Annemarie Kokosy
email: annemarie.kokosy@isen.fr

Mihai V Micea
email: mihai.micea@cs.upt.ro

Philippe Saey
email: philippe.saey@kuleuven.be

A Professional Project-Based Learning Method in Mobile Robotics*

Keywords: project-based learning, robotics, project management, industrial partnership, collaboration, team work

Due to its high potential and encouraging results, project-based learning emerges as a highly interesting paradigm in the education systems worldwide. Moreover, robotics is an interdisciplinary field where students could learn and apply their skills in mechanics, electronics, computer science, mathematics and control engineering. This paper presents a robotics projectbased learning methodology which focuses on collaborating with the industry to design, develop, evaluate, integrate and manage projects designated to be used in real-life applications. This learning method emphasizes and enables the students to apprehend the importance of fulfilling client requirements and the interactions with the client, the suppliers and with the other members of the team. The students, coached by a partner from industry, have the opportunity to apply and to improve their project management skills under a large-scale, highly complex project. This method is being applied since 2008 at ISEN Lille, France, with good results and significant impact.

INTRODUCTION

In professional applications, service robots are already having a significant impact in areas such as agriculture, surgery, logistics and underwater applications. Their importance is also growing in economic and social areas. The IFR (International Federation of Robotics) defines a service robot as a robot that performs useful tasks for humans or equipment excluding industrial automation application. As highlighted by the latest figures published by the IFR Statistical Department in 2013 [START_REF]World Robotics 2013 Service Robots[END_REF], the mobile platforms sold in 2012 featured a strong global growth, comparing with 2011 (see Fig. 1) and this growth will continue in the future years, going from 630 units in 2012 to 13000 in 2016 (as shown in Fig. 2), an increase of 20000%.

In this economic context, it is very important to train engineers in the technology in order to be able to meet the future challenges in service robotics.

The educational system in the university is still based primarily on lectures, lab practices and homework. Since several years, students in France also follow lectures on project management but, unfortunately, these are linked relatively weak to the scientific and engineering lectures. In the French academic educational system, students have to work in various companies, within internships ranging between 9 and 10 months. In the last year (Master 2 level), they usually spend the last 6 months prior to obtaining the diploma, working as trainees in a company. At ISEN, we propose to the students some small projects, for a group of 2 to 4 students, during the elective courses and a large project in Master 1 (about 250 to 300 hours) where they work in a group of 2 to maximum 8 students, depending on the subject. However, the occasions are extremely rare for the students to work in a large group in this kind of projects and to learn to apply all the project management skills.

Somewhat similar to the French system, undergraduate engineering studies in Romania are based on course lectures and lab workshops or projects, which substitute the lab workshops at some syllabuses. A total of around 270 hours are also allocated for practical internship in the industry. The internship hours, though, are evenly distributed over the first three years of study, thus resulting in a limited capability for developing very consistent projects and gaining deep, in situ, practical, skills on project management. Additionally, a total of 210 hours are allocated in the fourth year, for the development of the Diploma Project, which is the most consistent type of a project an engineering student in the Romanian system will face. In Belgium, the situation is very diverse, and large differences are observed between the regions, the universities and even (within) the engineering faculties. Moreover, the Flemish educational landscape is changing rapidly [START_REF] De Coninck | A distributed hands-on course on intelligent electrical systems[END_REF], and "flexibility" for students -not only choosing elective courses, but also deviating from the "model trajectory" by (re-)taking courses in different study years -heavily sollicitates both the educational system and the lecturers.

Engineering design exercises for small or larger groups of students, internal design contests, lab exchange sessions and so on, are both attractive to students and effective for staff and equipment allocation. Accent however is on "classic" lectures, labs and exercises. As an example, some aspects of project based engineering education on the Ghent campus of the KU Leuven (Faculty of Engineering Technology) are briefly described, mainly in the fields of automation and electronics. Bachelor 3 electronic engineering students have an annual design challenge for small autonomous vehicles having to find their way through a maze. Students do predesign, stay for a full project week on location at the North Sea coast with an intensive program of design related lectures, planning and designing, learn and use (initial) project management skills, and have a final challenge on the open day of the campus (which serves as project dead line). Bachelor 3 automation engineering students have two initial project design exercises: an electronic and an electrical engineering design, the latter in cooperation with industry.

Master 1 automation engineering students apply knowledge acquired in both lectures and lab exercises into a group (typically 10-12 students for one topic) control design of a realistic industrial control system. They use different tuning methods and controllers, choose the best design -control and disturbance behavior, robustness and so forth -using a set of performance indicators, and implement the chosen final design. In a group presentation, design steps and final results are presented and discussed.

As the Faculty of Engineering Technology has campuses on different locations, each with research groups having specific expertise, equipment and test-rigs, Master 1 automation engineering students profit from lab exchanges with different campuses throughout Flanders [START_REF] De Coninck | A distributed hands-on course on intelligent electrical systems[END_REF], [START_REF] Van Dessel | Guided Independent Learning Package for Advanced Topics in Electrical Engineering, Automation and Control Systems[END_REF]. This one day short student mobility has in the context of the Interreg 2 Seas project i-MOCCA even been extended to nearby foreign partners.

Robotics is an interdisciplinary field where students can apply their skills in mechanics, electronics, computer science, mathematics and control engineering, thus stimulating the student interest of learning in these fields. A constantly increasing number of national and international robotics challenges motivate the students to work in a team, within strict time frames and deadlines, and encourage their creativity. Examples of such contests are RobotChallenge [START_REF]RobotChallenge international championship for self-made, autonomous, and mobile robots[END_REF], founded in 2004, DARPA Grand Challenge [START_REF]DARPA Grand Challenge, prize competition for American autonomous vehicles[END_REF] and Robotics Challenge [START_REF]DARPA Robotics Challenge (DGC), prize competition aiming to develop semi-autonomous ground robots for complex tasks[END_REF], RoboCup [START_REF]RoboCup, competition for developing a team of fully autonomous humanoid robot fooball (soccer) players[END_REF], and the "Hard&Soft" International Computers Contest for Students, launched in 1994 [START_REF]Hard&Soft" International Computers Contest for Students[END_REF].

Project-based learning methods, with special focus on robotics, have gained a lot of attention recently. Universities around the world are developing and implementing robotics project-based environments to emphasize active, cooperative and problem-based learning [START_REF] Riek | Embodied Computation: An Active-Learning Approach to Mobile Robotics Education[END_REF]- [START_REF] Alimisis | Educational robotics: Open questions and new challenges[END_REF].

Modular robotic platforms based on LEGO components are used to develop remotely operated vehicle or autonomous underwater vehicle projects at Stevens Institute of Technology NJ, USA [START_REF] Stolkin | A simple ROV project for the engineering classroom[END_REF] and to study robotic design in creative ways, at the Chungnam National University Daejeon, Korea [START_REF] Jung | Experiences in Developing an Experimental Robotics Course Program for Undergraduate Education[END_REF]. LEGO Mindstorms NXT programmable bricks are also used in many project-based robotics learning approaches, for example in conjunction with the Matlab or LabVIEW programming and analysis environments [START_REF] Gomez-De-Gabriel | Using LEGO NXT Mobile Robots With LabVIEW for Undergraduate Courses on Mechatronics[END_REF].

Custom made modular robotic platforms are another frequent approach, due to its favorable design flexibility over project implementation time ratio. Such platforms have been developed to support project-based learning methods in mobile robotic design at ITESM Monterrey, Mexico Campus [START_REF] Josep | Mobile Robot Design in Education[END_REF], or in computer vision and robotic control fields at University of Lleida, Spain [START_REF] Palleja Cabre | Project-Based Learning Example: Controlling an Educational Robotic Arm With Computer Vision[END_REF]. To further increase the flexibility of the robotic platforms, Arduino or FPGA-based modules are used in many cases as programmable control elements [START_REF] Price | Project-Based Learning in Robotics and Electronics in Undergraduate Engineering Program Setting[END_REF].

Although the experience gained from these relatively new approaches needs to be further developed, the results are encouraging. There still are open issues and problems to be solved, such as the lack of proper evaluation frameworks and consistent research evidence to validate these methods [START_REF] Alimisis | Educational robotics: Open questions and new challenges[END_REF]. Another issue is the relatively low level of cooperation with the industry [START_REF] Pastor | Robotics and Control With Project-Based Learning Methodology[END_REF], which could provide the students with a very consistent, practical experience on real-life project development and management, which will be especially needed as future employees in companies.

From our point of view, it is essential to integrate the project management dimension on the project and also, to give to the students the possibility to design, construct and evaluate a robot designated to be used in the industry or services.

This point of view is also emphasized in [START_REF] Mills | Engineering education-Is problem-based or project-based learning the answer?[END_REF], which considers that a successful implementation of a project-based learning requires interest, cooperation and integration of faculty from at least the engineering, mathematics, science and business/management divisions of an institution. This is one of the main challenges that an institution has to handle for a successful implementation of this learning methodology.

