figure 9.15 Beam pattern of four (left) and six (right) element antenna arrays when d = λ /2.

–100 –80 –60 –40 –20 –100 –80 –60 –40 –200 20 40 60 80 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 20 40 60 80 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

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antenna elements. This situation creates mutual coupling between antenna elements. Figure 9.17 represents the effects of refl ections, which creates mutual coupling.

In this case, the array vector in Equation 9.3 does not any more represent the relationship between the signals received at different antennas. Indeed, the array vector represents the relative current or voltage of each antenna element with respect to another antenna element assuming no mutual effect between these elements. Now, if the mutual coupling exists, the received signal (voltage or current) over one antenna element is an addition of the received signal that is coming directly from the far - fi eld source and the refl ected signals from neighboring antenna elements; that is,

W

W

u u

u u

v

vM

M

M MM M

1 11 1

1

1

�…

� � �…

�⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

=⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥.. (9.6)

Figure 9.16 Antenna pattern when d = 2 λ . Many grating lobes avoid proper DOA estimation.

–100 –80 –60 –40 –20 0 20 40 60 80 1000

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Figure 9.17 Mutual coupling impacts the DOA performance.

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9.4 DOA ESTIMATION METHODS 293

In Equation 9.6 , W i refers to the received signal (voltage or current) at the antenna element i M∈{ }1 2, , ,… and u ii = 1 for i M∈{ }1 2, , ,… , and u ij represents the mutual coupling of antenna element j on antenna element i , and v i is the i th element of the array vector in Equation 9.3 . It is expected that uij < 1. In addition, usually, u u i j i mij im> − < −, if . In addition, it is expected that u i jij � 1 1, if − > . Thus, the effect of mutual coupling can be ignored for non - neighboring antenna elements in most scenarios.

Mutual coupling has a negative impact on DOA estimation techniques. This problem is usually mitigated by antenna calibration techniques. Many methods have addressed the calibration issue in smart antennas in the literature. Some works emphasize on calibrating mutual coupling between antenna elements [9 – 11] , while others emphasize on calibrating a receiver mismatch [12, 13] .

In a smart antenna system, the mutual coupling and the receiver mismatch exist simultaneously. The mutual coupling is altered by the antenna structure, dis-tances between antenna elements, and so on. The receiver mismatch is due to the independence of receivers and depends on receiver confi guration (amplifi er gain or attenuator attenuation).

If we compensate antenna element mutual coupling and receiver mismatch simultaneously, the best performance would be achieved. If we cannot compensate them simultaneously, we should compensate the one generating the largest error. The proposed techniques in References 10 and 11 all need a far - fi eld fi xed source implemented in a lab environment, that is, off - line calibration. The methods pre-sented in References 12 and 13 apply extra hardware in each receiver, which increases system cost and complexity.

9.4 DOA ESTIMATION METHODS *

Different DOA estimation methods have been proposed and discussed in the litera-ture (see Reference 1 ). In this section, we introduce the fundamentals of some techniques and compare them in terms of different measures. The section offers some MATLAB - based examples on the implementation of MUSIC and root MUSIC methods.

DOA Estimation Requirements The fi rst assumption in many DOA estimation techniques is the fact that the

source is located in the far fi eld. Usually, the far fi eld is considered a distance of about 10 λ . Thus, for a 3 - GHz frequency, the far fi eld is about 1 m. In many applica-tions, the usual distance of the antenna array and the target might be small. Thus, the implementation of DOA estimation for near - fi eld situations might be considered. Many papers have already discussed DOA estimation techniques for near - fi eld con-ditions [14, 15] .

In general, as shown in Figure 9.18 , DOA estimation needs an array of anten-nas located at a distance ( d ) from each other. Usually, d is selected to be half a wavelength to avoid any grating lobe and confusion in DOA (see Example 9.1). It

* Some of the materials in this section are partially published in Reference 3 .

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Figure 9.18 The general block diagram of a DOA estimator.

Element #M

Element #3

Element #2

Element #1

d PROCESSOR

DOA

Receiver

Receiver

Receiver

Receiver

should be noted that receiver synchronization is very critical in the DOA estimation problem as DOA estimation is computed by measuring the relative phase of signals received across a number of antenna elements.

Figure 9.19 represents the detailed block diagram of the WLPS (a single - node localization system introduced in Chapters 1 and 34 ), which needs DOA and TOA estimation. In this fi gure, DDC refers to digital down converter. As shown in Figure 9.19 , each antenna element is connected to an RF front end and a receiver. It is observed that both DOA estimation and BF are applied in the baseband, and after full amplitude, frequency and phase synchronization were applied across the signals received from antenna elements. This synchronization is required to extract the estimated DOA with minimal error.

Another important requirement of DOA estimation techniques is full calibra-tion of the system. Calibration is needed because

1. the RF front end of each antenna is not exactly similar to the other one and each may apply a different phase to the signal; and

2. even if the RF front end of antenna elements are fully similar, the interaction of propagation effects of antenna elements are not consistent. The latest effect is due to the antenna array mutual coupling. The impact of mutual coupling on the received phase is a function of DOA; that is, as the DOA varies, the induced phase due to mutual coupling changes.

To address the calibration problem, two general approaches have been proposed in the literature:

1. Antenna Calibration Techniques: These techniques are usually incorporated offl ine. They use lab experiments to fi nd error in DOA estimation due to the mutual coupling and receiver mismatch in order to fi x the DOA estimation problem. The receiver RF front and the specifi cations of antennas may change with time and temperature. Thus, offl ine calibration techniques may not perform over a long run and signal processing technique should also be used.

2. Signal Processing Methods: These techniques estimate the error in DOA estimation and apply corrections to the measured DOA estimation [16 – 18] .

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c09.indd 295c09.indd 295 8/5/2011 3:33:45 PM8/5/2011 3:33:45 PM


Examples of DOA estimation techniques proposed include spectral estimation methods (also called DAS), maximum likelihood (ML), and eigenstructure methods (include many types such as MUSIC, root MUSIC, and fused DAS and root MUSIC [3] ). The authors in References 1, 7 , and 19 have reviewed these techniques. In this chapter, we introduce some popular DOA estimation methods such as DAS, MUSIC, and root MUSIC. We will also introduce some new extensions of these techniques including fused DAS and root MUSIC and cyclostationary - based DOA estimation.

