Florence Nadal

Pascale Jardin

Keywords:

and Development of an Experimental Platform". A Multiple-Input Multiple-Output (MIMO) radar is a system employing multiple transmitters and receivers in which the waveforms to be transmitted can be totally independent. Compared to standard phased-array radar systems, MIMO radars offer more degrees of freedom which leads to improved angular resolution and parameter identifiability, and provides more flexibility for transmit beampattern design. The main issues of interest in the context of MIMO radar are the estimation of several target parameters (which include range, Doppler, and Direction-of-Arrival (DOA), among others). Since the information on the targets is obtained from the echoes of the transmitted signals, it is straightforward that the design of the waveforms plays an important role in the system accuracy. This document addresses the investigation of DOA estimation of non-moving targets and waveform design techniques for MIMO radar with colocated antennas. Although First, I would like to thank all the members of my thesis jury, Marc Lesturgie, Christophe Craeye, Daniel Roviras, Bernard Huyart, and Philippe Eudeline,

narrowband MIMO radars have been deeply studied in the literature, the existing DOA estimation techniques have been usually proposed and analyzed from a theoretical point of view, often assuming ideal conditions. This thesis analyzes existing signal processing algorithms and proposes new ones in order to improve the DOA estimation performance in the case of narrowband and wideband signals. The proposed techniques are studied under ideal and non-ideal conditions considering punctual targets. Additionally, we study the influence of mutual coupling on the performance of the proposed techniques and we establish a more realistic signal model which takes this phenomenon into account.

We then show how to improve the DOA estimation performance in the presence of distorted radiation patterns and we propose a crosstalk reduction technique, which makes possible an efficient estimation of the target DOAs. Finally, we present an experimental platform for MIMO radar with colocated antennas which has been developed in order to evaluate the performance of the proposed techniques under more realistic conditions. The proposed platform, which employs only one transmitter and one receiver architectures, relies on the superposition principle to simulate a real MIMO system.

Principle and Interest of MIMO Radar

Multi-antenna based radar systems are widely used in both military and civilian applications. One of the most implemented radar configurations is the phased-array radar system. Phased-arrays employ multiple transmitter and multiple receiver antenna elements which are usually colocated. The multiple transmitter elements are capable of cohering and steering the transmitted energy toward a desired direction by transmitting scaled and delayed versions of a single waveform. At the receiver array, the received signals can be steered in a given direction in order to maximize the probability of detection or the Signal-to-Noise Ratio (SNR). This can be done in two different ways: By performing analog beamforming via the use of phase shifters in the different receiver architectures, or by performing digital beamforming via adaptive processing. Digital beamforming offers several advantages over its analog counterpart, including the capability to steer multiple simultaneous beams [START_REF] Krim | Two decades of array signal processing research: The parametric approach[END_REF][2] and the possibility to implement single and multiple sidelobe cancelers [START_REF] Jeffrey | Phased-Array Radar Design: Application of Radar Fundamentals[END_REF].

Another type of multi-antenna radar system is the Multiple-Input Multiple-Output (MIMO) radar. A MIMO radar also employs multiple transmitter and multiple receiver elements, but unlike the phased-array systems, the different waveforms transmitted by a MIMO radar can be correlated or uncorrelated with each other [START_REF] Li | MIMO radar with colocated antennas[END_REF].

Compared to phased-array radars, MIMO radars offer more degrees of freedom which lead to improved angular resolution [5][6], improved parameter identifiability, and more flexibility for transmit beampattern design [START_REF] Li | MIMO radar with colocated antennas[END_REF]. Additionally, MIMO radars can synthesize larger virtual arrays which increases resolution and the number of targets that can be detected [START_REF] Bekkerman | Target detection and localization using MIMO radars and sonars[END_REF] [START_REF] Qu | Performance comparisons of MIMO and phased-array radar[END_REF]. There are several configurations of MIMO radar depending on the location of the transmitting and the receiving elements. One of them is the MIMO radar with widely separated antennas (or statistical MIMO radar) [8][9]. The separation between the different transceivers must be large enough (several wavelenghts) to receive uncorrelated echoes from the targets. This configuration allows exploiting the spatial diversity of the targets' Radar Cross Section (RCS) to improve the radar performance by addressing the problem similarly to a MIMO communications problem. Actually, by combining the different target echoes coming from different directions (see Figure 1.1) by non-coherent (or statistical) processing, a diversity gain is achieved, similarly to the diversity gain obtained in MIMO communications when data is transmitted over independent channels [START_REF] Haimovich | MIMO radar with widely separated antennas[END_REF].

Another type of MIMO radar, known as MIMO radar with colocated antennas, employs transmit and receive antenna arrays containing elements which are closely spaced relatively to the working wavelength (e.g. spaced by half the wavelength). In a receiver array of colocated antennas, the signals reflected by the targets have similar amplitude at each receive antenna element and the targets are usually modeled as punctual. While this configuration does not provide spatial diversity, spatial resolution can be increased by combining the information from all of the transmitting and receiving paths. This is done by coherent processing: By exploiting the different time delays and/or phase shifts, the received signals are coherently combined to form multiple beams. This configuration also has other benefits such as a good interference rejection and a good flexibility for transmitting a desired beampattern [START_REF] Li | MIMO radar with colocated antennas[END_REF]. MIMO radars with colocated antennas can be further classified into bistatic MIMO radars if the transmitter array is widely separated from the receiver array [START_REF] Yan | Multitarget identification and localization using bistatic MIMO radar systems[END_REF] (see Figure 1.2); or monostatic MIMO radars if the transmitter and receiver arrays are closely spaced or colocated [START_REF] Ma | Mono-static MIMO radar array design for interferences suppressing[END_REF] (see Figure 1.3).

One of the main issues of interest in the context of MIMO radar is the estimation of several target parameters which include range, Doppler, Direction-of-Arrival (DOA), and reflection coefficients, among others. Another main topic which attracts the interest of researchers is waveform design. In fact, the capability of transmitting different arbitrary waveforms by every element of the array allows having great flexibility when trying to transmit a desired beampattern. This capability can be exploited to improve the target parameter identification, to maximize the SNR, to improve angular and range resolution or to achieve interference rejection, among others. Moreover, additional improvement in resolution and interference rejection can be obtained by the use of wideband signals to synthesize the transmit beampatterns.

MIMO Radar Applications

Radar systems have been used in many fields of application in the last decades, including military and civilian areas, and the need of more sophisticated and accurate radar functions have been constantly increasing. Thanks to its improved capabilities in resolution, target parameter identification, and waveform design among others, MIMO radars might be widely used in the future and make possible the development of additional features such as communication by radar or intelligent signal coding [START_REF] Wiesbeck | The radar of the future[END_REF].

Today, MIMO radars can be used in many of the applications where other multi-antenna based radars are employed. Such applications include ground surveillance [START_REF] Lesturgie | Some relevant applications of MIMO to radar[END_REF] [START_REF] Martinez-Vazquez | UWB MIMO radar arrays for small area surveillance applications[END_REF], automotive [START_REF] Lutz | 77 GHz lens-based multistatic MIMO radar with colocated antennas for automotive applications[END_REF] [START_REF] Schuler | Array design for automotive digital beamforming radar system[END_REF] and interferometry [START_REF] Kim | Investigation of MIMO SAR for interferometry[END_REF] applications. Other interesting applications might be possible such as the detection of anti-personnel mines by ground penetrating radar measurements [START_REF] Fischer | Detection of antipersonnel mines by using the factorization method on multistatic ground-penetrating radar measurements[END_REF], the detection of tsunami waves [START_REF] Dzvonkovskaya | Simulation of tsunami signatures in ocean surface current maps measured by HF radar[END_REF] and maritime surveillance by employing a MIMO configuration of High Frequency Surface Waves Radar (HFSWR) [START_REF] Lesturgie | Some relevant applications of MIMO to radar[END_REF] [START_REF] Anderson | A MIMO technique for enhanced clutter selectivity in a multiple scattering environment: Application to HF surface wave radar[END_REF], or through-the-wall radar imaging applications for urban sensing [START_REF] Masbernat | An MIMO-MTI approach for through-the-wall radar imaging applications[END_REF]. MIMO radars also find applications in the medical area, e.g. for breast cancer detection [START_REF] Li | MIMO Radar Signal Processing[END_REF] or to monitor the water accumulation in the human body [START_REF] Pancera | Ultra wideband radar imaging: An approach to monitor the water accumulation in the human body[END_REF].

Problem Statement

This document addresses the investigation of DOA estimation of non-moving targets and waveform design techniques for monostatic MIMO radar with colocated antennas.

Although narrowband MIMO radars have been deeply studied in the literature, the existing DOA estimation techniques have been usually proposed and analyzed from a theoretical point of view, often assuming ideal conditions. Moreover, in the case of wideband signals, the assumptions done in the signal model no longer hold and narrowband detection techniques cannot be directly applied. The objective of this thesis is to study the existing DOA estimation and waveform design techniques and to develop new signal processing algorithms in order to improve the DOA estimation performance in the wideband case. The proposed techniques will be studied under ideal and non-ideal conditions considering punctual and non-moving targets. They will be validated by experimental results.

The thesis is divided into a theoretical and an experimental part which are described thereafter.

Theoretical Investigation

The thesis starts with the introduction of the signal model of MIMO radar with colocated antennas. Then, a review of the existing narrowband DOA estimation techniques is done. Since the use of wideband signals is gaining in importance, we investigate DOA estimation and waveform design in the wideband case. We propose new wideband DOA estimation techniques mainly based on a literature review on wideband array processing.

We also propose new waveform design algorithms.

Additionally, we study the electromagnetic interactions between the antenna elements in order to analyze their influence on the performance of the proposed techniques and to create a more realistic signal model. We then propose methods to overcome the undesired effects of mutual coupling such as the radiation pattern distortion and crosstalk.

Experimental Implementation

An experimental platform for MIMO radar with colocated antennas is developed in order to evaluate the performance of the proposed techniques under more realistic conditions. Since a real large MIMO system is particularly expensive and complex to develop, synchronize and calibrate, the proposed platform contains only one transmitting and one receiving Radio Frequency (RF) architectures. An automated mechanical system is used to simulate a real MIMO radar. By applying the superposition principle, the received signals are combined to construct the received signal matrix of the MIMO system.

Finally, a set of experimental results is presented which allows us to evaluate the real performance of some narrowband DOA estimation techniques.

Thesis Outline

The thesis is organized as follows. In Chapter 2, the narrowband far-field signal model is presented, followed by a review of some existing narrowband DOA estimation techniques.

Once the relevant theory has been introduced, the different techniques are compared via simulation results in order to highlight their strengths and weaknesses, including spatial resolution and robustness against noise and jammers. Moreover, the limit between the spherical-wave and plane-wave regions of the far field is also studied, showing when a target can really be assumed to be in the plane-wave region.

In

Publications

The work carried out in this thesis led to several publications, which are listed below.

International conferences with proceedings

MIMO Radar with Colocated Antennas

A MIMO radar with colocated antennas has many benefits compared to other MIMO radar architectures, such as a better spatial resolution and a good flexibility for transmitting a desired beampattern, among others [START_REF] Li | MIMO radar with colocated antennas[END_REF]. This configuration also allows the direct application of many adaptive array processing techniques for parameter estimation, including the well known Capon beamformer and MUSIC algorithm. In this chapter, the narrowband signal models for both the spherical-wave and plane-wave regions of the far field are described. Next, some existing detection techniques are presented and their performance is analyzed in the narrowband case with different MATLAB simulations. Other simulations are performed in order to establish when a target can be considered in the spherical-wave or in the plane-wave region of a MIMO radar. The obtained condition is compared to the one usually presented in the literature.

Far-Field Signal Model

In electromagnetic theory, the near field and the far field are two regions of the electromagnetic field radiated by a source, which are defined by relations between the distance from the source to the point where the field is measured, the wavelength of the transmitted signal and the aperture of the antenna. In the far-field region, the electric and magnetic components of the field radiated by a given antenna are orthogonal to each other [START_REF] Balanis | Advanced Engineering Electromagnetics[END_REF], and the field pattern does not change with the distance between the antenna and the point where the field is measured. The far-field condition is very useful to obtain a simple expression of a propagating wave. The condition for a point in the space to be considered in the far-field region of an antenna is given by [

24] R ≥ 2∆ 2 λ , (2.1)
where ∆ is the largest dimension of the antenna, λ is the wavelength, and R is the distance from the antenna to the point where the field is measured. In the same way, a target is considered to be in the far field of a MIMO radar if it is located in the farfield regions of both the transmitter and the receiver arrays. The far-field assumption is always used throughout this document.

Consider the complex representation of a time-varying current source signal given by

I(t) = c(t)e j2πfct (2.2)
where c(t) is the complex envelope and f c is the carrier frequency. Assuming that the source signal I(t) is incident at the input of a transmitting antenna, the far-field electric field at position p radiated by such antenna is given by [START_REF] Shlivinski | Antenna characterization in the time domain[END_REF]

E(p, t) = - µ 0 4π p [h t (u p , t) * I(t)] * δ t - k T p 2πf c = - µ 0 4π p h t (u p , t) * c t - k T p 2πf c e j(2πfct-k T p) , (2.3)
where k is the wave vector, u p = p/ p is the position unit vector,

µ 0 = 4π10 -7 T•m/A
is the permeability constant of free space, δ(t) is the Dirac delta function and h t (u p , t) is the effective height or far-field impulse response of the transmitting antenna. Operators

• and * denote respectively the vector Euclidean norm and the temporal convolution. Both the position and the wave vectors are in reality three-dimensional, however, they are assumed here to be two-dimensional since the arrays of interest in this work can only detect targets in the x -y plane. The wave vector is defined as

k = 2π λ sin θ cos θ T , (2.4)
where θ ∈ [-90 For simplicity, we will consider only the portion of the radiated field which is polarized in the u pol direction. Accordingly, the scalar electric field at position p is given by

E(p, t) = - µ 0 4π p h t (u p , t) * c t - k T p 2πf c e j(2πfct-k T p) , (2.5)
where

h t (u p , t) = h T t (u p , t) u pol . (2.6)
Consider now the transmitting Uniform Linear Array (ULA) shown in Figure 2.2. Note that the array is centered at the origin of the Cartesian coordinate system; for an odd number of antenna elements, the central element will be placed at the origin of the x -y plane. Then, the field radiated by the i th antenna element due to a source signal c i (t) at an arbitrary target position p of coordinates [D, H] is given by

E i (p, t) = - µ 0 4π r i h t,i (u r i , t) * c i (t -τ i ) e j(2πfct-k T i r i ) i = 0, ..., L t -1, (2.7)
where

h t,i (u r i , t) = h T t,i (u r i , t) u pol , (2.8)
h t,i (u r i , t) is the effective height of the i th transmitter antenna element,

τ i = k T i r i
2πfc is the time needed by the signal c i (t) to travel from the i th antenna element to the target, r i is the vector linking the i th antenna element with the target position p, u r i = r i / r i , and L t is the number of transmitting elements. The total field at the target location can be expressed as the superposition of the fields radiated by every antenna element as

E t (p, t) = - µ 0 4π Lt-1 i=0 1 r i h t,i (u r i , t) * c i (t -τ i ) e j(2πfct-k T i r i ) .
(2.9)

The field reflected by the target can be measured by a receiver array. Assuming that the transmitting and the receiving arrays are colocated1 , the reflected field measured at the l th receiving antenna is then given by

E r (x l , t) = -β 1 r l [h r,l (u r l , t) * E t (p, t)] * δ (t -τ l ) = µ 0 β 4π r l e -jk T l r l h r,l (u r l , t) * Lt-1 i=0 1 r i h t,i (u r i , t) * c i (t -τ i -τ l ) e j(2πfct-k T i r i ) l = 0, ..., L r -1, (2.10)
where

h r,l (u r l , t) = h T r,l (u r l , t) u pol , (2.11)
h r,l (u r l , t) and x l are respectively the effective height and the position of the l th receiving antenna element, τ l is the time needed by the reflected signal to travel from the target to the l th element, u r l = r l / r l , and β is the complex reflection coefficient of the target.

Narrowband Signals

A signal is said to be narrowband, wideband or ultra-wideband (UWB) depending on how large its bandwidth is. The condition for a bandpass signal to be narrowband is given by

B f c , (2.12)
where B is the bandwidth of the signal. The signal can be considered in practice as narrowband if its bandwidth is much smaller (at least ten times) than the median frequency, which is usually the carrier frequency [START_REF] Proakis | Digital Signal Processing: Principles, Algorithms, and Applications[END_REF]. In array signal processing theory, if the different complex envelopes c i (t) are narrowband, the baseband signal sampled at two different points in space by a receiving ULA does not change too much in amplitude and the different time delays τ i and τ l can be neglected. Hence, the set of signals at the receiver array can be approximated as [START_REF] Vaidyanathan | MIMO radar with broadband waveforms: Smearing filter banks and 2d virtual arrays[END_REF]

c i (t -τ i -τ l ) ≈ c i (t). (2.13)
Accordingly, (2.10) can be written as

E r (x l , t) = µ 0 β 4π r l e -jk T l r l h r,l (u r l , t) * Lt-1 i=0 1 r i h t,i (u r i , t) * c i (t) e j(2πfct-k T i r i ) l = 0, ..., L r -1, (2.14)
Moreover, if the system is narrowband, (2.14) can be simplified by expressing the electric field in terms of the antenna radiation patterns, which are parameters measurable in the frequency domain. To see this, consider the received electric field in the frequency domain (i.e. the Fourier transform of (2.14))

E r (x l , f ) = µ 0 β 4π r l e -jk T l r l H r,l (u r l , f ) Lt-1 i=0 1 r i H t,i (u r i , f ) C i (f -f c ) e -jk T i r i l = 0, ..., L r -1, (2.15)
where H r,l (u r l , f ), H t,i (u r i , f ), and C i (f ) are the Fourier transforms of h r,l (u r l , t), h t,i (u r i , t), and c i (t) respectively. The electric field is then proportional to the antenna transfer functions H r,l (u r l , f ) and H t,i (u r i , f ), which describe the antenna patterns as a function of the frequency, and the azimuth and elevation angles [START_REF] Licul | Unified frequency and time-domain antenna modeling and characterization[END_REF]. Note that if the system is narrowband, the transfer functions can be assumed to be constant within the working frequency band. The electric field radiated and/or received by a given narrowband antenna can then be assumed to be proportional to the antenna radiation pattern measured at the working frequency f c and at a given polarization. Accordingly, the reflected field (2.14) measured at the receiver array can be written under the narrowband assumption, as

E r (x l , t) = αβe j2πfct 1 r l e -jk T l r l g r,l (θ) Lt-1 i=0 1 r i g t,i (θ)c i (t) e -jk T i r i l = 0, ..., L r -1, (2.16)
where α is a proportionality constant and g t,i (θ) and g r,l (θ) are respectively the radiation patterns of the i th transmitter and the l th receiver elements, measured at frequency f c , at a given elevation angle, and for a polarization in the direction u pol .

Finally, the electric field at the receiver array can be written in vector notation as

E r (t) = αβe j2πfct a * r (θ, R) a H t (θ, R) c(t), (2.17)
where

E r (t) = E r (x 0 , t) • • • E r (x Lr-1 , t)
T is the set of electric fields measured at the receiver array, c(t

) = c 0 (t) • • • c Lt-1 (t)
T is the set of complex envelopes of the transmitted signals,

a t (θ, R) =      g * t,0 (θ) 1 r 0 e jk T 0 r 0 . . . g * t,Lt-1 (θ) 1 r L t -1 e jk T L t -1 r L t -1      , (2.18) a r (θ, R) =     g * r,0 (θ) 1 r 0 e jk T 0 r 0 . . . g * r,Lr-1 (θ) 1 r Lr -1 e jk T Lr -1 r Lr -1     , (2.19
)

r i = H 2 + D -i -Lt-1 2 d t 2 , r l = H 2 + D -l -Lr-1 2 d r 2 , H = R cos θ, D = R sin θ, (2.20)
and d t and d r are the inter-element spacings of the transmitting and the receiving arrays respectively.

The terms a t (θ, R) and a r (θ, R) are known as the transmit and receive steering vectors respectively. One may note that every wave vector k i is colinear with the corresponding vector r i and hence the dot product k T i r i is always 2π λ r i . Accordingly, we can write the transmit steering vector as

a t (θ, R) =      g * t,0 (θ) 1 r 0 e j 2π λ r 0 . . . g * t,Lt-1 (θ) 1 r L t -1 e j 2π λ r L t -1      . (2.21)
The target location is then defined by parameters [θ, R] where R is the radial distance between the origin of the Cartesian coordinate system and point p = [D, H], and θ is the angle between the radial vector and the y-axis.

The Plane-Wave Approximation

The electromagnetic theory states that the waves radiated by antennas of finite dimensions are spherical; their amplitudes are inversely proportional to the distance to the antenna [START_REF] Balanis | Advanced Engineering Electromagnetics[END_REF], which is consistent with the signal model (2.17). However, in many cases, the distance from the antenna to the target is large enough to assume that the wavefronts are locally plane. The far-field region can then be divided into a spherical-wave region and a plane-wave region, which are defined by relations between the distance from the antenna to the target position, the wavelength of the transmitted signal and the aperture of the antenna. The limit between the spherical-wave and plane-wave regions is evaluated in Section 2.3.4.

Although the steering vectors a r (θ, R) and a t (θ, R) in the model (2.17) are general and can always be used in the narrowband case, the plane-wave assumption allows us to do some simplifications in the signal model. First, since a plane wavefront propagates in a single direction, all the wave vectors are parallel as shown in Figure 2.3. Then, the dot product k T i r i is given by

k T i r i = 2π λ D -i - L t -1 2 d t sin θ + 2π λ H cos θ i = 0, . . . , L t -1.
(

Secondly, given that the inter-element spacing d t is much smaller than the distance from the array to the target, all the attenuation terms 1/ r i are approximately the same, i.e. 1/ r i ≈ 1/R. Thus, in the narrowband case, the transmit steering vector depends only on the direction θ of the wavefront and can be expressed as

a t (θ) = 1 R e j 2π λ (D sin θ+H cos θ) g * t,i (θ)e j 2π λ ( L t -1 2 -i)dt sin θ i=0,...,Lt-1
.

(2.23)

x

Plane wavefront

Transmitter array The term placed outside the vector is common to every element of the array and can be omitted. Therefore, the plane-wave transmit steering vector is finally given by

a t (θ) = g * t,i (θ)e j 2π λ ( L t -1 2 -i)dt sin θ i=0,...,Lt-1
.

(2.24)

In the same way, a plane-wave receive steering vector a r (φ) can be defined in terms of the direction φ of the wavefront traveling from the target to the receiver array. In general, the directions of the transmit and the receive wavefronts are known as Direction-Of-Departure (DOD) and DOA respectively. However, since the architecture of MIMO radar studied in this work has colocated transmitter and receiver arrays, only the term DOA and the angle θ will be used in the remainder of this document.

The Sampled Received Signals in the Narrowband Case

The previously presented signal model was written in terms of the radiated (and received) electric fields which are analog quantities. However, all the signal processing approaches are based on the observed data, obtained after down-conversion and sampling of the received signals.

x l (n) x l t = n Fs n = 0, . . . , N -1, l = 0, . . . , L r -1, (2.25)
where N is the number of samples, and F s is the sampling frequency which is taken equal to the bandwidth of the signal in the narrowband case. Then, the received signals due to the reflection from K targets in the plane-wave region can be written as

x(n) = K k=1 β k a * r (θ k )a H t (θ k )c(n) + z(n), (2.26)
where

x(n) = x 0 (n) • • • x Lr-1 (n) T , c(n) = c 0 (n) • • • c Lt-1 (n) T is the set of discrete-time complex envelopes of the transmitted signals, z(n) = z 0 (n) • • • z Lr-1 (n)
T represents the unmodelled interference and noise, and θ k and β k are respectively the DOA and the reflection coefficient of the k th target. The radiation patterns of every antenna element of a ULA are usually assumed to be identical, angle-independent and of unity gain, and hence, the plane-wave steering vectors a t (θ k ) and a r (θ k ) have the following simplified form

a t (θ) = e j 2π λ ( L t -1 2 -i)dt sin θ i=0,...,Lt-1 , (2.27)
a r (θ) = e j 2π λ ( Lr -1 2 -l)dr sin θ l=0,...,Lr-1 .

