Estimating the time and angle of arrivals in mobile communications

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Estimating the time and angle of arrivals in mobile

communications

Mémoire

Bahareh Elahian

Maîtrise en génie électrique

Maître ès sciences (M.Sc.)

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Résumé

Dans ce projet, nous présentons une méthode nouvelle et précise d’estimation de la direction et des délais d’arrivée dans un environnement à trajets multiples, à des fins d’estimation de canal. Récemment, les méthodes de super-résolution ont été largement utilisées pour l’es-timation à haute-résolution de la direction d’arrivée (DOA) ou de la différence de temps d’arrivée (TDOA). L’algorithme proposé dans ce travail est applicable à l’estimation d’un canal espace-temps pour des systèmes de traitement spatio-temporel qui emploient la tech-nologie hybride DOA / TDOA. L’estimateur est basé sur l’algorithme MUSIC classique pour trouver la DOA et en profitant d’un simple corrélateur, il est possible de trouver le retard de chaque arrivée. Il est pertinent d’associer chaque angle à son propre retard pour être capable d’estimer les caractéristiques du canal quand nous ne connaissons pas la séquence transmise par l’émetteur. Pour ce faire, nous proposons une formation de faisceaux (voix) très simple et optimale par l’application du MVDR (Maximum Variance Distortion-less Response). Cette formation de faisceaux maximise le signal desiré par rapport aux autres signaux.

Après détermination de l’angle d’arrivée par l’algorithme MUSIC, nous appliquons l’algorithme de formation de faisceaux MVDR pour obtenir le signal qui est reçu par le réseau d’antennes pour une direction. Ce signal est corrélé avec les autres signaux correspondants aux autres directions d’arrivée. Les pics dans les figures ainsi obtenues montrent le décalage tempo-rel de chaque source par rapport à celle obtenue par la formation de faisceaux MVDR. La soustraction du plus petit décalage, correspondant au premier signal reçu à chaque décalage temporel, nous donne le temps d’arrivée de chaque source. Pour être plus précis, nous pou-vons choisir la moyenne des vecteurs des délais estimés, chacun étant obtenu à partir d’une angle pour l’algorithme MVDR.

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Abstract

In this project, we present a novel and precise way of estimating the direction and delay of ar-rivals in multipath environment for channel estimation purposes. Recently, super-resolution methods have been widely used for high resolution Direction Of Arrival (DOA) or Time Difference Of Arrival (TDOA) estimation. The proposed algorithm in this work is applica-ble to space-time channel estimation for space-time processing systems that employ hybrid DOA/TDOA technology. The estimator is based on the conventional MUSIC algorithm to find the DOA and by using a simple correlator it is possible to find the delay of each arrival. It is of interest to associate each angle to its proper delay to be able to estimate the charac-teristics of the channel when we have no knowledge about the transmitted sequence. To do this, we suggest a very simple and optimal beamforming method by performing Maximum Variance Distortion-less Response (MVDR). This beamforming maximizes the desired signal in the desired direction compare to the other signals that come from other directions. After finding the DOAs by MUSIC algorithm and selecting our desired direction, we obtain the signal from this direction by applying MVDR beamforming. Then, we perform a corre-lation between this signal and the others incoming signals from other directions. The peaks in the simulation figures illustrate the delay between each source with the obtained signal from MVDR. If we subtract the delay of the first arrival (the smallest delay in time), from the delays indicated in the figures, we can obtain the delay of each arrival. To be more precise, the mean of these estimated TOAs vector follows the exact TOA of each source.

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List of Tables

2.1 Simulation parameters to compare different methods to find DOAs. . . 11

2.2 Simulation parameters to compare different methods to find DOAs with the same source power. . . 12

3.1 Non correlated signals M≤ N, K_→∞. . . 22

3.2 Non correlated signals M≤ N, K< ∞. . . 23

3.3 Non correlated signals, M_≥ N, K_→∞. . . 24

3.4 Correlated signals, M≥ N, K_→∞. . . 26

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List of Figures

2.1 Uniform array antenna with two impinging signals. . . 6 2.2 Comparison between Capon and conventional beamforming algorithms to find

the direction of arrivals with different source powers. . . 12 2.3 Comparison between Capon and conventional beamforming algorithms to find

the direction of arrivals with the same source power. . . 13 3.1 Space for M = 2 signals and an N = 3 elements antenna array and eigenspace

from Rxx[5]. . . 19 3.2 MUSIC pseudo-spectra to find the DOAs according to the parameters in Table 3.1. 22 3.3 MUSIC pseudo-spectra to find the DOAs according to the parameters in Table 3.2. 24 3.4 MUSIC pseudo-spectra to find the DOAs according to the parameters in Table3.3. 25 3.5 MUSIC pseudo-spectra to find the DOAs according to the parameters in Table 3.4. 27 3.6 Finding DOAs using the MUSIC algorithm. . . 28 3.7 Two possible subarrays from an eight-element linear antenna array usable with

ESPRIT. . . 30 4.1 Beamforming structure. . . 34 4.2 How MVDR works to maximized the desired beam while minimizing the others. 35 4.3 a) Maximizing the signal at 30◦and nulling the sources situated at 60◦and 90◦ b)

Maximizing the signal at 60◦and nulling the sources situated at 30◦and 90◦. . . . 36 5.1 MUSIC pseudo-spectrum from the simulation of the numerical example. . . 43 5.2 Correlation function curves between xn(t)(n = 1, 2, . . . 8) and the direct signal

sd(t)to find the delays. . . 44 5.3 All correlation functions curves between xn(t)(n= 1, 2, . . . 8) and ym(t)for each

m (m = 1, 2, 3, 4) after beamforming with wcapon to find the delays associated to the m-th arrival. . . 45 6.1 Correlation functions curves u1p(τ)(taking θ1 =−9.92◦, θ2= 30.12◦, θ3= 39.86◦

and θ4 =70.18◦) to find the associated delays. . . 50 6.2 Correlation functions curves u3p(τ)(θ1 = −9.92◦, θ2 = 30.12◦, θ3 = 39.86◦ and

θ4=70.18◦) to find the associated delays. . . 51 6.3 Correlation functions curves u2p(τ)(θ1 = −9.92◦, θ2 = 30.12◦, θ3 = 39.86◦ and

θ4=70.18◦) to find the associated delays. . . 52 6.4 Correlation functions curves u4p(τ)(θ1 = −9.92◦, θ2 = 30.12◦, θ3 = 39.86◦ and

θ4=70.18◦) to find the associated delays. . . 53 6.5 Variance of ˆθmversus SNR for all 4 sources at θ= [30◦ −10◦ 40◦ 70◦]. . . 55 6.6 Variance versus the changes of θ2−θ1when SNR1=15 dB, K =200. . . 57

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6.7 Variance versus the changes of θ2−θ1, when SNR1=10 dB and K =200. . . 57