As mentioned earlier, DOA estimation is possible via multiple antennas installed at the receiver. DAS applies several sets of complex weights to antenna elements for each angle. The delayed signals are summed and the power is com-puted. The DOA is determined by analyzing the output power from all sets of complex weights. DAS DOA estimation performance is low; however, its low com-plexity would allow online coarse DOA estimation. It uses simple arithmetic and can be easily implemented on fi eld programmable gate array (FPGA) systems.

MUSIC computes a spatial spectrum by estimating the noise subspace and determines the DOA from its dominant peak. Root MUSIC is similar to MUSIC in many aspects except that the DOA is computed by roots closer to the unit circle of a polynomial formed by the noise subspace.

In this section, fi rst, some criteria to compare different DOA estimation tech-niques are introduced. Next, DAS, MUSIC, root MUSIC, and a fusion of DAS and root MUSIC is introduced. The pros and cons of each technique are separately discussed.

Important Measures in DOA Estimation Techniques

• DOA Estimation Error. DOA estimation error is determined by the variance of the error in the DOA estimation process. This is the most important measure when comparing different DOA estimation techniques. This error itself is a function of different parameters that include signal - to - noise ratio ( SNR ), multipath, and antenna calibration. The estimation error is usually measured in terms of root mean square error ( RMSE ) and corresponds to

RMSE DOA= −( )=

∑1 2

1N

n

n

N

ˆ .θ θ

Here , , , ,θn n N∈{ }1 2 … , is the estimated DOA in the n th snap shot and θ DOA is the real DOA.

• Resolution. This represents the minimum angular separation of two incoming signals, which the DOA estimation method can resolve. This parameter itself is a function of SNR as well. As the SNR decreases, the resolution decreases as well. A technique that can resolve two sources with lower angular separation has a priority over other techniques.

• Sensitivity to Multipaths. As the number of multipaths in the environment increases, the DOA estimation error increases. The rate of change of perfor-mance with the number of multipaths is considered as a measure for the good-ness of a technique.

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• Sensitivity Sensor Errors Such as Calibration. We refer to the fact that calibra-tion is an important factor in creating a good DOA performance. Some DOA estimation techniques are heavily altered by antenna and RF component cali-bration, while other techniques are mildly altered.

• Sensitivity to SNR. SNR plays an important role in the performance of DOA estimation techniques. Specifi cally, performance results show that as SNR reduces, the sensitivity of all techniques increases. Lower SNR leads to higher sensitivity to SNR.

9.4.1 DAS

In DAS, an array of complex weights, � …w w w wM= [ ]1 2, , , ( M is the number of

antennas), is applied to the set of incoming signals over all antennas. These complex weights delay the signal by changing its phase. The delayed signals are then summed, and the output power is measured. If the complex weights are properly selected, the signals constructively interfere, resulting in a high output power. The relative phase delay ϕ i ( θ ) of the signal is a function of the DOA, θ DOA , wavelength λ , antenna spacing d = λ /2, and the antenna number i ; that is,

φ θ π θ λi i dDOA DOA( ) = − −( ) ⋅2 1 sin . (9.7)

This phase delay is relative to that of antenna 1. Ignoring noise effects, the received signal at antenna i is

r Re Aeij t o i= [ ]+ + ( )( )ω θ φ θDOA .

Here, ω = 2 π f o , f o is the center frequency, φ θi DOA( ) has been introduced in Equation 9.7 , and θ o refers to the phase shift in the signal due to the propagation time delay and the modulation phase. Considering a far - fi eld condition, the phase θ o would be the same for all received signals over all antennas.

The applied weight w i to the i th antenna element is

w e i dij

iwi

w

θ φ θ π θ λφ θ( ) = ( ) = −( ) ×( ), sin .2 1 (9.8)

The delayed (over all M antennas) and summed signal is

S s Ae ei

i

Mj t j

i

i iw

θ θ θ θ ω θ φ θ φ θ, ,DOA DOA

DOA( ) = ( ) ==

+( ) ( )+ ( )( )=

∑1 1

0

MM

∑ . (9.9)

In Equation 9.9 , φi and φiw are defi ned in Equations 9.7 and 9.8 , respectively. If

θ = θ DOA , then φ φiw

i= − ; hence, the relative phase would be zero and the power maximizes. The algorithm calculates the associated power, P ( θ l , θ DOA ), of L = 1 + (180/ Δ θ angles θ l ∈ [ − 90 ° , + 90 ° ], l ∈ {1, 2, … , L } and fi nds the angle that maximizes the power that is called the DOA. These angles are equally spaced with the resolution Δ θ . Hence, a power profi le for each DOA (normalized to the power of the fi rst antenna) is created that corresponds to

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PA

S l Ll lθ θ θ θ, , , , , , .DOA DOA( ) = ( ) ∈{ }11 2

2

2 … (9.10)

Here, | · | refers to the absolute value. If φ φi iw+ = 0, then S M Ae j t= ⋅ +( )ω θ0 , and the

associated maximum power corresponds to PA

S Mmax = =1

2

2 2. Mathematically,

ˆ argmax , .θ θ θθ

DOA DOA= ( )l

P (9.11)

The received signal r i experiences zero - mean Gaussian noise. Hence, in Equation 9.11 , P ( θ l , θ DOA ) estimation involves an (exponentially distributed) error. Here, we consider that N snapshots (e.g., N ≥ 25) of P ( θ l , θ DOA ) are measured in the time

domain. Hence, ˆ , ,P P ni l l iθ θ θ θDOA DOA( ) = ( ) + , i N∈{ }1 2, , ... , . Assuming indepen-dent measurements, ML estimation leads to temporal averaging over N snapshots; that is,

P PN

Pl l i l

i

N

out DOA DOA DOA,θ θ θ θ θ θ( ) = ( ) = ( )=

∑ˆ , , .1

1

(9.12)

The peak power in P out ( θ l , θ DOA ) should correspond to the DOA, but there are two problems associated with this approach: (1) Noise can easily throw off a simple peak search, and (2) this only leads to a resolution as fi ne as Δ θ .