For over 5 years, we are proposing this kind of projects to the students who choose the robotics electives of Master 2 at the Institut Supérieur d'Electronique et du Numérique (ISEN), Lille, France. This work presents the method of robotics project based learning applied at the ISEN. The project is an independent activity proposed to the students who choose to specialize in robotics at the end of their curricula. It replaces the traditional learning methods and is the only technical training the students are pursuing during the last 6 months of their learning period. While at the Politehnica University of Timisoara the students have robotics courses containing some projects and the Faculty of Engineering Technology from Gent proposes to the students a full project week with a challenge at the end, at ISEN Lille the teachers did an organizational and cultural change to implement the philosophy of the project oriented based learning [START_REF] Moesby | Reflections on making a change towards Project Oriented and Problem-Based Learning (POPBL)[END_REF].

The paper is organized as follows. Section II introduces the objectives of the project based learning methodology and explains how the project is organized and evaluated. Section III describes as a case study, the project realized in 2012/2013. Some discussions and the impact of this method are proposed in section IV. Section V concludes with a summary and direction for future work.

II. PROPOSED PROJECT BASED LEARNING METHODOLOGY

A. Main Objectives

Since 2008, the students who follow the robotics option in Master 2 at ISEN have to design, construct and test a mobile autonomous robot. The goal and main specifications of the robot are defined by the client, who can be a company or a start-up, a research laboratory or a specific department of the university or any institution which needs a specific mobile robot. The students have 5 months to complete the project, at a working schedule of 2.5 days per week. The project team is composed of around 16 students.

A coach who works in the industry helps the students to manage the entire project and a team of academic staff members with different skills (mechanics, informatics, electronics and automatic control) serves as technical consultants.

The learning objectives of such a project are the following:

• discover all the technical skills necessary to construct a mobile robot; • improve and increase technical knowledge in a specific field of robotics; • learn to do proper documentation and state of the art design, and choose the best solution for a technical problem by taking into account both the technical and the economical aspects;

• express ideas and technical solutions;

• writing of the technical reports;

• manage a project using the V-Cycle methodology [START_REF] Rook | Controlling software projects[END_REF];

• learn to use collaborative platforms and software to share information; • learn to meet the deadlines, to prepare purchase orders, to track the working progress and manage the delays; • learn to work in a team, to manage the human resources and the conflicts.

B. Organizational Elements

Different stakeholders are involved in the project, as depicted in Fig. 3. The first one is the client, which specifies the general requirements of the project.

A group of academic staff, with different fields of expertise, assists the students throughout the project in deciding the best technologies to use and how to choose the appropriate suppliers, according to the project needs and taking into account the delivery deadlines. It also anticipates the critical tasks of the project, helping the students to manage these tasks and to keep the established deadlines.

The coach from the industry has as main goals helping the student team to define the SPARC of the system (Speed, Power, Area, Reliability and Cost) and organizing the project work by following the V-Cycle methodology. The coach also assists in the specification of the interface between the different components of the system, in the testing and validation procedures, as well as in the project documentation and reporting.

A project manager is elected from the student members of the project team. He or she is in charge of the following key tasks:

• overall organization and planning of the project work;

• management of the periodical (weekly) technical meetings of the team with the academic staff group; • identification of critical tasks and the appropriate planning of the required resources, in agreement with the coach from the industry and the academic staff group; • supervision of the overall work progress; • management of project budget and acquisitions; • management of proper communication and information exchange between all the project stakeholders, and of the potential conflicts; • project documentation: technical specifications, periodic reports, technical documentation, tests etc.

Depending on the project specificity, different development topics (workgroups) are defined. In the case of robotics projects, the following main workgroups will typically be identified:

• mechanics, mechatronics and physical body;

• batteries and power management;

• motor driving and mobility;

• robotic perception, sensors and actuators;

• robotic software and control;

• human-machine interfaces.

Each workgroup has a student leader, in direct relation with the project manager. The group leaders establish, under the supervision of the project manager, the deadlines and the tasks of each workgroup, and monitor the progress of the group.

In order to achieve the goals of the project, the student team members affiliate themselves to the workgroups, based on their competences and points of interest. On the other hand, depending on the project dynamics and planning, students can temporarily be allocated to other workgroups. For example, students from the robotic software group can work with the colleagues in the mechanics or motor driving groups during the initial stages of the project, when the robotic hardware must be implemented prior to writing the control software.

As a vital element of the project management framework, communication between the stakeholders must be supported through timely, comprehensive information exchange mechanisms. Our approach included the following efficient and accessible means of communication:

• Dropbox file hosting service, for storing and sharing the project documents, files and reports; • SVN server, for managing the versions and revisions of the robotic software; • Facebook group, for the general presentation of the project through social networks; • Local directory service, to share the photos, contact information and personal schedules of the project team.

C. Project Assessment Method

Each year the project starts around middle of September and ends middle of February the next year. 15 ECTS points are allocated for it, thus involving between 375 and 450 hours of work for each student.

The project evaluation focuses on two main competences: the technical one and the communication one.

For the first competence, the evaluators are the coach from the industry and the academic staff. On one hand, the technical skills of the students in embedded systems are evaluated using a questionnaire. On the other hand, the technical work done by the students is assessed based on an evaluation grid which takes into account the following items: design (20%), implementation (20%), tests (40%) and autonomy of the student (20%). The final mark is the weighted average between the first mark (30%) and the second one (70%).

The second competence is evaluated by the academic staff, the client and the students themselves. The resulting mark is a weighted average between the marks given by the academic staff (20%), the student himself/herself (20%), the technical report (40%) and the final presentation of the work to the client (20%). Both professors and students use the same evaluation grid in order to have a homogenous evaluation. It takes the following items into account: the autonomy (25%), the involvement into the project (25%), the search of the best technical solutions (25%) and the implementation (25%).

III. CASE STUDY: ALICEN PROJECT

AliceN is an autonomous holonomic robot (see Fig. 4) requested by the communication department of ISEN in order to show to the visitors a relevant application of the skills learnt by the students from ISEN and also to welcome and guide the visitors.

It has been built in the period September 2012 to February 2013, by a team of 16 students, which has been organized according to the principles described in the previous section (see Fig. 5). The robot detects the visitors via a Kinect motion sensing device and moves autonomously towards the nearest detected person. Then, it is going to interact with the visitor through tablet device and offers general information about the company, or about the employees. If the visitor doesn't need help from the robot, it leaves the visitor and starts looking for another one. A modular hardware architecture has been designed for the AliceN robot, as depicted in Fig. 6. Its main control unit is an LS-373 embedded PC (Texas Instruments), running under the Ubuntu Linux platform.

Sensors producing a lot of data, such as the URG-04LX-UG01 Scanning Laser Rangefinder (Hokuyo) and the Xbox Kinect motion sensing device, are directly connected to the main control unit, through high bandwidth interfaces (USB). The other sensors such as the LV-MaxSonar-EZ0-3 ultrasonic (US) sensors (MaxBotix), the GP2Y0A02YKOF infra-red (IR) sensors (Sharp), accelerometer and electronic compass, are connected to a second control board, an Arduino Mega 2560 module.

Custom-made boards have been designed and implemented by the students, for the power management and motor driver subsystems of the robot. To implement the emergency stop module, a SN-SHK shock sensor (Cytron Technologies) and a manual emergency push button have been used.

To further increase the project reusability, a modular, classbased design has been employed for the main robotic control software, as seen in Fig. 7. IV. DISCUSSION ON THE RESULTS AND THE IMPACT OF THE PROPOSED METHODOLOGY A set of evaluations and analyses have been performed to assess the impact of the proposed robotic project based learning system. Table 1 shows the averages marks for the group of students who followed the Robotics option of the Master's studies at the ISEN Lille, France, between the years of study 2003/2004 and 2012/2013. The French grading system includes the marks between 0 and 20, 20 being the best one. Before 2008, the students following the robotic option were evaluated using a classic notation system, a weighted average between exam (60%) and labs (40%).

The results of the new method of learning, which has been applied since 2008/2009, are highlighted in TABLE I. We can observe that the student averages have been steadily higher since our project based learning method has been used. Detailed average marks just like underneath, for both technical and communication skills, are presented in TABLE II and TABLE III. It is interesting to notice that there is not a very large difference between the student and academic staff notations (see TABLE III). Moreover, in 2011/2012 the students evaluated themselves even more severely than the professors. The difference in the average marks given by the coach from the industry between the first two years and the others can be explained by the fact that at the beginning, the students had also some lectures, and the exam marks have been added to the marks given by the coach. We decided in 2010/2011 to stop the lectures because only the students who found a direct link between the topic of the lecture and their part of the project were really interested in these.

V. CONCLUSION

This paper presents a new way of learning, based on a project approach which is closely linked to a real need expressed by a client and is also managed as an industrial project. The robotic application requests various skills, allowing students to improve or acquire new competences on mechanics, electronics, informatics and/or automatic control. Moreover, the students will also significantly improve their project management skills, including the management of human resources and conflicts, management of equipment purchasing and of the budget. This kind of learning method helps students understand the importance of preparing good technical specifications in order to be able to point out and define the interfaces between the project groups, and of keeping the deadlines. Moreover, this method emphasizes and enables the students to apprehend the importance of fulfilling client requirements, along with the interactions with the client, the suppliers and the other members of the team. The students, coached by a partner from industry, have the opportunity to apply and to improve their management skills under a large-scale, highly complex project.