(2.28)

The equivalent spherical-wave model can be easily obtained by using the appropriate spherical-wave steering vectors.

Note that in practice the radiation patterns might not be identical from one element to the other and are always angle-dependent. These considerations should be taken into account in the signal model and will be discussed in Chapter 4.

The signal model can be described using matrix notation, by stacking the N received samples in the columns of a matrix X as

X = K k=1 β k a * r (θ k )a H t (θ k )C + Z. (2.29)
Both X and Z are of dimension L r × N while the matrix

C = c(0) • • • c(N -1) is of dimension L t × N .

Narrowband Direction-of-Arrival Estimation Techniques

Direction-of-Arrival estimation of narrowband sources using arrays of sensors is a topic which has been highly studied in the past years. To date, a variety of DOA estimation techniques have been proposed such as the Maximum Likelihood (ML) technique [START_REF] Ziskind | Maximum likelihood localization of multiple sources by alternating projection[END_REF],

Capon (also known as Minimum Variance) technique [START_REF] Capon | High-resolution frequency-wavenumber spectrum analysis[END_REF], and the MUSIC algorithm [START_REF] Schmidt | Multiple emitter location and signal parameter estimation[END_REF] among others. Many of those techniques can be directly applied to the radar context considering that the source signals to be detected are the signals reflected by the targets.

This section presents two relevant DOA estimation techniques (Capon and MUSIC) and the Generalized Likelihood Ratio Test (GLRT) technique adapted to the context of narrowband MIMO radar.

The Capon Beamformer

The Capon beamformer [START_REF] Capon | High-resolution frequency-wavenumber spectrum analysis[END_REF] is an array processing technique frequently used for DOA estimation. It uses a spatial filter w(θ) constrained to minimize the signal power coming from all directions but the desired ones. Let y(n) denote the output of the spatial filtering process as

y(n) = w H (θ)x(n).
(2.30)

The spatial filter w(θ) can be obtained by solving the following optimization problem [START_REF] Capon | High-resolution frequency-wavenumber spectrum analysis[END_REF][32] min

w P (θ) s.t. w H (θ)a * r (θ) = 1, (2.31)
where P (θ) is the output power defined as

P (θ) = E[|y(n)| 2 ] = w H (θ)R x w(θ), (2.32)
E[•] denotes the mathematical expectation, and

R x = E[x(n)x H (n)]
is the auto-covariance matrix of the received signals. The optimal vector w(θ) can be found using the method of Lagrange multipliers. It is given by (see Appendix A)

w(θ) = R -1 x a * r (θ) a T r (θ)R -1 x a * r (θ) . (2.33)
By replacing w(θ) by w(θ) in Equation (2.32), we obtain the Capon's spatial spectrum

P cap (θ) = 1 a T r (θ)R -1 x a * r (θ) , (2.34)
and the DOAs of the targets can be found by searching for the maxima of P cap (θ).

Note that in practice, the auto-covariance matrix R x is estimated using a finite set of N samples of x(n). This estimate can be found by calculating

Rx = 1 N XX H . (2.35)

MUSIC

The MUSIC algorithm is a subspace-based array processing technique originally proposed to estimate the DOAs of uncorrelated narrowband sources [START_REF] Schmidt | Multiple emitter location and signal parameter estimation[END_REF]. It uses eigenvalue decomposition to separate the auto-covariance matrix of the observed data into a signal and a noise subspace. The orthogonality between both subspaces is then exploited to locate the sources. The MUSIC algorithm is based on the following signal model:

x(n) = K k=1 a * r (θ k )s k (n) + z(n), (2.36)
where x(n) is the L r ×1 observed data vector, s k (n) is the k th source signal to be detected, and z(n) denotes a white additive noise which is independent from the transmitted signals s(n) and has a common variance σ 2 for all sensors. For notation simplicity, the steering vectors can be stacked in the columns of a matrix as A = a * r (θ 1 ) • • • a * r (θ K ) and the signal model becomes

x(n) = As(n) + z(n), (2.37)
where

s(n) = s 1 (n) • • • s K (n) T .
The auto-covariance matrix of the received signals

x(n) is given by R x = E[x(n)x H (n)] = AE[s(n)s H (n)]A H + E[z(n)z H (n)]. (2.38)
Then, denoting R s = E[s(n)s H (n)] as the source auto-covariance matrix, the auto-

covariance matrix R x becomes R x = AR s A H + σ 2 I, (2.39)
where I denotes the L r × L r identity matrix. Using a spectral decomposition, the autocovariance matrix of the received signals can be expressed as [1]

R x = UΛU H , (2.40)
where U = u 1 • • • u Lr is a matrix whose columns are the eigenvectors of R x , and Λ = diag(λ 1 , . . . , λ Lr ) is a diagonal matrix containing the eigenvalues of R x in decreasing order (i.e. λ 1 ≥ • • • ≥ λ Lr > 0). The latter spectral decomposition can be written as

R x = U s Λ s U H s + U n Λ n U H n , (2.41)
where Λ s = diag(λ 1 , . . . , λ K ) and Λ n = diag(λ K+1 , . . . , λ Lr ), and U s and U n are matrices composed by the eigenvectors of the signal-plus-noise subspace and the noise-only subspace respectively. The largest K eigenvalues λ i of R x correspond to the signal-plus-noise subspace and the L r -K smallest eigenvalues are equal to σ 2 .

Every eigenvector of the noise-only subspace is orthogonal to the signal-plus-noise subspace, and hence orthogonal to A. This can be seen from the eigenvalue equation of

R x (R x -λ i I) u i = 0 i = 1, 2, . . . , L r .
(2.42)

Then, using only the noise subspace eigenvectors and replacing R x by its expression (2.39), the latter equation becomes

AR s A H u i = 0 i = K + 1, . . . , L r .
(2.43)

Since the matrices A and R s are full rank (assuming that the source signals are non coherent), the matrix AR s A H is also full rank and the noise eigenvectors must be orthogonal to A, i.e. A H u i = 0. Then we have

a T r (θ)u i = 0 i = K + 1, . . . , L r .
(2.44) Note that {θ 1 , . . . , θ K } are the only possible solutions to Equation (2.44). Based on this orthogonality, a MUSIC spatial spectrum can be defined as

P M U SIC (θ) = 1 a T r (θ)U n U H n a * r (θ)
.

(2.45)

In the case of MIMO radar, the model described in Equation (2.26) is equivalent to (2.36) where the source signals s k (n) are the signals reflected by the targets, i.e.

s(n) =     s 1 (n) . . . s K (n)     =     β 1 a H t (θ 1 )c(n) . . . β K a H t (θ K )c(n)     .
(2.46)

Then, Equation (2.44) holds only if the equivalent of R s is still full rank. By defining

B =     β 1 a H t (θ 1 ) . . . β K a H t (θ K )     , (2.47)
Equation (2.46) can be written as

s(n) = Bc(n). (2.48)
Then, the auto-covariance matrix R s is given by

R s = E Bc(n)c H (n)B H = BR c B H . (2.49)
Given that B is full rank, R s will be full rank only if R c is also full rank, which is usually the case. Under this condition, the MUSIC algorithm can be applied to the MIMO radar signal model.

The Generalized Likelihood Ratio Test (GLRT)

The GLRT detection technique for narrowband MIMO radar was derived in [START_REF] Xu | Radar imaging via adaptive MIMO techniques[END_REF] assuming that the columns of the noise term Z of Equation (2.29) are i.i.d. circularly symmetric complex Gaussian random vectors with zero mean and unknown covariance matrix R z . The GLR is given by [START_REF] Xu | Radar imaging via adaptive MIMO techniques[END_REF][33]

ρ(θ) = 1 - max Rz f (X|β = 0, R z ) max β,Rz f (X|β, R z ) 1 N , (2.50)
where

f (X|β, R z ) = π -N Lr |R z | -N e -tr[R -1 z (X-βa * r (θ)a H t (θ)C)(X-βa * r (θ)a H t (θ)C) H ] (2.51)
is the probability density function (PDF) of the received signals given the parameters β (i.e. the target reflexion coefficient) and R z , and tr[•] and | • | denote the trace and the determinant of a matrix, respectively. The fractional part of Equation (2.50) is the ratio between two likelihood functions; the first one under the noise-alone hypothesis (without any target) and the second one under signal-plus-noise hypothesis (with a target in direction θ). If there is a target in a direction θ of interest, the denominator max β,Rz f (X|β, R z ) will be much greater than max Rz f (X|β = 0, R z ) and then the value of ρ(θ) will be close to one. Otherwise, if there is no target at θ, the value of ρ(θ)

will approach zero. After some derivations (see Appendix A), (2.50) can be written as

ρ(θ) = 1 - a T r (θ) R-1 x a * r (θ) a T r (θ) Q-1 a * r (θ) , (2.52)
where Q is defined as

Q = Rx - Rxc a t (θ)a H t (θ) RH xc a H t (θ) Rc a t (θ) , (2.53)
Rc is the estimated auto-covariance matrix of the received and the transmitted signals,

and Rxc is the estimated cross-covariance matrix between the received and the transmitted signals. The covariance matrices are estimated as

Rc = 1 N CC H , Rxc = 1 N XC H .
(2.54)

The use of the GLRT to detect the target DOAs is of particular interest since it is able to reject interference or jammers which are uncorrelated with the transmitted signals.

Narrowband Simulations

The previously described DOA estimation techniques for MIMO radar are investigated in this section using MATLAB simulations. The different strengths and weaknesses of each one are discussed, including spatial resolution and robustness against noise and jammers. Moreover, the limit between the spherical-wave and the plane-wave regions is also studied, showing when a target can really be assumed to be in the plane-wave region and avoid the errors which may occur due to a wrong use of the plane-wave approximation.

Simulation Parameters

Consider a MIMO radar with colocated antennas whose transmitter and receiver arrays are two ULA of L t = L r = L = 10 elements, and the inter-element spacings are

set to d t = d r = d = λ/2. The transmitted signals {c i (n)} Lt-1 i=0
are independent sequences of N = 512 Quadrature Phase Shift Keying (QPSK) symbols. Each symbol has a mean power of P s = 0.1. The carrier frequency is set to f c = 5.8 GHz. In the following simulations, the radiation patterns of the transmitting and receiving elements are assumed to be identical, angle-independent and of unity gain. the SNR (in dB) is then given by SNR = 10 log 10 However, in the case of a low SNR, such as -10 dB, the target DOAs can hardly be found in both Capon and MUSIC spectra as shown in Figure 2.5, and some peaks can appear as noise in this particular example. On the other hand, even though the values of the GLR are far from 1, the GLRT seems more robust to noise since the peaks corresponding to the targets can still be clearly seen. However, it is difficult to determine the minimum angular spacing between two targets that a given technique is able to detect since it depends on several parameters, including the number of antenna elements, the target positions, the SNR and the orthogonality of the transmitted waveforms.

Target Detection

x(n) = K k=1 β k a * r (θ k )a H t (θ k )c(n), x(n) = x0 (n) • • • xLr-1 (n) T , (2.55)
       N -1 n=0 Lr-1 l=0 |x l (n)| 2 N -1 n=0 Lr-1 l=0 |z l (n)| 2        . ( 2
As a conclusion, the MUSIC algorithm offers the best resolution among the three studied detection techniques and it is useful to detect closely spaced targets. As for the GLRT, even though it is not able to resolve two closely spaced targets, it is the most robust against noise and the least sensitive to the target reflection coefficients.

Detection in the Spherical-Wave Region

The signal model (2.29) is also applicable for a detection in the spherical-wave region by using the appropriate steering vectors, which are defined as (see Section 2.1.1)

a t (θ, R) =      1 r 0 e j 2π λ r 0 . . . 1 r L t -1 e j 2π λ r L t -1      , (2.57) a r (θ, R) =     1 r 0 e j 2π λ r 0 . . . 1 r Lr -1 e j 2π λ r Lr -1     .
(2.58)

Since such steering vectors depend on the direction θ k and the distance R k from the center of the array to the k th target (see Figure 2.8), the DOA estimation techniques must be performed along with a two-dimensional search. The target DOAs are then found by searching for the maxima of the three dimensional spatial spectra (P cap (θ, R), P M U SIC (θ, R), and ρ(θ, R)). Transmitter array Receiver array As shown in Figure 2.10, both Capon and MUSIC spectra show a high peak due to the jammer at 20 • , which could lead to a wrong detection. On the other hand, the GLRT totally rejects the strong jammer by showing only the peaks corresponding to the targets.

R 1 R 2 θ 1 θ 2 D 1 D 2 H 1 H 2 p 1 p 2
Despite its lower resolution compared to MUSIC and Capon, the GLRT is very robust against interference and jammers and can be used to estimate the number of targets.

Limit Between the Spherical-Wave and Plane-Wave Regions

The use of the general form of the steering vectors (2.57) and (2.58) (which takes into account the spherical shape of the wavefronts) may allow in theory the estimation of both parameters [θ k , R k ] of the targets even if they are located in the plane-wave region. However, this processing requires a bidimensional search and therefore a much higher processing time. Moreover, as the distance between the targets and the arrays increases, more distances should be included in the grid of distances R, making the processing even heavier. Hence, the use of the plane-wave approximation may be more appropriate in some cases to reduce the processing time at the expense of estimating only the directions θ k . The question is to determine when a target can be considered to be in the spherical-wave or in the plane-wave region of a MIMO radar. Even though the far-field condition (2.1) given by antenna theory is often used in signal processing as a plane-wave condition, the approximation done in the signal model might be inaccurate in a particular zone of the far-field region and the real plane-wave condition might differ from R > 2∆ 2 /λ (where ∆ is the largest dimension of the antenna). The limit between spherical-wave and plane-wave regions is studied herein from a signal processing point of view.

To evaluate the limit between spherical-wave and plane-wave regions, the Mean Square Error (MSE) of the estimated DOA (in degrees) has been computed for a target at 60 • located at different distances, using a grid of values of R going from 0.2 m to 10 m by a step of 0.05 m. For every distance, 100 different Monte Carlo trials have been performed.

The signal model used to simulate the signals going toward and coming from the target is

X = K k=1 β k a * r (θ k , R)a H t (θ k , R)C + Z, (2.59)
considering the spherical shape of the wavefronts. Then, the Capon, MUSIC, and GLRT techniques were performed to estimate the DOA using always the plane-wave steering vectors a t (θ) and a r (θ) defined by (2.27) and (2.28) respectively. This allows us to see the error generated by using the plane-wave approximation for different distances R.

The first simulation was performed for three different SNR levels (-10, 0, and 10 dB) considering a transmitter and a receiver antenna arrays of L t = L r = L = 10 elements.

The results are shown in Figure 2.11. As expected, the error is high for short distances R since the target is in the spherical-wave region, and it decreases as the distance becomes larger. We will consider that the target is in the plane-wave region once the MSE becomes stable, which means that the error introduced by the plane-wave approximation is no longer visible. From the MSE curves obtained with an SNR of 10 dB, the frequently used condition R = 2∆ 2 /λ does not hold for any of the three DOA estimation techniques, and the error actually stabilizes at R ≈ 5∆ 2 /λ, with ∆ = (L -1)d. However, for the case of lower SNR values the transition between the spherical-wave and plane-wave regions is less visible and the condition R > 2∆ 2 /λ is acceptable. If the DOA estimation techniques are performed using the spherical-wave steering vectors instead, the MSE should be relatively constant for all distances R (for a fixed SNR). This is illustrated by proving that the error due to the plane-wave approximation can be neglected.

R = 2∆ 2 λ R = 5∆ 2 λ (c)
A second simulation was performed to observe the evolution of the plane-wave condition as a function of the size of the antenna arrays. The simulation was done for a fixed SNR of 10 dB and three different antenna sizes (5, 10, and 15 elements). The results are shown in Figure 2.13. We can clearly verify that the plane-wave condition is dependent of the size of the antenna arrays. To consider that a target is located in the plane-wave region, the distance between the target and the center of the array must be all the more important as the arrays are large.

θ L = 5 L = 10 L = 15 R = 5∆ 2 λ L=5 R = 5∆ 2 λ L=10 (
θ L = 5 L = 10 L = 15 R = 5∆ 2 λ L=10 R = 5∆ 2 λ L=5 (b)
θ L = 5 L = 10 L = 15 R = 5∆ 2 λ L=10 R = 5∆ 2 λ L=5 (c)

Summary

The signal model of narrowband MIMO radar with colocated antennas was presented, followed by the description of the Capon, MUSIC, and GLRT methods as DOA estimation techniques of non-moving targets. The simulations showed that MUSIC offers the best angular resolution followed by Capon and GLRT. Because of its high resolution, MUSIC is able to resolve closely spaced targets. As for the GLRT, even though it is not able to detect closely spaced targets, it is the most robust against noise and the less sensitive to the target reflection coefficients among the three techniques. Moreover, the GLRT has the capability of rejecting strong interference or jammers which are uncorrelated with the transmitted signals.

Also, we investigated the limit between the spherical-wave and plane-wave regions from a signal processing point of view in order to determine when a target can be considered to be in the spherical-wave or in the plane-wave region of a MIMO radar. Even though the plane-wave condition R > 2∆ 2 /λ is acceptable in low SNR cases, the plane-wave region is actually farther away and some error might be introduced by a wrong use of the plane-wave approximation. Moreover, the actual SNR is often difficult to determine, so the plane-wave condition R > 5∆ 2 /λ seems to be more appropriate since it applies for low and high SNR levels, and for the three tested antenna sizes (5, 10, and 15 elements).

For other antenna sizes, the same study should be done in order to determine the right plane-wave condition.

Chapter 3

Wideband MIMO Radar with

Colocated Antennas

The use of wideband signals for radar systems has shown to provide many benefits.

According to radar theory, the radar range resolution is inversely proportional to the bandwidth of the transmitted signal [START_REF] Cohen | An overview of high range resolution radar techniques[END_REF], which means that the range resolution can be improved by the use of wideband and UWB signals. Unlike the narrowband case, where the target dimensions are smaller than the range resolution, in the wideband case the target dimensions can be larger than the range resolution and the targets can be modeled as multi-point targets. Then, target recognition can be achieved by exploiting the different short-time target echoes [35][36]. This feature is also useful for detecting several closely spaced targets. Additionally, wideband signals can increase the immunity of radar systems to external narrowband jammers and noise [START_REF] Shirman | Wideband radar (advantages and problems)[END_REF].

Wideband signals can be generated in several ways. The simplest wideband waveform is a pulse whose width is chosen to achieve a desired range resolution. Although reducing the pulse width allows increasing the range resolution, the average power is reduced (for a fixed pulse amplitude) making the targets harder to be detected. This problem might be solved by transmitting high peak-power pulses but this is in general expensive and difficult to achieve in practice. Nevertheless, high range resolution using low peak-power pulses is still attainable through the use of modern pulse compression techniques [START_REF] Mudukutore | Pulse compression for weather radars[END_REF].

Another type of wideband waveforms is the Continuous Wave (CW) signal (which contains many frequency components), such as linear Frequency Modulated (FM) chirps, stepped-frequency waveforms, and random (or pseudo-random) signals [START_REF] Sun | Ultra-wideband technology and random signal radar: An ideal combination[END_REF].

The use of random signals for radar has many advantages. Indeed, they allow transmitting a convenient average power while having a very high range resolution. Moreover, the adaptation of some DOA estimation techniques is presented in Section 3.5 along with a performance comparison.

Wideband Signal Model

In the narrowband case, the time delays in the signal can be neglected (as in (2.13))

and only the phase shifts in the propagating waves are taken into account. This approximation no longer holds in the case of wideband signals, and therefore a new signal model must be introduced. Consider a MIMO system configuration where the L t transmitting and L r receiving antennas are colocated (see Figure 2.2). In the case of one target located at (θ, R), the received baseband signal at the l th antenna element of the receiver array, derived from the electric-field model (2.10), is given by

x l (t) = µ 0 βe -jk T l r l 4π r l h r,l (u r l , t) * Lt-1 i=0 h t,i (u r i , t) r i * c i (t -τ i -τ l ) e j(2πfct-k T i r i ) e -j2πfct +z l (t) l = 0, ..., L r -1, (3.1)
where z(t) is the unmodelled interference and noise received by the l th antenna element.

Given that

τ i = k T i r i 2πfc and τ l = k T l r l 2πfc
, Equation (3.1) can be written as

x l (t) = µ 0 βe -jk T l r l 4π r l h r,l (u r l , t) * Lt-1 i=0 h t,i (u r i , t) r i * c i t - k T i r i 2πf c - k T l r l 2πf c e j(2πfct-k T i r i ) ×e -j2πfct + z l (t) l = 0, ..., L r -1. (3.2)
Assume that the signals c i (t) are wideband with a spectral support included in the interval -B 2 , B 2 . Then, Equation (3.2) can be written in frequency domain, after applying the Fourier transform, as

X l (f ) = µ 0 βe -jk T l r l 1+ f fc 4π r l H r,l (u r l , f c + f ) Lt-1 i=0 1 r i H t,i (u r i , f c + f ) e -jk T i r i 1+ f fc C i (f ) +Z l (f ), (3.3)
where

C i (f ), X l (f ), Z l (f ), H r,l (u r l , f ), and H t,i (u r i , f ) are the Fourier Transforms of c i (t), x l (t), z l (f ), h r,l (u r l , t
), and h t,i (u r i , t) respectively, e.g.

C i (f ) = ∞ -∞ c i (t)e -j2πf t dt. (3.4)
Given than the transfer functions H r,l (u r l , f ) and H t,i (u r i , f ) are proportional to the antenna radiation patterns [START_REF] Licul | Unified frequency and time-domain antenna modeling and characterization[END_REF], (3.3) can be expressed as

X l (f ) = αβ 1 r l e -jk T l r l 1+ f fc g r,l (θ, f c + f ) Lt-1 i=0 1 r i g t,i (θ, f c + f ) e -jk T i r i 1+ f fc C i (f ) +Z l (f ), (3.5)
where α is a proportionality constant and g r,l (θ, f c +f ) and g t,i (θ, f c +f ) are respectively the frequency-dependent radiation patterns of the l th receiver and the i th transmitter antenna elements.

The baseband signal model at frequency f can then be written in vector notation as

X(f ) = αβa * r (θ, R, f ) a H t (θ, R, f ) C(f ) + Z(f ), (3.6)
where

X(f ) = X 0 (f ) • • • X Lr-1 (f ) T , C(f ) = C 0 (f ) • • • C Lt-1 (f ) T , Z(f ) = Z 0 (f ) • • • Z Lr-1 (f ) T , and
a t (θ, R, f ) =        g * t,0 (θ, f c + f ) 1 r 0 e jk T 0 r 0 1+ f fc . . . g * t,Lt-1 (θ, f c + f ) 1 r Lt-1 e jk T L t -1 r L t -1 1+ f fc        (3.7)
and

a r (θ, R, f ) =        g * r,0 (θ, f c + f ) 1 r 0 e jk T 0 r 0 1+ f fc . . . g * r,Lr-1 (θ, f c + f ) 1 r Lr-1 e jk T Lr -1 r Lr -1 1+ f fc        (3.8)
are the wideband transmit and receive steering vectors respectively. Note that these steering vectors are frequency-dependent. Hence, in the spherical-wave case, the DOA estimation techniques should be performed using a three-dimensional grid of parameters [θ, R, f ] which would require an important processing time. In this chapter, we will only consider the plane-wave case and the DOA estimation techniques will be performed by a two-dimensional search over parameters [θ, f ].