A polynomial interpolator improves the performance of the peak fi nder of Equation 9.11 . Considering ( θ m , P m ) as the point with the maximum sampled power, the points {( θ m − n , P m − n ), … , ( θ m , P m ), … , ( θ m + n , P m + n )} are used for the polynomial fi t. The polynomial coeffi cients are found using a least squares fi t. DOA is the angle that maximizes the polynomial. We call this DOA estimator DAS1.

The second DAS algorithm (DAS2) fi ts all of the sampled data points to a theoretical set of data. For each given DOA, based on Equation 9.10 , a lookup table

is created off - line for the power profi le, P lz

l

Lk

true DOAθ θ,( )( )

=( ){ }

1, over all θ l ∈ [ − 90 ° , + 90 ° ]

and all true DOA, θDOAz k( )( ), z ( k ) ∈ {1, 2, … , Z ( k ) }, Z Qk k k( ) ( ) ( )= ΔθDOA, Q 0 180( ) = ,

Q k k+( ) ( )=1 2ΔθDOA, Δ Δθ θDOA DOAk k+( ) ( )<<1 . Here, k ∈ {0, 1, 2, … , K } refers to the iteration

number (an integer); K is selected large enough to achieve the desired performance. In each iteration k ≥ 1, a search is conducted around the priori estimated angle

θDOAk −( )1 . The angle ΔθDOA

k( ) is decreased stepwise to increase angle resolution with minimal computation cost. Based on minimum mean square error ( MMSE ) criteria,

ˆ argmin , ,θ θ θ θ θθ

DOA out DOA true DOA

DOA

kl l

z

z k

k

LP P( ) (= ( ) −

( )( )

( )1 ))=

( )⎡⎣⎢

⎤⎦⎥∑

2

1l

L

. (9.13)

The value of Δ θ specifi es the DOA estimation performance. Accordingly, DAS2 corresponds to the following steps: (1) The received power profi les are generated based on Equation 9.10 , over N snapshots; (2) P out ( θ l , θ DOA ) is calculated using

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Equation 9.12 ; (3) θDOAk( ) is estimated via ΔθDOA

k( ) , k = 0 by Equation 9.13 ; (4) k = k + 1,

and go to (3) for a new search around θDOAk( ) , that is, from θ θDOA DOA

k k( ) −( )− Δ 1 to θ θDOA DOAk k( ) −( )+ Δ 1 .

The performance of DAS2 is higher than that of DAS1. However, DAS2 suffers from higher computation cost.

The number of multiplications ( NOM ) is the complexity measure. For DAS1 with quadratic polynomial, NOM is

NOM = ⋅ ⋅ ⋅ + ⋅ +4 6 360N M L P . (9.14)

Using N = 5 snapshots, M = 6 antennas, Δ θ = 5 ° , and P = 5 sample points used in the quadratic fi t, NOM = 4830. For DAS2, NOM corresponds to

NOM DOA

DOA

= ⋅ ⋅ ⋅ + ⋅ +⎛⎝⎜

⎞⎠⎟

−( )

( )=

∑42

11

0

N M L Lk

kk

K ΔΔ

θθ

. (9.15)

Using N = 5 snapshots, M = 6 antennas, Δ θ = 5 ° , and K = 2, and ΔθDOAk o o o( ) = 10 1 0 1, , .

for k = 0, 1, 2, respectively, NOM = 6697. In general, DAS is a simple and low - complex DOA estimation technique.

However, its sensitivity with respect to SNR, calibration errors, and number of refl ections is high. Moreover, its resolution is low. These points will be presented and discussed via simulations in this section.

9.4.2 MUSIC and Root MUSIC

In these techniques, it is assumed that (1) a priori estimation of the number of sources is available and (2) the number of antenna elements is higher than the number of sources.

MUSIC : The received signal at each antenna element I ∈ {1, 2, … , M } is

x t w y t n ti i i i( ) = ( )⋅ ( ) + ( )θ . (9.16)

Here, wi θ( ) is defi ned in Equation 9.8 ; y i is the i th element of incident signal vector; and N i is the i th element of noise. We defi ne

�w θ( ), �y ,

�n, and X as arrays correspond-

ing to all M antenna elements. By applying temporal averaging over n ∈ {1, 2, … , N } snapshots, the

received signal sample covariance matrix R X is calculated. Each row of the eigenvec-tor matrix V N of R X consists of the eigenvectors corresponding to the set of smallest eigenvalues. These eigenvectors are called noise subspaces. The noise subspaces are orthogonal to the array vector

�w; that is,

�w VH

Nθ( )⋅ = 0. MUSIC estimates the DOA of incident signals by locating the peaks of MUSIC spectrum defi ned as

Hw V V wH

N NH

θθ θ

( ) =( )⋅ ⋅ ⋅ ( )( )

1� . (9.17)

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The orthogonality of �w θ( ) and V N minimizes the denominator of H ( θ ) and generates

the peaks of the MUSIC spatial spectrum. Searching for the directions that maxi-mizes H ( θ ) requires a substantial computational complexity.

Example 9.2 MUSIC Spectrum

MUSIC spectrum has been defi ned in Equation 9.17 . Use MATLAB and sketch this spectrum assuming two sources at 10 ° and 50 ° and six element antennas. Repeat the problem when the two sources are at 10 ° and 14 ° . Can this method resolve two sources when they are as close as 2 ° (e.g., 10 and 12)? Change the number of antenna elements to 12 and study the effect of increasing the number of antenna elements. Does it impact the resolution?

Solution

To generate the spectrum, we fi rst generate Equation 9.16 . Next, the spectrum is generated. The spectrum has been sketched in Figure 9.20 a. It is observed that even if the two sources are as close as 4 ° , they can be resolved; however, if the sources are not as close as 3 ° , they cannot be resolved.