The results obtained by the students who were involved in the project show that they are more motivated than by learning in a classical way. Moreover, the students spent a lot of time working on the project. Each year, since 2008/2009, they have been asking to work in the robotics laboratory during weekends and holidays. We had estimated that the last academic year (2012/2013), each student spent in total around 960 hours working on the project (including week-ends and holidays), which is the double compared with the number of hour defined by the ECTS.

In the future, as this experience proved to be very useful for the students, we would like to implement it also at the Politehnica University of Timisoara. Student exchanges through Erasmus programs have also been proposed for both Flemish and Romanians students, who are interested to follow this project, which provides the opportunity to work in a multicultural environment and, thereby, to gain new skills.

ACKNOWLEDGMENT

The authors would like to thank Maxime Agache, Head of Embedded system for the TGV, SNCF (France), the coach from industry, and the team of professors from ISEN, serving as technical consultants on the project: Jean-Marc Capron, Gérard Haw, Ferdinand Piette and Bruno Stefanelli.

Fig. 1 .

 1 Fig. 1. Service robots for professional use: sold units in 2012 and 2011 (source [1]).

Fig. 2 .

 2 Fig. 2. Service robots for professional use: forecast 2013-2016 (source [1]).

Fig. 3 .

 3 Fig. 3. Project stakeholders.

Fig. 4 .

 4 Fig. 4. AliceN, the visitor welcoming and guiding robot.

Fig. 5 .

 5 Fig. 5. Structure of the AliceN project team.

Fig. 6 .

 6 Fig. 6. Hardware architecture of the AliceN robot. Fig. 7. General software architecture of the main robotic control unit.

TABLE I .

 I STUDENT AVERAGES AT THE ROBOTICS OPTION OF MASTER'S STUDIES, SINCE 2003

	Year of Study	Number of Students in the Group	Average /20
	2003/2004	21	11.9
	2004/2005	32	12.3
	2005/2006	16	12.6
	2006/2007	11	13.8
	2007/2008	6	13.45
	2008/2009	13	15.44
	2009/2010	14	14.14
	2010/2011	12	15.95
	2011/2012	14	15.63
	2012/2013	16	15.96
	TABLE II.	AVERAGES FOR THE TECHNICAL SKILLS SINCE ACADEMIC
		YEAR 2008/2009

Year of study Coach from industry Academic staff Final average

		30%	70%
	2008/2009	12,61	15,84	14,87
	2009/2010	12,67	14,31	13,82
	2010/2011	16,23	16,75	16,59
	2011/2012	18,41	15,2	16,16
	2012/2013	18,71	15,13	16,20

TABLE III .

 III AVERAGES FOR THE COMMUNICATION SKILLS SINCE ACADEMIC YEAR 2008/2009

	Year of study Technical report Final presentation Academic staff Student Final Average
		40%	20%	20%	20%
	2008/2009	16,50	13,81	16,50	16,69	16,00
	2009/2010	14,02	15,29	14,33	14,65	14,46
	2010/2011	13,23	15,17	16,87	18,00	15,30
	2011/2012	14,96	15,29	16,38	15,46	15,41
	2012/2013	15,53	16,67	15,20	15,60	15,71

I.

* This research has been financially supported by EU Interreg IV A 2 Mers Seas Zeeën Cross-border Cooperation Programme under i-MOCCA project 07-022-BE and the i-MOSYDE cluster project 10-23-BE.


Olivier Rabaste

Laurent Savy

Quadratically Constrained Quadratic Programs for Mismatched Filter Optimization with Radar Applications

Keywords: MIMO ambiguity function, mismatched filter, quadratically constrained quadratic program

In this article, we consider the problem of optimal filter computation. The contribution of this paper is three-fold. First we propose a new method for finding the mismatched filter that minimizes the Peak-to-Sidelobe Ratio. While existing solutions solve this problem only approximately, the proposed method, based on a reformulation of the optimization problem as a convex quadratically constrained quadratic program (QCQP), insures that any found solution is a global solution of the problem. This formulation also permits to insert additional constraints, for instance a constraint on the loss in processing gain of the optimal filter. We study in this context the robustness of the proposed filter to phase errors. Second we show that the proposed QCQP can easily be adapted to deal with doppler shifts and provide results for specific applications. Third we investigate the possibility to use mismatched filters to reduce the high sidelobe level observed along the range axis of MIMO ambiguity functions produced by phase codes. We propose two MIMO receiver architectures that permit to integrate mismatched filters. The first architecture tries to minimize all autocorrelations and crosscorrelations of the transmitted phase codes, while the second architecture computes mismatched filters adapted to the signals transmitted in different angular direction. The latter approach permits to obtain MIMO ambiguity function very close to the perfect thumbtack shape.

I. INTRODUCTION

T HE detection theory states that the optimal filter for detecting a single target in gaussian noise is the matched filter that optimizes the Signal-to-Noise Ratio at the reception side [START_REF] Kay | Fundamentals of Statistical Signal Processing. Detection theory[END_REF]. Unfortunately this matched filter is optimal only in the single target case: the maximal processing gain that it provides is obtained at the cost of relatively high sidelobes that may penalize the detection in multitarget settings where weak targets are buried in the sidelobes generated by stronger targets or strong clutter. This problem arises in many applications such as classic radar range/doppler ambiguity functions, or more recently coherent MIMO radar ambiguity functions.

For linear chirp signals that are used in many radar applications, a possible solution to improve the sidelobe level in range is to apply weighting windows at the reception [START_REF] Levanon | Radar signals[END_REF]. Amoung the possible weighting windows are found the famous Hamming, Hann, Kaiser, Blackman, Dolph-Chebyshev windows and many others [START_REF] Harris | On the Use of Windows for Harmonic Analysis with the Discrete Fourier Transform[END_REF]. The main drawback of this approach is a broader mainlobe and some loss in the processing gain, since the optimal matched filter has in fact been replaced O. Rabaste and L. Savy are with ONERA, the French Aerospace Lab, Electromagnetic and Radar Department; BP 80100, 91123 Palaiseau, France. by a mismatched filter. For phase codes, weighting windows cannot be applied but one can find optimal filters that provide minimal Peak to Sidelobe Ratio (PSLR) [START_REF] Levanon | Radar signals[END_REF], [START_REF] Baden | Optimal peak sidelobe filters for biphase pulse compression[END_REF], [START_REF] Griep | Poly-Phase Codes and Optimal Filters for Multiple User Ranging[END_REF]. An approach proposed first in [START_REF] Baden | Optimal peak sidelobe filters for biphase pulse compression[END_REF] and followed in [START_REF] Griep | Poly-Phase Codes and Optimal Filters for Multiple User Ranging[END_REF] to find the optimal filter consists in an Iterated Reweighted Least Square (IRLS) method based on the computation of the optimal filter providing the best Integrated Sidelobe Level (ISL), that can be easily found by solving an L 2 -norm optimization problem under a linear constraint. It provides an approximate solution to the PSLR problem. A modified version of the RLS algorithm has also been proposed in [START_REF] Zrnic | Range sidelobe suppression for pulse compression radars utilizing modified rls algorithm[END_REF], [START_REF] Petrovic | Ecf filter design for radar applications[END_REF] with a minimax error criterion to reduce the PSLR. Finally a more recent approach aims at directly solving the PSLR problem by replacing the L 2 -norm by an L 2n -norm with n > 1 [START_REF] Cilliers | Pulse compression sidelobe reduction by minimization of Lp-norms[END_REF]. This new optimization problem, that provides again an approximate solution to the PSLR problem, must be solved numerically. Of course, in both cases the resulting mismatched filter presents some loss in the processing gain.

The contribution of this paper is three-fold. First we propose a new approach for finding the optimal mismatched filter in the PSLR sense. The proposed approach directly solves the PSLR problem by transforming it into an equivalent Quadratic Constrained Quadratic Program (QCQP). The proposed QCQP is shown to be convex so that any feasible solution is a global solution of the problem. In that sense, our approach finds the exact solution to the PSLR problem. A rich literature exists on QCQPs, and efficient solvers have been developed based on semidefinite programming relaxation [START_REF] Boyd | Convex Optimization[END_REF], [START_REF] Vandenberghe | Semidefinite Programming[END_REF] solved by interior point methods [START_REF] Karmarkar | A new polynomial-time algorithm for linear programming[END_REF], [START_REF] Nesterov | A general approach to polynomialtime algorithms design for convex programming[END_REF]. The proposed QCQP presents also the advantage to be easily extended with additional constraints. For instance, we show that it is possible to add constraints in order to control the loss in processing gain (LPG) of the resulting filter, the mainlobe shape or the ISL. In particular, controlling the LPG is interesting since it permits to ensure that a good PSLR is not obtained at the cost of a large LPG, or to limit the LPG to some predefined desired value. To our knowledge, such a constraint cannot easily be integrated in previous cited solutions and has therefore not been considered despite its importance. We provide some results on mismatched filters computed for Gold phase codes and chirp signals. For the phase codes, we also study on simulations the impact of phase errors or distorsions to the sidelobe level obtained with the mismatched filter. It comes that the mismatched filter is quite robust to such distorsions.