6.8 Standard deviation of the delay versus the changes of τm −τp, when SNR1 = 14 dB, SNR2=20 dB and K =200. . . 58

6.9 Effect of the SNR on the standard deviation of τ. . . . 59

6.10 Effect of the SNR on the average value of τ. . . . 59

6.11 TDOA estimation when there are no errors is in determining the DOA. . . 60

6.12 Effect of a 10◦error in determining the DOA on estimating τ. . . . 61

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List of Symbols

[]∗ Conjugate of a complex signal

[]H _{Hermitian transpose}

[]T _Transpose

∗ Correlation function

ai = e−j2πd sin θm (i=1,2,. . . M) Phase factor on different antenna elements

a(θm) Direction vector (N×1 dimensions for MUSIC) A= [a(θ1), . . . , a(θm)] Direction matrix (N×M for MUSIC)

bθ Sum of all non-linear vector components with a(θm)

c Light speed

C Symmetric Hermitian matrix in Root MUSIC algorithm which is equal to Pnin Conventional MUSIC algorithm

d The index shows the direct arrival’s property D Distance between two antenna elements D(z) Conventional Root Music polynomial

Es Principal eigenvector corresponding to the M largest eigenvalues

f0 Carrier frequency

I Identity matrix

J1, J2 Selection matrices in ESPRIT Algorithm

K Number of snapshots

M Number of incident signals on array antenna or number of sources

N Number of antenna array elements

Pcap Capon spatial spectrum

P(w) Average power of the array antenna

PMUSIC(θ) Spatial spectrum for MUSIC algorithm

Q Number of elements in sub-array

Rxx N×N array auto-correlation matrix Rss M_×M signal auto-correlation matrix sd(t) Direct signal from transmitter to receiver s(t) = [s1(t), . . . , sm(t)] m source signal vector

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ˆ

T Estimated time delay in correct ordering ˘

T Estimated time delay not in correct ordering

Ts Sampling time

u(t) Noise free signal space

Vn= [vM+1, . . . , vN] Noise subspace matrix whose columns are the noise eigenvectors v1,. . .,vM Eigenvector of Rxx

w= [w1, w2, . . . , wk]T Complex weight coefficient vector for correlated signals wcap Weight vector for Capon algorithm

wmvdr Weight vector for MVDR algorithm

xi(t) Captured signal in element i of array antenna

y(w) Received signal after beam forming with the weight vector w

α Complex weight coefficient, represents the time delay and amplitude

attenu-ation with respect to LOS

αθ Amplitude of the signal coming from direction θ

β Proportionality constant

β0 Phase constant

∆E = D Displacement of the elements in array antenna in ESPRIT theory ∆r_mi Distance between two different paths from source m on element i

σ_s2 Variance or power of the direct signal

σ_n2 Variance or power of the additive Gaussian noise signal

λi Eigenvalues of Rxx

θ Signal’s angle of arrival with respect to the array line

ˆθ Estimate of θ

µ Expectation of a function of realization for a probability model

µθ Variance of αθ

ϕ Phase difference between antenna elements

Φ Diagonal M_×M matrix containing the relative phase between adjacent time samples for each of the M sources

τm Delay taken by m-th signal path between two adjacent elements

Ψ Matrix whose eigenvalues are equal to diagonal elements of Φ and the columns of matrix P are the eigenvectors ofΨ

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Abbreviation

DOA Direction Of Arrival

ESPRIT Estimation of Signal Parameters via Rotational Invariance Techniques i.i.d. independent and identically distributed

JDTDOA Joint DOA and TDOA LMS Least Mean Squares LOS Line Of Sight

MVDR Minimum Variance Distortion-less Response MUSIC MUltiple SIgnal Classification

NLOS Non Line Of Sight

TDOA Time Difference Of Arrival

TLS-ESPRIT Total Least Squares-Estimation of Signal Parameters via Rotational Invari-ance Techniques

TOA Time Of Arrival

SIR Signal to Interference Ratio SNR Signal to Noise Ratio STD Standard Deviation ULA Uniform Linear Array

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To my parents in Iran and whom treat me like parents in Quebec

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Remerciements

My deepest gratitude goes first and foremost to my supervisor Dr. Dominic Grenier, for his constant support and guidance, and patience during the whole period of my thesis. His intuitive way of viewing things helped me grasp things about signal processing that were more than a mystery before. I will never forget his recommendation to not start out with something too difficult. He thought me to work my way up from something easy and that progressing step by step can make it a lot easier to proceed with the algorithms from the very beginning to the end point. Additionally, my special thanks also go to my co-supervisor Dr. Paul Fortier for his kind guidance and support to go in the right direction for my thesis. I would like to show my appreciation to another professor in LRTS, Jean-Yves Chouinard, for his interesting course in the theory of information which encouraged me to go forward to know more about codes and have another course in the theory of coding.

I would like to thank Emmanuel Racine for his friendly help and guidance and explanations during this thesis. Many thanks go to my friends in LRTS, especially Pascal Djiknavorian who when I talk to, gives me a feeling that I am talking to a real brother who I have never had in my life. I also appreciate the positive point of view of my friend, Mathieu Gallichand, who always encouraged me to view the glass half full and be more hopeful. Many thanks to all who participated in our coffee breaks, for making a friendly and enjoyable atmosphere for the laboratory.

Also special thanks to Suzanne Samson for her really kind emotional support and to be like a mother for me here in Quebec. I will never forget our coffee meetings and the snowshoes sport which we have done together.

And all my emotions, thanks, gratitude, feelings denotes to my lovely parents whose dis-tance support was the roots of this work. I do not know how I should express my love to both of you the angels of my life.

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Chapter 1

Introduction

1.1 Background and motivation

Nowadays, estimating the channel characteristics in mobile terminals and networks is among the most interesting topics in mobile radio research and development.

Joint parameter estimation plays an important role in several application technologies. There are a lot of algorithms which try to find the DOA and TDOA of the wireless users such as [1]. In addition, channel parameters are required to enhance the signal reception of the transmit-ted signal in space-time processing systems which is especially useful in multipath environ-ments. By estimating the channel parameters, the received signal from different paths can be weighted and shifted according to these channel parameters, to get the stronger signal, such as algorithms in [2], [3] or the algorithm which is used in Rake receivers [4].

Several method has been proposed to get angle or time delay of each path. Some of them considered an indoor environment to find TOA or TDOA [7]. Some other methods take DOA and TDOA.

Channel estimation is an important technique especially in wireless network systems. There are some undesirable effects of a wireless communication channel on the signals transmitted through it which are caused by its physical properties. These in turn always result in atten-uation, distortion, delays, and phase shifts of the signals arriving at the receiver end of the communication system. In multipath environments, what is needed is the direction and time delay of each arrival (direct signal and multiple reflected signals) for each user, so that the channel characteristics can be well estimated for each user. It is important to properly match the delays and directions of all arrivals and do so as efficiently as possible. Associating the directions to their proper delays can be very complicated. It is therefore desirable to seek a procedure that automatically yields and sorts the delay-direction pairs in a straightforward and highly precise way.

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For efficient transmission over wireless links, reliable channel estimation is critical. Channel estimation is based on the training sequence of bits that are unique for a certain transmitter and are repeated in every transmitted burst [23]. In other words, for each signal in transmis-sion we send also a certain known sequence to the receiver. In brief, we can state the reasons of channel estimation in wireless transmission as:

a) Allows the receiver to approximate the effect of the channel on the signal.

b) The channel estimate is essential for removing inter-symbol interference, noise rejec-tion, etc.

c) Also used in diversity combining, Maximum Likelihood (ML) detection, direction of arrival estimation, etc.