In the plane-wave case, the dot product k T i r i is given by (see Section 2.1.2)

k T i r i = 2πf c v D -i - L t -1 2 d t sin θ + 2πf c v H cos θ i = 0, . . . , L t -1, (3.9)
where v is the wave propagation speed. A similar expression can be obtained for the dot product k T l r l . Hence, by developing the dot products in (3.7) and (3.8), the plane-wave wideband steering vectors can be expressed as

a t (θ, f ) = 1 R e j 2π(fc+f ) v (D sin θ+H cos θ) g * t,i (θ, f c + f )e j2π(fc+f )( L t -1 2 -i) d t sin θ v i=0,...,Lt-1 , (3.10) a r (θ, f ) = 1 R e j 2π(fc+f ) v (D sin θ+H cos θ) g * r,l (θ, f c + f )e j2π(fc+f )( Lr -1 2 -l) dr sin θ v l=0,...,Lr-1 . (3.11)
Then, omitting what is common to every element of the arrays, the plane-wave wideband steering vectors are given by

a t (θ, f ) = g * t,i (θ, f c + f )e j2π(fc+f )( L t -1 2 -i) d t sin θ v i=0,...,Lt-1 , (3.12)
and

a r (θ, f ) = g * r,l (θ, f c + f )e j2π(fc+f )( Lr -1 2 -l) dr sin θ v l=0,...,Lr-1 . (3.13)
The wideband signal model in the case of K targets located in the plane-wave region is then expressed in frequency domain as

X(f ) = K k=1 β k a * r (θ k , f )a H t (θ k , f )C(f ) + Z(f ). (3.14)
Note that the constant α has been omitted for notational simplicity.

Before applying digital signal processing to the received signals, it is convenient to consider the sampled baseband signals

x l (n) x l t = n Fs n = 0, . . . , N -1, l = 0, . . . , L r -1. (3.15)
Also, the DOA estimation techniques must be performed over the whole frequency band using in practice a grid of discrete frequencies. Hence, it is suitable to consider the

Discrete Fourier Transform (DFT) of {x l (n)} N -1 n=0 X l (p) = N -1 n=0 x l (n)e -j2π n N (p-N 2 )
p = 0, . . . , N -1.

(3.16)

Accordingly, Equation (3.14) can be written in discrete-frequency domain as

X(p) ≈ K k=1 β k a * r (θ k , p)a H t (θ k , p)C(p) + Z(p) p = 0, . . . , N -1, (3.17)
with

a t (θ k , p) = g * t,i (θ k , f c + f p )e j2π(fc+fp)( L t -1 2 -i) d t sin θ k v i=0,...,Lt-1 (3.18) and a r (θ k , p) = g * r,l (θ k , f c + f p )e j2π(fc+fp)( Lr -1 2 -l) dr sin θ k v l=0,...,Lr-1 , (3.19)
where

f p = pF s N - F s 2 , (3.20)
and

X(p) = X 0 (p) • • • X Lr-1 (p) T , C(p) = C 0 (p) • • • C Lt-1 (p) T , and
Z(p) = Z 0 (p) • • • Z Lr-1 (p)
T are the DFT (element-wise) of x(n), c(n), and z(n)

respectively. Note that the use of the DFT makes (3.14) and (3.17) only approximately equivalent.

In this chapter, the radiation patterns of the antenna elements are assumed to be identical, angle-independent and of unity gain within the working frequency band.

Accordingly, the plane-wave steering vectors are finally given by In all the simulations presented in this chapter, we consider a wideband MIMO radar with colocated antennas whose transmitter and receiver arrays are two ULA of L t = L r = L = 10 elements, and the inter-element spacings are set to half the minimum wavelength, i.e.

a t (θ k , p) = e j2π(fc+fp)( L t -1 2 -i) d t sin θ k v i=0,...,Lt-1 (3.
d t = d r = d = v/(2(f c + F s /2)). Both arrays operate at a carrier frequency f c = 1 GHz. The signals {c(n)} N -1 n=0 of N = 512 symbols, are normalized such that Rc i,i = 1 L i = 0, . . . , L -1, (3.23)
where Rc i,i denotes the (i, i) th element of Rc . Note that the normalization implies that the mean power of the signal transmitted by every antenna element equals 1/L.

Existing Waveform Design Techniques

Waveform design is one of the most attractive research topics about MIMO radars. In fact, the capability of transmitting different arbitrary waveforms by every element of the array allows having great flexibility when trying to transmit a desired beampattern. The waveform design problem can be addressed in several ways: The set of waveforms can be chosen in order to obtain a desirable MIMO ambiguity function to improve the angular and range resolution [START_REF] Chen | MIMO radar ambiguity properties and optimization using frequency-hopping waveforms[END_REF]; alternatively, by having previous knowledge on the target and clutter statistics, the waveforms can be adapted to the target signatures to improve the detection and/or interference rejection performance [START_REF] Friedlander | Waveform design for MIMO radars[END_REF]; also, the waveforms can be synthesized by designing a covariance matrix such that the power is transmitted to a desired range of angles, which has shown to improve the DOA estimation performance in the narrowband case [START_REF] Stoica | On probing signal design for MIMO radar[END_REF].

In this section, two recently proposed wideband waveform design techniques are described followed by the proposition of a multiband waveform design which satisfies a low Peak-to-Average Power Ratio (PAPR) constraint.

Wideband Beampattern Formation via Iterative Techniques (WBFIT)

The Wideband Beampattern Formation via Iterative Techniques (WBFIT) algorithm was proposed in [START_REF] He | Wideband MIMO systems: Signal design for transmit beampattern synthesis[END_REF] to design low PAPR sequences for transmit beampattern synthesis in wideband MIMO systems. The signals to be transmitted are designed such that their DFT are the solution of a beampattern matching problem.

Considering that the transmitted signals at angle θ are given, in discrete-frequency

domain, by C(θ, p) = a H t (θ, p)C(p) p = 0, . . . , N -1, (3.24)
the spatial power distribution at frequency f p + f c is given by

P (θ, f p + f c ) = | C(θ, p)| 2 = |a H t (θ, p)C(p)| 2 . ( 3.25)
A waveform design problem can then be formulated with the goal of designing a set

of signals {c(n)} N -1 n=0 such that the power distribution {P (θ, f p + f c )} N -1
p=0 matches a desired beampattern. Letting δ(θ h , p) denote the desired beampattern, where {θ h } H h=1 represents a grid of angles covering the interval [-90 • , 90 • ], the beampattern matching problem can be formulated as

min {c(n)} H h=1 N -1 p=0 δ(θ h , p) -a H t (θ h , p)C(p) 2 s.t. PAPR {c i (n)} N -1 n=0 ≤ , i = 0, . . . , L t -1, (3.26)
where the PAPR of the i th signal {c i (n)} N -1 n=0 is defined as

PAPR {c i (n)} N -1 n=0 = max n |c i (n)| 2 1 N N -1 n=0 |c i (n)| 2 , ( 3.27)
and ≥ 1 is a predefined threshold.

The authors propose in [START_REF] He | Wideband MIMO systems: Signal design for transmit beampattern synthesis[END_REF] a two-stage design approach to solve the optimization problem (3.26). In stage 1, the problem is solved with respect to {C(p)} N -1 p=0 , which are considered to be general complex vectors. In stage 2, the DFT of {c(n)} N -1 n=0 is fitted to the obtained {C(p)} N -1 p=0 subject to the PAPR constraint. Both stages are described in Appendix B.

Spectral Density Focusing Beampattern Synthesis Technique (SFBT)

The Spectral Density Focusing Beampattern Synthesis Technique (SFBT) is an approach proposed in [START_REF] Jardin | On wideband MIMO radar: Extended signal model and spectral beampattern design[END_REF] to transmit the power directly to the targets of interest. It follows from (3.25) that we can transmit the power in a direction θ 0 by choosing a signal whose DFT is

C(p) = a t (θ 0 , p) p = 0, . . . , N -1.
(3.28)

Then, the beampattern is given by

P (θ, f p + f c ) = a H t (θ, p)a t (θ 0 , p) 2 = Lt-1 i=0 e j2π(fc+fp)( L t -1 2 -i) d t (sin θ 0 -sin θ) v 2 . (3.29)
It can be noted that the beampattern will have a maximum for θ = θ 0 . Then, the signals to be transmitted can be obtained by performing the Inverse Discrete Fourier Transform

(IDFT) of {C(p)} N -1 p=0 , i.e. c i (n) = 1 N N -1 p=0 C i (p)e j2π n N (p-N 2 ) n = 0, . . . , N -1, i = 0, . . . , L t -1. (3.30)
The signals resulting from (3.30) are given by

c i (n) = 1 N e j2πfcu i e -j π N (u i Fs+n) sin (π(u i F s + n)) sin π N (u i F s + n) n = 0, . . . , N -1, i = 0, . . . , L t -1, (3.31)
where p=0 . This method is not optimal since it is not the solution of an optimization problem such as (3.26). However, the beampattern created by the signals obtained in (3.33) will present maxima around the directions { θk } K k=1 .

u i = L t -1 2 -i d t sin θ 0 v . (3.32) Note that each signal {c i (n)} N -1 n=0 is a phase-shifted

Comparison Between WBFIT and SFBT

The previously described waveform design techniques, WBFIT and SFBT, are compared in this section via several MATLAB simulations.

Mono-Target Case

Consider that we have an initial DOA estimate θ = 40 • of one target (K = 1) located in the plane-wave region. The WBFIT is used to synthesize a set of signals {c(n)} N -1 n=0 such that the power is directly transmitted to the target by defining the following desired beampattern:

δ(θ, p) = 1, θ = θ 0, other θ p = 0, . . . , N -1.
(3.34)

Note that the beampattern is aimed to have a main lobe at 40 • over the whole range of Nevertheless, a maximum of power is still transmitted in the desired direction θ while every sequence {c i (n)} N -1 n=0 perfectly meets the PAPR constraint (see Figure 3.5).

discrete frequencies f c -Fs 2 , f c + Fs 2 -Fs N .
A transmit beampattern is also synthesized by the SFBT. The set of signals {c

(n)} N -1 n=0
is choosen such that the power is transmitted in the direction θ by calculating the IDFT (element-wise) of (3.33). As shown in Figure 3.6, the beampattern synthesized by the SFBT is very smooth, similar to the WBFIT beampattern of Figure 3.2. However, we must note that the mainlobe is wider at low frequencies. Actually, in the case of one target at θ, the SFBT beampattern is given by

P (θ,

f p + f c ) = a H t (θ, p)a t ( θ, p)e jφ(p) 2 =   sin πL t (f c + f p ) dt(sin θ-sin θ) v sin π(f c + f p ) dt(sin θ-sin θ) v   2 p = 0, . . . , N -1, (3.35)
which implies that the mainlobe will be wider as f p decreases. On the other hand, the design of the {c(n)} N -1 n=0 by the SFBT can lead to high PAPR signals, as shown in Figure 3.7, which are usually undesirable since either they require a large range of dynamic linearity of the circuits resulting in high power consumption or they generate spectral regrowth in adjacent channels.

F s = f c /2 = 500 MHz.
We observe that the WBFIT is degraded as the bandwidth becomes large: The mainlobe is centered at θ at high frequencies and is deviated at low frequencies. Moreover, even though a maximum of power can be transmitted around θ, comparable amount of power is transmitted in other directions as shown in Figure 3.9, where we have plotted the WBFIT beampattern at frequency f c (i.e. P (θ, f c )). Indeed, the mainlobe level is only around 1.7 and 3.6 dB higher than the greatest sidelobes. On the other hand, the SFBT works relatively well: The mainlobe has an approximately constant amplitude at θ over the whole range of frequencies. However, the mainlobe becomes wider as the frequency decreases, which is coherent with (3.35). In contrast to WBFIT, SFBT allows transmitting much more power to direction θ than to other directions as shown in Figure 3.11: The mainlobe level is at least 10 dB higher than the sidelobe levels.

Multi-Target Case

Suppose that we have initial estimates of K = 3 target DOAs: θ1 = -60 • , θ2 = 0 • , and θ3 = 40 • .

The desired beampattern used in the WBFIT design is given by The WBFIT ( = 2) and the SFBT beampatterns for F s = f c /5 = 200 MHz are illustrated by Figures 3.12 and 3.13 respectively. We can see that both beampatterns have three high lobes around the targets directions. While the WBFIT beampattern has a smooth shape in almost the whole range of frequencies, the SBFT beampattern is more irregular due to the different random phase associated to every target in (3.33).

δ(θ, p) = 1, θ = θk , k = 1, . . . , K 0
However, if the bandwidth is increased, the WBFIT may be highly deteriorated. This is the case for F s = f c /2 = 500 MHz, as shown in Figure 3.14. We can see that the lobes are deformed as |θ| becomes large; moreover, a parasite lobe appears around -40 • .

As for the SFBT, the beampattern shown in Figure 3.15 is not deformed and allows transmitting the power directly to the three targets. As expected, the lobes are wider at low frequencies.

The WBFIT allows designing low PAPR sequences while matching a desired beampattern. As discussed before, the WBFIT works very well for relatively small bandwidths, such as F s = f c /5, but its performance is seriously degraded as the bandwidth increases. On the other hand, even though the SFBT does not provide low PAPR signals, it works relatively well in both small and large bandwidth cases.

Moreover, since the SFBT is not an iterative process, it is much faster to compute than WBFIT and hence more suitable for real-time applications. In the previous simulations, the SFBT was at least 2000 times faster than WBFIT.

Proposition of a Multiband Waveform Design

In the MIMO radar context, every target reflects the superposition of all of the transmitted signals, and hence some correlation may exist between the reflected

Principle of the Multiband Spectral Focusing Beampattern

Synthesis Technique (M-SFBT)

So far SFBT allows transmitting the power directly to the targets of interest in the whole range of frequencies. However, we could exploit the frequency diversity to have more flexible beampatterns using several frequency bands. The use of multiple bands has other benefits, such as the reception of uncorrelated target echoes.

In order to make the signals reflected by the targets uncorrelated, we proposed in [START_REF] Gómez | Multiband waveform synthesis and detection for a wideband MIMO radar[END_REF] the Multiband Spectral Focusing Beampattern synthesis Technique (M-SFBT) which is a modified version of the original SFBT. The M-SFBT consists in allocating one or more non-overlapping frequency bands to each target. Letting δ k (p) denote the desired power distribution allocated to the k th target, the set of signals to be transmitted can be constructed in frequency domain as

C(p) = K k=1 a t ( θk , p)δ k (p)e jφ k (p) p = 0, . . . , N -1.
(3.37)

Then, the signals in time domain {c(n)} N -1 n=0 can be found by performing the IDFT (element-wise) of (3.37). Note that such signals may have a relatively high PAPR (as shown in Section 3.3.3 in the case of SFBT) which is usually undesirable.

A low PAPR can be achieved by adding a constraint in the design of the waveforms, following the same idea as WBFIT. The set of signals {c (n)} N -1 n=0 of low PAPR can then be designed by solving the following minimization problem:

min {c i (n)} N -1 n=0 c i (n) -c i (n) 2 s.t. PAPR {c i (n)} N -1 n=0 ≤ i = 0, . . . , L t -1. (3.38)
This problem can be easily solved by performing the "nearest-vector" algorithm described in [START_REF] Tropp | Designing structured tight frames via an alternating projection method[END_REF] and recalled in Appendix B.

Numerical Examples

Assume that there are K = 3 targets located in the plane-wave region and that we have initial DOA estimates θ1 = -60 • , θ2 = 0 • , and θ3 = 40 • . Then, the signals to be transmitted can be synthesized via M-SFBT with a suitable choice of {δ k (p)} in order to generate a multiband beampattern. A simple choice of {δ k (p)} is to allocate a single frequency band to each target as where f ck and B k are respectively the center frequency and the bandwidth of the band allocated to the k th target. In this example we use non-overlapping frequency bands so that the signals reflected by the targets are uncorrelated. In the following simulations, the {δ k (p)} are chosen as

δ k (p) = 1 for f ck -B k 2 ≤ f p ≤ f ck + B k 2 0 otherwise p = 0, . . . , N -1 k = 1, . . . , K, (3.39
δ k (p) = 1 for (k -1) N K ≤ p ≤ k N K -1 0 otherwise p = 0, . . . , N -1 k = 1, . . . , K, (3.40)
where x denotes the largest integer less or equal to x.

The signals {C(p)} N -1 p=0 are then generated in frequency domain using (3.37) and the corresponding beampattern P (θ,

f p + f c ) = a H t (θ, p)C(p)
2 is shown in Figure 3.16 for

F s = f c /5 = 200 MHz.
We can see that the power is directly transmitted to the targets using a different frequency band for each one. As expected, the signals {c(n)} N -1 n=0 have relatively high PAPR with values varying from 5.8 to 9.2 (see Figure 3.17). Let us impose the PAPR constraint = 2. The set of signals {c (n)} N -1 n=0 satisfying the PAPR constraint is then found by solving the problem (3.38). The new beampattern is given by P (θ,

f p + f c ) = a H t (θ, p)C (p) 2 , where {C (p)} N -1 n=0 is the DFT (element-wise) of {c (n)} N -1
n=0 . As shown in Figure 3.18 the beampattern now has an uneven aspect but the spectral and spatial distribution of the new waveforms {c (n)} N -1 n=0 are close to those of the initial {c(n)} N -1 n=0 , i.e. the power is transmitted to the targets while keeping the desired frequency allocation. Moreover, the new waveforms perfectly meet the PAPR constraint as shown in Figure 3 We must note that a multiband beampattern can also be synthesized by WBFIT. Similar to (3.40), the WBFIT desired beampattern is chosen as

δ(θ, p) = 1 for (k -1) N K ≤ p ≤ k N K -1 and θ = θk , k = 1, . . . , K 0 otherwise p = 0, . . . , N -1, (3.41)

Wideband Direction-of-Arrival Estimation Techniques

To date, most of the DOA estimation techniques for MIMO radar have been proposed

in the case of narrowband signals as described in Chapter 2. However, given that the approximations done in the narrowband model no longer hold with wideband signals, those techniques cannot be directly applied in the wideband context. In this section we propose the adaptation of some wideband array processing techniques to the context of wideband MIMO radar.

Wideband Array Processing Techniques

In wideband array processing, a variety of techniques have been proposed in the literature to estimate the DOAs of wideband sources. In most of the cases, the received signals are decomposed in several narrowband components using the DFT, leading to the following signal model:

X(p) ≈ K k=1 a * r (θ k , p)S k (p) + Z(p) p = 0, . . . , N -1, (3.42)
where

{S k (p)} N -1 p=0
is the DFT of the k th source signal. Note that taking

S k (p) = β k a H t (θ k , p)C(p)
, the signal model (3.42) becomes equivalent to the MIMO radar signal model (3.17), which means that, in a MIMO radar context, the sources to be detected are the targets reflecting the signals to the receiver array. Thus, many source DOA estimation techniques can be applied in the context of MIMO radar. Some of these techniques are described thereafter.

Incoherent Methods

A low-complexity method consists in applying narrowband DOA estimation techniques, such as Capon or MUSIC, at every frequency component of the signal model (3.42). Some form of averaging procedure is then used to combine the individual results and obtain the final DOA estimates [START_REF] Wax | Spatio-temporal spectral analysis by eigenstructure methods[END_REF] [START_REF] Pham | Adaptive wideband aeroacoustic array processing[END_REF]. These methods are said to be incoherent since they do not combine the results obtained at every frequency bin in a coherent way. Although the incoherent methods are simple to implement and effective in high SNR cases, their performance is deteriorated in low SNR cases or when the SNR is not constant in the whole frequency band of interest. Indeed, the DOA estimates can be very bad at some frequencies, and the averaging procedure can also lead to inaccurate estimates.

Coherent Methods

A more sophisticated method, called Coherent Signal Subspace Method (CSSM), exploits the fact that the signal subspace at a given frequency is different from that at another frequency [START_REF] Wang | Coherent signal-subspace processing for the detection and estimation of angles of arrival of multiple wide-band sources[END_REF]. Actually, the signal-plus-noise subspaces at different frequencies can be coherently combined to form a single signal subspace which can be used to estimate the DOAs of wideband sources. This is achieved by using a focusing procedure: The covariance matrices of the received signals at several frequency bins are transformed into a general covariance matrix at a reference frequency f ref by using different transformation (or focusing) matrices. The reference frequency, which is often chosen to be equal to the center frequency, can be optimally selected by the method presented in [START_REF] Valaee | The optimal focusing subspace for coherent signal subspace processing[END_REF]. The general covariance matrix at frequency f ref is then given by [START_REF] Wang | Coherent signal-subspace processing for the detection and estimation of angles of arrival of multiple wide-band sources[END_REF]

R(f ref ) = N -1 p=0 α p Φ(f p ) Rx (f p )Φ H (f p ), (3.43)
where α p is a weighting coefficient proportional to the SNR in the p th frequency bin, Rx (f p ) is the estimated covariance matrix of the received signals at frequency f p , and Φ(f p ) is the focusing matrix which is able to transform the signal-plus-noise subspace at frequency f p to the corresponding subspace at frequency

f ref . The so-obtained R(f ref )
can then be used along with subspace-based methods, such as MUSIC, to estimate the source DOAs. A method to estimate the covariance matrices is presented in Appendix B.

The focusing matrices can be found by solving the following optimization problems [START_REF] Hung | Focussing matrices for coherent signal-subspace processing[END_REF]:

min Φ(fp) Â(f ref ) -Φ(f p ) Â(f p ) F s.t. Φ(f p ) H Φ(f p ) = I p = 0, . . . , N -1 (3.44) with Â(f p ) = a * r ( θ1 , p) • • • a * r ( θK , p) , (3.45)
where the { θk } K k=1 are initial estimates of the source DOAs (also called focusing angles), I is the L r × L r identity matrix, and • F denotes the Frobenius norm [START_REF] Horn | Matrix Analysis[END_REF] defined as

A F = tr A H A 1/2 .
The CSSM has shown to have better performance than the incoherent methods in low SNR cases [47][48]. However, it requires initial focusing angles which must be very close to the true DOAs to form the focusing matrices, and its estimation performance is sensitive to those initial values even in high SNR cases [START_REF] Swingler | Source location bias in the coherently focused highresolution broad-band beamformer[END_REF].

Test of Orthogonality of Projected Subspaces (TOPS)

The Test of Orthogonality of Projected Subspaces (TOPS) was proposed in [START_REF] Yoon | TOPS: New DOA estimator for wideband signals[END_REF] as an alternative to the CSSM to estimate the DOAs of uncorrelated sources. TOPS does not suffer from bias at high SNR and it has an improved performance at low SNR compared to the incoherent methods. Even though TOPS also uses transformation matrices to exploit multiple frequency components, it does not require the use of initial focusing angles. In fact, the signal subspace at a reference frequency f ref is transformed over all of the other frequencies for every hypothesized angle θ. Then, an orthogonality test between the transformed signal-plus-noise subspaces and the noise subspaces is performed at every frequency component. The orthogonality will be preserved only when an hypothesized angle θ corresponds to a true DOA.