Second we propose to extend the proposed QCQP to deal with possible doppler shifts, a feature which has, to our knowledge, never been considered in the literature, although the mismatched filter is not very tolerant to doppler shifts. We show that additional doppler constraints are still quadratic convex and can therefore easily be inserted in the optimization program. The proposed extended QCQP permits to obtain good performance in a range of interesting radial velocities. Of course, it should still be dedicated to specific applications, for instance in digital communications, for the detection of slow-moving targets, or for the case of airborne radars. These two last cases are studied here on simulations.

The third contribution of this paper considers the specific setting of the MIMO radar. In this MIMO framework, the different transmitting antennas or subarrays transmit different signals. We consider here a MIMO system composed of colocated antennas that takes advantage of the transmitting signal orthogonality to separate the different signal contributions at the reception and retrieve the transmission phase [START_REF] Li | MIMO Radar -Diversity means Superiority[END_REF], [START_REF]MIMO Radar with Colocated Antennas[END_REF], [START_REF] Chen | MIMO Radar Ambiguity Properties and Optimization Using Frequency-Hopping Waveforms[END_REF], [START_REF] Abramovich | Noncausal Adaptive Spatial Clutter Mitigation in Monostatic MIMO radar: Fundamental Limitations[END_REF]. At the reception side, the optimal MIMO processing consists of a general matched filter in range, angles and doppler. The output of such a matched filter may be summarized through the use of the so-called MIMO ambiguity function. It has been defined and extensively studied in [START_REF] Antonio | MIMO Radar Ambiguity Functions[END_REF], [START_REF] Chen | MIMO Radar Ambiguity Properties and Optimization Using Frequency-Hopping Waveforms[END_REF], [START_REF] Chen | Signal Processing Algorithms for MIMO Radar[END_REF]. One fondamental feature of the MIMO ambiguity function is that range and angular space cannot be decoupled unless perfectly orthogonal waveforms are transmitted. Unfortunately such families of waveforms do not exist and one must face a certain amount of range/angle coupling [START_REF] Rabaste | Signal Waveforms and Range/Angle Coupling in Coherent Colocated MIMO Radar[END_REF]. However for phase code families such as those codes used in digital communications, for instance the Gold codes [START_REF] Gold | Optimal binary sequences for spread spectrum multiplexing[END_REF], the Kasami codes [START_REF] Kasami | Weight Distribution Formula for Some Class of Cyclic Codes[END_REF] or some chaotic codes [START_REF] Mazzini | Chaotic complex spreading sequences for asynchronous DS-CDMA -Part I: System modeling and results[END_REF], relative decoupling is achieved at the cost of a spreading of the energy in the whole ambiguity domain, with relatively high sidelobe level. The use of beamforming at the reception side permits to decrease to some extent these sidelobes along the angular dimension, but high sidelobes remain along the range axis.

We therefore propose here to adapt the mismatched filter strategy to improve the MIMO ambiguity function for phase codes. We remain here in the context of the detection of slow-moving targets and do not consider any doppler shifts. Two approaches are considered. First the mismatched filter is considered to improve the global orthogonality of the phase code family by improving the sidelobe levels of the autocorrelations and crosscorrelations of the codes. This approach, which is in the principle similar to the one proposed in [START_REF] Griep | Poly-Phase Codes and Optimal Filters for Multiple User Ranging[END_REF] for multi-user communications, leads to a receiver architecture identical to the standard MIMO system, where the classic matched filters to each transmitted waveforms are replaced by their mismatched counterparts. The second approach consists in applying the mismatched filter on the signal received by the target instead of the different signals transmitted by the phased array antennas. Since the signal received by a target differs with the direction, several mismatched filters must be computed, one per each considered direction. This implies a certain change in the receiver architecture. This second strategy seems to provide a good solution to the sidelobe level for the MIMO ambiguity function of phase codes. Examples based on the MIMO ambiguity function computation for good chaotic codes are presented.

This paper is organised as follows. In section II, we present the mismatched filter framework. The strategy proposed to find the optimal PSLR filter by solving a QCQP is developed in section III, and some simulation results are provided. In section IV, we propose an extension of the QCQP that permits to integrate doppler constraints, and we present again simulation results on two study cases. Then we turn our attention to the MIMO context. The general MIMO ambiguity function is presented in section V, as well as two strategies to adapt the mismatched filter framework to the MIMO ambiguity function. Simulation results are provided in this same section.

II. MISMATCHED FILTER

Let us consider a signal vector s containing N samples:

s = [s 1 , s 2 , . . . , s N ] T .
In the absence of noise, the classic matched filter consists in computing the correlation vector y of length 2N -1 given by:

y = Λ N (s)s (1)
where the correlation matrix

Λ K (s) of size K + N -1 × K is defined by Λ K (s) =                       s N 0 . . . .
                     

Kcolumns

and K represents the number of columns of Λ K (s). This matched filter is known to be optimal for single target problems, i.e. it maximizes the energy at the peak response. However it does not care about the sidelobe level. Therefore it is not necessarily fit for multitarget problems, where the sidelobe level is of primary importance. Instead of computing (1), a mismatched filter will compute the convolution between signal s and a different filter sequence q in a similar way:

y = Λ K (s)q, (2)
where the output vector y is of length K + N -1. As we can already notice from this expression, the length K of the filter sequence q may be larger than the signal length N . In the following, we will assume for simplicity that K = N + 2p so that the length of y is odd, and we symetrically add the same number p of samples on both sides of the code.

Since the matched filter maximizes the Signal to Noise Ratio (SNR) at the peak response, any other filter will result in a Loss in Processing Gain (LPG) at the peak response. This LPG can be defined as the ratio between the SNR gathered with the mismatched filter and the optimal SNR obtained with the matched filter:

LPG = 10 log 10 SNR mismatched SNR matched = 10 log 10 |q H s| 2 (q H q)(s H s) ≤ 0. (3)
In the literature, exact formula has been provided for computing the optimal sequence q that minimizes the ISL (Integrated Sidelobe Level) [START_REF] Baden | Optimal peak sidelobe filters for biphase pulse compression[END_REF], [START_REF] Griep | Poly-Phase Codes and Optimal Filters for Multiple User Ranging[END_REF]. It is obtained by solving the following optimization problem:

min q y H Fy = min q Fy 2 2 s.t. s H q = s H s, ( 4
)
where F is a diagonal matrix defined by vector [1, . . . , 1, 0, 1, . . . , 1] with ones everywhere except for a 0 at entry N + p, x n represents the L n norm of vector x, and the constraint s H q = s H s is mainly included to discard the null solution. The analytic solution for this problem, simply obtained by using Lagrange multipliers, is

q ISL = (s H s) Λ K (s) H FΛ K (s) -1 s s H (Λ K (s) H FΛ K (s)) -1 s . (5)
Such a solution is optimal regarding the ISL criteria but not the PSLR (Peak to SideLobe Ratio) criteria which may be of greater interest depending on the application. Therefore an iterative algorithm based on solution [START_REF] Griep | Poly-Phase Codes and Optimal Filters for Multiple User Ranging[END_REF] has been proposed to obtain an approximate solution for the PSLR problem [START_REF] Baden | Optimal peak sidelobe filters for biphase pulse compression[END_REF], [START_REF] Griep | Poly-Phase Codes and Optimal Filters for Multiple User Ranging[END_REF]. This iterative approach is very similar to Iterated Reweighted Least Squares (IRLS) methods [START_REF] Green | Iteratively reweighted least squares for maximum likelihood estimation, and some robust and resistant alternatives (with discussion)[END_REF]. This solution provides quite interesting results, but it relies on an heuristic and therefore leads only to an approximate solution to the PSLR problem. Another approach has recently been considered to solve the PSLR problem. It consists in replacing the L 2 norm in the optimization problem (4) by an L 2n norm with n greater than 1 [START_REF] Cilliers | Pulse compression sidelobe reduction by minimization of Lp-norms[END_REF]. Analytical solution is not available anymore and numerical methods must then be considered to solve this problem. Again this approach does not provide the exact solution to the PSLR problem.

III. QCQP FOR MISMATCHED FILTER

In this section, we propose a new approach for solving the PSLR problem. This solution consists in building a convex Quadratically Constrained Quadratic Program (QCQP) that can then be efficiently solved by numerical methods. The convexity of the proposed program insures that any found feasible solution is a global optimum of the PSLR problem [START_REF] Boyd | Convex Optimization[END_REF]. Besides the flexible structure of this formulation permits to include additional constraints in the problem that may be of interest.

A. QCQP solution to the PSLR problem

Let us first define the PSLR problem. It consists in solving the following optimization problem:

min q Fy ∞ s.t. s H q = s H s, (6)
where x ∞ = max i |x i | represents the infinity norm of vector x. Again the constraint is introduced in order to discard the null solution. This problem cannot be solved analytically. However it can be equivalently cast as the following QCQP:

min q t s.t. s H q = s H s, q H λ λ λ H N +p+i (s)λ λ λ N +p+i (s)q ≤ t for |i| ≥ 1, (7)
where λ λ λ n (s) is the n th row of matrix Λ K (s) (to lighten the notation, we discard here the index K in the notation λ λ λ n (s)).