There are several articles that try to find simultaneously DOA and TDOA of the arrival sig-nals in multipath environments. In [8] - [12], the authors show how we can use different simple algorithms such as MUSIC [5] to find the DOAs and in each they propose an algo-rithm for also finding the TDOAs. The problem in many of these articles are the calculations which are massive or they need a known transmitted sequence for their algorithm and if not, they are not capable to associate each DOA to its proper TOA.

In this work, a practical method to estimate the TDOA associated with DOA information in the presence of multipath is proposed. By using high resolution, we focus on eliminating the DOA and TDOA of each arrival, which are two closely related aspects of array processing.

1.2 Objectives and novel contributions

It is clear that the accuracy of DOA and TDOA measurements are important for better chan-nel estimation.

There are numerous works that have been done to estimate the direction and time difference of arrival of signals in multipath areas. Some of the main disadvantages of these works are:

a) In most of them we need to have some knowledge about the transmission channel and most of all we need the information related to the transmitted signal or at least a known sequence of the transmitted signal.

b) We can obtain the DOAs and the TDOAs, but we are not capable to associate them together.

c) There is a massive amount of work and complicated calculations to do to overcome one of the aforementioned drawbacks.

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To do so, we propose to use the conventional MUSIC algorithm [5] and Capon beamforming [18], [19] in conjunction with the correlation function. The latter is made between a copy of signal which is coming from specific direction and other copy of received signals from other DOAs observed. The proposed new numerical method is then able to associate the DOA from the MUSIC algorithm and the TDOA from the special correlator.

Thesis outline

There are 6 chapters in this dissertation. The subsequent chapters are as follows:

Chapter 2 is an overview on the model of the system and signal which we use in this work. In all other chapters we will respect the same notations and definitions mentioned in this chapter.

Chapter 3 provides a brief background of the MUSIC algorithm and the equations which we need to implement in our program to find the DOAs in the multipath environment with an array antenna. In the same chapter, we show different assumptions to perform the best estimation of direction of arrivals by varying the main parameters in the MUSIC algorithm, namely the number of elements and the number of sources as well as the SNR. We compare the MUSIC algorithm with Capon beamforming. In addition in this chapter, we introduce the Root-MUSIC and ESPRIT algorithms, which are MUSIC enhancements and perform some simulations to better illustrate their performances.

In Chapter 4, we show the main properties of MVDR beamforming as an optimal beam-forming in multipath environments, which is able to maximize the signal to interference ratio (SIR) so as to specify one direction of our arrival signals from the others.

In the following chapter, Chapter 5, we introduce our proposed algorithm to deal with the problems which exist for the association of DOAs to their proper TDOAs, as discussed in the previous chapter. Then we explain the mathematical development and completely describe the proposed algorithm. In this chapter we proposed two versions of our algorithm. The simulation of the first version is done in this chapter and we discuss about how we can find the DOAs and TDOAs and associate them together even more simply.

In Chapter 6, we perform simulations to show that the second version of the proposed al-gorithm works well. In the same chapter, we find the limitations of our alal-gorithm in the direction of arrivals or the time difference of each arrival. Then, we search for the effect of the SNR on the variance of the directions and the delays of all arrival on the array antenna. We also show results on the affect of the errors in estimating the direction of arrivals on the TOAs.

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Chapter 2

System model

2.1 Introduction

This chapter is a general introduction of the basic theory of the developments presented in this document. The main objective of this chapter is to cover part of the theory which is essential and also the popular processing methods which already exist in the channel estimation domain. In addition in this chapter, we will introduce the principal notations which we will use in the rest of this document.

2.2 Array antenna modeling

The system model in this work is a linear array antenna consisting in N identical elements on which we have impinging signals send from the users. At first, we consider one user with M signals (where M< N) consists of one direct signal and M₋1 reflected ones. Figure 2.1 illustrates only the direct signal and one reflection from this user. It will be possible to extend the proposed algorithm to consider several users with different sources giving more direct signals and more reflections.

Assuming no noise, the m-th signal received at the n-th element at time t can be written as: xnm(t) = <{sm(t)e

jω0t_} ₁_≤_n_≤ _{N; 1}_≤_m_≤ _M _(2.1)

where sm(t)is the analytic complex envelope of the signal. Each reflection signal is in fact a delayed and weighted copy of the direct signal. So the envelope in (2.1) is now expressed as: sm(t) =αmsd(t−Tm) (2.2) where αmand Tmare the (arbitrary) amplitude and delay of each signal path, respectively. In our assumption sd(t)is the direct signal received at the first element with αd =1 and Td =0; d is the index of the direct signal.

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x1 x2 x3 xN θ ∆ 2∆ ∆ = D sin θ τ = ∆/c D

Figure 2.1: Uniform array antenna with two impinging signals.

In the case of the m-th signal alone, each output of the N elements of the array antenna is also an advanced or delayed copy of the output taken at the first element by a delay of τnm:

xnm(t) = <{sm(t−τnm)e jω0(t−τnm)_} _(2.3) = _<{sm(t−τnm)e− jω0τnm | {z } snm(t) }ejω0(t)_. _(2.4)

The narrowband assumption can be used since the carrier frequency f0 is assumed much larger than the bandwidth of the source. Moreover, the total length of the antenna array involved a maximum delay τnm which is always small with respect to the variation on the envelope (or equivalently the inverse of the signal bandwidth). So, we can write:

sm(t−τnm) ≈ sm(t). (2.5) The delay τnm causes only a phase shift between different element outputs e−

jω0τnm _{as seen} in (2.4). This is the reason why it is more convenient to work in the complex plane with the analytic complex envelope of each signal.

The next step is to express the delay τnm in terms of array geometry. This delay is related to the difference of path length followed by the plane wave from element 1 to element n in such a way to have:

τnm = ∆nm

c (2.6)

where c is the speed of light. Then the phase shift becomes:

ϕnm = ω0τnm =

ω

c∆nm

= β0∆nm (2.7)

where β0 = 2π_λ represents the phase constant and λ is the wavelength. Considering a linear array antenna with interelement spacing D,∆mnis linked to the direction of arrival θmof the

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m-th signal (shown in Figure 2.1), and to the distance between the n-th element and the first one, as:

∆nm = (n−1)D sin θm. (2.8) So, for a linear array antenna, we have:

ϕnm = (n−1)ϕm (2.9)

ϕm = βDsin θm (2.10)

The total length of the array is then(N₋1)D.

The third step is to write equations in a matrix form considering now the M signals. We define at first the vector x(t)as a snapshot of the N outputs (one output per element of the array antenna) sampled at the same time t. The analytic received signal is composed of the sum in the complex plane of the M signals arriving from different directions and at different times and is expressed as:

x(t) = M

∑

m=1 sm(t) +n(t) (2.11) where: sm(t) =       s1m(t) s2m(t) .. . sNm(t)       . (2.12)

Using (2.4) in conjonction with (2.6), (2.7) and (2.8), we can rewrite (2.12) as:

sm(t) =       1 e−jϕm .. . e−j(N−1)ϕm      sm(t). (2.13)

The vector in (2.13) is an N×1 vector which is known as the direction vector or steering vector and is expressed as:

a(θm) = h

1, e−jϕm_{, . . . , e}−j(N−1)ϕmi> _. _(2.14) The symbol>denotes transpose and n(t)is the N×1 complex additive noise vector. The noise at different elements of the array antenna can be considered as zero-mean Gaussian stationary random processes and independent from each element. So the noise at the an-tenna elements is mutually uncorrelated and is also uncorrelated with the signals.