Figure 9.20 (a) Six - element antenna MUSIC spectrum for 10 ° and 50 ° (left) and for 10 ° and 14 ° (right). (b) MUSIC Spectrum for sources at 10 ° and 12 ° for 6 - element (left) and for 12 - element (right) antennas.

–100 –80 –60 –40 –20 0 20 40 60 80 1000

5

10

15

20

25

30

35

40

45

–100 –80 –60 –40 –20 0 20 40 60 80 1000

5

10

15

20

25

30

35

40

45

50

–100 –80 –60 –40 –20 0 20 40 60 80 1000

5

10

15

20

25

30

35

40

–100 –80 –60 –40 –20 0 20 40 60 80 1000

5

10

15

20

25

30

35

40

45

(a)

(b)

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Figure 9.20 b sketches the scenario that the sources are as close as 2 ° . It is observed that the MUSIC algorithm is not able to resolve the two sources any more. Thus, the resolution of this algorithm for a six - element antenna is in the order of 4 ° . Now, if the number of antenna elements is increased to 12, higher resolution can be attained. Typically, it is observed that a resolution of 2 ° is achievable. This conclu-sion is consistent with our discussions in Example 9.1: As the number of antenna elements increases, lower HPBW is attained. However, it should be noted that increasing the number of antenna elements from 6 to 12 improves the resolution from 4 ° to 2 ° , while the complexity of the antenna element system as well as the MUSIC algorithm highly increases. It should be noted that increasing the number of antenna elements as well as the complexity associated with MUSIC leads to higher power consumption as well. Thus, the number of antenna elements should be wisely selected. The associated codes have been provided in the fi le Chapter_9_Example_2.m. MATLAB codes can be found online at ftp://ftp.wiley.com/public/sci_tech_med/matlab_codes .

The block diagram of the MUSIC has been sketched in Figure 9.21 a. In general, the MUSIC algorithm is a complex algorithm as it needs to fi nd the eigen-values of a matrix. In addition, it usually needs the priori information of the number of sources (or refl ections). However, its sensitivity with respect to the number of refl ections, calibration error, and SNR is relatively lower compared to almost all DOA estimation techniques. As will be discussed later in this section, root MUSIC performs better than MUSIC, and its complexity is relatively lower.

Root MUSIC : MUSIC calculates the spectrum for a fi nite number of directions based on the selection of the angle increment. Such searching method has large computational and storage requirements [20] . To refi ne DOA estimation and to alleviate the algorithm complexity, root MUSIC has been proposed by Barabell [5] . As summarized in Figure 9.21 b, root MUSIC fi nds the roots of the polynomial represented by

Figure 9.21 Block diagram of the (a) MUSIC and (b) root MUSIC algorithms.

DOA from Spectral Peaks

Spatial Spectrum P(θ )

Noise Subspace VN

Eigenvalue Decomposition

Signal Covariance Matrix Rx

Observed Signal X

(a)

Signal Covariance Matrix Rx

Noise Subspace VN

Coefficient Vector C

Polynomial f (Z )

Find All Roots

Roots with Unit Magnitude

DOA from Phase of Roots

(b)

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f z H z( ) = ( )−1 . (9.18)

Its complexity is lower compared with MUSIC [7] . In Equation 9.18 , z refers to w i ( θ ) defi ned in Equation 9.8 . Equation 9.18 can be rewritten as

H z C zkk

k L

L−

=− +

−

( ) = ⋅( )∑1

1

1

, (9.19)

where C k is the sum of the elements of the matrix C V VN NH= along the k th

diagonal

C Ck mn

m n k

=− =∑ . (9.20)

Note that the magnitude of z corresponds to one, and its phase contains information of DOA.

Root MUSIC evaluates all the roots of H − 1 ( z ) and assigns those roots that are closer to the unit circle and the remaining roots that are farther from the unit circle to noise. Among all roots, the closer a root is to the unit circle, the more likely it stands for DOA, which can be found from

θλ

π=

− ⋅ ( )⋅

⎛⎝⎜

⎞⎠⎟

−sinarg

.1

2

z

d (9.21)

Hence, to estimate DOA using root MUSIC algorithm, it calculates 2 N – 2 roots of the polynomial H − 1 ( z ), where N represents the number of antennas, then it fi nds the root, which lies on or is closest to the unit circle. Based on the defi nition of z , it should be noted that unlike MUSIC, root MUSIC is only applicable in the case of uniform linear arrays ( ULA s). This is considered a limitation of root MUSIC algorithm.

Example 9.3 Root MUSIC

Repeat Example 9.2 for root MUSIC assuming a source at 30 ° .

Solution

The roots of the root MUSIC in the z - plane and the amplitude of the roots are sketched in Figure 9.22 . It is depicted that only 30 ° leads to the amplitude of unity, and it is considered as the DOA. The MATLAB program of root MUSIC is in the fi le Chapter_9_Example_3.m. MATLAB codes can be found online at ftp://ftp.wiley.com/public/sci_tech_med/matlab_codes .

Complexity Analysis: Complexity is defi ned as the NOM required to execute the algorithm. Both MUSIC and root MUSIC compute the received signal sample covariance matrix R X (with NOM A ), as well as eigenvalues and eigenvectors of R X (with NOM B ):

NOMA M N N= ⋅ ⋅ +4 2 2 (9.22)

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and

NOMB N=26

33, (9.23)

where M is the number of snapshots and N is the number of antennas. Specifi cally, MUSIC computes spatial spectrum H − 1 ( z ) over a certain range of angles, which leads to the following NOM:

NOMC M M M f stp= ⋅ − ⋅ +( )⋅ ( )4 2 1 23 2 θ θ, , . (9.24)

In Equation 9.24 ,

f stp stpθ θ ππ θ π θ

1 22

180

1

180, , sin sin( ) = ⋅

⋅⎛⎝⎜

⎞⎠⎟ −

⋅⎛⎝⎜

⎞⎠⎟

⎡⎣⎢

⎤⎦⎥

÷ ,, (9.25)

where [ θ θ1 2, ] is the angular range to compute the MUSIC spectrum and stp is the number of angular steps used to compute the MUSIC spectrum. Hence, the total NOM of MUSIC is

NOM NOM NOM NOM1

2 3 2 3 2426

34 2 1 2

= + +

= ⋅ ⋅ + ⋅ + + ⋅ − ⋅ +( )⋅

A B C

M N M M M M M f θ θ, , sstp( ). (9.26)

On the other hand, root MUSIC specifi cally computes the roots of a polynomial, which costs

NOMD N N N= − + −220

3212 206 783 2 . (9.27)

Figure 9.22 Root MUSIC roots in z - plane and the amplitude.