This QCQP contains exactly 2K -1 constraints: the first linear constraint permits to discard the trivial solution, while each of the 2(K -1) other quadratic constraints limits the level for each correlation sample. Note that the first constraint is linear and the other constraints are quadratic convex (since matrix λ λ λ H N +p+i (s)λ λ λ N +p+i (s) is rank 1 with positive non zero eigenvalue and is therefore positive semidefinite), while the objective is constant. Thus we have indeed defined a QCQP that is exactly identical to the PSLR problem (6) whose solution therefore exactly matches the PSLR solution.

An interesting feature of this formulation is that it permits to easily include additional constraints, as long as they remain quadratic convex. For instance it is possible to add a constraint regarding the Loss in Processing Gain (LPG) induced by the mismatched filter. Indeed, inserting in (3) the constraint of equal energies s H q = s H s, it comes that LPG = 10 log 10 s H s q H q .

Constraining the LPG to be better than some -10 log 10 α dB with α ≥ 1 implies then: LPG ≥ 10 log 10 α ⇔ q H q ≤ α s H s.

The right hand side of the last equation is a constant, and therefore such a constraint is quadratic convex and can be inserted in the QCQP. This additional constraint is of primary importance because it permits to ensure that the good PSLR provided by the optimal filter q is not obtained at the cost of a large LPG, or conversely because it is possible to limit the LPG to some reasonable predefined loss.

It is also possible to add constraints on the shape of the mainlobe. Indeed in some applications it may be desired to get very low sidelobes at the cost of a larger mainlobe. It is the case for instance when using weighting windows [START_REF] Levanon | Radar signals[END_REF]. In such a case, an easy solution consists in adding zeros in the diagonal of matrix F to eliminate constraints. But such a strategy does not constrain anymore the peak level in the mainlobe, that may distort the mainlobe shape. A possible remedy consists in adding fixed value constraints for the peak level in the mainlobe. More generally, it is possible to add some fixed value constraints to any samples of the filter output.

Constraining the filter output sample i to be less than a given value b i can simply be achieved by adding the quadratic convex constraint:

q H λ λ λ H N +p+i (s)λ λ λ N +p+i (s)q ≤ b i
Finally, the general optimization problem to solve when adding LPG constraints and mainlobe constraints is the following:

min q t s.t. s H q = s H s, q H λ λ λ H N +p+i (s)λ λ λ N +p+i (s)q ≤ t for |i| ≥ N M L , q H q ≤ α s H s, q H λ λ λ H N +p+i (s)λ λ λ N +p+i (s)q ≤ b i for |i| < N M L , (8)
where N M L is the number of samples in the right side of the mainlobe. Of course additional constraints on other correlation samples could be added if desired. Finally we can remark that a constraint on the level of ISL can also be added to the QCQP since such a constraint, that can be written as q H Λ K (s) H FΛ K (s)q ≤ c ISL where c ISL is a predefined constant, is also quadratic convex. However we will not use this kind of constraint in the remainder of the article.

Note that these additional constraints may not always be inserted as simply in the L 2 of L 2n solutions. In particular, for the L 2 problem, these additional constraints lead to non analytical solution for the ISL problem which is the keypoint for the iterative procedure to solve the PSLR problem.

Since the proposed QCQP is convex, it can be solved very efficiently and any found solution is insured to be globally optimal. A rich literature exists for solving QCQPs, with solutions based on semidefinite programming relaxation [START_REF] Boyd | Convex Optimization[END_REF], [START_REF] Vandenberghe | Semidefinite Programming[END_REF] solved by efficient interior point methods [START_REF] Karmarkar | A new polynomial-time algorithm for linear programming[END_REF], [START_REF] Nesterov | A general approach to polynomialtime algorithms design for convex programming[END_REF]. Many softwares have been developped for solving this kind of problems. In this work, we used CVX, a MATLAB package for specifying and solving convex programs [START_REF] Research | CVX: Matlab software for disciplined convex programming, version 2.0 beta[END_REF], [START_REF] Grant | Graph implementations for nonsmooth convex programs[END_REF], to solve the general convex program [START_REF] Cilliers | Pulse compression sidelobe reduction by minimization of Lp-norms[END_REF]. Note that the optimization procedure is iterative and may therefore be costly, depending on the size of the problem. However in the case of mismatched filter, the computational cost is not of primary importance since the computation can be done offline.

B. Some results

In this section, we present a few results obtained with the QCQP formalism.

1) Phase code: First we study the performance obtained with a Gold sequence [START_REF] Gold | Optimal binary sequences for spread spectrum multiplexing[END_REF] of length N = 127. Figure 3.a) presents the output of the classic matched filter for such a sequence as well as the output obtained for a mismatched filter computed with the QCQP defined by equation ( 7) with a constrained LPG of -1 dB and a mismatched filter of the same length as the matched filter, i.e. K = N . The PSLR for the classic matched filter is equal to 14.5 dB while the PSLR for the mismatched filter is 23.7 dB, inducing a gain of 9.2 dB. The measured LPG for the mismatched filter reaches the constraint of -1 dB. Figure 3.b) presents the result obtained for the same sequence and K = N when setting the LPG constraint to -2 dB. In that case, the PSLR for the mismatched filter is 24.6 dB, inducing a gain of 10.1 dB. Again the measured LPG reaches the objective of -2 dB. Note that for this first case, removing the constraint on the LPG provides an optimal filter with PSLR equal to 25.1 dB, i.e. a gain of 10.6 dB, at the price of an LPG of -3.7 dB. We also present in Figure 2, for completeness, a comparison between the optimal filter computed with respect to the ISL criteria and the optimal filter computed with the proposed QCQP for the PSLR optimization.

In this first setting, the length of the mismatched filter matches the length of the matched filter. It is of course possible to get improved performance by increasing the length of the mismatched filter. We present in Figure 3.a) and b) the results obtained when setting K = 3N . For the constraint LP G = -1 dB, the PSLR obtained is 26.6 dB, providing a gain of 12.1 dB, while for the constraint LP G = -2 dB, the PSLR obtained is 30.9 dB, providing a gain of 16.5 dB compared to the matched filter sidelobe level. For this second case, removing the constraint on the LPG provides an optimal filter with PSLR equal to 37.0 dB, i.e. a gain of 22.5 dB, at the price of an LPG of -4.7 dB.

2) Linear chirp signal: We now study the performance obtained with the QCQP formalism when applied to a linear chirp signal. The application of the mismatched filter to this type of signal was for instance considered in [START_REF] Cilliers | Pulse compression sidelobe reduction by minimization of Lp-norms[END_REF], [START_REF] Cilliers | On the trade-off between mainlobe width and peak sidelobe level of mismatched pulse compression filters for linear chirp waveforms[END_REF]. Note that this study case is shown here only for illustration purpose: we believe that the mismatch filter is too sensitive for chirp signals with phase varying continously to be of practical interest in real applications, contrary to phase codes. We present in Figure 4 the result for a chirp signal of bandwidth B = 2 MHz with time-bandwidth product BT = 50, oversampled at a sampling frequency f s = 3B, so that there are 150 samples. The length of the mismatched filter is fixed to K = 350, the LPG constraint is set to -1 dB and N M L is set to 4, i.e. we do not put any constraint on samples included in the matched filter mainlobe. The PSLR measured is equal to 48.1 dB. For information is presented in the same figure the output when a Hamming window is used at the reception. For this Hamming window, the LPG measured is -1.4 dB. The gain obtained by the mismatched filter PSLR compared to the Hamming PSLR is equal to 11.7 dB with a mainlobe width narrower and a better LPG. Note that even better gain can be obtained by increasing the length of the mismatched filter to K = 450, that is exactly thrice the length of the chirp: in that case, we obtain a PSLR of 57.3 dB, i.e. a gain of 20.9 dB compare to the Hamming PSLR.

We also present in Figure 5 the result obtained when we add constraints on the mainlobe shape; here these constraints are set so that the mainlobe of the mismatched filter matches the mainlobe obtained with the Hamming window. The LPG is still constrained to -1 dB. In that case, the PSLR measured becomes 56.3 and the gain compared to the Hamming case is increased to 20.0 dB.

Remark that we provide here results for oversampled chirp signals (sampling frequency larger than B). Indeed for chirp signals, the mismatched filter should be computed for oversampled signals since the phase of the chirp varies continuously. If applied only to signals sampled at the bandwidth, larger sidelobes will appear as soon as the targets are located at non integer delays. Moreover this oversampling should be chosen large enough (at least a factor 8) in order to mitigate this phenomena. We chose here a smaller factor of 3 to speed up the computation of the mismatch filter, knowing that the sidelobe level does not change much with the oversampling. Note that this problem does not arise for phase code since the phase code is constant over the chip duration so that any non integer delay will provide the same phase.

3) Robustness to phase errors: In this section we investigate the robustness of such mismatched filters to phase errors. This is important since phase errors will occur in real applications due to the imperfections of the transmitters and/or the receivers, and to some filtering steps that must be applied to the transmitted signal. We propose here to consider the robustness of the mismatched filter with respect to two kinds of errors.