Finally, the snapshot x(t)can be written in matrix from equations (2.11) to (2.14) like this: x(t) = As(t) +n(t) (2.15)

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where A is the steering matrix:

A= [a(θ1), a(θ2), . . . , a(θM)] (2.16) and s(t)is the source vector:

s(t) = [s1(t), s2(t), . . . , sM(t)]> . (2.17)

Because the construction of the steering vector a(θ) in (2.14), the steering matrix A has a Vandermonde structure for a linear array antenna.

We recall that narrowband assumption implies that the delays produced by interelement spacing (τm) are much smaller than signal variations so they are related to phase shifting. On the other hand, the delays (Tm) between the components of arrival signals are much larger than the duration of autocorrelation of emitted signal (or just the direct signal). There-fore these components are non correlated between themselves and we can consider them as independents sources.

2.3 Covariance matrix

Many signal processing applied to array antennas involve second order statistical estima-tion. The second order statistics applied to the signal vector x(t)gives the auto-covariance matrix Rxx(t)(or simply covariance matrix). This is an N×N matrix having some special properties that can be used to develop specific estimator. For our project, all sources are as-sumed stationary in the strict sense, so we can drop the t parameter and Rxxcan be written as:

Rxx = E n

x(t)xH(t)o

= ARssAH+σ_n2IN (2.18)

The symbol H denotes the Hermitian transpose. The variables σ_n2 and IN are the variance of the additive noise and identity matrix, respectively, because the noises are independent from one element to the others; Rssdenotes the M×M auto-correlation matrix for the source signals: Rss = Ens(t)sH(t)o s(t) =       s1(t) s2(t) .. . sM(t)       . (2.19) 8

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The matrix Rssis a diagonal matrix since the arrivals are considered as independent sources. The variable σ_m2 then corresponds to the power of the m-th arrivals. This matrix, called the source auto-covariance matrix, is written as:

Rss=       σ_s2₁ 0 · · · 0 0 σ_s2₂ · · · 0 .. . ... ... 0 0 _{· · · ·} σ_s2_M       . (2.20)

On the other hand Rnnis the auto-covariance matrix of the noise. Since we suppose that the noise is i.i.d. (independently and identically distributed), we can write

Rnn= σ_n2I. (2.21)

where I is the identity matrix and σ_n2is the power of the noise.

Note that because σ_n2 ≥ 0, Rxx is now positive definite; all of its eigenvalues are greater or equal to zero. In addition, Rsscan be classified in three cases [5], [6]:

1. Diagonal, non-singular and positive definite when the signals are uncorrelated; 2. Non-diagonal, non-singular and positive definite when some signals are partially

cor-related;

3. Non-diagonal, singular and semi-positive definite when some signals are correlated, which means that some eigenvalues are equal to zero.

2.4 Estimation of DOAs

This section points out some of the numerous ways to find the direction of arrival on array antennas in narrowband systems. Here we just mention some algorithms which we will use in the rest of the thesis.

We can regroup all DOA’s estimation algorithms in three main groups:

1. Classic methods which are related to conventional beamforming or beam-steering 2. The methods based on maximum likelihood.

3. The method based on calculation of subspaces.

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We begin to define a single output of an array antenna which is a weighted sum of each element output:

y(w) =wHx (2.22)

where w is the weight vector. The average power of this single output for a given weight vector is calculated by:

P(w) =E_|ym(w)|2=wHRxxw. (2.23) We can see the covariance matrix appearing.

2.4.1 Conventional beamforming or beam-steering

Many choices are available for w. Depending on the choice, the antenna has some properties. For example, by a simple phase shift constructed in a similar manner to the steering vector defined in (2.14), it is possible to steer the main lobe of the antenna in the direction θ0, putting

θm =θ0.

The most simple beamforming is the conventional beamforming. It consists of making a beam steering in direction θ0putting the weight vector:

wbs(θ0) = h 1, e−jϕ0_{, . . . , e}−j(N−1)ϕ0 i_> (2.24) where ϕ0is given by (2.10).

The estimated direction of arrivals are obtained by plotting the power of P(w(θ))given by (2.23) in all directions. By doing so, the directions which have more power are considered as the directions where the sources are located.

The principle of beam steering is just to compensate the phase shift produced by the differ-ence of path length.

2.4.2 Capon or Maximum Likelihood method

The Maximum Likelihood (ML) algorithm [16] is also a high resolution method. The true ML solution is computationally prohibitive, especially when multiple sources exist, although it is more robust and has a smaller estimation error as compared with any other DOA estimators. The Capon algorithm [18], [19] is a variant of ML. It does not really realize a maximum likelihood estimation in a wide sense but considers only the other sources not present in the analysis direction as interference; the noise is not taken into account.

The algorithm starts by constructing a real-life signal model as described in the previous section. The peaks in the Capon spectrum occur when a source exists in the direction θ0 by minimizing the contribution of the undesired interferences while maintaining the gain

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along the look direction to be constant, usually unity. Then the output power, without noise, is equal to the power of the source in that direction. That is,

min E|y(w(θ))|2₌_{min w}H_Rxxw_, _{subject to w}H_a_{(θ) =}_{1 .} _(2.25)

Using Lagrange multiplier, the weight vector that solves (2.25) can be shown to be:

wcapon =

R−_xx1a(θ)

aH_(θ)_R−1

xxa(θ)

. (2.26)

The output power of the array antenna as a function of θ gives the DOA estimation using the Capon method:

Pcapon = 1

aH_(θ)_R−1

xxa(θ)

. (2.27)

The directions of arrival correspond to the peaks in this angular spectrum and the high of the peaks correspond to the power of the sources.

Example 2.1: Comparison between Capon and conventional

beamforming

To compare the Capon algorithm with conventional beamforming, we perform a simulation according to the parameters given in Table 2.1.

Figure 2.2 illustrate a comparison between these two algorithms. It is evident that the Capon algorithm is more efficient, since in the Capon case, all sources are estimated with more precision and hence the peaks are narrower. We can observe that for these two methods, the amplitude of each peak corresponds to the power level of the arrivals.

Table 2.1: Simulation parameters to compare different methods to find DOAs. Parameters Values N 7 θ [30◦, 60◦, 90◦] σ_s2 [6 4 10] K ∞ D λ/2

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Table 2.2: Simulation parameters to compare different methods to find DOAs with the same source power. Parameters Values N 7 θ 30◦, 60◦, 90◦ σ_s2 11 σ_n2 1 K ∞ D λ/2 0 20 40 60 80 100 120 1 2 3 4 5 6 7 8 9 10 θ [°] σ s 2 [dB] Conventional Beamforming Capon

Figure 2.2: Comparison between Capon and conventional beamforming algorithms to find the direction of arrivals with different source powers.

In another simulation, we assume that the noise power σ_n2 = 1 and the power of all the arrivals is σ_s2 = 11, i.e. all arrivals have the same power which is 11 times the power of the noise. The results are illustrated in Figure 2.3.

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0 20 40 60 80 100 120 1 2 3 4 5 6 7 8 9 10 11 θ [°] σ s 2 [dB] Conventional Beamforming Capon

Figure 2.3: Comparison between Capon and conventional beamforming algorithms to find the direction of arrivals with the same source power.