TOPS uses diagonal transformation matrices of the form

Φ(∆f p , θ) = diag e j2π(fc+∆fp)( Lr -1 2 -l) dr sin θ v l=0,...,Lr-1 (3.46)
where

∆f p = f p -f ref . (3.47)
It is easy to see that the matrix Φ(∆f p , θ) allows transforming the steering vector (3.48)

Similarly, the transformation matrix can be used to transform the signal-plus-noise subspace at the reference frequency U s (f ref ) into the signal-plus-noise subspace at any other frequency by performing the following operation [START_REF] Yoon | TOPS: New DOA estimator for wideband signals[END_REF]:

F s (f p ) = Φ(∆f p , θ)U s (f ref ) p = 0, . . . , N -1, (3.49)
where F s (f p ) is the transformed signal-plus-noise subspace at frequency f p . Note that the different signal-plus-noise and noise subspaces at every frequency can be obtained by eigenvalue decomposition of the covariance matrices of the received signals { Rx (f p )} N -1 p=0 . The quality of the estimated subspaces depends on the estimation of the covariance matrices, which is linked to the SNR and the number of samples. However, it is possible to reduce some errors by projecting the transformed signal-plus-noise subspaces onto the null space of a r (θ, p) (see [START_REF] Yoon | TOPS: New DOA estimator for wideband signals[END_REF] for details). By defining the projection matrix P(θ, f p ) as

P(θ, f p ) = I -a T r (θ, p)a * r (θ, p) -1 a * r (θ, p)a T r (θ, p) p = 0, . . . , N -1, (3.50)
the projected signal-plus-noise subspaces are given by

F s (θ, f p ) = P(θ, f p )F s (f p ) p = 0, . . . , N -1. (3.51)
Given that the signal-plus-noise and the noise subspaces are in theory orthogonal (as described in Section 2.2.2), an orthogonality test can be performed at every hypothesized angle θ for every frequency f p . This is done by defining a matrix D(θ) as

D(θ) = F H s (θ, f 0 )U n (f 0 ) • • • F H s (θ, f N -1 )U n (f N -1 ) , (3.52)
where U n (f p ) is the noise subspace at frequency f p . Note that the matrix D(θ) should become rank deficient when θ corresponds to a true DOA. In practice, the matrix D(θ)

might not be rank deficient but close to singular. Hence, the source DOAs can be found by searching for the maxima of the TOPS spatial spectrum

P T OP S (θ) = 1 σ min (θ) , (3.53)
where σ min is the smallest singular value of D(θ).

Incoherent Techniques for MIMO Radar

Similar to the incoherent methods used in wideband array processing, many narrowband detection techniques can be performed to estimate the target DOAs in the context of wideband MIMO radar. Indeed, since the signal model (3.17

The Wideband Capon Beamformer

In the case of wideband signals, Capon spatial filters can be obtained at every frequency component by solving N optimization problems

min w w H (θ, f p )R x (f p )w(θ, f p ) s.t. w H (θ, f p )a * r (θ, p) = 1 p = 0, . . . , N -1, (3.54)
where R x (f p ) is the covariance matrix of the received signals at frequency f p .

As presented in Section 2.2.1 (see also Appendix A), the resolution of every problem in (3.54) leads to the following narrowband Capon spatial spectra

P cap (θ, f p ) = 1 a T r (θ, p)R -1 x (f p )a * r (θ, p) p = 0, . . . , N -1. (3.55)
In order to have a general Capon spectrum, we propose to average the results obtained in every frequency bin as follows:

Pcap (θ) = 1 N N -1 p=0 P cap (θ, f p ). (3.56)
The target DOAs can then be estimated by searching for the maxima of Pcap (θ).

The Wideband MUSIC Algorithm

In Section 2.2.2 we described the narrowband MUSIC algorithm, which exploits the fact that the noise eigenvectors are orthogonal to the receive steering vectors a r (θ) when θ corresponds to a true DOA (see (2.44)). This orthogonality also holds true in the wideband case, i.e.

a T r (θ, p)u i (f p ) = 0 i = K + 1, . . . , L r p = 0, . . . , N -1, (3.57)
where {u i (f p )} Lr i=1 are the eigenvectors of R x (f p ). Then, a MUSIC spatial spectrum can be calculated at every frequency component as

P M U SIC (θ, f p ) = 1 a T r (θ, p)U n (f p )U H n (f p )a * r (θ, p) . (3.58)
Finally, a general wideband MUSIC spatial spectrum can be calculated as [START_REF] Yoon | TOPS: New DOA estimator for wideband signals[END_REF] PMUSIC (θ) = 1

N -1 p=0 a T r (θ, p)U n (f p )U H n (f p )a * r (θ, p) . (3.59)
The target DOAs can be found by searching for the maxima of PMUSIC (θ).

The Wideband GLRT Technique

The narrowband GLR at every frequency f p is given by

ρ(θ, f p ) = 1 - a T r (θ, p) Rx (f p ) -1 a * r (θ, p) a T r (θ, p) Q-1 (f p )a * r (θ, p) p = 0, . . . , N -1, (3.60)
where Q(f p ) is defined as The GLR obtained at every frequency can be averaged as

Q(f p ) = Rx (f p ) - Rxc (f p )a t (θ, p)a H t (θ, p) RH xc (f p ) a H t (θ, p) Rc (f p )a t (θ, p) , (3.61)
ρ(θ) = 1 N N -1 p=0 ρ(θ, f p ), (3.62)
and the target DOAs can be estimated by searching for the maxima of ρ(θ).

Numerical Examples

Simulations are performed as described in Section 3. In order to evaluate the DOA estimation performance, the MSE for the target at -30 • is computed using 500 Monte Carlo trials for both the omnidirectional and the SFBT stage. after the SFBT stage. In contrast, the resolution of GLRT is significantly reduced after transmitting the SFBT beampattern, i.e. the lobes are spatially larger. This is due to the non-orthogonality of the signals synthesized by SFBT. Indeed, the best spatial resolution can be obtained after transmitting orthogonal signals [START_REF] Bekkerman | Target detection and localization using MIMO radars and sonars[END_REF]. Even though the signals transmitted in the omnidirectional stage are not perfectly orthogonal, the use of independent sequences will always offer the best spatial resolution.

Omnidirectional Stage

After the SFBT stage, the three techniques have similar DOA estimation performance at high noise levels (-10 log 10 σ 2 ≤ 5), as shown in Figure 3.26. However, we must notice that MUSIC outperforms both Capon and GLRT at low noise levels (-10 log 10 σ 2 ≥ 10).

Comparison of Omnidirectional Probing and SFBT Performance

The concentration of the power in the target directions using SFBT allows improving the DOA estimation performance. This is illustrated by the MSE curves shown in Figure 3.27 for the target at -30 • . As we can see, the estimation performance of Capon and MUSIC are significantly improved after the SFBT stage. As for GLRT, despite the loss in resolution after the SFBT stage, the DOA estimation performance is slightly improved compared to the omnidirectional stage.

Adaptation of TOPS to the MIMO Radar Context

The TOPS technique was originally developed to estimate the DOAs of uncorrelated sources. In the case of MIMO radar, the signals reflected by the targets can be partially correlated, which can degrade the performance of TOPS. This can be observed by applying TOPS as described in Section 3.5.1.3 after transmitting an omnidirectional pattern.

In order to successfully apply TOPS in the context of MIMO radar, the signals to be transmitted can be synthesized using a multiband beampattern as described in Section 3.4. TOPS must then be performed in every frequency band, where in each case the reference frequency will be the center frequency of the corresponding band.

Accordingly, we derive thereafter a general expression for the use of TOPS in multiple frequency bands, called M-TOPS.

Description of M-TOPS

Let define the transformed signal-plus-noise subspace of the k th band at frequency f p as

F s k (f p ) = Φ(∆f p k , θ)U s (f ck ) p = 0, . . . , N -1, (3.63)
where

∆f p k = f p -f ck , (3.64)
and U s (f ck ) is the signal-plus-noise subspace at the center frequency of the k th band f ck . The projected signal-plus-noise subspaces of each frequency band are then given by

F s k (θ, f p ) = P(θ, f p )F s k (f p ) p = 0, . . . , N -1, (3.65)
with P(θ, f p ) defined in (3.50). Then, a test of orthogonality must be performed for every frequency band by defining K matrices D k (θ) as

D k (θ) = F H s k (θ, f 0 )U n (f 0 ) • • • F H s k (θ, f N -1 )U n (f N -1 ) . (3.66)
Note that every matrix D k (θ) will be close to singular when the hypothesized θ corresponds to a true DOA. Therefore, the target DOAs can be estimated by searching for the maxima of

P M -T OP S (θ) = max 1 σ k (θ) K k=1 , (3.67)
where σ k (θ) is the smallest singular value of D k (θ).

Numerical Examples

Consider K = 3 targets located in the plane-wave region at θ 1 = -30 • , θ 2 = 0 • , and

θ 3 = 60 • with reflection coefficients {β k } K k=1 equal to 1.
Given that we do not assume any prior knowledge on the target DOAs, an initial We can see that now only the peaks corresponding to the target DOAs appear in the spectrum. However, the incoherent methods fail when the SNR at each frequency varies, as discussed in [START_REF] Yoon | TOPS: New DOA estimator for wideband signals[END_REF]. In that case, the use of alternative methods such as M-TOPS might be convenient. Chapter 4

Summary

Effects of Mutual Coupling on MIMO Radar Performance

The signal models of MIMO radar for both the narrowband and the wideband cases have been presented in Chapters 2 and 3 respectively. While these models are general, some assumptions were done to simplify the development and simulation of the detection techniques and waveform design algorithms. As it is usually done in the literature, the antenna elements of the ULAs used for transmission and reception were assumed to have identical characteristics, i.e. equal gain, radiation pattern, and bandwidth among others. However, such characteristics can significantly differ from one element to the other in real antenna arrays due to the existence of mutual coupling: The electromagnetic characteristics of every antenna element are influenced by the neighboring elements. Indeed, due to the proximity between the antenna elements, part of the signal radiated by every single element is received by the surrounding elements, even if they are all transmitting elements [START_REF] Allen | Mutual Coupling in Array Antennas[END_REF]. Moreover, the coupled signal might be re-radiated or scattered. As for the receiver elements, they might reflect part of the incident waves and thus act like small transmitters even if they are supposed to "receive" only.

The mutual coupling in antenna arrays depends on several factors, including the type of antennas, the inter-element spacings, the antenna orientation, the bandwidth, the directivity, and the feeding network among others. The existence of mutual coupling leads to several negative effects on the array performance. The electromagnetic interactions between different antenna elements cause changes (in magnitude and phase) in the current distributions of every antenna element which leads to an alteration of the different input impedances [START_REF] Kelley | Array antenna pattern modeling methods that include mutual coupling effects[END_REF]. This usually produces impedance mismatches at the transmitters, receivers, and transmission lines. Also, in many array configurations, the change in the current distributions produces distortions in the radiation patterns. Actually, the radiation of every antenna element might totally differ from that of an isolated element. This is non-negligible and can significantly affect the DOA estimation performance [START_REF] Gupta | Effect of mutual coupling on the performance of adaptive arrays[END_REF]- [START_REF] Vargas | Mutual coupling effects correction in microstrip arrays for direction-of-arrival (DOA) estimation[END_REF].

Another negative effect is the direct coupling between the transmitter and the receiver elements. In fact, part of the transmitted signals can be directly received by the receiver elements depending on the separation between the transmitter and receiver arrays. Herein, this phenomenon will be referred to as "crosstalk".

The effects of mutual coupling (radiation pattern distortion and crosstalk) in the performance of narrowband MIMO radar with colocated antennas are studied in this chapter. In Section 4.1 the different radiation patterns are taken into account in order to improve the DOA estimation performance in the presence of mutual coupling. In Section 4.2 we propose a crosstalk reduction technique based on a signal processing approach.

Radiation Pattern Distortion due to Mutual Coupling

The effects of mutual coupling on the radiation patterns of antenna arrays have been highly studied in the literature. In [START_REF] Kelly | An adaptive detection algorithm[END_REF], different array modeling methods are presented and compared in the case of phased array systems. The authors also present a pattern prediction method for small and medium-sized arrays of equally spaced elements.

Various approaches for reducing mutual coupling can be found in the literature. In many cases, parasitic structures are added between the antenna elements to reduce the coupled power [START_REF] Wang | Mutual coupling reduction between two conformal microstrip patch antennas[END_REF]- [START_REF] Min | Improved MIMO antenna by mutual coupling suppression between elements[END_REF]. Other methods based on antenna design are presented in [START_REF] Ouyang | Reducing mutual coupling of closely spaced microstrip MIMO antennas for WLAN application[END_REF]- [START_REF] Farsi | Mutual coupling reduction between planar antennas by using a simple microstrip U-section[END_REF]. Also, instead of modifying the antenna structures, the distortions in the radiation patterns can be compensated by designing compensation networks based on the mutual impedances of the antenna elements [START_REF] Niow | Compensate for the coupled radiation patterns of compact transmitting antenna arrays[END_REF] or the scattering parameter (S-parameter) matrix [START_REF] Wallace | Mutual coupling in MIMO wireless systems: A rigorous network theory analysis[END_REF]. This compensation can also be achieved in post-processing by including S-parameter-based compensation matrices in the array processing algorithms [START_REF] Lau | Minimum norm mutual coupling compensation with applications in direction of arrival estimation[END_REF][60][61] [START_REF] Qun | Analysis of DOA and adaptive beam forming including mutual coupling[END_REF].

In this section, we analyze the influence of mutual coupling on the radiation patterns of a narrowband MIMO radar with colocated antennas via electromagnetic simulations.

Unlike the approaches mentioned above, we do not try to compensate or reduce mutual coupling. We show that taking into account the radiation pattern of every antenna element allows reducing the DOA estimation errors without need of using compensation matrices or parasitic structures.

The work persented in this section is the result of a collaboration with Prof. Vincent Fusco at Queen's University Belfast and the Institute of Electronics, Communications and Information Technology (ECIT).

Radiation Pattern of an Isolated Element

Consider the single linearly polarized patch antenna shown in

Taking the Radiation Patterns into Account

The simulations presented in Chapter 2 were performed using the standard steering vectors a t (θ) = e

x(n) = K k=1 β k ã * r (θ k )ã H t (θ k )c(n) + z(n), (4.3)
where The pattern distortions can be taken into account by using the general form of the steering vectors to compute the spatial spectra. We can see that the lobe in the GLRT and MUSIC spectra is now re-centered at the target DOA. In contrast, a high regrowth appears in the Capon spectrum at angles close to -90 • and 90 • . The regrowth is so large that the target lobe can hardly be seen. A similar but less important regrowth also appears in the MUSIC spectrum; however, it does not affect the lobe around -40 • . This regrowth is related to the definition of the Capon and MUSIC spectra, and the weak magnitudes of the radiation patterns at angles close to -90 • and 90 • . Actually, the Capon and MUSIC spectra, computed using the general steering vectors are given by (see Section 2.2)

ãt (θ) = g * t,i (θ)e j 2π λ ( L t -1 2 -i)dt sin θ i=0,...,Lt-1 (4.
P cap (θ) = 1 ãT r (θ)R -1 x ã * r (θ) (4.6)
and Then, the spatial spectra can be computed using α t (θ) and α r (θ) instead of ãt (θ) and ãr (θ) respectively.

P M U SIC (θ) = 1 ãT r (θ)U n U H n ã * r (θ) . ( 4
Consider the same target at -40 • (β = 1) and the same noise level (-10 log 10 σ 2 = -10).

The Capon, MUSIC, and GLRT spectra obtained using the general phase-only steering vectors are shown in Figure 4.12. As we can see, there is no regrowth in any of the spectra given that the magnitudes of the radiation patterns are not included in the steering vectors. Moreover, the lobes are still centered close to the target DOAs. The impact of the different steering vectors on the DOA estimation performance is evaluated in Section 4.1.4. • Ideal case: The signal propagation is simulated without pattern distortion by using the signal model x(n) = βa * r (θ 1 )a H t (θ 1 )c(n) + z(n), and the Capon, MUSIC, and GLRT spectra are computed using the standard steering vectors a r (θ) and a t (θ).

• Standard processing case: The signal propagation is simulated including pattern distortion by using the signal model x(n) = βã * r (θ 1 )ã H t (θ 1 )c(n) + z(n), but the Capon, MUSIC, and GLRT spectra are computed using the standard steering vectors a r (θ) and a t (θ).

• Mutual-Coupling (MC) based processing case: The signal propagation is simulated including pattern distortion by using the signal model

x(n) = βã * r (θ 1 )ã H t (θ 1 )c(n) + z(n)
, and the Capon, MUSIC, and GLRT spectra are computed using the general steering vectors ãr (θ) and ãt (θ).

• Mutual-Coupling (MC) based phase-only processing case: The signal propagation is simulated including pattern distortion by using the signal model

x(n) = βã * r (θ 1 )ã H t (θ 1 )c(n) + z(n)
, and the Capon, MUSIC, and GLRT spectra are computed using the general phase-only steering vectors α r (θ) and α t (θ). The results are shown in Figure 4. [START_REF] Lesturgie | Some relevant applications of MIMO to radar[END_REF]. We can see that the minimum MSE is obtained in the ideal case, however this is an unrealistic simulation given that the signal propagation is modeled assuming that the radiation patterns are all identical and angle independent.

We can also see that important errors are obtained using the standard processing given that pattern distortion is not taken into account in the steering vectors a r (θ) and a t (θ).

In contrast, we can clearly see that the use of the MC-based processing allows reducing the errors introduced by the distortions in the radiation patterns: The MSEs in the DOA estimated by Capon, MUSIC, and GLRT are greatly reduced and approach the ideal MSE curves. We must however note that MC-based method increases the MSE in the DOA estimated by Capon at high noise levels (-10 log 10 σ 2 ≤ -5), which is due to the high regrowth present in the Capon spectrum at high noise levels. Nevertheless, we can see that this problem is solved by using the general phase-only steering vectors to compute the spatial spectra: The MSE in the DOA estimated by Capon is reduced and is always lower than the MSE of the standard processing case. As for MUSIC and GLRT, there is no visible difference between both MC-based cases (the MSE curves are overlapped), which means that excluding the magnitudes of the radiation patterns does not introduce any significant error.

Crosstalk

Besides the distortion of the radiation patterns, the existence of crosstalk is another negative consequence of the small separation between the antenna arrays. In the case of MIMO radar, the signals reflected by the targets and received by the receiver array are corrupted by a part of every transmitted signal which is directly transferred from the transmitter elements to the receiver ones. This can be seen as noise or interference which is correlated with the transmitted signals and can significantly degrade the DOA estimation performance.

Crosstalk Modeling

The signals directly transmitted from the transmitter to the receiver elements can be modeled as a mixture of the set of the transmitted signals. In the case of narrowband signals, the baseband signal received by the l th receiver element (l = 0, . . . , L r -1) due to the signals directly transmitted by the L t transmitter elements is given by

Lt-1 i=0 m l,i c i (n), (4.10)
where m l,i is a complex transmission coefficient between the i th transmitter element and the l th receiver element.

By placing the set of transmission coefficients in a L r × L t crosstalk matrix M, the MIMO radar narrowband signal model in the presence of crosstalk is given by

x(n) = K k=1 β k ã * r (θ k )ã H t (θ k )c(n) + Mc(n) + z(n), (4.11)
where

M =     m 0,0 • • • m 0,Lt-1 . . . . . . . . . m Lr-1,0 • • • m Lr-1,Lt-1     .
(4.12)

Crosstalk Reduction

The crosstalk matrix can be estimated from a first transmission in an environment without any target. In that case, the narrowband received signal is

x(n) = Mc(n) + z(n). (4.13)
We seek to determine the matrix M which minimizes the MSE criterion

J = E x(n) -Mc(n) 2 . (4.14)
By denoting m l the l th row of M and xl (n) the l th element of x(n), the criterion (4.14) can be written as

J = E Lr-1 l=0 |x l (n) -m l c(n)| 2 . (4.15)
Finally, the optimal l th row of M will be the one that minimizes

J l = E |x l (n) -m l c(n)| 2 . (4.16)
The solution of (4.16) is that of a classical Wiener filtering:

ml = E xl (n)c H (n) E c(n)c H (n) -1 . (4.17)
Consequently, the optimal crosstalk matrix according to the MSE criterion is given by

M = R xc R -1 c , (4.18)
where

R xc = E x(n)c H (n) .
Once this matrix has been estimated, the contribution of crosstalk to the received signals can be reduced by calculating

x sc (n) = x(n) -Mc(n) (4.19)
where M is an estimate of M, computed from the estimated versions of R xc and R c .

Numerical Examples

In this section, the influence of crosstalk on the MIMO radar performance and the Although this is not a realistic crosstalk matrix, it is useful to evaluate the system performance in the presence of correlated interference.

A first simulation is done considering the ideal signal model Capon, MUSIC, and GLRT spectra are re-computed using α r (θ) and α t (θ). As shown in Figure 4.17, the secondary lobes are once again totally suppressed from the GLRT spectrum, and the lobes in the Capon and MUSIC spectra are re-centered at the target DOAs. We can then see that both crosstalk and pattern distortion can together be taken into account in order to reduce the negative influence of mutual coupling on the MIMO radar performance.

x(n) = K k=1 β k a * r (θ k )a H t (θ k )c(n) + Mc(n) + z(n), (4.20

Summary

In this chapter, the influence of mutual coupling on the DOA estimation in a MIMO radar system has been described and studied by combining signal processing with electromagnetic simulations. In Section 4.1, we showed that in antenna arrays, the radiation patterns differ from one element to another. As a consequence, the resolution and the performance of the DOA estimation algorithms are highly degraded. Those problems are caused not only by mutual coupling, but also by the use of the standard steering vectors assuming that the radiation patterns are all identical and angle-independent, as it is commonly done in the literature. We then showed the importance of using a more exact expression of the steering vectors: When the different radiation patterns (which take into account the effects of mutual coupling) are included in the steering vectors, the DOA estimation performance is greatly improved and gets close to the performance obtained in the ideal mutual-coupling-free case.

Moreover, we showed that including only the phase of the radiation patterns allows improving the DOA estimation performance of Capon (by suppressing the regrowth at the spectrum edges) without degrading the GLRT and MUSIC performance.

In Section 4.2, we studied the influence of crosstalk in MIMO radar performance. We

Hardware Description

An experimental platform has been developed in order to study the actual performance of a narrowband MIMO radar with colocated antennas. In a conventional MIMO system, each antenna element is associated with a separate RF architecture and the overall system has to be synchronized. The requirements for such implementations are difficult to fulfill especially for large MIMO systems and lead to high cost and complex hardware at the RF level.

The proposed platform uses only one transmitter (Tx) and one receiver (Rx) RF architectures. Actually, a single transmitter antenna element is used to transmit a chosen waveform and a single receiver antenna element is used to receive the signals reflected by the targets. An automated mechanism containing two rails, one for each antenna, places the transmitter and the receiver element in every position of a ULA (see Figure 5.1). A series of measurements is performed for a given position of the transmitter, while the receiver takes the different positions of a ULA. The same procedure is repeated for the different positions of the transmitter antenna. In this way, all possible configurations between the transmitter and the receiver in a MIMO system are covered. By applying the superposition principle, the received signals at each position can be wisely combined to construct the received signals matrix X of the MIMO system. As the superposition principle is valid provided that the environment is stationary, the measurements are carried out in an anechoic chamber as shown in The distance between the rails is about 8λ. In order to reduce the crosstalk level, the space separating the transmitter and the receiver elements was filled with absorbent material. Actually, the crosstalk reduction technique presented in Section 4.2 should work with any crosstalk level in theory; however, a high crosstalk level might increase the noise floor at the analog-to-digital conversion stage, and in practice, the weak reflected signals might be undetectable or inaccurately converted by the ADC (Analog-to-Digital Converter) which has a resolution of 14 bits. Moreover, a high crosstalk level might saturate the receiver and produce non-linear effects. Some pictures of the experimental platform are shown in Figure 5.4.

Targets θ 1 θ 2 Absorbent material 0 L r -1 l 0 R x T x i L t -1

Synchronization

In order to synchronize the transmitter and the receiver architectures, both the AWG and the VSA are first linked by the same 10 MHz reference clock. Then, the signal acquisition done by the VSA is triggered by an external "trigger signal" directly transmitted from the AWG. As shown in Figure 5.5, the trigger signal has the same length as the transmitted signal and has a single pulse: Only the first symbol is set to "1" while all the others are set to "0". The receiver is then triggered by the positive slope of the trigger signal so that the signal acquisition starts at the same instant that the first symbol is transmitted.