First we consider random phase errors uniformly drawn in a given interval [-θ max , θ max ] where θ is the phase argument. The erroneous phase code generated can be written s e = Es where E is a diagonal matrix with n th element expressed as e jǫn where ǫ n is a random phase error. The mismatched filter is computed on the theoretical perfect phase code and applied to the erroneous code. The result is presented in Figure 6 for θ max = 5 • . As we can see, the mismatched filter output remains very flat. The mismatched filter can therefore be considered to be robust to such kind of errors, as long as they remain reasonable.

We consider then more realistic distorsions due to the application of a low-pass filter on the transmitted signal that cuts frequencies larger than the code bandwidth. Spectrums for the phase code and the low-pass filter output are shown in Figure 7. The mismatched filter is again computed on the theoretical perfect code and applied to the distorted code. The result is presented in Figure 8. Again the sidelobe level is very flat and we can conclude that the mismatched filter is robust to distorsions due to filtering steps. Note however that it is still better to compute the mismatched filter on the filtered signal, if the knowledge of this signal is available. This is especially true for chirp signals.

IV. MISMATCHED FILTER WITH DOPPLER CONSTRAINTS

In the previous section, we have proposed a convex QCQP to obtain the optimal filter in the zero doppler case. Since the mismatched filter is by nature optimized for that particular doppler value, it is intrinsically not robust to doppler shifts, especially when these doppler shifts take high values. In particular, when applying the optimal filter to a doppler-shifted signal with increasing doppler values, one can that the maximum sidelobe level rapidly increases with the dopplershift, reaching the level provided by the matched filter or even worse.

We propose now a modified convex QCQP that permits to take into account possible moderate doppler shifts, and therefore to mitigate this effect. Note that to our knowledge, this problem has never been considered in the literature, perhaps because the proposed solutions could not be fit easily to these additional constraints. As we will see, in the proposed framework, taking into account doppler shifts only implies to insert additional convex quadratic constraints into the convex QCQP.

A. Convex QCQP with doppler constraints

Let us denote by s ν the signal vector s modified by a doppler shift ν. The k th component s ν k of s ν is provided by

s ν k = s k e j2πνkT s ,
where T s is the sampling period. To keep the previous notation for zero-doppler signals, we simply set s 0 = s. The output of the optimal filter q applied to this doppler shifted signal is given as previously by y ν = Λ N (s ν k )q. Let us then consider a set of D doppler shift values [ν 1 , ν 2 , . . , ν D ]. Finding the optimal filter q that minimizes both the sidelobe level for the zero doppler signal s and the doppler-shifted signals s ν1 , s ν2 , . . . , s νD can be written as the following infinity norm minimization:

min q Fy T , (F ν1 y ν1 ) T , . . . , (F νD y νD ) T T ∞ s.t. s H q = s H s, (9)
where the matrix F ν d is a diagonal matrix similar to the previously defined matrix F, with the difference that it takes into account a possible shift in delay of the maximum of the output signal y ν d , i.e. the zero values in the diagonal will be located around entry N + p + c(ν d ), where c(ν d ) represents the number of samples corresponding to the delay shift of the maximum of the ambiguity function for the doppler shift ν d . This is necessary for chirp signals whose ambiguity function presents a distance-doppler coupling where the maximum of the ambiguity function for a given doppler shift ν d is located at a non zero delay. For phase codes, this will generally not be the case, i.e. c(ν d ) = 0. The presence of additional doppler outputs in the objective of problem ( 9) can be easily taken into account as additional convex quadratic constraints in the proposed convex QCQP, leading to the following equivalent QCQP:

min q t s.t. s H q = s H s, q H q ≤ α s H s, q H λ λ λ H N +p+c(ν d )+l (s ν d )λ λ λ N +p+c(ν d )+l (s ν d )q ≤ t for |l| ≥ N M L and d ∈ {0, . . . , D}, q H λ λ λ H N +p+c(ν d )+l (s ν d )λ λ λ N +p+c(ν d )+l (s ν d )q ≤ b ν d l
for |l| < N M L and d ∈ {0, . . . , D}.

(10) As we can see from this program, the constraints for signal s and doppler-shifted signals s ν d are very similar, the only change appearing from the possible delay shift taken into account through the integer term c(ν d ). Clearly, each additional doppler value considered induces a large increase of the size of the problem since it involves a large amount of constraints (2N + 2p -2 -2(N M L -1) constraints if no constraints on the mainlobe values are considered, 2N + 2p -2 constraints if constraints on the mainlobe values are used). Besides, since more constraints are considered, it is obvious that the optimal value of the PSLR will be worse at zero doppler for the optimal filter with doppler constraints that for the filter optimized only for the zero doppler.

B. Application on two study cases

We study in this section the application of the optimal filter, both as defined by QCQP (8) (no doppler consideration) and by QCQP [START_REF] Vandenberghe | Semidefinite Programming[END_REF] (that takes into account additional doppler constraints), on two particular applications. Note that we consider here dedicated applications, first the detection of slow-moving targets with relatively low doppler-shift using the phase code previously discussed, and second a specific case of airborne radar with very small bandwidth and pulse duration.

1) Phase code with slow-moving targets: We consider in this section the Gold code studied in the previous section, and an application to the detection of slow-moving targets. The parameters used are the same as in section III-B1. Besides we consider here that the carrier frequency is f c = 3 GHz and the duration of the signal is T p = 100 µs. With such parameters, the ambiguity function [START_REF] Levanon | Radar signals[END_REF] for the Gold code at zero delay reaches its first null for a radial velocity equal to 499.6 m.s -1 , i.e. a target with this radial velocity cannot be detected with such a code. We will therefore consider here only targets in the range [-500, 500] m.s -1 . Note that in the range [0, 500] m.s -1 , the level of the ambiguity function maximum decreases continuously, but at the same time, the sidelobes provided by the optimal filter are expected to increase.

The optimal filter with doppler constraints is applied with four doppler constraints, at radial velocities equal to -450 m.s -1 , -250 m.s -1 , 250 m.s -1 and 450 m.s -1 . The matched filter, the optimal filter with no doppler constraints and the optimal filter with doppler constraints are then used to process signals with different doppler shifts in the range [-500, 500] m.s -1 . An example of the output observed for a signal with a doppler shift of 250 m.s -1 is presented in figure 9. Clearly on this figure, the sidelobe level insured by the optimal filter with doppler constraints is better than the one provided by the optimal filter without these constraints. Performance is measured in terms of PSLR with respect to the radial velocity, and presented in figure 10. We can observe that the optimal filter with no doppler constraints is not very robust to doppler shift: its PSLR rapidly increases with the radial velocity of the target, and becomes even worse than the PSLR of the matched filter for velocities larger than 200 m.s -1 . On the contrary, the PSLR provided by the optimal filter with mismatched constraints remains relatively flat, and better than the matched filter in the whole velocity range.

2) Airborne radar: In airborne radar, the transmitted signal is usually a chirp with a very small time-bandwidth product, modulated by a carrier in X-band. Due to the very limited time-bandwidth product, classic windowing such as Hamming window is not fully efficient. On the contrary, due to the relatively high carrier frequency and the very short pulse duration, equivalent phase code would be relatively doppler tolerant, and thus the mismatched filter is expected to be useful in this setting.

Let us therefore consider a chirp signal of bandwidth B = 2.66 MHz and pulse duration T p = 3.75 µs, passed first through a low-pass filter, and then transmitted with a carrier at frequency f c = 10 GHz. The sampling frequency is set to f s = 5B. We compute the optimal filter for this signal with a loss in processing gain constrained to -1 dB, i.e. less smaller than the LPG induced by the Hamming window (equal to -1.56 dB for that particular chirp signal), and a mainlobe width similar to that of the matched filter (i.e. narrower than the one provided by the Hamming window). Then this optimal filter is used to process signals with different doppler shifts corresponding to target radial velocities from 0 to 1200 m.s -1. We compute also the optimal filter with doppler constraints. For this solution, two additional doppler constraints are used, one at ν = 600 m.s -1 and the other at ν = -600 m.s -1 . Note that with this specific chirp signal, the maximum of the ambiguity function for a doppler shift of v r = 1200 m.s -1 is shifted by a delay 2v r f c T p /(Bc) equal to 109.8 ns, that is almost one time sample since the sampling period is here T s = 123.5 ns. Therefore, in this case, it is necessary to take into account this delay shift in the QCQP.

An example of the output of the two mismatched filters is presented in figure 11 for a target with a radial velocity of 600 m.s -1. The tolerance to doppler shifts of the two mismatched filters is computed in terms of PSLR and compared to the matched filter and to the Hamming window. Results are presented in figure 12 with respect to the radial velocity. We can observe that the PSLR of the output of the mismatched filter is strongly dependent on the doppler shift: it becomes worse when the radial velocity of the target increases. In that sense, it is not very robust to the doppler shift. However, it should be noted that it remains better than the Hamming window for all the velocities considered, with the same LPG and a mainlobe width constrained here to the mainlobe width of the matched filter, thus narrower than the mainlobe obtained with the Hamming window (as can be observed in figure 11). As for the optimal filter with doppler constraints, it provides an almost identical level of sidelobes in the range [-600, 600] m.s -1 that corresponds to the introduced constraints, with a level better than the matched filter and the Hamming window, but worse than the optimal filter with no doppler constraints between -500 m.s -1 and 500 m.s -1 . Outside the specified doppler range, its PSLR starts increasing but remains better than that of the optimal filter with no doppler constraints.