2.5 Conclusion

In this chapter, we reviewed the signal representation to model the system. We also pre-sented the basic theory to model an array antenna which we will use in this project. We pointed out some known algorithms to find direction of arrivals (DOAs) in multipath areas. A very short comparison between them was given. The Capon or Maximum Likelihood method beats the conventional beamforming method to find precisely the DOAs in multi-path environments. There is one more method which is more precise than Capon, namely the MUSIC algorithm, which will be introduced in the next chapter.

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Chapter 3

MUSIC Algorithm

3.1 Introduction

In this chapter, a specific estimator called the MUSIC (MUltiple SIgnal Classification) rithm for DOA estimation is fully investigated. The original or conventional MUSIC algo-rithm was proposed by Schmidt [1] to estimate the parameters of multiple uncorrelated or at most partially correlated signals incident on an antenna array. It can provide asymptotically unbiased estimates of the following parameters:

1. The number of signals;

2. The angle of arrivals in the space domain (or the frequency of arrivals in the time domain).

Because the resolution given by the MUSIC algorithm exceeds the Rayleigh resolution cri-terion, it is classified as a high-resolution or super-resolution algorithm. This criteria was developed following works made by Lord Rayleigh in 1879. It states that two arrivals are considered resolved when the first minimum angular spectrum value of one signal coincides with the maximum angular spectrum value of the other signal. In addition to the presenta-tion of convenpresenta-tional MUSIC, we will give some useful and basic details about ESPRIT and also Root-MUSIC. Both are based on MUSIC.

3.2 Array signal space definition

Before going too far about eigen-analysis, it is necessary to have more knowledge about array data space and the relationship between the array signal vector x(t)and the steering matrix A or the steering vector a(θ).

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We assume M signals impinging on an N elements (with N> M) antenna array. In the noise-less case, the eigenvalues (and eigenvectors) of the covariance matrix Rxx can be grouped into two sets:

1. N₋M null eigenvalues and the corresponding eigenvectors construct the kernel (null space) also called noise subspace;

2. M non-null eigenvalues and the corresponding eigenvectors construct the signal sub-space.

As a result, an N dimensional space is spanned by N eigenvectors and this space is parti-tioned into the above two subspaces: the noise and the signal subspaces.

In the case of M > N, the signals cannot be resolved by an antenna array since the system has more unknowns (M) than equations (N).

As we will see later in this chapter, the noise eigenvectors and the signal eigenvectors are orthogonal, hence the inner product of the signal eigenvectors with the noise eigenvectors should be zero. Based on the above eigen-decomposition technique and the partition into two orthogonal subspaces, the parameters of multiple signals can be estimated. The follow-ing development can help to figure out the array data space and the relationship between array signal vector x(t), steering matrix A or steering vector a(θM).

The expression of the array signal vector x(t)in (2.11) can be rewritten as: x(t) =As(t)

| {z } u(t)

+n(t). (3.1)

The matrix A is already defined in (2.16). The vector u(t) represents the noise free signal vector. It can be seen that u(t)is the linear combination of the steering vectors a(θm)where the coefficients are the source signals sm(t). The covariance matrix Rxxof x(t)can now be written as:

Rxx = ARssAH+σ_n2I

= Ruu+σ_n2IN . (3.2)

As introduced in [1], there are N−M eigenvectors associated to the kernel (null space) of Rxxin the noiseless case. But this is only valid when all source signals are independent or not fully correlated together to keep the full rank of Rss. To be sure that we have independent sm(t), the path delay Tm in (2.2) should be greater than the autocorrelation of sd(t)for all 1_≤ m_≤ M but d ₆₌m (we recall that T_d =0 since the index d is reserved to the direct path). With this condition, we have

E_{s_d(t)sm(t)}

σ_d2 =

E_{s_d(t)s_d(t₋Tm)}

σ_d2 1 (3.3)

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where σ_d2is the power of the direct signal as seen in (2.20).

3.3 Eigen-analysis for array signal space

To begin the analysis, we have to organize the eigenvalues and corresponding N_×1 eigen-vectors of Rxxin a decreasing order:

{λ1, λ2, . . . λN}, (λ1 ≥λ2≥ . . . λM >0)⇐⇒ {v1, v2, . . . , vM}. (3.4)

In the noisy case, having the covariance matrix of the noise in the form of (2.18), the following results can be obtained:

λ1 ≥λ2≥ . . . λM >σ_n2

λM+1 = λM+2 = . . . = λN = σ_n2 (3.5)

The corresponding eigenvectors have the following properties since Rxxhas Hermitian sym-metry:

vi⊥vj or viHvj =0 (i, j=1, 2, . . . M, i6=j) (3.6) and

v_iHvi =1 . (3.7)

Equation (3.6) shows well this fact that all eigenvectors of a covariance matrix are always orthogonal.

A matrix Vnis made from the eigenvectors viassociated with the N−M smaller eigenvalues of Rxxas:

Vn= [vM+1, vM+2, . . . , vN] . (3.8) Since all column of Vnare orthonormal, they span a space which corresponds to the noise space. We can use this matrix to project a given vector in the noise space. If the vector lies in the source space, the projection will be null since the source space is orthogonal to the noise space. The derivation can be made as follows.

The noise space eigenvectors correspond to the kernel of Ruu, so we can write:

ARssAHvi =0 i= M+1, . . . N . (3.9) By applying a left multiplication AH to (3.9) and continuously multiplying(AHA)−1 _{to the} resulting equation, one can get the following expression:

(AHA)−1₍_AH_A₎_RssAH_v

i =0 i= M+1, M+2, ..., N or

RssAHvi =0 i= M+1, M+2, ..., N

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The covariance matrix of sources Rsshas full rank (under the assumption of independent or not fully correlated sources). Then its inverse exists. By a left multiplication by R−_ss1 on the above equation, we conclude by these important relations:

AHvi = 0 i= M+1, . . . N so aH(θj)vi = 0 or viHa θj =0 ( j = 1, 2, . . . M i = M+1, M+2, . . . N (3.11)

The following important facts are deduced:

- There are N−M minimum eigenvalues that are equal to σ2.

- All direction vectors a(θj) (j = 1, 2, . . . M)in the Vanderemonde-structure steering ma-trix A are orthogonal to the N−M noise eigenvectors corresponding to the N−M minimum eigenvalues.

The concept of eigen-analysis-based signal space is illustrated in Figure 3.1. As shown in the figure, M= 2 signals are impinging on an array antenna with N=3 elements. The two dimensional signal subspace can be visualized as a plane spanned by the signal eigenvectors v1and v2; the one dimensional noise subspace is orthogonal to the plane, in the direction of eigenvector v3.

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a(θ1) a(θ2)

x1 x2

x3

The continuum of direction v1

v3 v2

The signal subspace spanned by the signal eigenvectors

signal eigenvector

noise eigenvector

Figure 3.1: Space for M = 2 signals and an N = 3 elements antenna array and eigenspace from Rxx[5].