However, this synchronization procedure was found to be inaccurate. In fact, the trigger signal provided by the AWG exhibits jitter on the rising and falling edges which leads to phase synchronization errors. where φ denotes the phase error term and α is an attenuation constant. The phase error can be estimated by searching for the φ which minimizes the Least Squares (LS) criterion

J = N -1 n=0 r(n) -αc(n)e jφ 2 .
(5.2)

The phase φ which minimizes the criterion is such that

∂J ∂φ = jα N -1 n=0 r(n)c * (n)e -jφ -c(n)r * (n)e jφ = 0. (5.3)
Finally, the optimal phase according to the LS criterion is given by

φ = arg N -1 n=0 r(n)c * (n) . (5.4)
The evolution of the estimated phase error φ can be observed by performing different acquisitions of the signal {r(n)} N -1 n=0 for a same transmitted signal {c(n)} N -1 n=0 . Accordingly, we continuously transmit the signal {c(n)} N -1 n=0 , which is a sequence of N = 512 QPSK symbols, during 24 hours. The carrier and sampling frequencies are set to f c = 5.88 GHz and F s = 1.28 MHz respectively. The received signal {r(n)} N -1 n=0 is measured every 2 minutes (i.e. 720 times). The different phase errors are calculated using (5.4) and are shown in Figure 5.7. As we can see, important phase errors occur during the whole measurement process. This can highly degrade the DOA estimation performance of MIMO radar with colocated antennas given that the phase of the signals is a critical parameter in the DOA estimation problem. We found that, with this configuration, it is impossible to estimate the target DOAs unless the phase errors are compensated.

Adopted synchronization configuration

In order to overcome the phase synchronization problem, we use the directional coupler and the SPDT shown in Figure 5.3. Every transmitted frame {s(n)} Ns-1 n=0 is composed of a reference signal {s ref (n)} N -1 n=0 and the useful signal {c(n)} N -1 n=0 as shown in Figure 5.8. Even if the whole frame is transmitted by the Tx antenna, only the useful part {c(n)} N -1 n=0 is used to estimate the target DOAs. The reference signal, which is directly transmitted from the AWG to the VSA via the directional coupler, is used to estimate the phase synchronization error φ (according to (5.4)) present at every signal acquisition. The so obtained φ is then used to compensate the phase error present in the reflected useful signal.

At the receiver, an SPDT is used to switch between the reference signal and the useful signal reflected by the targets. As shown in Figure 5.8, the transmitted frame is composed of two idle symbols at the beginning, followed by the useful signal {c(n)} N -1 n=0 , two other idle symbols, and the reference signal

{s ref (n)} N -1
n=0 at the end. The idle symbols are used to avoid any switch bouncing that may affect either the useful signal or the reference signal. The "sync" signal has two purposes, its positive slope is used to trigger the signal acquisition and its high and low levels are used to control the states of the switch: When the "sync" signal is at "1" the useful signal will pass through the switch, and when it is at "0" the reference signal will pass. At the receiver, the reference and useful signals are found from the received frame by correlation process.

The amplitude of the reference signal is a parameter which must be carefully chosen in order to reduce the quantization errors. A method to optimally adjust the reference signal amplitude is presented in Section 5.3.

System Calibration

The experimental platform is calibrated before and during the measurement process.

The AWG and the VSA are automatically calibrated before the measurement process starts.

A parameter which must be adjusted during the measurement process is the reference voltage of the ADC in the VSA. If the reference voltage is set too low, the input signal may overload the ADC circuitry which introduces distortion into the measurements. If the reference voltage is set too high, the conversion accuracy is decreased and the noise floor is increased. The reference voltage should then be set equal to the maximum input signal amplitude in order to have the best possible accuracy in the analog-to-digital conversion stage. This value is also used to configure all the internal amplifiers and attenuators of the VSA to maximize the dynamic range and minimize the signal distortion caused by the non-linearities of the circuits. The flow chart of the developed process for the configuration of the ADC reference voltage ADC ref is shown in Figure 5.9(a). The reference voltage must be previously set to a value higher than the maximum input signal amplitude. Then, the mean power P s of the input signal (i.e. the received frame) is measured by the VSA, and the reference voltage ADC ref is calculated from this value. Note that the so obtained ADC ref may be lower than the maximum input signal which would overload the system. If the VSA detects an overload warning, the reference voltage is multiplied by 10 0.1 (increase of 2 dB) until the warning disappears.

Another parameter which must be adjusted is the amplitude of the reference signal

{s ref (n)} N -1 n=0
. This value must be set as close as possible to the amplitude of the reflected useful signal so that they have equal quality of quantization. Indeed, if there is a big difference between the reference and the reflected useful signal amplitudes, the signal of the smallest amplitude might be inaccurately converted or even be under the noise floor. The reference signal amplitude must then be controlled in order to match the reflected signal amplitudes, which can change depending on the target locations. This is done using a variable attenuator which is controlled by the PC following the procedure shown in Figure 5.9(b). First, the power P x of the useful reflected signal and the power P ref of the reference signal are measured (in dBm) by the VSA. The total attenuation (in dB) must then be set to the nearest integer to ∆P = P ref -P x . Note that if an attenuation value "last att" was set previous to the measurement of P x and P ref , the new total attenuation (in dB) must be set to att = ∆P + last att. Validations are made in order to ensure that the total attenuation is always set between 0 and 58 dB according to the specifications of the employed attenuator. However, it has been observed that for targets located at more that 1 m from the antennas, P ref is always greater than P x for an attenuation of 0 dB and hence, "att" should never take negative values. It is important to note that both the ADC reference voltage and the amplitude of the reference signal must be adjusted together. Actually, the powers P x and P ref might be inaccurately measured if ADC ref had not been properly set. Moreover, once the amplitude of the reference signal has been changed, the reference voltage of the ADC must be readjusted. A "flag" is set to True to inform the system that the attenuation value has been changed and that it must be reverified after readjusting the ADC. The "flag" is set to False when no verification of the attenuation value is needed. As shown in Figure 5.10, a cyclic procedure to optimally set both the reference signal amplitude and the ADC reference voltage has been developed. Note that ADC ref is first set to 0.3162 V, which corresponds to an input signal power of -10 dBm, to avoid overloading the system.

Regarding the antennas displacement, the positioning system must be calibrated prior to any measurement process. The antennas are displaced over the rails by a slider which is actuated by a step motor. The rails have sensors at the ends, one of which is used to calibrate the origin of the positioning system. This system is then able to place the antennas to every desired position with a precision of ±0.1 mm.

Experimental Results

In this section we present some experimental results obtained using the proposed platform. The number of Tx and Rx positions is L = L t = L r = 10. The set of transmit signals {c i (n)} Lt-1 i=0 are independent sequences of N = 512 QPSK symbols. The same reference signal s ref (n), which is a sequence of N QPSK symbols, is transmitted at every position i of the Tx antenna along with the corresponding useful signal c i (n). Every transmitted frame {s i (n)} Lt-1 i=0 is of length N s = 2N + 4 symbols (including the idle symbols). The symbol frequency is set to 64 kHz while the sampling frequency is set to 1.28 MHz. The different positions of the antenna elements are

separated by d = d t = d r = λ/2.
Every target is placed using a rotating arm whose rotational axis is in the middle of the rails as shown in Figure 5

Repeatability Test

A repeatability test of the experimental platform has been performed using 70 successive trials for one target located at -6.5 • and at a distance of 1.8 m of the center of the rails. We observe a maximum fluctuation of about ±1 • on the DOA estimated by Capon around the true target DOA which does not represent a high variation. As for MUSIC, we obtain even better results with a fluctuation of about ±0.4 • . The small fluctuations can be due to noise, to nonlinearities of the circuits, and to residual calibration errors.

Narrowband Detection

In a first measurement process, we place two targets at θ 1 = -15 • and θ 2 = 18 • both at a distance of 1.7 m from the center of the rails. The tested targets are two metallic cylinders of diameter of 6 cm and height of 1.5 m. As presented in Chapter 2, the planewave condition for a MIMO radar of L = 10 transmitting and receiving elements is R > 5∆ 2 /λ (with ∆ = (L -1)d), which gives R > 5.17 m. Our targets are then in the spherical-wave region and hence the spherical-wave steering vectors must be used to compute the spatial spectra.

Once the whole measurement process is finished, the narrowband Capon, MUSIC and GLRT spectra are computed using a grid of angles θ ∈ [-90

[θ 1 , R 1 ] = [-15 • , 1.7 m] and [θ 2 , R 2 ] = [18 • , 1.7 m]).
usually unknown, might also deteriorate the GLRT detection performance either if the noise level is too low (leading to ill-conditioning issues) or if the noise is not Gaussian.

Crosstalk Reduction

The negative effects of crosstalk can be overcome by using the crosstalk reduction technique proposed in Chapter 4. First, the crosstalk matrix M must be estimated in a first measurement process in the environment without any target. Then, the crosstalk term is subtracted from the received signals.

Estimation of the Crosstalk Matrix

In order to test the reliability of the estimation of the crosstalk matrix, the measurement process in the target-free environment has been performed 100 times, using a different indicates that the estimates of the coefficients of M are reliable. We must also note that the use of different sets of transmit signals does not influence the estimation of M in a significant way.

x sc (n) = x(n) -Mc(n). (5.6)
In this particular case, we use as crosstalk matrix the average of M over the 100 trials.

Then the Capon, MUSIC, and GLRT spectra are computed again from the signals ). However, only a few secondary lobes are attenuated in the GLRT spectrum after the crosstalk reduction, and we are still unable to estimate the target DOAs from this spectrum. The remaining secondary lobes may be due to the characteristics of the actual noise present in our measurement system and environment. Indeed, the GLRT technique was developed assuming the presence of white Gaussian noise (see Appendix A). However, the characteristics of the actual noise present in the anechoic chamber and the RF architecture might differ from the Gaussian assumption which might deteriorate the detection performance. Moreover, the noise level might also be very low which would lead to ill-conditioning problems.

x sc (n). As shown in Figures 5.16
-90°- 70°- 50°- 30°- 10°1 0°3 0°5 0°7 0°9 0°1 1 1.3 1.3 1.6 1.6 1.9 1.9 2.2 2.2 2.5 2.5 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 x 10 -5 (a)
-90°- 70°- 50°- 30°- 10°1 0°3 0°5 0°7 0°9 0°1 1 1.3 1.3 1.6 1.6 1.9 1.9 2.2 2.2 2.5 2.5 (b)
-90°- 70°- 50°- 30°- 10°1 0°3 0°5 0°7 0°9 0°1 1 1.3 1.3 1.6 1.6 1.9 1.9 2.2 2.2 2.5 2
[θ 1 , R 1 ] = [-15 • , 1.7 m] and [θ 2 , R 2 ] = [18 • , 1.7 m]).

Addition of White Gaussian Noise and Crosstalk Reduction

To observe the influence of noise on the performance of GLRT, let us add white Gaussian noise to the received signals x(n) before crosstalk reduction (without subtracting the

crosstalk term Mc(n)) as x +noise (n) = x(n) + z(n) n = 0, . . . , N -1, (5.7)
where the {z(n)} N -1 n=0 are complex Gaussian random vectors with zero mean. The power of this additive noise is chosen to be of 70 dB below the received signals power (the latter being of -36 dBm).

The GLRT is then computed from the signals {x +noise (n)} N -1 n=0 and is shown in Figures 5. [START_REF] Fischer | Detection of antipersonnel mines by using the factorization method on multistatic ground-penetrating radar measurements[END_REF] Let us now reduce the crosstalk term from x +noise (n) by computing

x sc+noise (n) = x +noise (n) -Mc(n) (5.8)
and compute again the GLRT from x sc+noise (n). As shown in Figures 5.20

[θ 1 , R 1 ] = [-15 • , 1.7 m] and [θ 2 , R 2 ] = [18 • , 1.7 m]).
crosstalk reduction, which allows us to clearly identify the target directions.

Summary

We developed an experimental measurement platform of MIMO radar with colocated antennas using a single Tx/Rx RF architecture. The proposed platform is much less complex and expensive than a real MIMO system. It is also reconfigurable since the

-90°- 70°- 50°- 30°- 10°1 0°3 0°5 0°7 0°9 0°1 1 1.3 1.3 1.6 1.6 1.9 1.9 2.2 2.2 2.5 2
[θ 1 , R 1 ] = [-15 • , 1.7 m] and [θ 2 , R 2 ] = [18 • , 1.7 m]).
inter-element spacings and the number of antenna elements can be easily modified, which allows performing different kinds of test.

The repeatability test showed fluctuations in the estimated DOAs of maximum ±1 • around the true DOA, which proves that the platform is reliable.

The obtained measurements allowed us to validate some detection techniques usually studied from a theoretical point of view. The results showed that the performance of the GLRT is highly affected by the noise characteristics. We remedied this matter by adding white Gaussian post-processing noise on the received signals. We also demonstrated the effectiveness of the crosstalk reduction technique in reducing the secondary lobes from the GLRT spectrum.

Our contributions presented in this chapter are the development of a reconfigurable experimental platform for narrowband MIMO radar with colocated antennas, and the investigation of the narrowband DOA estimation and crosstalk reduction techniques from an experimental point of view.

Chapter 6

Conclusions and Perspectives

The signal model of narrowband MIMO radar with colocated antennas was presented in Chapter 2 followed by the description of the Capon, MUSIC, and GLRT methods as DOA estimation techniques of non-moving targets. The simulations showed that MUSIC offers the best angular resolution followed by Capon and GLRT. Because of its high resolution, MUSIC is able to resolve closely spaced targets. As for the GLRT, even though it has the lowest angular resolution, it is the most robust against noise and the less sensitive to the target reflection coefficients among the three techniques. Moreover, the GLRT has the capability of rejecting strong interference or jammers which are uncorrelated with the transmitted signals.

Additionally, we showed that particular attention must be paid while using the planewave assumption. Actually, the plane-wave condition which must be considered from a signal processing point of view differs from the far-field condition R > 2∆ 2 /λ established in antenna theory. Indeed, considering the wavefront as plane for R > 2∆ 2 /λ may introduce additional errors in the DOA estimation. We found that for antenna arrays of 5, 10, and 15 elements, the condition R > 5∆ 2 /λ is more appropriate and the error introduced by the plane-wave approximation can be neglected.

In Another consequence of mutual coupling is crosstalk, which can significantly degrade the DOA estimation performance: We observed that the resolution of Capon and MUSIC is highly decreased and the GLRT spectrum presents several secondary lobes which do not allow the estimation of the target DOAs. In order to overcome this problem, we proposed a crosstalk reduction technique based on a signal processing approach:

The crosstalk matrix is first estimated (by solving a minimum MSE criterion) from a first transmission in an environment without any target, and the crosstalk term is then subtracted from the received signals when targets are present.

From an experimental point of view, we developed a platform for narrowband MIMO radar with colocated antennas, as presented in Chapter 5. Since a large MIMO system is particularly expensive and complex to develop, the proposed platform employs only one transmitter and one receiver RF architectures. An automated mechanism places both the transmitter and the receiver elements in every position of a ULA, and the received signal matrix is constructed by applying the superposition principle. This platform is not only easier to calibrate and synchronize than a real MIMO system would be, but it is also reconfigurable since the number of antenna elements and the inter-element spacings can be easily changed.

The experimental results allowed us to validate some narrowband DOA estimation techniques, which are usually studied from a theoretical point of view. We observed that the performance of the GLRT is affected by the noise characteristics and can be improved by adding white Gaussian post-processing noise. We also demonstrated the validity of the proposed crosstalk reduction technique.

Suggestions for future work are presented below: 5. In the future, the experimental platform should allow taking the whole effects of mutual coupling into account. This can be done by using real transmitter and receiver arrays, instead of single mobile Tx and Rx elements. The low complexity and cost of the actual platform can be maintained by using two RF switches (one for each array), so that only one transmitter and one receiver elements are active at each time, while the other elements are terminated with matched impedances.

6. The platform needs to evolve in the future so that it can deal with wideband signals. This will require replacing the actual narrowband RF architecture (which includes the antennas, the power amplifiers, the signal generator, and the signal analyzer) with a wideband one.

7. Finally, the ambiguity functions in the case of wideband MIMO radar could be investigated in order to estimate Doppler and range parameters of moving targets.

The feasibility of an experimental implementation has to be explored.

APPENDICES

Given a complex variable w = w R + jw I , the complex derivative of w is

∂ ∂w = ∂ ∂w R + j ∂ ∂w I .
As a consequence ∂w ∂w = 1+ j 2 = 0

∂w * ∂w = 1+ j(-j) = 2. (A.4)
Using the method of Lagrange multipliers to optimize gives

∂f ∂w -λ ∂g ∂w = 0 2R x w -2λa * r (θ) = 0 w = λR -1
x a * r (θ).

(A.5)

Then, applying the constraint

w H a * r (θ) = 1 λa T r (θ)R -1 x a * r (θ) = 1 λ = 1 a T r (θ)R -1 x a * r (θ) . (A.6)
Therefore, the Capon weights are

w(θ) = R -1 x a * r (θ) a T r (θ)R -1 x a * r (θ)
.

(A.7)

A.2 The GLRT

This section details the derivation of the GLRT presented in [START_REF] Xu | Radar imaging via adaptive MIMO techniques[END_REF].

For the derivation of the GLRT it is assumed that the columns of the residual term Z in (A.1) are i.i.d circularly symmetric complex Gaussian random vectors. All the columns are assumed to have zero mean and unknown but equal covariance matrix R z .

Before defining the GLRT, the Probability Density Function (PDF) of the residual term,

Z will be defined. The PDF for Z i , the i th complex Gaussian random column of Z, is

f (Z i ) = 1 π Lr |R z | e -[Z H i R -1 z Z i] . (A.8)
Now, replace R z with the equivalent expression Rx R-1

x R z . Then, ignoring the factor π Lr , the objective of the minimization in (A. [START_REF] Lesturgie | Some relevant applications of MIMO to radar[END_REF]) is

|R z | e tr[R -1 z Rx] = Rx R-1 x R z e tr[R -1 z Rx] . (A.14)
Then, let A = R -1 z Rx . Also, diagonalize A so that A = PΛP -1 with Λ = diag(λ i ) i=1,...,Lr , where λ i are the eigenvalues of A such that

R -1 z Rx u i = λ i u i i = 1, . . . , L r , (A.15)
where u i are the eigenvectors of A. It should be noted that

|A| = |Λ|, A -1 = Λ -1
and also that tr

[A] = tr[Λ].
Note that R z is a covariance matrix and therefore positive semidefinite. Then, R -1 z is also positive semidefinite, which allows to write

u H i Rx R -1 z Rx u i ≥ 0. (A.
f (X | β, R z ) = (πe) -N Lr Rx -N a T r (θ) R-1 x a * r (θ) a T r (θ) Q-1 a * r (θ) -N . (A.40)
Thus, substituting (A.21) and (A.40) into (A.10) gives

ρ(θ) = 1 -        (πe) -N Lr Rx -N (πe) -N Lr Rx -N a T r (θ) R-1 x a * r (θ) a T r (θ) Q-1 a * r (θ) -N        1 N (A.41)
and the GLR is

ρ(θ) = 1 - a T r (θ) R-1 x a * r (θ) a T r (θ) Q-1 a * r (θ) . (A.42)
Step 0: Given initial arbitrary values of {φ hp } in [0, 2π], repeat steps 1 and 2 until convergence.

Step 1: For {φ hp } at their last values, denoted as { φhp }, let

A(p) =     a H t (θ 1 , p) . . . a H t (θ H , p)     , b(p) =     δ(θ 1 , p)e j φ1p . . . δ(θ H , p)e j φHp     . (B.10)
Then, the criterion (B.9) can be written as

H h=1 N -1 p=0 |δ(θ h , p)e jφ hp -a H t (θ h , p)C(p)| 2 = N -1 p=0 b(p) -A(p)C(p) 2 . (B.11)
The minimizer {C(p)} N -1 p=0 is given by the least-squares estimate:

Ĉ(p) = (A H (p)A(p)) -1 A H (p)b(p). (B.12)
Step 2: For {C(p)} N -1 p=0 set at their last values, the minimizer {φ hp } of the criterion (B.9) is given by φhp = arg{a H t (θ h , p) Ĉ(p)}. (B.13)

Stage 2

In stage 2, a set of signals {c(n)} N -1 n=0 is synthesized (under the PAPR constraint), so that its DFT approximates the { Ĉ(p)} N -1 p=0 obtained in stage 1.

It can be noted that the {C(p)} N -1 p=0 solving the beampattern matching problem stated in (B.5) have a phase ambiguity, i.e. C(p) and C(p)e jψp lead to the same value of (B.5) for any ψ p . To exploit this phase ambiguity, let introduce the auxiliary variables {ψ p } N -1 p=0 . Then, the signals {c(n)} N -1 n=0 whose DFT approximate the previously obtained { Ĉ(p)} N -1 p=0 can be found by solving the following optimization problem:

min {c(n)},{ψp} N -1 p=0 ĈT (p)e jψp -1 e -j2π 1 N (p-N 2 ) • • • e -j2π (N -1) N (p-N 2 ) C T 2 , (B.14)
where

C =        c 0 (0) • • • c 0 (N -1) c 1 (0) • • • c 1 (N -1) . . . . . . c Lt-1 (0) • • • c Lt-1 (N -1)        . (B.15)
By defining

e H p = 1 e -j2π 1 N (p-N 2 ) • • • e -j2π (N -1) N (p-N 2 )
,

F H =     e H 0 .
. . The problem can be solved using the following cyclic algorithm:

Step 0: Given arbitrary initial values for {ψ p } N -1 p=0 and the { Ĉ(p)} N -1 p=0 obtained from stage 1, repeat steps 1 and 2 until convergence.

Step 1: For {ψ p } N -1 p=0 set at their last values, the minimization (B.18) with respect to {c(n)} N -1 n=0 depends on the PAPR constraint. The resolution of (B.18) can be done by solving L t separate optimization problems (i.e. for i = 0, . . . , L t -1): [START_REF] Dzvonkovskaya | Simulation of tsunami signatures in ocean surface current maps measured by HF radar[END_REF] where v i denotes the i th column of 1 N FG T . This problem can be solved using the "nearest-vector" algorithm proposed in [START_REF] Tropp | Designing structured tight frames via an alternating projection method[END_REF] and recalled in Appendix B.2. overlapped with 3 4 N f samples. The covariance matrices are then estimated as

min {c i (n)} N -1 n=0 v i -c i (0) • • • c i (N -1) T 2 s.t. PAPR {c i (n)} N -1 n=0 ≤ , (B.
Rx (f p ) = 1 M M m=1 X m (p )X H m (p ) Rc (f p ) = 1 M M m=1 C m (p )C H m (p ) Rxc (f p ) = 1 M M m=1 X m (p )C H m (p ) p = 0, . . . , N f -1, (B.24)
where X m (p ) and C m (p ) denote respectively the received and the transmitted sequences at frequency f p for the m th block (m = 1, . . . , M ), i.e.

X m (p ) = N f -1 n=0 x n + m -1 4 N f e -j2π n N f p - N f 2 C m (p ) = N f -1 n=0 c n + m -1 4 N f e -j2π n N f p - N f 2 . (B.
x(n) = K k=1 β k a * r (θ k , R k )a H t (θ k , R k )c(n) + z(n), (C.1) où a t (θ, R) =      g * t,0 (θ) 1 r 0 e j 2π λ r 0 . . . g * t,Lt-1 (θ) 1 r L t -1 e j 2π λ r L t -1      (C.2) et a r (θ, R) =     g * r,0 (θ) 1 r 0 e j 2π λ r 0 . . . g * r,Lr-1 (θ) 1 r Lr -1 e j 2π λ r Lr -1     (C.
k T i r i = 2π λ D -i - L t -
x(n) = K k=1 β k a * r (θ k )a H t (θ k )c(n) + z(n). (C.7)
En notation matricielle, ce modèle s'écrit

X = K k=1 β k a * r (θ k )a H t (θ k )C + Z, (C.8) avec X = x(0) • • • x(N -1) , C = c(0) • • • c(N -1) , Z = z(0) • • • z(N - 1
(θ) = 1 - max Rz f (X|β = 0, R z ) max β,Rz f (X|β, R z ) 1 N , (C.11)
où R z est la matrice de covariance du bruit et f (X|β, R z ) est la densité de probabilité des signaux reçus connaissant β et R z .