V. MISMATCHED FILTER FOR COHERENT MIMO AMBIGUITY FUNCTION

In this section, we will first present the coherent MIMO ambiguity function and then propose two different strategies permitting to use mismatched filters in this framework.

A. Coherent MIMO ambiguity function

Let us consider a transmitting array composed of N E colocated antennas. In classic phased array, all antennas transmit the same waveform, and only the phase of the waveform is different to form the beam in a given direction. In coherent colocated MIMO radar, each of the N E antennas transmits a different waveform s m (t), so that the signal transmitted by the array in direction θ is given by

s(t, θ) = NE m=1 e jx T E,m k(θ) s m (t), (11)
where x E,m is a vector representing the position of the m th antenna, k(θ) is the wave vector. Note here that in the MIMO framework, overall signals transmitted in two different directions differ, since they result from the summation of the same codes with different phase coefficients.

Let us now assume that the reception array is composed of N R antennas located at positions x R,n . At the reception, the optimal processing consists in computing the MIMO ambiguity function, provided by the expression [START_REF] Chen | MIMO Radar Ambiguity Properties and Optimization Using Frequency-Hopping Waveforms[END_REF]:

A(τ, ν, θ, θc) = Beamforming on receive N R n=1 e jx T R,n (k(θc)-k(θ)) × N E m ′ =1 N E m=1 e jx T E,m k(θc)-jx T E,m ′ k(θ)
Beamforming on transmit

sm(t)s * m ′ (t -τ )e -j2πνt dt Cross ambiguities . ( 12
)
where τ is the delay, ν is the doppler shift, θ is the reception angle and θ c is the true direction of the target. In this expression, we considered only one angular direction θ (azimuth or elevation) and the ambiguity function is a function of four parameters because the reception angle θ and the target angle θ c cannot be summarized with only one angle parameter. If two angular directions were to be considered, then the ambiguity function would become a function of six parameters. The architecture for such a processing is provided in Figure 13. It consists in three steps: first, classic reception beamforming is applied to the N R reception antenna inputs, producing N θ outputs z θj RBF corresponding to different angular directions outputs. Then to each z θj RBF are applied N E matched filters corresponding to the N E transmitted waveforms; there are thus N E N θ matched filtering operations. The output of filter i applied to signal z θj RBF is denoted z ij . Each input z θj RBF leads to N E outputs (z ij ) i=1,...,NE . Finally for each direction θ j , transmission beamforming is applied to the corresponding N E matched filter outputs, producing one output z θj for each considered direction. Note that the order of these three steps can be changed, and reception beamforming applied only as the final stage of the processing; however such a strategy will generally lead to more computations than the one proposed here.

As can be noticed from Equation ( 12), the reception processing is completely decoupled in the MIMO ambiguity function, and we can therefore reduce the problem to one single reception antenna. For pulse trains, if the classic radar approximation of negligible phase rotation within the pulse stands, then the doppler effect can also be decoupled [START_REF] Chen | MIMO Radar Ambiguity Properties and Optimization Using Frequency-Hopping Waveforms[END_REF]. It remains to study the following reduced

(t + τ ) dt. (13)
In this expression, no further decoupling between range and angles can be achieved unless the transmitted waveforms are perfectly orthogonal with the same autocorrelation function λ s (τ ) [START_REF] Rabaste | Signal Waveforms and Range/Angle Coupling in Coherent Colocated MIMO Radar[END_REF], i.e.

s m (t)s * m ′ (t + τ ) dt = λ s (τ )δ m,m ′ ,
where δ m,m ′ is the Kronecker operator. Unfortunately this perfect orthogonality cannot be achieved in practice although it would be desired. Some waveform families may however approach this condition [START_REF] Rabaste | Signal Waveforms and Range/Angle Coupling in Coherent Colocated MIMO Radar[END_REF]. Among these families, an interesting case is provided by the CDMA (Code Division Multiple Access) codes used in digital communications. These families are composed of phase codes with good autocorrelation and crosscorrelation properties. Among the well-known families used or proposed for CDMA are found the Gold codes [START_REF] Gold | Optimal binary sequences for spread spectrum multiplexing[END_REF], the Kasami codes [START_REF] Kasami | Weight Distribution Formula for Some Class of Cyclic Codes[END_REF] or more recently the chaotic codes [START_REF] Mazzini | Chaotic complex spreading sequences for asynchronous DS-CDMA -Part I: System modeling and results[END_REF]. Although Gold and Kasami codes are optimized with respect to their periodic auto and crosscorrelations, they still present acceptable aperiodic correlations. On the other hand chaotic codes are directly designed for presenting good aperiodic correlations. Besides, while Gold and Kasami families are limited to sequences of length 2 n -1, with for the Kasami family the additional limitation that n must be even, and limited also in the number of possible sequences for a given length, chaotic families are very flexible in terms of length and number of sequences, which can take any value. Therefore in the following we will consider chaotic sequences. Note that there exist many ways to generate chaotic sequences. We use here spatiotemporal chaotic sequences defined in [START_REF] Meherzi | A family of spatiotemporal chaotic sequences outperforming gold ones in asynchronous ds-cdma systems[END_REF] that were specifically designed for DS/CDMA systems in digital communications.

The interesting feature with such phase codes is that approximate decoupling between range and angles can be achieved, as can be seen in Figure 14 that presents the range/reception angle cut of the MIMO ambiguity function for chaotic codes of length N = 127 with N E = 4 transmission antennas and no reception beamforming (or equivalently only N R = 1 reception antenna). We observe a thumbtack type of MIMO ambiguity function. There is no noticeable coupling effect but the corresponding energy is spread over the entire parameter space with relatively high sidelobe level. This spreading comes from the summation of the non zero crosscorrelations of the different codes. It can partly be weighted by the reception processing, as can be seen in Figure 15 that present the MIMO ambiguity function of the same codes with the reception processing. However relatively high sidelobes remain along the range axis for the good angle hypothesis. There is therefore a need to decrease this level. propose in the following sections two strategies based on the computation of mismatched filters that aim at lowering this sidelobe level.

B. Mismatched MIMO receiver with transmitted sequences

The first proposed strategy consists in replacing each matched filter in architecture of Figure 13 by a mismatched filter that provides the lowest autocorrelation and crosscorrelation sidelobes; note that we use abusively here and in the following the term autocorrelation for the output of a mismatched filter corresponding to sequence s i when applied to sequence s i and the term crosscorrelation for the output of this mismatched filter when applied to sequence s j with j = i. Let us denote by s m = [s m,1 , s m,2 , . . . , s m,N ] T the phase code transmitted by antenna m, by q m = [q m,1 , q m,2 , . . . , q m,K ] T the mismatched filter corresponding to code s m and by y i,j = Λ(s j )q i the output of mismatched filter q i applied to transmitted sequence s j . The problem of finding the optimal filter q i that minimizes both sidelobe level of the autocorrelation output y i,i and the sidelobe level of the crosscorrelation outputs y ij for j = i can be written as the following infinity norm minimization: min qi y T i,1 , . . . , y T i,i-1 , (Fy i,i ) T , y i,i+1 , . . . , y i,NE T ∞ s.t. s H i q i = s H i s i . As previously, this difficult minimization problem can be cast as the following equivalent QCQP:

minq i t s.t. s H i qi = s H i si, q H i λ λ λ H N +p+l (si)λ λ λ N +p+l (si)qi ≤ t for |l| ≥ 1, q H i λ λ λ H N +p+l (sj)λ λ λ N +p+l (sj)qi ≤ t for |l| ≥ 0 and j = i,
where λ λ λ l (s j ) is the l th row of matrix Λ(s j ). Of course it is still possible to introduce additional constraints for filter

q i H i si, ≤ α s H i si, q H i λ λ λ H N +p+l (si)λ λ λ N +p+l (si)qi ≤ t for |l| ≥ NML, q H i λ λ λ H N +p+l (si)λ λ λ N +p+l (si)qi ≤ b l for |l| < NML, q H i λ λ λ H N +p+l (sj)λ λ λ N +p+l (sj)qi ≤ t for |l| ≥ 0 and j = i.
Such a program is quadratic convex and can therefore be efficiently solved using CVX. Note that it contains many additional constraints compared to the single signal problem studied in section III. Thus it is not expected to provide a sidelobe level as low as one would get when solving the single signal problem.