3.4 Rank calculation

When the signals impinging on the antenna array are uncorrelated or partially correlated, Rss = E{s(t)sH(t)}is non-singular with rank{Rss} = N. Moreover, the Vandermonde-structure of the steering matrix A is known to have full rank over the smaller dimension. The matrix Rxx= ARxxAHhas a rank of:

rank{Rxx} = ranknARssAHo

≤ minnrank_{A_}, rank_{Rss}, rank{AH} o

= min(rank_{A_}, rank_{Rss}) . (3.12) In general, inequality of rank{Z} ≤min(rank{X}, rank{Y})for a matrix product of Z=XY is encountered in special circumstanced where X or Y have at least one row or one column completely equal to zero. Here, by considering the natural property of matrices A and Rss, this probability will be discarded and we can write:

rank_{Rxx} =min{rank{A}, rank{Rss}} . (3.13)

When we have

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• each source is in a direction different from the others then, from (2.14), the first M columns of matrix A make an ensemble which is linearly independent;

the resulting rank calculation gives:

rank{Rxx} =min(min(M, N), M) =min(M, N). (3.14)

3.5 Estimation of the covariance matrix

The expression of the covariance matrix in (2.18) or in (3.2) is an ideal situation. In reality, the expectation requires an infinite number of samples in the time domain.

In fact, we have to made an estimation of the covariance matrix from a limited K number of snapshots taken at discrete time(kTs)where Tsis the sample period. The estimation is given by: ˆ Rxx ≈ 1 K K

∑

k=1 x(kTs)xH(kTs). (3.15)

With an estimation rather than the exact value, the smaller eigenvalues of Rxxare not equal to σ_n2, and the two subspaces are approximated. So the products in (3.11) are not completely equal to zero.

3.6 MUSIC spectrum

We recall from (3.11) that all steering vectors a(θm)are orthogonal to all vectors in Vn. Con-ventional MUSIC is based on this fact and we can write:

aH(θm)VnVHna(θm) =0 (m=1, 2, ..., M) (3.16) where index m indicates the source signal index (m= 1 for direct, m=2, ..., M for reflected ones).

Therefore, by exploiting the orthogonality between the steering vector and the null space in (3.16), the MUSIC spectrum for spatial estimation is expressed as [6]:

PMUSIC(θ) = 1

aH_(θ)_VnVH

na(θ)

(3.17)

where aH(θ)is constructed as in (2.14) with different values of θ.

The denominator of (3.17) gives the norm of the resulting projection of the scanning vector a(θ) on the noise subspace. If the angle θ is equal to one in the set θm, then a(θ) = a(θm) which lies in the source subspace, orthogonal to the noise subspace. The projection is null.

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Consequently, peaks of the MUSIC spectrum correspond to the direction-of-arrival of the signals impinging on the antenna array. The relationship between the number of elements N in the linear array and the number M of uncorrelated signals should be as follows:

N_≥ M+1 (3.18)

This is because there should be at least one vector space for noise among N eigenvectors to scan out M= N₋1 signals.

The steps in the conventional MUSIC algorithm are summarized as follows:

1. Choose N (the number of linear array elements) sufficiently large (N> M, the number of uncorrelated signals);

2. Estimate the correlation matrix Rxxof the K measured snapshots;

3. Calculate the eigenvalues of the correlation matrix Rxx;

4. Estimate the number of uncorrelated sources M by an analysis of the eigenvalues;

5. Construct the noise matrix Vnbased on noise eigenvectors;

6. Evaluate the spatial “pseudo-spectrum” using (3.17);

7. Pick the M peaks of PMUSIC(θ)and obtain the DOAs.

3.7 Simulation of the classical method to find the DOAs

3.7.1 Non correlated signals M

_≤

N, K

→

∞

This is an ideal case where the noise covariance has exactly the form σ_n2I. Figure 3.2 shows the pseudo-spectrum from the MUSIC algorithm with the given values indicated in the caption. All sources are located correctly and there are well observed maxima in the curves for these two θ.

If we compare this pseudo-spectrum to the one obtained with the Capon estimator, there is no doubt that MUSIC estimations are more precise in the spatial dimension since the peaks are very sharp. In part (b) a lower SNR in dB for all sources is used. The SNR for the m-th source corresponds to the σ_m2/σ_n2ratio. The same remarks than in part (a) are made but the performance impairment is more accentuated. The MUSIC estimator in this part shows very good performances in estimating the DOAs of the sources.

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0 20 40 60 80 100 120 140 160 180 −10 0 10 20 30 40 50 60 θ [deg] p( θ ) [dB]

(a) SNR=10 dB for each source.

0 20 40 60 80 100 120 140 160 180 −10 0 10 20 30 40 50 θ [deg] p( θ ) [dB]

(b) SNR=0 dB for each source.

Figure 3.2: MUSIC pseudo-spectra to find the DOAs according to the parameters in Table 3.1.

Table 3.1: Non correlated signals M≤ N, K→∞. Parameters Values N 7 θ 30◦, 50◦, 70◦, 90◦, 120◦, 150◦ K ∞ D λ/2 22

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3.7.2 Non correlated signals M

_≤

N, K

<

∞

In this subsection, we decrease the number of snapshots K. Therefore as it is explained, the dominator in (3.17) is not exactly zero with an approximation of the covariance matrix. Therefore, we expect lower performances than in the previous subsection. Figure 3.3 shows the result of this assumption with K =100. The indicated SNR is applied to all arrivals, i.e. each arrival has the same power level which is 10 times or equal to the noise power.

The same case but with the Capon estimator shows a little change compare to the ideal co-variance matrix as stated in Example 2.1. On the other hand, the MUSIC estimator shows a lower performance, especially when the SNR = 0 dB; we can see that the first two sources are not well estimated by the sharp peak in the pseudo-spectrum. The inverse of the co-variance matrix is not so affected as the two orthogonal subspaces. In general, however, we observe that the DOA estimation performances of the MUSIC algorithm, even in less than ideal conditions, remain higher than those of conventional estimators or Capon estimators.

Table 3.2: Non correlated signals M≤ N, K<∞. Parameters Values

N 7

θ 30◦, 50◦, 70◦, 90◦, 120◦, 150◦

K 100

D λ/2

3.7.3 Non correlated signals, M

_≥

N, K

→

∞

In this subsection, the number of elements are less than the number of sources. As it is pointed out in [20], it is easy to conceive that such a situation may be encountered in radio communications where the number of incident front wave in the receiving antenna becomes significant. As it is shown in Figure 3.4, the pseudo-spectrum can just estimate 4 sources of a total of 6 sources. In addition, even with K _→∞, the peaks are not well estimated even in the MUSIC case. Considering non correlated signals and also M≥ N, as we stated in (3.14), we have rank_{Rxx} = N and therefore N−rank{Rxx} =0: no eigenvalues able to span the noise subspace and no vector a(θ)satisfying aH(θ)vi =0.

However, only the eigenvector associated with the smaller eigenvalue is taken as the noise subspace here. The pseudo-spectrum PMUSIC(θ)on θ = θm will be discrete. As it is shown in Figure 3.4, the denominator of (3.17) has a small value but is not zero. The exact DOAs cannot be extracted. If we want to estimate the exact values of DOAs in the case of non correlated signals and also M ≥ N, we have to go deeper in the details which is out of the perspective of this thesis. Here we just point out the result of this case.