−4 −3 −2 −1 0 1 2 3 4

−1

0

1

2

10

Real Part

Imagin

ary

Part

−60 −40 −20 0 20 40 60 800

1

2

3

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Hence, the total NOM of root MUSIC is

NOM NOM NOM NOM2

2 3 24 82 211 206 78

= + += ⋅ ⋅ + ⋅ − ⋅ + ⋅ −

A B D

M N M M M . (9.28)

Figure 9.23 compares the complexity of MUSIC and root MUSIC algorithms. As it is observed, the complexity of MUSIC is higher than root MUSIC. This is due to the fact that MUSIC requires exhaustive search through all possible steering vectors to estimate DOA.

Comparison of MUSIC and Root MUSIC : Sensitivity to SNR : Figure 9.24 compares the RMSE in the DOA estimation of MUSIC and root MUSIC. RMSE is defi ned as

RMSE DOA= −( )=

∑1 2

1N

i

i

N

ˆ ,θ θ (9.29)

where N is the number of DOA estimations, θi is the i th estimated DOA, and θ DOA is the real DOA.

Figure 9.24 is sketched assuming one signal source, six antennas, 50 snap-shots, 50 estimations, and DOA = 30 ° .

It is observed that as SNR increases, the DOA RMSE of both MUSIC and root MUSIC algorithms improves. Using the same NOM, root MUSIC is more robust to SNR. MUSIC costs four times more NOM to attain comparable perfor-mance with root MUSIC.

Figure 9.25 illustrates the accuracy of MUSIC and root MUSIC within a DOA range of − 80 ° to 80 ° . Here, it is assumed that one signal source, six antennas, 50 snapshots, and 50 estimations are available. In addition, SNR = 20 dB. The RMSE

Figure 9.23 Numbers of snapshots versus NOM for MUSIC; step size = 0.001.

0 500 1000 1500 2000 2500 30000

1

2

3

4

5

6

7

8

9

Numbers of Snapshots

Com

ple

xity (

# o

f M

ultip

lications)

MUSIC, 8 antennas

Root MUSIC, 8 antennas

MUSIC, 6 antennas

Root MUSIC, 6 antennas

× 105

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of MUSIC and root MUSIC are signifi cantly low within a DOA range of − 50 ° to 50 ° , beyond which the RMSE increases drastically. With comparable NOM, root MUSIC has lower RMSE and stable performance over a wide range of angles.

In general, the sensitivity of root MUSIC with respect to SNR, number of refl ections, and calibration error (see Reference 21 ) is lower than that of MUSIC.

Sensitivity to Multipaths: Figure 9.26 shows the impact of a multipath on RMSE. In this fi gure, we have considered a line - of - sight ( LOS ) signal and single refl ection that is considered a non - line - of - sight ( NLOS ) signal, and the power of NLOS to LOS is considered 0.1. In addition, six antennas, 50 snapshots, 50 estima-tions are assumed. It is observed that the errors take on a fringe pattern with respect to the angular difference between LOS and NLOS signals. This occurs because the

Figure 9.24 RMSE of DOA versus SNR.

5 10 15 20 25 30 35 40 45 500

0.05

0.1

0.15

0.2

0.25

0.3

0.35

SNR (dB)

RM

SE

of D

OA

(degre

e)

MUSIC: NOM = 17,088, step size = 0.01°

MUSIC: NOM = 84,918, step size = 0.001°

Root MUSIC: NOM = 18,438

Figure 9.25 RMSE of DOA versus true DOA.

−100 −80 −60 −40 −20 0 20 40 60 80 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

True DOA (degree)

RM

SE

of D

OA

(degre

e)

MUSIC: NOM = [15,000,18,000], step size = 0.01

MUSIC: NOM = [60,000,100,000], step size = 0.001

Root MUSIC: NOM = 18,438

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main lobe of the refl ected signal interferes with the side lobes of the LOS signal. When this interference occurs, the power in one of the side lobes increases enough to be considered the main lobe, generating an incorrect DOA.

In general, for a comparable DOA estimation performance, root MUSIC is more computationally effi cient than MUSIC.

9.4.3 DAS and Root MUSIC Fusion

We can merge DAS and root MUSIC to support the localization process with lower complexity: We modify root MUSIC to accept a priori knowledge of the DOA pro-vided by DAS. Classical root MUSIC computes a polynomial and fi nds all of its roots. The DOA is a function of the phase of one of these roots. To reduce the com-putational complexity, we modify classical root MUSIC to search for a single root using Newton ’ s method [9] . The magnitude of the initial root z 0 is 1; its phase is – (2 π d / λ ) · sin θ coarse . Here, θ coarse refers to the DOA estimated via DAS.

First, the tangent line of the polynomial is found, and then the root of that tangent line is computed. The zero of the tangent is a better approximation of the polynomial root. The iteration equation is

z f z f z zi i i i+ = ( ) ′ ( )[ ] −1 . (9.30)

Here, f ( z ) is defi ned in Equation 9.18 . Once the iterative method converges to a fi nal root, z Final , the phase of that root would be the fi ne DOA estimation: ˆ sin argθ λ πFine Final= − ⋅ ( ) ⋅( )−1 2z d . Here, arg( z Final ) corresponds to the phase of z Final .

If the error in the coarse DOA estimation is large, and the seed root for Newton ’ s method is beyond some threshold, the algorithm may converge to an incorrect root. This problem can be easily detected. Classical root MUSIC operates by fi nding the roots close to 1. This is used to check the validity of Newton ’ s method: If the root found has a magnitude near 1, it would be valid. The roots far from 1 (outside of some threshold) would be invalid.

Figure 9.26 RMSE of DOA under multipath conditions.