En supposant que les colonnes du terme de bruit Z sont des vecteurs indépendants et identiquement distribués, Gaussiens, complexes, circulairement symétriques, de moyenne nulle, et tous de même matrice de covariance, le GLRT est donné par [START_REF] Xu | Radar imaging via adaptive MIMO techniques[END_REF] ρ WBFIT est une technique qui permet, par une méthode itérative, la conception de formes d'ondes respectant un diagramme de rayonnement désiré et une contrainte de PAPR (Peak to Average Power Ratio) [START_REF] He | Wideband MIMO systems: Signal design for transmit beampattern synthesis[END_REF].

(θ) = 1 - a T r (θ) R-1 x a * r (θ) a T r (θ) Q-1 a * r (θ) , (C.12) où Q = Rx - Rxc a t (θ)a H t (θ) RH
[θ 1 , R 1 ] = [-65 • , 0.35 m], [θ 2 , R 2 ] = [5 • , 0.45 m] et [θ 3 , R 3 ] = [
f p = pF s N - F s 2 , (C.17) F s est la fréquence symbole, X(p) = X 0 (p) • • • X Lr-1 (p) T , C(p) = C 0 (p) • • • C Lt-1 (p) T et Z(p) = Z 0 (p) • • • Z Lr-1 (
WBFIT se déroule en deux étapes : dans la première, les signaux dans le domaine fréquentiel {C(p)} 0-1 p=0 sont conçus de façon à approximer un diagramme de rayonnement désiré δ(θ h , p); dans la deuxième étape, les signaux dans le domaine temporel {c(n)} N -1 n=0 sont calculés à partir des {C(p)} 0-1 p=0 tout en respectant une contrainte de PAPR.

Les signaux à émettre {c(n)} N -1 n=0 sont alors la solution du problème d'optimisation problème min C. • Cas idéal : la propagation des signaux est simulée sans inclure les diagrammes de rayonnement dans le modèle du signal x(n) = βa * r (θ 1 )a H t (θ 1 )c(n) + z(n), et les spectres de Capon, MUSIC et GLRT sont calculés en utilisant les vecteurs directionnels classiques a r (θ) et a t (θ).

min {c(n)} H h=1 N -1 p=0 δ(θ h , p) -a H t (θ h , p)C(p) 2 s.c. PAPR {c i (n)} N -1 n=0 ≤ , i = 0, . . . , L t -1, (C.18) où PAPR {c i (n)} N -1 n=0 = max n |c i (n)| 2 1 N N -1 n=0 |c i (n)|
{c i (n)} N -1 n=0 c i (n) -c i (n) 2 s.c. PAPR {c i (n)} N -1 n=0 ≤ i = 0, . . . , L t -1.
• Cas du traitement classique : la propagation des signaux est simulée en incluant les diagrammes de rayonnement dans le modèle du signal x(n) = βã * r (θ 1 )ã H t (θ 1 )c(n) + z(n), mais les spectres de Capon, MUSIC et GLRT sont calculés en utilisant les vecteurs directionnels classiques a r (θ) et a t (θ).

3. 1 4 . 2 ccE

 142 Flow diagram of the wideband simulations. . . . . . . . . . . . . . . . . . 3.2 The WBFIT beampattern in dB with = 2 and F s = f c /5 = 200 MHz (1 target at θ = 40 • ). (a) 2D plot, (b) 3D plot. . . . . . . . . . . . . . . . . . 3.3 PAPR of every {c i (n)} N -1 n=0 synthesized by WBFIT ( = 2). . . . . . . . . 3.4 The WBFIT beampattern in dB with = 1 and F s = f c /5 = 200 MHz (1 target at θ = 40 • ). (a) 2D plot, (b) 3D plot. . . . . . . . . . . . . . . . 3.5 PAPR of every {c i (n)} N -1 n=0 synthesized by WBFIT ( = 1). . . . . . . . . 3.6 The SFBT beampattern in dB with F s = f c /5 = 200 MHz (1 target at θ = 40 • ). (a) 2D plot, (b) 3D plot. . . . . . . . . . . . . . . . . . . . . . . 3.7 PAPR of every {c i (n)} N -1 n=0 synthesized by SFBT. . . . . . . . . . . . . . . xi List of Figures xii 3.8 The WBFIT beampattern in dB with = 2 and F s = f c /2 = 500 MHz (1 target at θ = 40 • ). (a) 2D plot, (b) 3D plot. . . . . . . . . . . . . . . . 3.9 The WBFIT beampattern in dB at f c ( = 2). . . . . . . . . . . . . . . . 3.10 The SFBT beampattern in dB with F s = f c /2 = 500 MHz (1 target at θ = 40 • ). (a) 2D plot, (b) 3D plot. . . . . . . . . . . . . . . . . . . . . . . 3.11 The SFBT beampattern in dB at f c . . . . . . . . . . . . . . . . . . . . . . 3.12 The WBFIT beampattern in dB with = 2 and F s = f c /5 = 200 MHz ( θ1 = -60 • , θ2 = 0 • , θ3 = 40 • ). (a) 2D plot, (b) 3D plot. . . . . . . . . . . 3.13 The SFBT beampattern in dB with F s = f c /5 = 200 MHz ( θ1 = -60 • , θ2 = 0 • , θ3 = 40 • ). (a) 2D plot, (b) 3D plot. . . . . . . . . . . . . . . . . . 3.14 The WBFIT beampattern in dB with = 2 and F s = f c /2 = 500 MHz ( θ1 = -60 • , θ2 = 0 • , θ3 = 40 • ). (a) 2D plot, (b) 3D plot. . . . . . . . . . . 3.15 The SFBT beampattern in dB with F s = f c /2 = 500 MHz ( θ1 = -60 • , θ2 = 0 • , θ3 = 40 • ). (a) 2D plot, (b) 3D plot. . . . . . . . . . . . . . . . . . 3.16 The M-SFBT beampattern in dB with F s = f c /5 = 200 MHz ( θ1 = -60 • , θ2 = 0 • , θ3 = 40 • ). (a) 2D plot, (b) 3D plot. . . . . . . . . . . . . . . . . . 3.17 PAPR of every {c i (n)} N -1 n=0 synthesized by M-SFBT. . . . . . . . . . . . . 3.18 The M-SFBT beampattern in dB with = 2 and F s = f c /5 = 200 MHz ( θ1 = -60 • , θ2 = 0 • , θ3 = 40 • ). (a) 2D plot, (b) 3D plot. . . . . . . . . . . 3.19 PAPR of every {c i (n)} N -1 n=0 synthesized by M-SFBT with = 2. . . . . . . 3.20 The WBFIT multiband beampattern in dB with = 2 and F s = f c /5 = 200 MHz ( θ1 = -60 • , θ2 = 0 • , θ3 = 40 • ). (a) 2D plot, (b) 3D plot. . . . . 3.21 Stacked narrowband spectra of (a) Capon, (b) MUSIC, and (c) GLRT after the omnidirectional stage (θ 1 = -30 • , θ 2 = 0 • , θ 3 = 60 • , F s = f c /5 = 200 MHz). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.22 Incoherent spatial spectra of (a) Capon, (b) MUSIC, and (c) GLRT after the omnidirectional stage (θ 1 = -30 • , θ 2 = 0 • , θ 3 = 60 • , F s = f c /5 = 200 MHz). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.23 MSE in θ for the target at -30 • using the Capon, MUSIC, and GLRT estimates after the omnidirectional stage (F s = f c /5 = 200 MHz). . . . . . 3.24 Stacked narrowband spectra of (a) Capon, (b) MUSIC, and (c) GLRT after the SBFT stage (θ 1 = -30 • , θ 2 = 0 • , θ 3 = 60 • , F s = f c /5 = 200 MHz). 3.25 Incoherent spatial spectra of (a) Capon, (b) MUSIC, and (c) GLRT after the SFBT stage (θ 1 = -30 • , θ 2 = 0 • , θ 3 = 60 • , F s = f c /5 = 200 MHz). . . 3.26 MSE in θ for the target at -30 • using the Capon, MUSIC, and GLRT estimates after the SFBT stage (F s = f c /5 = 200 MHz). . . . . . . . . . . 3.27 MSE in θ for the target at -30 • using the estimates given by (a) Capon, (b) MUSIC, and (c) GLRT (F s = f c /5 = 200 MHz). . . . . . . . . . . . . 3.28 The TOPS spectrum after the omnidirectional stage (θ 1 = -30 • , θ 2 = 0 • , θ 3 = 60 • , F s = f c /5 = 200 MHz). . . . . . . . . . . . . . . . . . . . . . . . 3.29 The M-SFBT beampattern in dB with = 2 and F s = f c /5 = 200 MHz (θ 1 = -30 • , θ 2 = 0 • , and θ 3 = 60 • ). . . . . . . . . . . . . . . . . . . . . . . 3.30 The TOPS spectrum after the M-SFBT stage (θ 1 = -30 • , θ 2 = 0 • , θ 3 = 60 • , F s = f c /5 = 200 MHz). . . . . . . . . . . . . . . . . . . . . . . . 3.31 MSE in θ for the target at -30 • using the Capon, MUSIC, and GLRT estimates after the SFBT stage and the M-TOPS estimates after the M-SFBT stage (F s = f c /5 = 200 MHz). . . . . . . . . . . . . . . . . . . . . . 4.1 Linearly polarized patch antenna. . . . . . . . . . . . . . . . . . . . . . . . List of Figures xiii Normalized radiation pattern (in magnitude) of an isolated patch antenna at 5.8 GHz in (a) 3D and (b) 2D (cutting plane phi = 0 • ). . . . . . . . . . 4.3 Transmitter and receiver arrays of patch antennas resonating at 5.8 GHz (L = L t = L r = 6). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Normalized radiation patterns (in magnitude) of the receiving elements at 5.8 GHz (ports 1 to 6). . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Normalized radiation patterns (in magnitude) of the transmitting elements at 5.8 GHz (ports 7 to 12). . . . . . . . . . . . . . . . . . . . . . 4.6 Capon, MUSIC, and GLRT spectra for three targets at θ 1 = -40 • , θ 2 = 20 • , and θ 3 = 40 • , neglecting the pattern distortions (-10 log 10 σ 2 = 20). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Capon, MUSIC, and GLRT spectra for three targets at θ 1 = -40 • , θ 2 = 20 • , and θ 3 = 40 • , taking the radiation patterns into account (-10 log 10 σ 2 = 20). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8 Capon, MUSIC, and GLRT spectra for three targets at θ 1 = -40 • , θ 2 = -5 • , and θ 3 = 5 • , neglecting the pattern distortions (-10 log 10 σ 2 = 20). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.9 Capon, MUSIC, and GLRT spectra for three targets at θ 1 = -40 • , θ 2 = -5 • , and θ 3 = 5 • , taking the radiation patterns into account (-10 log 10 σ 2 = 20). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.10 Capon, MUSIC, and GLRT spectra for one target at -40 • , neglecting the pattern distortions (-10 log 10 σ 2 = -10). . . . . . . . . . . . . . . . . . . . 4.11 Capon, MUSIC, and GLRT spectra for one target at -40 • , taking the radiation patterns into account (-10 log 10 σ 2 = -10). . . . . . . . . . . . . 4.12 Capon, MUSIC, and GLRT spectra for one target at -40 • , taking only the phase of the radiation patterns into account (-10 log 10 σ 2 = -10). . . 4.13 MSE in θ for one target at -40 • using the estimates given by (a) Capon, (b) MUSIC, and (c) GLRT. . . . . . . . . . . . . . . . . . . . . . . . . . . 4.14 Capon, MUSIC, and GLRT spectra for two targets at θ 1 = -20 • , and θ 2 = 20 • before crosstalk reduction (ideal case). . . . . . . . . . . . . . . . 4.15 Capon, MUSIC, and GLRT spectra for two targets at θ 1 = -20 • , and θ 2 = 20 • after crosstalk reduction (ideal case). . . . . . . . . . . . . . . . . 4.16 Capon, MUSIC, and GLRT spectra for two targets at θ 1 = -20 • , and θ 2 = 20 • before crosstalk reduction (taking only the phase of the radiation patterns into account). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.17 Capon, MUSIC, and GLRT spectra for two targets at θ 1 = -20 • , and θ 2 = 20 • after crosstalk reduction (taking only the phase of the radiation patterns into account). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Scheme of the antenna configuration of the measurement platform. . . . . 5.2 Experimental measurement configuration. . . . . . . . . . . . . . . . . . . 5.3 Tx/Rx RF architecture block diagram. . . . . . . . . . . . . . . . . . . . . 5.4 System overview. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Example of timing diagram of the baseband transmit signal and the trigger signal (only the first 20 symbols are shown). . . . . . . . . . . . . . 5.6 Configuration of a first synchronization test. . . . . . . . . . . . . . . . . . 5.7 Phase error computed from 720 acquisitions. . . . . . . . . . . . . . . . . 5.8 Example of timing diagram of the baseband transmit signal using a reference signal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Complex envelope of a transmitted signal [V] D x-component of a target position vector [m] d r Inter-element spacing of the receiver array [m] d t Inter-element spacing of the transmitter array [m] Electric field reflected by a target [V/m] E t Total electric field at a target location [V/m] f Frequency [Hz] f c Carrier frequency [Hz] g r,l Radiation pattern of the l th receiver antenna element g t,i Radiation pattern of the i th transmitter antenna element H y-component of a target position vector [m] Number of antenna elements of a receiving array L t Number of antenna elements of a transmitting array m l,i Transmission coefficient between the i th transmitter element and the l th receiver elements n Discrete-time index N Number of temporal samples of a signal N s Total number of temporal samples of a transmitted frame p Discrete-frequency index P cap Capon's spatial spectrum P M U SIC MUSIC spatial spectrum R Radial distance between the origin of the Cartesian coordinate system and a target position [m] Vector of complex envelopes of the transmitted signals [V] E r Vector of electric fields measured by a receiver array [V/m] p Position vector [m] k Two-dimensional wave vector [rad/m] r Vector linking one element of an antenna array with a target position [m] s Set of source signals to be detected x Vector of received signals Auto-covariance matrix of the transmitted signals R x Auto-covariance matrix of the received signals R xc Cross-covariance matrix between the received and the transmitted signals R z Auto-covariance matrix of noise X Matrix of received signals Z Matrix of noise and interference Determinant of a matrix or absolute value of a scalar number • Euclidean norm of a vector • F Frobenius norm of a matrix x Largest integer less than or equal to x arg{•} Argument of a complex number diag(v) Diagonal matrix with the elements of vector v on the main diagonal

Figure 1 . 1 :

 11 Figure 1.1: MIMO radar with widely separated antennas.

Figure 1 . 2 :Figure 1 . 3 :

 1213 Figure 1.2: Bistatic MIMO radar with colocated antennas.

  Chapter 3, the signal model is extended to the case of wideband signals. Then, a review and comparison of two existing wideband waveform design techniques is performed. Based on these existing techniques, we propose a new multiband waveform design technique which allows decorrelating the signals reflected by the targets. Moreover, we propose an adaptation of the previously reviewed narrowband DOA estimation techniques to the wideband context. Additionally, a wideband array processing technique is adapted to the context of wideband MIMO radar. The performance of the studied techniques is analyzed and compared through simulation. In Chapter 4, the electromagnetic interactions between the different elements of the antenna arrays are taken into account in order to establish a more realistic signal model. The influence of mutual coupling on the DOA estimation performance is then studied by combining signal processing with electromagnetic simulations. Moreover, we show how to improve the DOA estimation performance in the presence of distorted radiation patterns and propose a crosstalk reduction technique, which makes possible an efficient estimation of the target DOAs. Finally, the developed experimental platform is fully described in Chapter 5, including the synchronization and calibration procedures. Then, experimental results are presented in order to analyze the real performance of the discussed narrowband algorithms, including the DOA estimation and crosstalk reduction techniques. Additional mathematical developments regarding the reviewed algorithms are provided in Appendices A and B. An extended summary in French is given in Appendix C. The different notations are defined at their first appearance and are common to the entire document.

•

  O. Gómez, P. Jardin, F. Nadal, B. Poussot, and G. Baudoin. "Multiband waveform synthesis and detection for a wideband MIMO radar". In 2011 IEEE International Conference on Microwaves, Communications, Antennas and Electronics Systems (COMCAS), pages 1-5, 2011. • O. Gómez, F. Nadal, P. Jardin, G. Baudoin, and B. Poussot. "On wideband MIMO radar: Detection techniques based on a DFT signal model and performance comparison". In 2012 IEEE Radar Conference (RADAR), pages 0608-0612, 2012. • O. Gómez, B. Poussot, F. Nadal, P. Jardin, and G. Baudoin. "An Experimental Platform for MIMO Radar with Colocated Antennas". In 2012 IEEE Asia-Pacific Microwave Conference Proceedings (APMC), pages 1085-1087, 2012. National conference with proceedings • O. Gómez, B. Poussot, F. Nadal, P. Jardin, and G. Baudoin. "Radar MIMO cohérent : développement d'une plateforme expérimentale". In 18 èmes Journées Nationales Micro-ondes (JNM), 2013. Paper presented at a workshop • O. Gómez, B. Poussot, F. Nadal, P. Jardin, and G. Baudoin. "A reconfigurable experimental platform for coherent MIMO radar". In SONDRA Workshop, 2013.Chapter 2

Figure 2 . 1 :

 21 Figure 2.1: An antenna transmitting a signal of spherical wavefront.

Figure 2 . 2 :

 22 Figure 2.2: A transmitter Uniform Linear Array.

Figure 2 . 3 :

 23 Figure 2.3: Far-field plane wavefront transmitted by a ULA.

2. 3 . 2 . 1

 321 Detection in the Plane-Wave Region Consider K = 3 targets located in the plane-wave region at θ 1 = -60 • , θ 2 = 0 • , θ 3 = 40 • with reflection coefficients β 1 = β 2 = β 3 = β = 1. The received signals are constructed using the plane-wave narrowband signal model (2.29) where the noise term is modeled as white Gaussian noise such that the SNR equals to 10 dB. Here the SNR is defined as the ratio between the mean power of the signals reflected by the targets (measured by the receiver array) and the mean power of the noise term. By denoting the signals reflected by the targets as

Figure 2 . 4 :

 24 Figure 2.4: Capon, MUSIC, and GLRT spectra for three targets located in the planewave region at θ 1 = -60 • , θ 2 = 0 • , and θ 3 = 40 • , and an SNR of 10 dB.

. 56 )Figure 2 . 4

 5624 Figure 2.4 shows the Capon, MUSIC and GLRT spectra constructed using a grid of angles θ ∈ [-90 • , 90 • ] with a mesh grid size of 0.1 • . The MUSIC spectrum has the best resolution, showing the sharpest peaks, followed by Capon. Since the SNR is relatively good (10 dB), the three techniques can clearly identify the DOAs of the targets.

Consider now the three targets (θ 1 =

 1 -60 • , θ 2 = 0 • , θ 3 = 40 • ) with different reflection coefficients β 1 = 1, β 2 = 0.5, and β 3 = 0.2.Figure 2.6 shows the spatial spectra of Capon, MUSIC, and GLRT for an SNR of 10 dB. As we can see, both Capon and MUSIC are very sensitive to the β k coefficients: The peaks corresponding to targets of β 2 = 0.5 (at θ 2 = 0 • ) and β 3 = 0.2 (at θ 3 = 40 • ) can hardly be seen. On the other hand, the three

Figure 2 . 5 :

 25 Figure 2.5: Capon, MUSIC, and GLRT spectra for three targets located in the planewave region at θ 1 = -60 • , θ 2 = 0 • , and θ 3 = 40 • , and an SNR of -10 dB.

Figure 2 . 6 :

 26 Figure 2.6: Capon, MUSIC, and GLRT spectra for three targets located in the planewave region (θ 1 = -60 • , θ 2 = 0 • , θ 3 = 40 • ) with different reflection coefficients (β 1 = 1, β 2 = 0.5, β 3 = 0.2).

Figure 2 . 7 :

 27 Figure 2.7: Capon, MUSIC, and GLRT spectra for two closely spaced targets located in the plane-wave region (θ 1 = 17 • , θ 2 = 22 • ).

Figure 2 . 8 :

 28 Figure 2.8: MIMO radar with colocated transmitter and receiver arrays, and two targets.

Figure 2 .Figure 2 . 9 :

 229 Figure 2.9 shows the spatial spectra of Capon, MUSIC, and GLRT, constructed using a grid of angles θ ∈ [-90 • , 90 • ] with a mesh grid size of 0.1 • and a grid of distances R ∈ [0.1 m, 1.1 m] with a step of 0.01 m. The SNR was set to 10 dB. As shown, the high resolution of MUSIC allows an easy estimation of the parameters [θ k , R k ] of the targets; however, the lobes become larger as the distance R k increases. The GLRT can also be used to estimate both parameters by searching for the maxima in angular and

2. 3 . 3

 33 Jammer Rejection In real world environments, the signals received by a radar might be mixed with several undesired waves coming from sources external to the radar, such as communication signals (satellite, GPS, WiFi, ...), signals coming from other radars, or jammers. To observe the influence of this kind of interferences on the DOA estimation techniques let us consider K = 3 targets located in the plane-wave region at θ 1 = -60 • , θ 2 = 0 • , θ 3 = 40 • with β = 1. A strong jammer located in the plane-wave region at 20 • transmits an unknown QPSK modulated signal of magnitude 1000, i.e. 70 dB above the transmitted signal power P s . The SNR is set to 10 dB as defined in (2.56), where the noise terms z l (n) do not include the jamming signal.

Figure 2 . 10 :

 210 Figure 2.10: Capon, MUSIC, and GLRT spectra in the presence of a jammer at 20 • .

Figure 2 . 11 :

 211 Figure 2.11: MSE in θ for (a) Capon, (b) MUSIC, and (c) GLRT, for different SNR (plane-wave processing).

Figure 2 .

 2 Figure 2.12, where the MSEs of the DOA estimated by the GLRT using spherical-wave and plane-wave processing are compared in the case of an SNR of 10 dB. Note that both MSEs have approximately the same values after the plane-wave condition R > 5∆ 2 /λ,

Figure 2 . 12 :

 212 Figure 2.12: MSE in θ for GLRT after spherical-wave or plane-wave processing (SNR = 10 dB).

Figure 2 . 13 :

 213 Figure 2.13: MSE in θ for (a) Capon, (b) MUSIC, and (c) GLRT, for different antenna array sizes (plane-wave processing).

1 .Figure 3 . 1 :

 131 Figure 3.1: Flow diagram of the wideband simulations.

  version of the other ones, which leads to a set of correlated signals and a rank deficient auto-covariance matrix R c . However, the use of uncorrelated signals is usually desired before applying adaptive processing algorithms. In order to get a full-rank auto-covariance matrix, a random phase {φ(p)} N -1 p=0 can be included in the design of the {C(p)} N -1 p=0 . Indeed, {C(p)} N -1 p=0 and {C(p)e jφ(p) } N -1 p=0 lead to the same value of P (θ, f p + f c ) in the mono-target case. The idea of the SFBT is then to construct the signals in discrete-frequency domain {C(p)} N -1 p=0 , in the case of multiple targets, as C(p) = K k=1 a t ( θk , p)e jφ k (p) p = 0, . . . , N -1, (3.33) where { θk } K k=1 are initial estimates of the target DOAs, and {φ k (p)} take random values in [0, 2π]. Finally, the signals to be transmitted are obtained by performing the IDFT of {C(p)} N -1

Figure 3 .

 3 Figure 3.2 shows the beampattern (calculated as (3.25)) synthesized by WBFIT with = 2. The beampattern is designed for a symbol frequency F s = f c /5 = 200 MHz and using a grid of angles θ ∈ [-90 • , 90 • ] with a mesh grid size of 0.1 • . We can see that the beampattern has a mainlobe in the direction θ = 40 • . The mainlobe has approximately the same amplitude and width over the whole range of frequencies. As shown in Figure 3.3, the signals synthesized by WBFIT satisfy the PAPR constraint: The PAPR of every sequence {ci (n)} N -1 n=0 is equal to 2.If a very low PAPR constraint is used instead, the beampattern may be degraded. As shown in Figure3.4, the beampattern synthesized for = 1 becomes uneven.