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0 20 40 60 80 100 120 140 160 180 −10 0 10 20 30 40 50 θ [deg] p( θ ) [dB]

0 20 40 60 80 100 120 140 160 180 −10 −5 0 5 10 15 20 25 30 35 θ [deg] p( θ ) [dB]

Table 3.3: Non correlated signals, M≥ N, K→∞. Parameters Values N 5 θ 30◦, 50◦, 70◦, 90◦, 120◦, 150◦ K ∞ D λ/2 24

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0 20 40 60 80 100 120 140 160 180 0 10 20 30 40 50 θ [°] P( θ ) [dB]

0 20 40 60 80 100 120 140 160 180 −5 0 5 10 15 20 25 30 35 θ [°] P( θ ) [dB]

Figure 3.4: MUSIC pseudo-spectra to find the DOAs according to the parameters in Table3.3.

3.7.4 Correlated signals M

_≥

N, K

→

∞.

In this subsection, we change our assumption and consider that the signals are fully corre-lated. This situation occurs when a signal is a copy of another one like a very close echo.

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Suppose that the first and second signals at 30◦and 50◦are correlated. Therefore we have: s2(t) =αs1(t) (3.19) where α is the complex weight coefficient, which represents the phase shift and amplitude attenuation with respect to the LOS signal, for example.

As it is well illustrated in Figure 3.5, it is impossible to estimate our first two sources.

Table 3.4: Correlated signals, M ≥N, K→∞. Parameters Values

N 7

θ 30◦, 50◦, 70◦, 90◦, 120◦, 150◦

K 100

D λ/2

To explain more about this case, suppose we have just two source signals impinging on a linear three element array antenna, and suppose a noiseless case. The exact covariance matrix Rxxof the data x(t) = [a(θ1)s1(t)a(θ2)s2(t)]>with σ₁2=1, can be expressed as:

Rxx = E{x(t)xH(t)} = A " 1 α∗ α α2 # AH (3.20)

It is evident that Rssis not a full rank with rank{Rss} =1. Then the source subspace from Rxx has one dimension; only one scanning vector can (by chance) be perpendicular to the noise subspace. Therefore, DOAs cannot be estimated by the MUSIC spatial pseudo-spectrum directly by (3.17).

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0 20 40 60 80 100 120 140 160 180 −5 0 5 10 15 20 25 30 35 40 45 θ [°] P( θ ) [dB]

0 20 40 60 80 100 120 140 160 180 −5 0 5 10 15 20 25 30 35 40 θ [°] P( θ ) [dB]

Example 3.1: Comparison between MUSIC, Capon and

beamforming

To show that MUSIC works much better than the Capon and conventional beamforming methods, we simulate the same example as in Section 2.5.1, with the same parameters as in

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Table 2.2 using the MUSIC algorithm. Figure 3.6 shows the results. Compared to Figure 2.3, we see that the peaks of the pseudo spectrum (3.17) are sharper and higher than with the other methods. 0 20 40 60 80 100 120 140 160 180 −10 0 10 20 30 40 50 θ [deg] p( θ ) [dB]

Figure 3.6: Finding DOAs using the MUSIC algorithm.

3.8 Root-MUSIC

To evaluate DOAs with the MUSIC algorithm, a scanning vector is needed to scan over all possible angle of arrivals. To obtain a finer resolution, more sample points must be taken. Moreover, we need a visual analysis of the pseudo-spectrum to retrieve the peaks and finally find the DOAs. Consequently, we need a lot of processing resources. The Root-MUSIC algorithm is a modification of the MUSIC algorithm without using a scanning vector. The pseudo-spectrum of MUSIC in (3.17) can be rewritten as [6]:

P(θ) = 1

aH_(θ)_Ca_(θ) (3.21)

where C=VnVHn.

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For a linear array antenna, the element of the steering vector in (2.14) can be written as: amn =e−

j(n−1)ϕm _(3.22)

which is related to the n-th element of the m-th source. So we have:

P_MUSIC−1 = N

∑

p=1 N

∑

q=1 ej(p−1)ϕ_c pqe−j(q−1)ϕ (3.23)

where cpqis the entry in the p-th row and q-th column of C. Combining the two summations into one, (3.23) can be simplified as:

P_MUSIC−1 =

N−1

∑

n=₋₍N−1)

cnej nϕ (3.24)

where cn≡∑p−q=ncpq, that is cnis the sum of the elements in the n-th diagonal of C. Because of the Hermitian symmetry of C, it is evident that c₋n=c∗nand c0is real. A polynomial D(z) similar to (3.24) can be constructed as follows:

D(z) =

N+1

∑

n=₋₍N+1)

cnz−n. (3.25)

Evaluating the MUSIC spectrum P(θ)becomes equivalent to the polynomial evaluation of D(z)on the unit circle: the peaks in the MUSIC spectrum are where the roots of D(z)lie close to the unit circle. In the noiseless case, the poles lie exactly on the unit circle at locations determined by the DOAs. The M roots zm (having unity modulus) should correspond to those of (3.24) posing z−_mn = ejnϕm_{. In the complex plane, the roots on the unit circle are the} positions of the sources ϕm. It is enough for a linear array antenna to solve the following equation to find the angular position of the sources:

ˆθi =arcsin −arg(zmi) βD . (3.26)

Example 3.2: Root-MUSIC

With the same parameters as in examples 2.1 and 3.1, if we want to show the performance of Root-MUSIC, with the same parameters as in Table 2.2, we can obtain the polynomial:

D(z) = (0.0725+0.0992i)z6+ (−0.3768−0.4189i)z5+ (−0.7178+0.3522i)z4

+ (−0.4940+0.0192i)z3+ (−0.4475+0.6261i)z2+ (−0.0364−1.2191i)z+4.0000

+ (₋0.0364+1.2191i)z−1+ (₋0.4475₋0.6261i)z−2+ (₋0.4940₋0.0192i)z−3

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To find the position of the sources, first we need the zeros of D(z):

zm = [(5.2383−1.5974i) (−1.2255+1.3016i) (0.4481+1.7629i) (−0.9127−0.4086i)

(₋0.9127₋0.4086i) (₋1.0000i) (₋1.0000i) (1.0000)

(1.0000) (₋0.3835+0.4072i) (0.1354+0.5328i) (0.1747₋0.0533i)]

The angle we obtain by Root-MUSIC are [00.0000◦ 60.0000◦ 30.0000◦] which shows how much this algorithm is precise in finding the angle of the different directions, without the necessity to scan the direction space.

3.9 ESPRIT

In the case of high resolution direction-finding algorithms, ESPRIT [13] is attractive for real-time applications due to its computational and implementation advantages. The basic idea behind this algorithm is to exploit the so called array displacement invariance structure, i.e. two identical subarrays separated by a common distance∆E.

Subarray 1 Subarray 2 Q = N− 1 Q = N_{− 1} ∆E = D Subarray 1 Subarray 2 Q = N− 2 Q = N_{− 2} ∆E= 2D

Figure 3.7: Two possible subarrays from an eight-element linear antenna array usable with ESPRIT.