MUSIC, PNLOS/PLOS = 0.5

Root MUSIC, PNLOS/PLOS = 0.5

MUSIC, PNLOS/PLOS = 0.1

Root MUSIC, PNLOS/PLOS = 0.1

−100 −80 −60 −40 −20 0 20 40 60 80 1000

0.5

1.0

1.5

2.0

2.5

Direction Difference (degrees)

RM

SE

of D

OA

(degre

e)

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There are several ways to proceed from here: (1) Discard the result and wait for a new estimate from the fi rst stage; (2) discard the result, modify the phase of the seed value, and try again; or (3) discard the result and use the coarse DOA approximation as the fi nal DOA. The choice made depends on the importance of a fi ne DOA approximation, how often a DOA approximation must be generated, and the power constraints of the system. Figure 9.27 summarizes the fusion algorithm.

By merging Newton ’ s method with classical root MUSIC, only one root has to be calculated for DOA estimation with comparable accuracy, while classical root MUSIC requires the calculation of 2 M – 2 roots. Thus, the computational burden of the polynomial rooting step is reduced. For modifi ed root MUSIC, NOM is reduced to

NOM = ⋅ + ⋅ ⋅ − ⋅ + ⋅ +38 3 4 3 148 263 2 2M N M M M . (9.31)

For improved performance in the second stage, N = 50 snapshots are used with modifi ed root MUSIC. Considering M = 6 antennas, NOM = 10,742 multiplications is needed, while for the same performance, classical root MUSIC requires 18,522 multiplications. Hence, fi ne estimation with modifi ed root MUSIC is more cost - effective than with classical root MUSIC. It should be considered that the offl ine DOA estimation does not need an updating rate as high as online DOA estimation (required for detection). This will reduce the overall computation cost as well.

Figure 9.27 Flowchart of delay and sum + root MUSIC.

Fine DOA

End

No

Yes

Yes

No

Is the root’s magnitude

close enough to 1

(threshold 1)?

Coarse DOAas

an initial root

Delay and sumModified

root MUSIC

Find the tangent line

of the polynomial.

Compute the root of

that tangent line.

Is the root’s magnitudecloser enough to 1

(threshold 2)?

Compute the phaseof the final root.

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Simulations and Performance Analysis: Monte Carlo simulations are con-ducted to evaluate the proposed algorithms assuming ULA, single source, and Gaussian noise. For DAS1, quadratic polynomial, and for DAS2, K = 2, and ΔθDOA

k o o o( ) = 10 1 0 1, , . for k = 0, 1, 2, respectively, are considered. For the fusion method, the error of the coarse DOA estimation is taken uniform over [ − 2 ° , 2 ° ]. Figure 9.28 illustrates that the RMSE increases as the DOA moves away from the boresight (0 ° ). DAS has an error between 0.2 ° and 0.5 ° when the DOA is between − 40 ° and + 40 ° . In addition, the stability of DAS1 is lower than DAS2. The RMSE of root MUSIC and the modifi ed root MUSIC are practically identical. Their RMSE is less than half of that of DAS methods. In addition, their stability over DOA is higher than DAS. Finally, Figure 9.28 shows that increasing SNR improves the performance.

Figure 9.29 shows the accuracy of DOA estimation methods as a function of complexity. For all techniques, the RMSE decreases drastically with complexity, until a certain threshold. This threshold is unique for each method. Beyond this threshold, the impact of adding more complexity on the RMSE is negligible. Considering six antennas, this threshold is about 20,000, 12,000, 5000, and 4000 for DAS2, root MUSIC, DAS1, and fusion methods, respectively. Considering the cost of online (coarse) DOA (DAS1), the combined cost of both coarse and fi ne estima-tions is still less than that of a single estimation with root MUSIC. The fi gure also shows that for DAS (root MUSIC), as the number of antennas increases, for the same performance, the complexity would decrease (increase).

The performance of DOA estimation techniques and their ability to discrimi-nate signals coming from different directions is very important. In Figure 9.30 , the corresponding performance of modifi ed root MUSIC is illustrated. The incoming signal is simulated as the sum of LOS and NLOS signals. The error is measured as a function of Δ Φ (the angular difference between LOS and NLOS signals) and the ratio of the power of the NLOS and LOS signals ( P NLOS / P LOS ). The error takes on a fringe pattern with respect to Δ Φ . This occurs because the main lobe of the refl ected

Figure 9.28 RMSE of estimations versus true DOA: one signal source, six antennas, 50 snapshots, and 500 estimations.

−80 −60 −40 −20 0 20 40 60 800

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Direction of arrival (degrees)

RM

S e

rror

(degre

es)

DAS1, 5 dBDAS1, 15 dBDAS2, 5 dBDAS2, 15 dBRoot MUSIC, 5 dBRoot MUSIC, 15 dBFusion method, 5 dBFusion method, 15 dB

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signal interferes with the side lobes of the LOS signal. When this interference occurs, the power in one of the side lobes increases enough to be considered the main lobe, generating an incorrect DOA.

9.4.4 Comparison

Table 9.1 compares different DOA estimation methods in terms of different mea-sures. In general, it is depicted that root MUSIC has much better properties compared

Figure 9.29 RMSE of estimation techniques versus the number of multiplications, DOA = 30 ° , 500 estimations, and SNR = 15 ° dB.

0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.00

0.5

1.0

1.5

Complexity (# of multiplications)

RM

S E

rror

(degre

es)

× 104

DAS1, 4 AntennasDAS1, 6 AntennasDAS2, 4 AntennasDAS2, 6 AntennasRoot MUSIC, 4 AntennasRoot MUSIC, 6 AntennasFusion method, 4 AntennasFusion method, 6 Antennas

Figure 9.30 RMSE of the fusion method under multipath conditions LOS signal + single refl ection DOA = 30 ° , six antennas, 50 snapshots, and 500 estimations.

−100 −80 −60 −40 −20 0 20 40 60 80 1000

0.5

1.0

1.5

2.0

2.5

Direction Difference (degrees)RM

SE

of D

ela

y a

nd S

um

+ R

oot M

US

IC (

degre

es)

PNLOS/PLOS = −5 dB




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to MUSIC. In addition, this table depicts that the proposed fusion method leads to much lower complexity compared with other methods proposed in the literature.