Figure 3 . 2 :

 32 Figure 3.2: The WBFIT beampattern in dB with = 2 and F s = f c /5 = 200 MHz (1 target at θ = 40 • ). (a) 2D plot, (b) 3D plot.

Figure 3 . 3 :

 33 Figure 3.3: PAPR of every {c i (n)} N -1 n=0 synthesized by WBFIT ( = 2).

Figure 3 . 4 :

 34 Figure 3.4: The WBFIT beampattern in dB with = 1 and F s = f c /5 = 200 MHz (1 target at θ = 40 • ). (a) 2D plot, (b) 3D plot.

Figure 3 . 5 :

 35 Figure 3.5: PAPR of every {c i (n)} N -1 n=0 synthesized by WBFIT ( = 1).

Figure 3 . 6 :

 36 Figure 3.6: The SFBT beampattern in dB with F s = f c /5 = 200 MHz (1 target at θ = 40 • ). (a) 2D plot, (b) 3D plot.

Figure 3 . 7 :

 37 Figure 3.7: PAPR of every {c i (n)} N -1 n=0 synthesized by SFBT.

Figure 3 . 8 :Figure 3 . 9 :

 3839 Figure 3.8: The WBFIT beampattern in dB with = 2 and F s = f c /2 = 500 MHz (1 target at θ = 40 • ). (a) 2D plot, (b) 3D plot.

  , other θ p = 0, . . . , N -1.

Figure 3 . 10 :

 310 Figure 3.10: The SFBT beampattern in dB with F s = f c /2 = 500 MHz (1 target at θ = 40 • ). (a) 2D plot, (b) 3D plot.

Figure 3 . 11 :

 311 Figure 3.11: The SFBT beampattern in dB at f c .

Figure 3 . 12 :Figure 3 . 13 :

 312313 Figure 3.12: The WBFIT beampattern in dB with = 2 and F s = f c /5 = 200 MHz ( θ1 = -60 • , θ2 = 0 • , θ3 = 40 • ). (a) 2D plot, (b) 3D plot.

Figure 3 . 14 :Figure 3 . 15 :

 314315 Figure 3.14: The WBFIT beampattern in dB with = 2 and F s = f c /2 = 500 MHz ( θ1 = -60 • , θ2 = 0 • , θ3 = 40 • ). (a) 2D plot, (b) 3D plot.

Figure 3 . 16 :

 316 Figure 3.16: The M-SFBT beampattern in dB with F s = f c /5 = 200 MHz ( θ1 = -60 • , θ2 = 0 • , θ3 = 40 • ). (a) 2D plot, (b) 3D plot.

Figure 3 . 17 :

 317 Figure 3.17: PAPR of every {c i (n)} N -1 n=0 synthesized by M-SFBT.

  .

  [START_REF] Dzvonkovskaya | Simulation of tsunami signatures in ocean surface current maps measured by HF radar[END_REF]

  .

Figure 3 . 18 :

 318 Figure 3.18: The M-SFBT beampattern in dB with = 2 and F s = f c /5 = 200 MHz ( θ1 = -60 • , θ2 = 0 • , θ3 = 40 • ). (a) 2D plot, (b) 3D plot.

Figure 3 . 19 :

 319 Figure 3.19: PAPR of every {c i (n)} N -1 n=0 synthesized by M-SFBT with = 2.

Figure 3 .Figure 3 . 20 :

 3320 Figure 3.20 shows the WBFIT multiband beampattern under the PAPR constraint = 2. As expected, the WBFIT beampattern is smoother than the M-SFBT's counterpart. It also has a narrower lobe at -60 • . However, WBFIT is much more computationally expensive than M-SFBT.

a 1 2 1 = 1 =

 111 r (θ, ref ) at the reference frequency f ref into the corresponding steering vector at any other frequency f p , i.e. Φ(∆f p , θ)a r (θ, ref ) = e j2π(fc+f ref +∆fp)( Lr --l) dr sin θ v l=0,...,Lre j2π(fc+fp)( Lr -1 2 -l) dr sin θ v l=0,...,Lra r (θ, p).

  ) uses the DFT to decompose the signals into several narrowband components, the narrowband Capon, MUSIC, and GLRT techniques described in Section 2.2 can be applied in every frequency bin. The individual spatial spectra can then be averaged to obtain the DOA estimates as we proposed in[START_REF] Gómez | On wideband MIMO radar: Detection techniques based on a DFT signal model and performance comparison[END_REF].

  Rc (f p ) is the estimated auto-covariance matrix of the transmitted signals at frequency f p , and Rxc (f p ) is the estimated cross-covariance matrix between the received and the transmitted signals at frequency f p .

  2 for F s = f c /5 = 200 MHz. The covariance matrices are estimated for N f = 65 frequency components f p as described in Appendix B. For all the simulations, K = 3 targets located in the plane-wave region at θ 1 = -30 • , θ 2 = 0 • , and θ 3 = 60 • are considered, and the {β k } K k=1 are set to 1. An initial omnidirectional stage is performed by transmitting independent random sequences {c(n)} N -1 n=0 of N = 512 QPSK symbols. The narrowband Capon, MUSIC, and GLRT techniques are then applied in every frequency bin. In a second stage, SBFT is used to match a desired beampattern using the DOA estimates obtained in the omnidirectional stage.

Figure 3 .

 3 Figure 3.21 shows the Capon, MUSIC, and GLRT angle and frequency dependent spatial spectra (P cap (θ, f p ), P M U SIC (θ, f p ), and ρ(θ, f p )), obtained after the transmission of independent signals. The reciprocal of noise level is set to -10 log 10 σ 2 = 0. As expected, MUSIC has the best resolution over the whole bandwidth. The results obtained at every frequency are averaged to calculate the general spectra Pcap (θ), PMUSIC (θ), and ρ(θ) according to (3.56), (3.59), and (3.62) respectively. As shown in Figure 3.22, the target DOAs can be found by searching for the maxima in any of the three averaged spectra.

Figure 3 .

 3 Figure 3.23 shows the MSEs of the estimates obtained using Capon, MUSIC, and GLRTfor the target at -30 • . As we can see, the GLRT, despite its apparent lower resolution, has better performance than Capon and MUSIC in the omnidirectional stage. MUSIC has slightly better performance than Capon in the whole range of noise levels and has even better performance than GLRT at low noise level (-10 log 10 σ 2 = 15).

Figure 3 . 21 :Figure 3 . 22 :

 321322 Figure 3.21: Stacked narrowband spectra of (a) Capon, (b) MUSIC, and (c) GLRT after the omnidirectional stage (θ 1 = -30 • , θ 2 = 0 • , θ 3 = 60 • , F s = f c /5 = 200 MHz).

Figure 3 . 23 :Figure 3 . 24 :

 323324 Figure 3.23: MSE in θ for the target at -30 • using the Capon, MUSIC, and GLRT estimates after the omnidirectional stage (F s = f c /5 = 200 MHz).

Figure 3 . 25 :

 325 Figure 3.25: Incoherent spatial spectra of (a) Capon, (b) MUSIC, and (c) GLRT after the SFBT stage (θ 1 = -30 • , θ 2 = 0 • , θ 3 = 60 • , F s = f c /5 = 200 MHz).

Figure 3 . 26 :

 326 Figure 3.26: MSE in θ for the target at -30 • using the Capon, MUSIC, and GLRT estimates after the SFBT stage (F s = f c /5 = 200 MHz).

Figure 3 . 27 :

 327 Figure 3.27: MSE in θ for the target at -30 • using the estimates given by (a) Capon, (b) MUSIC, and (c) GLRT (F s = f c /5 = 200 MHz).

  omnidirectional stage is performed by transmitting independent sequences {c(n)} N -1 n=0 of N = 512 symbols. The symbol frequency is set to F s = f c /5 = 200 MHz. Let us directly apply TOPS to the received signals as described in Section 3.5.1.3.

Figure 3 .

 3 Figure 3.28 shows the TOPS spatial spectrum obtained using (3.53), for a reciprocal of noise level of -10 log 10 σ 2 = 10. Since the received signals are not uncorrelated, several false peaks appear around the true DOAs which could lead to a wrong estimation of the target directions.

Figure 3 . 28 :Figure 3 . 29 :

 328329 Figure 3.28: The TOPS spectrum after the omnidirectional stage (θ 1 = -30 • , θ 2 = 0 • , θ 3 = 60 • , F s = f c /5 = 200 MHz).

Figure 3 .

 3 Figure 3.30 shows the TOPS spatial spectrum P M -T OP S (θ) calculated from (3.67).

Figure 3 .Figure 3 . 30 :

 3330 Figure 3.31 shows the MSE of M-TOPS for the target at -30 • (computed using 500 Monte Carlo trials) compared with the previously obtained MSE curves of the incoherent Capon, MUSIC, and GLRT techniques after the SFBT stage. As we can see, the incoherent methods have better DOA estimation performance than M-TOPS.

Figure 3 . 31 :

 331 Figure 3.31: MSE in θ for the target at -30 • using the Capon, MUSIC, and GLRT estimates after the SFBT stage and the M-TOPS estimates after the M-SFBT stage (F s = f c /5 = 200 MHz).

A

  wideband signal model of MIMO radar was presented in Section 3.1. In this model, the received signals are decomposed into several narrowband components by using the DFT. Next, two recently proposed wideband waveform design techniques, WFBIT and SFBT, were described and compared in Section 3.3. WBFIT can synthesize low PAPR sequences and beampatterns that are usually smoother than those obtained by SFBT. However, the performance of WBFIT is seriously degraded when the bandwidth is relatively large (F s = f c /2): The beams are deformed and parasitic lobes might appear around the directions of interest. Moreover, WFBIT is an iterative process that requires a considerable amount of computing time. On the other hand, SFBT works well even in the case of relatively large bandwidths (F s = f c /2) and is much faster to compute than WFBIT (at least 2000 times). However, the sequences synthesized by SFBT usually have high PAPR. For this reason, a multiband waveform design technique (M-SFBT) based on SFBT was proposed in Section 3.4. The M-SFBT is used to transmit the power directly to the targets while allocating a different non-overlapping frequency band to each one. The use of multiband beampatterns allows receiving uncorrelated signals from the targets. Moreover, the signals synthesized by M-SFBT meet a PAPR constraint similar to WBFIT. In Section 3.5 some target DOA estimation techniques were presented based on the existing wideband array processing techniques. The incoherent methods consist in applying narrowband techniques (Capon, MUSIC, and GLRT) at each frequency component and averaging the results over frequency to obtain a general spectrum. The simulations showed that the use of SFBT improves the DOA estimation performance of the incoherent techniques compared to an omnidirectional probing. Even though the three incoherent techniques, Capon, MUSIC, and GLRT have similar estimation performance after the omnidirectional stage, the incoherent MUSIC algorithm has the minimum MSE after the SFBT stage. TOPS is a more sophisticated method which exploits the orthogonality between the signal-plus-noise and the noise subspaces at different frequencies. However, TOPS cannot be successfully applied when the received signals are correlated: Several false peaks appear around the target directions in the TOPS spectrum, which could lead to a wrong detection. As described in Section 3.5.3, after transmitting a multiband beampattern, the received signals are uncorrelated and the target DOAs can be successfully estimated by performing TOPS in each frequency band. Our contributions presented in this chapter are the proposition of a multiband waveform design technique (M-SFBT) which allows receiving uncorrelated signals from the targets, the adaptation of narrowband DOA estimation techniques to the wideband case, and the adaptation of TOPS to the context of wideband MIMO radar with colocated antennas.

Figure 4 . 1 .

 41 The antenna, designed to resonate at 5.8 GHz, is coaxially fed and has a RT5880 substrate of dielectric constant ε r = 2.2. The normalized radiation pattern of the patch antenna at 5.8 GHz, obtained by electromagnetic simulations in CST Microwave Studio, is shown in Figure4.2. We do not see any particular distortion in the radiation pattern since there is no mutual coupling in an isolated element. For simplicity and given that the arrays of interest in this document detect targets only in azimuthal directions (θ), in the following simulations the radiation patterns are assumed to be two-dimensional, but only the component at phi = 0 • is of interest.

Figure 4 . 1 :

 41 Figure 4.1: Linearly polarized patch antenna.

Figure 4 . 2 :

 42 Figure 4.2: Normalized radiation pattern (in magnitude) of an isolated patch antenna at 5.8 GHz in (a) 3D and (b) 2D (cutting plane phi = 0 • ).

Figure 4 . 3 :

 43 Figure 4.3: Transmitter and receiver arrays of patch antennas resonating at 5.8 GHz (L = L t = L r = 6).

4. 1 . 2

 12 Radiation Patterns of the Elements of a Transmitter and a Receiver Array Consider now the transmitter and receiver arrays of L = L t = L r = 6 elements shown in Figure 4.3. The elements are identical to the patch antenna of Figure 4.1, the interelement spacings are d t = d r = λ/2 and the arrays are separated by a distance of 2λ. The receiving elements are placed at ports 1 to 6 while the transmitting elements are at ports 7 to 12.

Figures 4 .

 4 Figures 4.4 and 4.5 show the radiation patterns at 5.8 GHz of the antenna elements of the receiver and the transmitter arrays respectively. As we can see, all the radiation patterns are deformed and totally differ from that of an isolated element. Moreover, even if the arrays are two ULAs, the radiation patterns are all different. We must however note that the radiation patterns of the elements of a same array are symmetrical because of the geometry of the array. Also, the radiation patterns of the receiving elements are very similar to those of the transmitting elements.

Figure 4 . 4 :

 44 Figure 4.4: Normalized radiation patterns (in magnitude) of the receiving elements at 5.8 GHz (ports 1 to 6).

Figure 4 . 5 :

 45 Figure 4.5: Normalized radiation patterns (in magnitude) of the transmitting elements at 5.8 GHz (ports 7 to 12).

  figure 4.3 and using the signal model

1 ( 4 . 5 )

 145 ) = g * r,l (θ)e j 2π λ ( Lr -1 2 -l)dr sin θ l=0,...,Lrare the general steering vectors which include the different radiation patterns. The radiation patterns used here are those obtained by electromagnetic simulations and shown in Figures 4.4 and 4.5. In practice, the radiation patterns can be measured or computed using the active element pattern method [57][72]. Consider K = 3 targets located in the plane-wave region at θ 1 = -40 • , θ 2 = 20 • , θ 3 = 40 • with reflection coefficients β 1 = β 2 = β 3 = β = 1. Let us apply the narrowband Capon, MUSIC, and GLRT techniques neglecting the pattern distortions: The different spatial spectra are computed using the standard steering vectors a t (θ) and a r (θ). The spectra obtained for a reciprocal of noise level of -10 log 10 σ 2 = 20 are shown in Figure 4.6. As we can see, the peaks in the spectra are not centered in the target directions in none of the three techniques, which leads to biased DOA estimates. We can also note that the resolution of MUSIC is significantly reduced: The lobes in the MUSIC spectrum are almost as wide as those in the Capon spectrum.

Figure 4 .Figure 4 . 6 :

 446 Figure 4.6: Capon, MUSIC, and GLRT spectra for three targets at θ 1 = -40 • , θ 2 = 20 • , and θ 3 = 40 • , neglecting the pattern distortions (-10 log 10 σ 2 = 20).

Figure 4 . 7 :Figure 4 . 8 :

 4748 Figure 4.7: Capon, MUSIC, and GLRT spectra for three targets at θ 1 = -40 • , θ 2 = 20 • , and θ 3 = 40 • , taking the radiation patterns into account (-10 log 10 σ 2 = 20).

Figure 4 . 9 :Figure 4 . 10 :

 49410 Figure 4.9: Capon, MUSIC, and GLRT spectra for three targets at θ 1 = -40 • , θ 2 = -5 • , and θ 3 = 5 • , taking the radiation patterns into account (-10 log 10 σ 2 = 20).

Figure 4 . 11 :

 411 Figure 4.11: Capon, MUSIC, and GLRT spectra for one target at -40 • , taking the radiation patterns into account (-10 log 10 σ 2 = -10).

Figure 4 .

 4 [START_REF] Ma | Mono-static MIMO radar array design for interferences suppressing[END_REF] shows the Capon, MUSIC, and GLRT spectra obtained using the steering vectors ãt (θ) and ãr (θ) for the same noise level (-10 log 10 σ 2 = -10).

. 7 )L t - 1 2 1 ( 4 1 .

 71141 Given that the magnitudes of the radiation patterns are close to zero for angles close to -90 • and 90 • (seeFigures 4.4 and 4.5), P cap (θ) and P M U SIC (θ) may have high values when |θ| tends to 90 • . This regrowth seems to be accentuated at high noise levels.This problem might be solved by including only the phase of the radiation patterns in the steering vectors. Let us define the general phase-only steering vectors as α t (θ) = e -jarg{g t,i (θ)} e j 2π λ ( -i)dt sin θ i=0,...,Ltθ) = e -jarg{g r,l (θ)} e j 2π λ ( Lr -1 2 -l)dr sin θ l=0,...,Lr-

Figure 4 . 12 :

 412 Figure 4.12: Capon, MUSIC, and GLRT spectra for one target at -40 • , taking only the phase of the radiation patterns into account (-10 log 10 σ 2 = -10).

4. 1 . 4 DOA

 14 Estimation Performance in the Presence of Distorted Radiation Patterns In order to evaluate the DOA estimation performance of narrowband Capon, MUSIC, and GLRT techniques in the presence of distorted radiation patterns, the MSE for one target located at θ 1 = -40 • (β = 1) is computed using 500 Monte Carlo trials. The MSE in the DOA estimated in degrees is computed for four different cases:

Figure 4 . 13 :

 413 Figure 4.13: MSE in θ for one target at -40 • using the estimates given by (a) Capon, (b) MUSIC, and (c) GLRT.

  effectiveness of the crosstalk reduction technique are presented via MATLAB simulations. The validity of this technique is illustrated in Chapter 5 by experimental results using real hardware. Consider a narrowband MIMO radar with colocated antennas whose transmitter and receiver arrays are the two ULAs of L = L t = L r = 6 elements. The simulations are performed using the signal model (4.11) considering K = 2 targets located in the planewave region at θ 1 = -20 • and θ 2 = 20 • , and a reciprocal of noise level of -10 log 10 σ 2 = 20. Both the real and the imaginary parts of the coefficients of the crosstalk matrix M are randomly generated and uniformly distributed in the open interval (-1/ √ 2 , 1/ √ 2).

  )which does not take the different radiation patterns into account.The Capon, MUSIC, and GLRT spatial spectra computed from the received signals x(n) (using a r (θ) and a t (θ)) are shown in Figure4.14. As we can see, the GLRT spectrum is highly affected by crosstalk as several secondary lobes appear around the true DOAs, which can lead to a wrong detection. Actually, the crosstalk term Mc(n) can be seen as a noise correlated with the transmitted signals, which is not consistent with the assumptions made in the definition of the GLRT and explains the sensitivity of the latter to crosstalk. In contrast, Capon and MUSIC are less sensitive to this phenomenon given that the crosstalk term Mc(n) is not expressed in terms of a steering vector and hence it is included in the "noise-only" subspace. However, the resolution of Capon and MUSIC is degraded and the estimated DOAs are actually biased.The crosstalk matrix is then estimated using (4.18) after simulating a target-free environment. Next, the crosstalk is reduced from the received signals by computing(4.19). Finally, the Capon, MUSIC, and GLRT spatial spectra are computed from x sc (n). As shown in Figure4.15, after the crosstalk reduction the resolution of Capon and MUSIC is significantly improved. Moreover, no secondary lobes appear in the GLRT spectrum, which allows an appropriate estimation of the target DOAs.

Figure 4 . 14 :Figure 4 . 15 :Figure 4 . 16 :Figure 4 . 17 :

 414415416417 Figure 4.14: Capon, MUSIC, and GLRT spectra for two targets at θ 1 = -20 • , and θ 2 = 20 • before crosstalk reduction (ideal case).

5 Experimental

 5 showed that the resolution of Capon, MUSIC, and GLRT is affected by crosstalk or by interference which is correlated with the transmitted signals. It is clear that the GLRT is much more sensitive to this phenomenon than Capon and MUSIC, and is unable to detect the targets. In order to overcome this problem, we presented a more realistic signal model(4.11) which takes mutual coupling into account and should always be used in the case of narrowband MIMO radars with colocated antennas. We then proposed a crosstalk reduction technique: The crosstalk matrix is first estimated (based on a minimum MSE approach) from a transmission in an environment without any target, and the crosstalk term is finally subtracted from the received signals. The simulation results showed that after crosstalk reduction, there are no longer secondary lobes in the GLRT spectrum and the resolution of Capon and MUSIC is improved, which makes possible an efficient estimation of the target DOAs.Our contributions presented in this chapter are the proposition of a more realistic signal model which takes mutual coupling into account, the introduction of the phase-only steering vectors which deal with distorted radiation patterns, and the proposition of a crosstalk reduction technique.Chapter Platform for MIMO Radar with Colocated AntennasSeveral DOA estimation techniques for a narrowband MIMO radar with colocated antennas have been studied and compared in Chapters 2 and 4. However, all of these techniques have been developed and simulated from a theoretical point of view assuming ideal conditions, e.g. punctual targets, transmission on an additive white Gaussian noise channel, and absence of multi-path phenomenon. In this chapter, we present an experimental platform for MIMO radar which allows testing the previously proposed DOA estimation techniques in nearly real conditions. The developed measurement platform is described in Section 5.1, while the synchronization and calibration procedures are described in Sections 5.2 and 5.3 respectively. Finally, some measurement results are presented in Section 5.4, which includes a repeatability test of the proposed platform, DOA estimation, and crosstalk reduction.

Figure 5 . 1 :Figure 5 . 2 :

 5152 Figure 5.1: Scheme of the antenna configuration of the measurement platform.

Figure 5 . 2 .

 52 Figure 5.2. Note that this platform does not allow taking the whole effects of mutual coupling into account since we have the same radiation pattern at every position of the Tx and Rx antennas.

Figure 5 . 3 :

 53 Figure 5.3: Tx/Rx RF architecture block diagram.

Figure 5 . 4 :Figure 5 . 5 :

 5455 Figure 5.4: System overview.

5. 2 . 1

 21 Evaluation of the phase synchronization error in a wired transmission To evaluate the phase errors introduced by an inaccurate synchronization we consider the wired transmission shown in Figure 5.6.

Figure 5 . 6 :Figure 5 . 7 :

 5657 Figure 5.6: Configuration of a first synchronization test.

Figure 5 . 8 :

 58 Figure 5.8: Example of timing diagram of the baseband transmit signal using a reference signal.

Figure 5 . 9 :

 59 Figure 5.9: Flow charts of (a) the ADC reference configuration process and (b) the attenuation configuration process.

Figure 5 . 10 :

 510 Figure 5.10: Flow chart of ADC/attenuation calibration process.

Figure 5 . 11 :

 511 Figure 5.11: Target positioning scheme.

Figure 5 .

 5 Figure 5.12 shows the DOA estimated by Capon and MUSIC as a function of the trial index.

Figure 5 . 12 :

 512 Figure 5.12: Repeatability test of the DOA estimation using Capon and MUSIC (one target at [θ, R] = [-6.5 • , 1.8 m]).

  , 90 • ] with a mesh grid size of 0.1 • and a grid of distances R ∈ [1 m, 2.5 m] with a step of 0.01 m. The results are shown in Figure 5.13. As we can see, the Capon and MUSIC spectra show two lobes close to the target DOAs (the estimated DOAs are [ θ1 , θ2 ] Capon = [-15.3 • , 18.5 • ] and [ θ1 , θ2 ] MUSIC = [-15.2 • , 18.5 • ]). In contrast, we are unable to detect the targets by using the GLRT since several secondary lobes appear around the target DOAs. The secondary lobes can be seen more clearly by plotting the component of the spectra at

Figure 5 . 13 :Figure 5 . 14 :

 513514 Figure 5.13: (a) Capon, (b) MUSIC, and (c) GLRT spectra (experimental measurements with two targets at [θ 1 , R 1 ] = [-15 • , 1.7 m] and [θ 2 , R 2 ] = [18 • , 1.7 m]).