Without loss of generality we suppose the linear array case. As shown in Figure 3.7, the first Q and last Q elements can be used to form two identical sub-arrays with displacement ∆E =D(N−Q), 1≤ Q≤ N−1 (we must have Q> M). Thus, the array manifold vectors for these two sub-arrays can be expressed as A1 = J1Aand A2 = J2A, where J1and J2are selection matrices: J1 = [ IQ |{z} Q×Q 0 |{z} Q×N−Q ], J2= [ _|{z}0 Q×N−Q IQ |{z} Q×Q ]. (3.27) 30

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It is clear that J1Apicks the first Q rows of A, whereas J2Achooses the last Q rows of A. Since A is a Vandermonde matrix, it can be shown that:

J1A=J2AΦH (3.28)

whereΦ is:

Φ=diag{e−jβ∆Esin θ1_{, . . . , e}−jβ∆Esin θM}_. _(3.29)

It is obvious that the matrix A is in the signal space spanned by Vswith Vs = I−Vnsince the signal space is orthogonal to the noise space spanned by Vn. Thus, there exists a matrix Tso that A can be represented as A=VsT. Thus, we can rewrite (3.28) as:

J1VsT=J2VsTΨH (3.30)

where

Ψ= TΦT−1. (3.31)

The matrixΨsis created from the matrixΦ by a transformation that preserves the eigenval-ues. In other words, if we know the eigenvalues ofΨ, we also know the eigenvalues of Φ. SinceΦ is a diagonal matrix, its eigenvalues are directly the element in the diagonal. As a consequence, the eigenvalues ofΨ and the angles of arrivals are connected via (3.29) by the expression e−jβ∆Esin θm_{, where m}₌_{1, . . . M.}

3.10 Conclusion

In this chapter we reviewed the important points of the MUSIC algorithm and its two exten-sions: the Root-MUSIC and ESPRIT algorithms.

MUSIC performs using the eigen-decomposition of the signal covariance matrix Rxx. This matrix generates two orthogonal subspaces called signal and noise subspaces. The latter is spanned by the eigenvectors associated with the N₋M smaller eigenvalues. The principle is to project the scanning vector on the noise subspace and the pseudo-spectrum gives the length of the projection. The peaks of this pseudo-spectrum are related to the DOAs. Also, by simulations, we showed some problems which can occur with the MUSIC algorithm when the number of sources is larger than the number of elements in the array antenna or when we have correlated source signals.

The Root-MUSIC algorithm eliminates the need for performing a computationally intensive search, replacing it with a root-solving problem. The implementation of the root-MUSIC algorithm is easier than the MUSIC algorithm which requires careful determination of the peaks and interpolation of the array manifold.

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The other mentioned algorithm in this chapter is ESPRIT. The algorithm can successfully estimate the direction of arrivals by exploiting a displacement property of the array. The dis-placement translates into an underlying rotational invariance of signal subspaces spanned by two data vectors received by two subarrays.

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Chapter 4

MVDR Beamforming

4.1 Introduction

The objective of beamforming is to resolve the direction of arrival of spatially separated sig-nals within the same frequency band. Among all forms of beamforming, Minimum Variance Distortion Response (MVDR) is well known as the optimum beamformer to maximize the signal to interference ratio. In this chapter, we will have a glance at the main properties and equations of this beamformer.

4.2 Specification of a beam between different arrival beams

The advantage of array antennas is to have direct access to each elements so it is possi-ble to create spatial filtering by weights adjustment assigned to each element output. The weighting is made by an amplitude and phase control to form an electronically-steerable beamformer [21]. The element outputs are multiplied by the weights and summed to pro-duce the beamformer output. Having the sensor outputs x(t)at time t, the array weights w(θm)where θm is the direction of interest, the beamformer can be expressed as:

ym(t) =wH(θm)x(t). (4.1)

The adaptive beamformer based on the MVDR criterion selects the array weights w(θm) by minimizing the variance (i.e. average power) of the beamforming output y(t), subject to maintaining unity response in the look direction (the direction of the source of interest). This principle is also retrieved in the Capon estimator seen in Chapitre 1. This explains why MVDR is also referred to as Capon beamforming. Figure 4.1 illustrates a basic adaptive beamformer with adjustable weights able to put the main lobe in the direction of interest while eliminating other signals which are considered as interference as in Figure 4.2. In this manner, the signal-to-interference ration (SIR) goes to infinity in theory, or is maximized in practice.

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sensor 1 sensor N x1 xN w1 wN P y = w∗_x

Figure 4.1: Beamforming structure.

We have to estimate the complex N×1 weight vector w by maximizing the SIR. The input signal vector x(t) in the noiseless case can be separated into two parts: the signal coming from the desired direction θ and all other signals. It can be rewritten as:

x=αθ(t)a(θ) +bθ . (4.2)

where αθ is complex amplitude of the signal coming from direction θ, bθ is the sum of all vector components non-linear with a(θ), i.e. all signals coming from others directions than

θ. If no signal is coming from this direction, then αθ = 0, while if a signal comes from this direction (i.e. θ = θm) then αθm(t) = sm(t), using the same notation as in (2.13). The SIR is now expressed by:

SIR= αθwH(θ)a(θ) 2 aH_(θ)_R bba(θ) (4.3) where Rbb = Rxx−µθa(θ)a H_(θ) _(4.4) and µθ =var{αθ(t)}. (4.5) Applying Schwartz’s inequality to (4.3) yields:

SIR_{≤ |}αθ| 2

aH(θ)R−_bb1a(θ). (4.6) The maximum of this ratio is obtained for:

wmvdr(θ) =βR−_bb1a(θ) (4.7)

where β is a proportionality constant. Using the matrix inversion lemma and (4.4), we can write: R−_bb1= R−_xx1+µθ Rxxa(θ)aH(θ)R−_xx1 1−µθaH(θ)R− 1 bb a(θ) . (4.8)

Applying (4.8) in (4.7), we retrieve the well-known expression of the MVDR beamforming as: wmvdr(θ) = R−_xx1a(θ) aH_(θ)_R−1 xxa(θ) . (4.9) 34

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β0D

ϕ

Desired Signal Interfering signal

Figure 4.2: How MVDR works to maximized the desired beam while minimizing the others.

This is exactly the same expression as the weight vector for the Capon estimator given by (2.26).

As illustrated in Figure 4.2, the radiation diagram of the antenna shows a main lobe in the direction of the desired signal while the zeros of the diagram are in the direction of the two other signals. In this regard, there is only one direction which has the maximum power and the others are approximately zero.

Figure 4.3 shows better how beamforming nulls the interfering signals while maximizing the desired one. In this example, we suppose the sources are at [30 60 90]◦, and the other parameters are according to Table 2.1. Using MVDR beamforming, the directions can be obtained as shown in Figure 4.3:

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0 20 40 60 80 100 120 140 160 180 −8 −6 −4 −2 0 2 4 6 θ [°] σ 2 [dB] (a) 0 20 40 60 80 100 120 140 160 180 −8 −6 −4 −2 0 2 4 6 θ [°] σ 2 [dB] (b)

Figure 4.3: a) Maximizing the signal at 30◦and nulling the sources situated at 60◦and 90◦ b) Maximizing the signal at 60◦and nulling the sources situated at 30◦and 90◦.

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4.3 Conclusion

In this chapter we reviewed one of the popular beamformers called MVDR. The weight vector helps us to maximize the signal to interference ratio. Thus, MVDR ensures that the signal passes through the beamformer undistorted. Therefore, the output signal power is the same as the look-direction source power alone. The minimization process then minimizes the total noise, including interferences and uncorrelated noise.

The MVDR will be used in this project to keep only the signal impinging on the array from a specific direction. The output will then be equivalent to retrieving a copy of this signal alone, without any interference.

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Estimating the time and angle of arrivals in mobile communications