9.5 DOA ESTIMATION FOR PERIODIC SENSE TRANSMISSION *

In many applications including the WLPS, the transmission of the signal is periodic. As mentioned in Chapter 1 and shown in Figure 9.31 , WLPS consists of two types of nodes: (1) a multiantenna radio called dynamic base station (DBS), and (2) a single - antenna radio called TRX (transceiver). The localization process starts at the DBS (see Fig. 9.31 ). It transmits a signal and requests the availability of TRX in its coverage area. Each TRX transmits a unique ID code back to the DBS as soon as it detects a request sent by the DBS and informs its availability.

The DBS receiver estimates the round - trip time and thus the distance between TRX and DBS. The round trip is calculated by estimating the TOA of the TRX ’ s response signal (ID code) with reference to the starting time of the DBS ’ s IDR signal (see Fig. 9.31 ). To calculate the range, the TRX response time delay T d should be taken into account. DBS uses an antenna array to fi nd the DOA of the TRX. Thus,

TABLE 9.1. Comparison of Different DOA Estimation Techniques

Sensitivity to SNR

Sensitivity to Calibration

Sensitivity to Multipath

Resolution Complexity

DAS High Moderate High Low Low Max entropy Moderate Moderate Moderate Moderate Moderate MUSIC Low Higher Low High High Root MUSIC Lower High Lower Higher High ESPRIT Low Low Low High Very high Fusion Low Moderate Lowest Very high Moderate

Figure 9.31 DBS and TRX interaction and positioning process.

T = IRT: transmission period.

The ID code signal received by DBS (from TRX)

R, TOA

, DOA

ID request signal transmitted by a DBS

TOA

θ

τ

τt TRX

DBS

* The materials in this section have been partially published in Reference 24 .

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9.5 DOA ESTIMATION FOR PERIODIC SENSE TRANSMISSION 311

DBS is capable of localizing the TRX independently via DOA – TOA estimation (see Fig. 9.31 ). When multiple DBS nodes perform DOA – TOA positioning, a fusion across those DBS nodes improves the positioning performance [22, 23] .

As shown in Figure 9.31 , the DBS transmits the ID request signal periodically with the period of ID request repetition time ( IRT ) to all mobiles in its coverage area. This periodic transmission will create a cyclostationary property at the received signals at both TRX and DBS receivers, which can be exploited to improve TOA, DOA, channel estimation, and, accordingly, receiver performance.

This periodic sense transmission may avoid stationarity of received signals, which in turn avoids a good estimation of sample autocorrelation of observed signals. Autocorrelation of observed signals should be calculated for applications such as optimal BF, TOA, and DOA estimation (e.g., in MUSIC algorithm). Typically, Figure 9.32 represents the error in the estimation of the sample covariance matrix defi ned as

MSE = ( ) − ( )==

∑∑ ˆ , , .R m u R m uu

M

m

M

11

Here, M is the number of antenna array elements R m u R m u, ,( ) ( )( ), is the ( m , u ) element of the real (estimated) covariance matrix. It is observed that this error is a function of the number of users (nodes) and the duty cycle of the periodic sense signals that is denoted by d c = τ / T . The parameters τ and T have been depicted in Figure 9.31 .

Periodic sense transmission of the signals creates a cyclostationary process in the received signal over each band. As shown in Figure 9.33 , the cyclostationary period can be interpreted as the time over which the interference experienced by each transmitted symbol remains unchanged. Cyclostationarity enables high - performance covariance matrix estimation, which leads to optimum BF, detection, and localization. The cyclostationary process is available for a time period, T cy , called

Figure 9.32 Error in autocovariance calculation due to lack of stationary assumption for different duty cycles.

0.5

0.4

0.3

0.2

0.1

0

MS

E

10 20 30 40 50 60

Number of Users

dc = 0.1

dc = 0.01

dc = 0.001

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cyclostationary coherence time, which is a function of the bandwidth ( B ) over which the signal is transmitted, the beamwidth of antenna array ( θ B ), the distance between two nodes that are involved in the localization of each other ( d ), the speed of nodes ( v ), and the speed of light ( c ) [24] . T cy is approximated by the inverse of cyclosta-tionary Doppler spread , B d , cy , and

BBv v

dd cy

B, max , .=

⎛⎝⎜

⎞⎠⎟2

2

c θ (9.32)

Theoretical and simulation T cy results have been sketched in Figure 9.34 [24] . These results are computed by defi ning and allocating a probability to the cyclostationary time (see Reference 24 for details). As expected, by increasing the speed of vehicle or the number of TRX sets, the cyclostationary time decreases.

Figure 9.34 Cyclostationarity duration in terms of distance (left) and number of TRXs and their average speed (right).

0.8

0.6

0.4

0.2

0

Cyc

locohere

nce t

ime (

second)

100 101 102 103

101

100

10−1

10−2

Distance (m) Number of TRXs

Airport securityapplication

Vehicle collisionavoidance application

v = 2 m/s

v = 5 m/sv = 10 m/s

v = 30 m/s

v = 60 m/s

0 10 20 30 40 50

mv = 2 m/smv = 2 m/s (simul.)mv = 5 m/smv = 5 m/s (simul.)

mv = 10 m/smv = 10 m/s (simul.)mv = 30 m/smv = 30 m/s (simul.)

T cs (

second)

Figure 9.33 Representation of the cyclostationary process and its role in improving localization performance.

The same interference over the cyclostationarity coherence time enables optimal beamforming and localization.

Received signal in period T Received signal in period T + 1 Received signal in period

Fusion

Desired signal

Interfering signal

Interfering signal

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9.5 DOA ESTIMATION FOR PERIODIC SENSE TRANSMISSION 313

An important issue of the cyclostationary - based autocorrelation estimator is the maximum possible value of the number of periods T (see Fig. 9.33 ) over which the cyclostationarity holds (i.e., the ratio of T cy and T ). A larger value N cy = T cy / T leads to better covariance matrix estimation. Typically, T cy is in the order of 1 second for a speed of 2 m/s and the distance of 10 m between nodes, 300 - MHz bandwidth, and 27 ° of HPBW. Now, in Figure 9.33 , if the period of transmission over each band is 1 ms, then within N cy = 1000 periods, the channel would remain unchanged over each transmitted symbol.