We apply this QCQP to the chaotic sequences used for Figure 14. We first present in Figure 16 the resulting autocorrelation and crosscorrelations obtained for the mismatched filter q 1 computed with respect to sequence s 1 , with a constrained LPG of -2 dB. Similar results are obtained for the other mismatched filters q 2 , q 3 and q 4 . We notice that the sidelobe level is the same for the autocorrelation and the crosscorrelations; we get a PSLR equal to 21.0 dB when the PSLR for the matched filter ranges from 14.0 dB to 15.7 dB depending on the auto or crosscorrelation considered. The gain ranges between 5.3 dB and 7.0 dB, i.e. it is, as expected, much smaller than if only one sequence were considered, due to the large number of constraints (in the single sequence case, we obtain for this chaotic sequence a PSLR equal to 31.5 dB). Note that applying a constraint on the LPG is here very important. Indeed, the resulting mismatched filter with an unconstrained LPG provides a PSLR of 21.5 dB only at the price of an LPG of -5.3 dB.

We now insert the computed mismatched filters into the MIMO receiver architecture in place of the classic matched filter. The resulting MIMO ambiguity function is presented in Figure 17. The measured along the range axis for the good angle hypothesis θ = 0 is equal to 20.0 dB which represent a gain of only 2.7 dB compared to the classic MIMO ambiguity function at the price of an LPG of -2 dB. Note that the PSLR for the mismatched MIMO ambiguity is larger than the PSLR obtained on the sole autocorrelation and crosscorrelations presented in Figure 16; this is because the MIMO ambiguity function mixes all auto and crosscorrelations. From this result, it appears that this first solution does not provide interesting performance.

C. Mismatched MIMO receiver with sequence received by target

As we have seen in previous section, optimizing the auto and crosscorrelation of each transmitted phase code s i with respect to all other codes s j for j = i does not seem appropriate in the MIMO context. This is mainly due to the fact that the signal received by the target is a combination of all transmitted codes, so that the MIMO ambiguity function does not represent the particular output of one code with its particular autocorrelation and crosscorrelation, but rather the summation of all autocorrelations and crosscorrelations of all codes. A better strategy consists in computing directly the mismatched filter to the signal received by the target.

Let us therefore consider the signal received by a target located in direction θ c . This signal is a linear combination Fig. 18. Architecture at the receiver side for the proposed mismatched MIMO ambiguity (MMF q θ i : Mismatched filter for direction θ i ).

of the codes transmitted by the different antennas and is given by Equation [START_REF] Karmarkar | A new polynomial-time algorithm for linear programming[END_REF]. It is straighforward to compute the optimal mismatched filter for this signal since it is provided by the solution of problem [START_REF] Cilliers | Pulse compression sidelobe reduction by minimization of Lp-norms[END_REF]. However note that in this case the signal received by the target depends on the direction of the target. Therefore, different mismatched filters should be computed for each considered direction. This leads to a modified receiver architecture. The proposed new receiver architecture is presented in Figure 18. It consists first in applying the reception beamforming to the signals received at the different reception antennas, thus forming N θ output signals (z θi RBF ) i=1,...,N θ . Then to each filter q θi , computed from the signal transmitted in direction θ i . The output is then N θ time vectors (z θi ) i=1,...,N θ , one for each considered direction. Note that in this new receiver architecture, the range matched filters and the transmission beamforming steps are somehow gathered in only one mismatched filter for a specific angle.

The result obtained with such a strategy using the same chaotic sequence as previously is presented in Figure 19. It appears that the obtained MIMO ambiguity function presents very low sidelobes along the range axis at the good angle hypothesis θ = 0. The measured PSLR along that axis is equal to 35.0 dB. Further away from that range axis, the sidelobe level increases and is spread again uniformly. This result is interesting because these high sidelobes can be decreased with the reception beamforming while keeping low PSLR along the range axis. The resulting mismatched MIMO ambiguity function after reception beamforming is presented in Figure 20. We can observe that most of the sidelobes have indeed been reduced, only a fraction remaining in the mainlobe at a quite low level. This effect can even be decreased by using more antennas at the reception than at the transmission, thus reducing the reception mainlobe, as is shown in Figure 21. Finally we obtain a very clean mismatched MIMO ambiguity function.

Note that in all these we only presented range/reception angle cuts of the MIMO ambiguity function. However recall that there is another angular dimension represented by the true target direction. The good behavior of the mismatched filter should be checked along that direction too. We therefore present in figure 22 beamforming. Fortunately the resulting cut is similar to the range/reception angle cut, which can be explained by the fact that the effect of the target angle is similar to the effect of the reception angle.

VI. CONCLUSION

In this article, we have considered the design of a filter providing the best PSLR. Contrary to methods in the literature that use heuristics to solve this problem, we define a convex quadratic constrained quadratic program equivalent to the PSLR problem that can be efficiently solved by interior point methods, and provides the optimal solution to the optimization problem. We have shown that this QCQP can be easily extended with a number of additional constraints, for instance on the loss in processing gain, the shape of the mainlobe or some doppler constraints. In particular for the doppler constraints, we have shown on simulations that, whereas the classic mismatch filter is not robust to doppler shifts, such doppler constraints can improve a lot the PSLR for non zero radial velocities. We have finally shown that such an optimal filter can be used in the MIMO radar framework. We have in particular proposed a modified processing architecture where the optimal filter is applied to the signal received by the target in different directions. This new architecture permits to improve the MIMO ambiguity function of phase codes.

It should be noted that the applications considered here are mainly radar applications, but the proposed QCQP framework for the mismatched filter is very general and could be applied to any application where a PSLR constraint is desired, for instance in digital communications, or in antenna arrays to improve the digital beamforming for nonlinear or sparse arrays. Besides many other constraints than those presented here can be added. In that sense, the proposed QCQP framework is very flexible.

Fig. 1 .

 1 Fig. 1. Matched and mismatched filters for a Gold sequence of length N = 127 and a constrained LPG of: a) Top: -1 dB; b) Bottom: -2 dB. Here the length of the mismatched filter is K = N .

Fig. 2 .Fig. 4 .

 24 Fig.2. Comparison of mismatched filters for ISL and for PSLR criteria for the same Gold sequence as in Figure3.a) for an LPG of -1 dB. Here the length of the mismatched filter is K = N . The gain here is the difference in sidelobe level between the two mismatched filters.

Fig. 5 .

 5 Fig. 5. Matched, hamming mismatched and optimal mismatched filters for a chirp signal of bandwidth B = 2 MHz with time-bandwidth product BT = 50, with a constrained LPG of -1 dB. The length of the mismatched filter is K = 350. The small figure inside represents a zoom on the mainlobe: the mainlobe width for the mismatched filter is similar to that obtained with the hamming window.

Fig. 6 .Fig. 7 .Fig. 8 .

 678 Fig. 6. Matched and mismatched filters for a Gold sequence of length N = 127 and a constrained LPG of -2 dB when the phase code is subject to uniform random phase errors drawn in the interval [-5, 5] • .

Fig. 9 .Fig. 10 .

 910 Fig. 9.Matched and mismatched filters for a Gold sequence of length N = 127 and a constrained LPG of -1 dB for a target with a radial velocity of 200 m.s -1 . Carrier frequency: fc = 3 GHz; pulse duration: Tp = 100 µs.

Fig. 11 .Fig. 12 .

 1112 Fig. 11. Matched filter, Hamming output, and mismatched filters for a linear chirp signal of bandwidth B = 2.66 MHz with time-bandwidth product BT = 10 oversampled at a sampling frequency f s = 5B, with a constrained LPG of -1 dB, and a target with radial velocity of 600 m.s -1 . The signal has been filtered by a low-pass filter at the chirp bandwidth. The length of the mismatched filter is 150.

Fig. 13 .

 13 Fig. 13. Architecture at the receiver side for a MIMO radar system.

Fig. 14 .Fig. 15 .

 1415 Fig. 14. Range/reception angle cut of the MIMO ambiguity function for spatiotemporal chaotic codes of length N = 127; N E = 4 transmission antennas, no reception beamforming. Pulse duration Tp = 100 µs. PSLR along the range axis for θ = 0: 17.3 dB.

Fig. 16 .

 16 Fig. 16. Output of the mismatched filter q 1 applied to sequences s 1 , s 2 , s 3 , and s 4 (from left to right and top to bottom). N = 127 chips, K = 3N , LPG= -2 dB.

Fig. 17 .

 17 Fig. 17. Range/reception angle cut of the mismatched MIMO ambiguity function for spatiotemporal chaotic codes of length N = 127; N E = 4 transmission antennas, no reception beamforming. Pulse duration Tp = 100 µs. PSLR along the range axis for θ = 0: 20.0 dB.

  the range/target angle cut of the mismatched MIMO ambiguity function with the reception

Fig. 19 .Fig. 20 .

 1920 Fig. 19. Range/reception angle cut of the mismatched MIMO ambiguity function, second strategy, for spatiotemporal chaotic codes of length N = 127; N E = 4 transmission antennas, no reception beamforming. Pulse duration Tp = 100 µs. PSLR along the range axis for θ = 0: 35.0 dB.

Fig. 21 .Fig. 22 .

 2122 Fig. 21. Range/reception angle cut of the mismatched MIMO ambiguity function, second strategy, for spatiotemporal chaotic codes of length N = 127; N E = 4 transmission antennas, reception beamforming with N R = 10 antennas. Pulse duration Tp = 100 µs. PSLR along the range axis for θ = 0: 35.0 dB.