Figure 5 . 15 .Figure 5 . 15 :

 515515 Figure 5.15. As we can see, the relative standard deviations of | ml,i | are lower than 3.5% and the circular standard deviations [73] of arg { ml,i } are lower than 0.035 which

5. 4 . 3 . 2

 432 Subtraction of the Crosstalk Term Let us consider the previous configuration of two targets at [θ 1 , R 1 ] = [-15 • , 1.7 m] and [θ 2 , R 2 ] = [18 • , 1.7 m]. We can now reduce the crosstalk term Mc(n) from the received signals by computing

  and 5.17, the resolution of Capon and MUSIC seems to be improved since their corresponding spectra exhibit sharper lobes after the crosstalk reduction (the estimated DOAs are now [ θ1 , θ2 ] Capon = [-15.5 • , 18 • ] and [ θ1 , θ2 ] MUSIC = [-15.3 • , 18.1 • ]

Figure 5 . 16 :

 516 Figure 5.16: (a) Capon, (b) MUSIC, and (c) GLRT spectra after crosstalk reduction (experimental measurements with two targets at [θ 1 , R 1 ] = [-15 • , 1.7 m] and [θ 2 , R 2 ] = [18 • , 1.7 m]).

Figure 5 . 17 :

 517 Figure 5.17: Component of the Capon, MUSIC, and GLRT spectra at R = 1.7 m after crosstalk reduction (experimental measurements with two targets at [θ 1 , R 1 ] = [-15 • , 1.7 m] and [θ 2 , R 2 ] = [18 • , 1.7 m]).

  and 5.19. As we can see, several secondary lobes are significantly attenuated by the addition of white Gaussian noise only. However, some secondary lobes are still present around the target DOAs because of crosstalk.

Figure 5 . 18 :Figure 5 . 19 :

 518519 Figure 5.18: GLRT spectrum after the addition of white Gaussian noise and before crosstalk reduction (experimental measurements with two targets at [θ 1 , R 1 ] = [-15 • , 1.7 m] and [θ 2 , R 2 ] = [18 • , 1.7 m]).

Figure 5 . 20 :Figure 5 . 21 :

 520521 Figure 5.20: GLRT spectrum after the addition of white Gaussian noise and after crosstalk reduction (experimental measurements with two targets at [θ 1 , R 1 ] = [-15 • , 1.7 m] and [θ 2 , R 2 ] = [18 • , 1.7 m]).

Chapter 3 ,

 3 we extended the signal model to the case of wideband signals. Next, we investigated and compared two recently proposed wideband waveform design techniques, the WBFIT and the SFBT. The WBFIT is an iterative technique which can synthesize low PAPR sequences with transmitted beampatterns usually smoother than those obtained with the SBFT. However, the performance of WBFIT is poor for relatively large bandwidth, such as F s = f c /2, leading to deformed beampatterns. On the other hand, the SFBT works well even in the case of relatively large bandwidths and is at least 2000 times faster to compute than the WBFIT. However, the sequences synthesized by the SFBT usually have high PAPR. Based on those two techniques, we proposed a modified version of SFBT, called M-SFBT, which meets a PAPR constraint and allows transmitting the power directly to the targets while allocating a different non-overlapping frequency band to each one. The use of multiband beampatterns is advantageous since it allows receiving uncorrelated signals from the targets, which makes possible the adaptation of wideband array processing techniques (such as TOPS) to the context of wideband MIMO radar. Note that, although the simulations presented in Chapter 3 allowed us to obtain relevant information on the performance of the waveform design techniques, they were done assuming ideal conditions, i.e.omitting the radiation patterns of the antennas' elements and neglecting mutual coupling. In fact, the actual beampatterns of real antenna arrays may differ from those shown in the simulation results. Such characteristics should be taken into account in further research in order to investigate the performance of the wideband waveform design techniques under more realistic conditions.We also presented some wideband DOA estimation techniques based on the existing wideband array processing techniques. We first introduced the incoherent methods for wideband MIMO radar, which consist in applying narrowband DOA estimation techniques (such as Capon, MUSIC and GLRT) at several narrowband frequency components and averaging the results to obtain a general spatial spectrum. The performance tests showed that the incoherent Capon, MUSIC, and GLRT techniques have similar performance after transmitting an omnidirectional pattern; however, the GLRT gives the minimum MSE on the DOA. After transmitting a beampattern synthesized by SFBT, the performance of the three techniques is improved and MUSIC outperforms Capon and GLRT at low noise levels (-10log 10 σ 2 > 10). Also, we proposed an adaptation of TOPS to the context of wideband MIMO radar.TOPS was originally developed to estimate the DOAs of uncorrelated sources and cannot be successfully applied if the signals reflected by the targets are correlated. We then proposed an adaptation of TOPS to the context of MIMO radar. When used along with a multiband beampattern (generated for instance by M-SFBT), we showed that the targets can be properly detected.In Chapter 4, we took into account the electromagnetic interactions between the different antenna elements in order to introduce a more realistic signal model. By combining electromagnetic simulations with signal processing, we were able to evaluate the performance of the narrowband DOA estimation techniques in the presence of mutual coupling. We showed that the existence of mutual coupling introduces distortions in the radiation patterns which degrade the DOA estimation performance of Capon, MUSIC, and GLRT. We then showed that taking into account the different radiation patterns in the expressions of the steering vectors allows improving the DOA estimation performance. Given that some techniques might present regrowth at the spectrum edges (e.g. Capon and MUSIC) caused by low magnitudes of the radiation patterns, we can alternatively omit the magnitudes and use only the phase of the radiation patterns without introducing any significant error. On the other hand, the weak amplitudes of the radiation patterns at large angles (in absolute value) make targets located at angles close to -90 • or 90 • difficult to detect. A new challenge involving antenna and waveform designs is then to improve the DOA estimation of targets located at absolute large angles.

1 . 2 . 4 .

 124 The performance of the wideband waveform design techniques (WBFIT, SFBT, and M-SFBT) has to be investigated and compared considering the radiation pattern of every transmitter antenna element. From a theoretical point of view, this can be done by simulating wideband antenna arrays to obtain the radiation patterns at every frequency component of interest. Then, the so-obtained radiation patterns could be included in the wideband steering vectors to simulate more realistic transmit beampatterns. The effects of mutual coupling have to be investigated in the case of wideband signals by combining electromagnetic simulations with signal processing, similarly to the narrowband case. The different radiation patterns obtained at different frequency components have to be included in the transmit and receive steering vectors to evaluate the influence of the pattern distortion on the wideband DOA estimation performance. Additionally, crosstalk has to be investigated also in the case of wideband signals, and new methods of crosstalk reduction have to be explored. One method may rely on the estimation of a different crosstalk matrix at every frequency component via a signal processing approach. The crosstalk matrices might also be obtained by measuring the transmission S-parameters between the transmitter and the receiver elements.3. The possibility of improving DOA estimation of targets located close to -90 • or 90 • has to be explored. This problem might be addressed by exploiting the antenna pattern diversity and/or the waveform diversity. The possibilities of improving the DOA estimation performance by exploiting the wave polarization diversity could be explored. Indeed, different kinds of targets may produce different types of reflections depending on the polarization of the impinging waves as it is often seen in polarimetric and weather-type radars[START_REF] Carter | Geologic studies of planetary surfaces using radar polarimetric imaging[END_REF]-[START_REF] Galletti | Degree of polarization at horizontal transmit: Theory and applications for weather radar[END_REF].

  .14) can be written asmin {c(n)},{ψp} G T -F H C T 2 .(B.17)Given that (1/ √ N )F is a unitary matrix, the optimization problem can finally be written

3 )Figure C. 1 :

 31 Figure C.1: réseau d'antennes linéaire et uniforme.

Figure C. 2 :Figure C. 3 :

 23 Figure C.2: spectres spatiaux de Capon, MUSIC et GLRT en présence d'un brouilleur à 20 • .

Figure C. 4 : 1 2

 41 Figure C.4: MSE en θ de la DOA estimée par (a) Capon, (b) MUSIC et (c) GLRT, pour différents SNR (traitement d'ondes planes).

C. 2 . 2 . 3

 223 figures C.5 et C.6 respectivement. On peut constater que dans les deux diagrammes de rayonnement, un maximum de puissance est bien émis dans la direction de la cible dans toute la bande de fréquence. Les diagrammes de WBFIT et de SFBT sont très similaires, cependant, le lobe du diagramme de SFBT est un peu plus large que celui de WBFIT pour les basses fréquences.

Figure C. 5 :Figure C. 6 :

 56 Figure C.5: diagramme de rayonnement WBFIT en dB avec = 2, F s = f c /5 = 200 MHz (1 cible à θ = 40 • ). (a) Tracé en 2D, (b) tracé en 3D.

Figure C. 7 :Figure C. 8 :

 78 Figure C.7: PAPR de chaque {c i (n)} N -1 n=0 synthétisé par (a) WBFIT ( = 2) et (b) SFBT.

(C. 23 )

 23 figure C.8. Dans ce cas, nous avons décidé d'envoyer la puissance dans les directions de K = 3 cibles dont les premières estimées sont θ1 = -60 • , θ2 = 0 • et θ3 = 40 • , tout en utilisant des bandes de fréquence distinctes suivant l'allocation de puissance désirée

Figure C. 9 :

 9 Figure C.9: PAPR de chaque {c i (n)} N -1 n=0 synthétisé par M-SFBT avec = 2.

Figure C. 10 : 1 2 1 , 11 -Figure C. 11 :Figure C. 12 :Figure C. 13 :

 101111111213 Figure C.10: spectres spatiaux incohérents de (a) Capon, (b) MUSIC et (c) GLRT (θ 1 = -30 • , θ 2 = 0 • , θ 3 = 60 • , F s = f c /5 = 200 MHz).

12 Figure C. 15 :

 1215 Figure C.15: diagrammes de rayonnement normalisés (en amplitude) des éléments récepteurs à 5.8 GHz (ports 1 à 6).

12 Figure C. 16 :L t - 1 2 1 ( 1 (C. 41 )- 5 •Figure C. 17 :

 121611141517 Figure C.16: diagrammes de rayonnement normalisés (en amplitude) des éléments émetteurs à 5.8 GHz (ports 7 à 12).

Figure C. 18 :L t - 1 2 1 ,

 1811 Figure C.18: spectres de Capon, MUSIC et GLRT dans le cas de trois cibles à θ 1 = -40 • , θ 2 = -5 • et θ 3 = 5 • , avec prise en compte des diagrammes de rayonnement (-10 log 10 σ 2 = 20).

•

  Cas du traitement basé sur le couplage mutuel (CM) : la propagation des signaux est simulée en incluant les diagrammes de rayonnement dans le modèledu signal x(n) = βã * r (θ 1 )ã H t (θ 1 )c(n) + z(n), et les spectres de Capon, MUSIC et GLRT sont calculés en utilisant les vecteurs directionnels généraux ãr (θ) et ãt (θ).• Cas du traitement basé sur le couplage mutuel (CM) utilisant la phase uniquement : la propagation des signaux est simulée en incluant les diagrammes de rayonnement dans le modèle du signalx(n) = βã * r (θ 1 )ã H t (θ 1 )c(n) + z(n),et les spectres de Capon, MUSIC et GLRT sont calculés en utilisant les vecteurs directionnels généraux (phase uniquement) α r (θ) et α t (θ). Les résultats sont présentés figure C.19. On peut constater que le cas de simulation idéal donne les meilleurs résultats, cependant il s'agit d'un cas non réaliste. En revanche, le cas du traitement classique donne des erreurs importantes (fortes valeurs de la MSE pour tous les niveaux de bruit) car les diagrammes de rayonnement incluant les effets du couplage mutuel ne sont pas pris en compte pour calculer les spectres spatiaux. D'autre part, les résultats obtenus pour Capon montrent que dans le cas du traitement basé sur le couplage mutuel, la MSE a de fortes valeurs quand le niveau du bruit est élevé (-10log 10 σ 2 ≤ -5), conséquence de la remontée spectrale observée pour des angles proches de -90 • et 90 • . Lorsque les vecteurs directionnels incluant uniquement la phase des diagrammes de rayonnement sont utilisés, la MSE diminue et on obtient les résultats qui s'approchent le plus des résultats du cas idéal. Quant à MUSIC et au GLRT, les deux traitements basés sur le couplage mutuel donnent des résultats similaires : les courbes des MSE sont superposées, ce qui indique que le fait d'utiliser seulement la phase des diagrammes de rayonnement n'introduit pas d'erreurs importantes dans l'estimation de la DOA.

Figure C. 19 :mC. 4 . 2

 1942 Figure C.19: MSE en θ de la DOA estimée par (a) Capon, (b) MUSIC et (c) GLRT, pour une cible à -40 • .

Figure C. 24 :Figure C. 25 :C. 5 Conclusions

 24255 Figure C.24: spectres de Capon, MUSIC et GLRT avant réduction du crosstalk (mesures expérimentales avec deux cibles à θ 1 = -15 • et θ 2 = 18 • ).

  Even though narrowband MIMO radars have been deeply investigated in recent years, the assumptions done in the signal model no longer hold in the case of wideband signals, and narrowband detection techniques cannot be directly applied. The chapter is organized as follows: The wideband signal model of MIMO radar is described in Section 3.1, followed by the description of the simulation procedure in Section 3.2.

Additionally, random signals are suitable for military applications since they are difficult to detect by other radar systems, and offer good interference suppression capabilities if correlation processing is done at the receiver. Moreover, random signals are also very appropriate for civilian applications since they provide excellent electromagnetic compatibility (EMC) performance

[START_REF] Sun | Ultra-wideband technology and random signal radar: An ideal combination[END_REF]

[START_REF] Liu | Random signal radar -A winner in both the military and civilian operating environments[END_REF]

. In this chapter, only the random-based wideband signals are taken into account. Next, two existing wideband waveform design techniques are investigated in Section 3.3. In Section 3.4, we propose a multiband waveform design technique aiming at decorrelating the signals reflected by the targets, which allows applying many wideband array processing detection techniques in the context of MIMO radar.

  [START_REF] Shlivinski | Antenna characterization in the time domain[END_REF] étroite. En particulier, nous étudions l'influence du couplage direct entre les réseaux d'antennes d'émission et de réception (appelé "crosstalk") sur les performances des techniques proposées. Nous établissons un modèle du signal permettant de prendre en compte ce phénomène et proposons une technique de réduction du "crosstalk" qui On considère un système MIMO dont les L t antennes d'émission et les L r antennes de réception sont colocalisées. Sous l'hypothèse d'avoir K cibles localisées en champ lointain aux angles {θ k } K k=1 , le signal bande étroite reçu par le réseau de réception s'écrit en bande de base (voir chapitre 2)

	permet une estimation efficace des DOA des cibles. Nous montrons par ailleurs
	comment améliorer les performances d'estimation de DOA en présence de diagrammes
	de rayonnement incluant le couplage entre antennes. Le dernier apport principal de
	cette thèse est la conception et réalisation d'une plateforme expérimentale comportant
	une seule architecture d'émetteur-récepteur, qui permet de simuler un système MIMO
	utilisant des antennes colocalisées en appliquant le principe de superposition. Cette
	plateforme nous a permis d'évaluer et valider les performances des techniques
	proposées dans des conditions plus réalistes.
	C.1 Radar MIMO bande étroite
	C.1.1 Modèle du signal

  p)

	T
	sont
	respectivement l'ensemble des signaux reçus, l'ensemble des signaux émis et l'ensemble
	des termes de bruit à la fréquence f p .

C.2.2 Techniques existantes de conception de formes d'ondes

Dans cette section, deux techniques de conception de formes d'ondes large bande existant dans la littérature sont brièvement décrites. Ces techniques nécessitent de disposer de premières estimées des DOA des cibles.

C.2.2.1 WBFIT (Wideband Beampattern Fromation via Iterative

Techniques)

  ≥ 1 est la limite supérieure de PAPR prédéfinie, et {θ h } H h=1 est une grille d'angles couvrant l'intervalle [-90 • , 90 • ].

	C.2.2.2 SFBT (Spectral density Focusing Beampattern synthesis

2 , (C.19)

  cap (θ, f p ) est le spectre de Capon obtenu à la fréquence f p .

	Ainsi, le spectre général de Capon Pcap (θ) est obtenu en calculant la moyenne
	arithmétique des spectres obtenus à chaque fréquence
	Pcap (θ) =	1 N	N -1 p=0	P cap (θ, f p ),	(C.25)
	où P Quant à MUSIC, la moyenne est plutôt effectuée sur le test d'orthogonalité entre les
	sous-espaces signal et les sous-espaces bruit. Le spectre général de MUSIC est alors
	donné par
	PMUSIC (θ) =	N -1	1 r (θ, p)U n (f p )U H a T n (f p )a * r (θ, p)	,	(C.26)
		p=0
	où U n (f p ) est une matrice contenant les vecteurs propres du sous-espace bruit à la
	fréquence f p .

2.4.1 Adaptation des techniques d'estimation de DOA bande étroite au cas large bande (méthodes incohérentes) Etant donné que notre modèle du signal large bande (C.14) est un modèle à fréquence discrète, il est possible d'appliquer les techniques bande étroite Capon, MUSIC et GLRT, à chaque fréquence f p . La question est de savoir combiner convenablement les résultats obtenus à des fréquences distinctes, de façon à obtenir un spectre spatial général permettant de localiser les cibles. Une méthode simple consiste à calculer une moyenne des résultats obtenus sur l'ensemble des fréquences. Ces techniques sont appelées méthodes incohérentes.

Note that Equation (2.10) is general: While the transmitting and receiving arrays are both assumed to be centered at the origin of the x -y plane, the different transmitting elements may not be colocated with the receiving elements, and both arrays may have a different number of elements (i.e. Lt = Lr).

Note that the signal model (4.11) also includes the different radiation patterns.

Acknowledgements

I would like to thank the laboratory engineers of Université Paris-Est for all the help and facilities they provided me during the development of the experimental part of this project. A special thanks to David Delcroix, whose involvement and invaluable help in the development of the experimental platform were essential in the success of this project.

I also want to thank Prof. Vincent Fusco, the Queen's University Belfast and the Institute of Electronics,

Abbreviations

ADC

Analog-to-Digital Converter AWG Appendix A

Narrowband Derivations

Consider a narrowband MIMO radar system with L t transmitting antennas and L r receiving antennas. According to the signal model (2.29), the received signal due to the reflection from one target located in the plane-wave region is given by

where C and X are the matrices of the transmitted and the received signals respectively, and Z is a residual term which includes the unmodelled noise and interference. Each row of C, X, and Z contains N temporal samples. a t (θ) is the L t × 1 plane-wave transmit steering vector, a r (θ) is the L r ×1 plane-wave receive steering vector and β is the complex reflection coefficient of the target.

The mathematical developments of the narrowband Capon and GLRT techniques are described thereafter.

A.1 The Capon Beamformer

The Capon minimization is min where the derivatives ∂f ∂w and ∂g ∂w are obtained using the following derivative rule:

But then, since Z 1 , . . . , Z N are all independent from each other,

The GLR is defined as [START_REF] Kelly | An adaptive detection algorithm[END_REF] ρ

where

is the PDF of the received signal matrix X, given parameters β and R z (which is equivalent to the PDF of the residual term Z = X -βa * r (θ)a H t (θ)C) as derived in (A.9).

Therefore, the GLR exploits the difference in the PDF when there is no target and the PDF when a target with complex coefficient β is present. When a target is present, the denominator max β,Rz f (X|β, R z ) will be much greater than max Rz f (X|β = 0, R z ) and then the value of ρ(θ) will be close to one. When there is no target, the value of ρ(θ) will approach zero.

Consider the maximization in the numerator of the second term of the GLR. This

Given that Rx = 1 N XX H , and using the properties of the trace this becomes

The lower bound of this inequality will be reached when λ i = 1 for i = 1, . . . , L r . In this case, Λ = I and therefore, A = PP -1 = I, where I denotes the L r × L r identity matrix.

For this to be true, it is required that R -1 z Rx = I so R z = Rx . Finally, substituting (A.20) into (A.13) gives

Similarly, the maximum value of the denominator of the second term in the GLR can be found:

which is of the same form as (A.12). Thus, this maximum can be shown to be

It is now necessary to analyze the minimization in (A.24). Let Q be defined as

where Rc = 1 N CC H . Then, it can be shown that

Now, let µ = (a H t (θ) Rc a t (θ)) and move Q out of the determinant so the equation becomes

Consider the property on matrix determinants given by

Note that this property allows the dimensions of the matrix whose determinant is being evaluated to change.

Therefore, letting Y = βa * r (θ) -XC H at(θ) N µ , (A.27) can be written as

Therefore, the matrix argument of the second determinant has been transformed to a scalar.

Let define the scalars

Note that µ and σ are real and positive because Rc and Q are Hermitian positive semidefinite.

Considering that 1

we have

Note that the equality holds for β = η N µσ .

Then, taking into account that | Q| ≥ 0 (| Q| being Hermitian positive semidefinite), it

which can be rewritten as

XC H a t (θ) .

(A. [START_REF] Kelly | An adaptive detection algorithm[END_REF] 123

Note that the left-hand side of this inequality corresponds to (A.29), and that inequality (A.33) can achieve equality for

Now using the identity given in (A.28) to rewrite the right-hand side of inequality (A.33)

gives

Replacing µ and Q with their expressions in the equation gives

Rx is then moved out of the equation giving

Once again, the matrix whose determinant is being calculated is reshaped by the identity (A.28), by moving the matrix R-1

x a * r (θ) to the back, to give

Wideband Derivations

Consider a wideband MIMO radar system with L t transmitting antennas and L r receiving antennas. According to the signal model (3.17), the received signals due to the reflection from K targets located in the plane-wave region are given by

with

where

F s is the sampling frequency, and

T are the DFT (element-wise) of x(n), c(n), and z(n) respectively.

B.1 The WBFIT Algorithm

The Wideband Beampattern Formation via Iterative Techniques (WBFIT) algorithm was proposed in [START_REF] He | Wideband MIMO systems: Signal design for transmit beampattern synthesis[END_REF]

where the PAPR of the i th signal {c i (n)} N -1 n=0 is defined as

and ≥ 1 is a predefined threshold.

In order to solve the optimization problem (B.5), the authors propose in [START_REF] He | Wideband MIMO systems: Signal design for transmit beampattern synthesis[END_REF] a two-stage design approach, which is recalled thereafter. which minimize the following criterion:

The criterion (B.9) can be minimized with respect to {C(p)} N -1 p=0 and {φ hp } using the following cyclic algorithm:

Step 2: For {c(n)} N -1 n=0 set at their latest values, the minimizer {ψ p } N -1 p=0 in (B.18) is given by ψp = arg{ ĈH (p)q p } p = 0, . . . , N -1, (B.20)

where q T p is the (p + 1) th row of F H C T .

B.2 The Nearest-Vector Algorithm

The nearest-vector algorithm was proposed in [START_REF] Tropp | Designing structured tight frames via an alternating projection method[END_REF], and recalled in [START_REF] He | On aperiodic-correlation bounds[END_REF], to design sequences of low PAPR. Given an initial sequence y = y(0)

where ≥ 1 is a predefined threshold. Additionally, we impose the following energy constraint

C.3.2.2 Proposition d'une technique de réduction du "crosstalk"

La matrice de crosstalk peut être estimée à partir d'une première transmission dans un environnement sans cible. Dans ce cas, le modèle du signal est

On cherche la matrice de crosstalk M qui minimise le critère MSE

Ce critère d'optimisation peut être décomposé en L r problèmes de filtrage de Wiener classique (voir chapitre 4). Après un simple développement mathématique, la matrice

Le terme de crosstalk peut alors être soustrait des signaux reçus dans un cas de fonctionnement normal du radar en calculant

où M est une estimée de M, obtenue à partir de versions estimées de R xc et R c .

C.3.2.3 Simulation

On considère un radar MIMO de