Therefore, as depicted in Figure 9.33 , based on the cyclostationary principles, the interference across a symbol remains unchanged over multiple periods deter-mined by T cy . Hence, the estimated autocorrelation of the observed signal

�y n, ω( )

(considering all antenna elements) for each symbol, n , and across multiple periods, ω , corresponds to

ˆ , , .R nN

y n y ncy

H

Ncy

( ) = ( ) ( )=

∑1

1

� �ω ωω

(9.33)

As shown in Figure 9.33 , using Equation 9.33 , DOA and TOA can be estimated across each symbol, n . Next, these DOA and TOA are fused to improve the position-ing performance. Moreover, in optimal BF, Equation 9.33 is used to compute the weight matrix applied to the antenna elements.

We incorporate cyclostationarity and estimate DOA using each bit of the received signal in all frames via the MUSIC method and then fi nd the best estimate of DOA based on those observations (see Fig. 9.34 ). ML estimator is applied across DOA estimations, which correspond to

ˆ

ˆ

.ω

ωσ

σ

ML

k

kk

K

kk

K= =

=

∑

∑

21

21

1 (9.34)

The optimal DOA is computed from the optimal spatial frequency ωML via

ˆ sin ˆ .θλπ

ω= ⎛⎝⎜

⎞⎠⎟

−1

2 dML (9.35)

Here, primary DOA estimates are obtained from different bits of consecutive frames. In other words, after receiving signals within all IRT frames within T cs , we apply MUSIC to the bth , b ∈{1, 2, … , N b } bit of all frames and calculate DOA ( N b is the number of bits in an ID frame); that is,

ˆ ; , , , ..ω ω ωb b bb N= + =Δ 1 2 …

Similarly, for a statistically large number of samples, the estimation errors are zero - mean Gaussian variables; that is, ∼ Δω σb b bN b N∼ …0 1 22, , , , ,( ) = .

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Figures 9.35 and 9.36 represent the MSE performance simulation results of the estimated DOA. The assumptions include the following: (1) Two TRXs are present at 15 ° and 25 ° , respectively; (2) a DBS receiver equipped with a ULA with six elements and half - wavelength element spacing; (3) binary phase - shift keying ( BPSK ) modulation; (4) 20 bits for each ID frame; (5) the difference between the TOAs of the two TRXs is 5 T b ; and (6) T cs = 20 IRT ; this assumption would be rea-sonable selecting IRT ≈ 50 μ s [6] , as the T cs calculated in Figure 9.34 is in the order

Figure 9.35 Performance comparison of proposed approaches in terms of SNR for two TRXs.

1 2 3 410

–4

10–3

10–2

10–1

100

NocombComb-app1Comb-app2

Mea

n S

qu

are

Err

or

(MS

E)

# of TRX

Figure 9.36 Performance comparison of proposed approaches in terms of number of TRXs at SNR = 20 ° dB.

10 15 20 2510

–4

10–3

10–2

10–1

100

NocombComb-app1Comb-app2

SNR (dB)

Mea

n S

qu

are

Err

or

(MS

E)

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ACKNOWLEDGMENTS 315

of 100 ms. Figure 9.35 shows that at MSE ≈ 10 − 2 ; about 10 - dB (15 - dB) improvement is achievable via the fi rst (second) estimation approach compared to a system without combining.

Figure 9.36 represents the variation of MSE with the number of TRXs, assum-ing SNR = 20 ° dB and the difference between TOAs of the desired TRX and j th TRX is 5 T b , 10 T b and − 5 T b for j = 2, 3, 4, respectively. It is observed that (1) when only one TRX is available, the performance of both combining approaches is identical (as expected); (2) considering two TRXs, the second approach leads to better MSE performance; and (3) as the number of TRXs increases, the difference between MSE performances increases. It should be pointed out that better performance of the second combining approach is achievable with a minimal expense of computing the estimation error variance.

The results in Figures 9.35 and 9.36 represent three scenarios. The worst performance is associated with the scenario that fusion is not available and DOA is estimated by applying MUSIC across all bits of one IRT period (no combining). In this case, it is clear that there is a major error in the covariance matrix estimation of MUSIC that reduces the performance of DOA estimation. In the second scenario (comb - app1), fusion is applied across multiple estimated DOAs (in a number of IRT periods within cyclostationary time). Here, each DOA is estimated by applying MUSIC to all bits of the same IRT period. In this case, the performance improves; however, since covariance estimation involves error, the performance is ideal.

The third scenario (comb - app2) is consistent with Figure 9.33 where covari-ance matrix estimation is applied across the same bit of multiple ITR periods. Next, fusion is applied across all bits. In this case, covariance is ideally estimated and the best performance is achievable. The results represent a major difference in perfor-mance for a higher number of TRXes. It is due to the fact that as the number of TRXes increases, the MSE in the autocorrelation matrix increases (see Fig. 9.32 ).

9.6 CONCLUSION

The chapter presented the basics of antenna arrays and DOA estimation techniques. In addition, it presented a DOA fusion method that merges a simple DOA estimation method, which is DAS, and a complex method, which is root MUSIC. It is observed that this merger leads to higher performance and lower complexity. This makes this DOA estimation technique a good candidate for the implementation on software - defi ned radios. In addition, we proposed a DOA estimation method for periodic sense transmissions that is inherent in localization systems such as RADAR and WLPS. It is depicted that by exploiting cyclostationarity, we can improve the performance of DOA estimation methods.

ACKNOWLEDGMENTS

Many former students in my group have contributed in the preparation of pictures and MATLAB programs for this chapter. Specifi cally, I would like to thank Dr. Zhonghai Wang and Ms. Xiaofeng Yang.

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Documents

CHAPTER AN INTRODUCTION TO DIRECTION - OF - ARRIVAL