Valentino Marini Govigli

Sophie Alkhaled

Tor Arnesen

Carla Barlagne

Mari Bjerck

Catie Burlando

Mariana Melnykovych

Carmen Rodrí Guez Fernandez-Blanco

Patricia Sfeir

Elena Górriz Mifsud

Testing a framework to co-construct social innovation actions. Insights from seven marginalized rural areas

Keywords: Transdisciplinarity, knowledge co-construction, feasibility assessment, participatory approaches, forestry, rural development, multi-actor approach

Innovation actions within European H2020 projects aim at testing research results in practice. When supporting social innovations in rural areas, such testing requires the alignment of several rural actors in order to entail behavioral changes beyond the individual level. Recently, social innovation has been recognized as an important tool for rural areas as well, developing new solutions to respond to wicked problems for improving local living conditions at the grassroots level. In this study, we present an operational framework for the early governance of social innovation actions. This framework was applied to co-construct seven innovation actions across Europe and the Mediterranean basin applied to forestry, agriculture, and rural development. Our results show that the connection of the research teams with the local context, their degree of involvement in the action, as well as the types of relationships developed with the local actors played a key role for identifying a suitable social action. Additionally, we show that the process of defining a social innovation is complex and requires recursive engagement which might lead to evolution through time, especially in the first phases of the process. Lastly, conducting the feasibility assessment enabled the action implementers to reflect upon what they could 'offer' the local actors both in term of their own skills as social innovation practitioners (i.e. facilitation, mentoring, planning skills), but also in terms of the boundaries/limits of the project (e.g. time, funding). Such findings help us to draw general lessons for the development and governance of social innovation actions in rural areas, potentially applicable to any rural sector.

Introduction and Theoretical background

Current societies in Europe and beyond are facing complex, multi-scale and interrelated challenges. These include rural depopulation, demographic changes such as aging, in-and outmigration, youth unemployment, increasing poverty and inclusion of vulnerable groups, as well as global issues connected to climate change, energy and natural resources provision, and food security [START_REF]State and Outlook 2015 the European Environment[END_REF][START_REF] Fra | Together in the EU. Promoting the participation of migrants and their descendants[END_REF][START_REF] Godfray | Food security: the challenge of feeding 9 billion people[END_REF]. These trends require systemic and holistic interventions -often missing or curtailed -, leading to innovative solutions. These global issues are named Societal Challenges under the European framework of research Horizon 2020 (H2020) [START_REF]Societal Challenges -European Commission[END_REF]. Transdisciplinary frameworks and assessment approaches can help develop innovative policies and evaluate investments models that can efficiently tackle these challenges and generate effective long-term solutions at local and global scales [START_REF] Leijten | Investing in Research and Innovation for Grand Challenges[END_REF].

Technological, economic and social innovations may provide different perspectives to analyses and address complex problems, complementing Business As Usual solutions, and allowing the development of policy, scientific or technological advances [START_REF] Leijten | Investing in Research and Innovation for Grand Challenges[END_REF][START_REF] Kuhlmann | The challenge of addressing Grand Challenges[END_REF]. In particular, interest in social innovation for transforming or 'reconfiguring' existing social, cultural and economic arrangements has increased worldwide in practice and in the political arena (e.g. the social innovation fund established in the US in 2009 [START_REF]Social Innovation Fund[END_REF]; EU recommendations [START_REF]Social investment: Commission urges Member States to focus on growth and social cohesion[END_REF]).

Available Social Innovation frameworks have highlighted the role of human agency in powering actions for collective benefits [START_REF] Cajaiba-Santana | Social innovation: Moving the field forward. A conceptual framework[END_REF][START_REF]Social innovation and territorial development[END_REF][START_REF]The International Handbook on Social Innovation: Collective Action, Social Learning and Transdisciplinary Research[END_REF][START_REF] Mulgan | Social innovation: what it is, why it matters and how it can be accelerated[END_REF][START_REF]Creating Economic Space for Social Innovation[END_REF]. Agency "seeks to change practices to respond to specific needs" [START_REF] Secco | Set of methods to assess SI implications at different levels. Instructions for WP5 & 6[END_REF] by "…imagin [ing] new ways of being, new relationships and new ways of doing" [START_REF] Haxeltine | A framework for transformative social innovation[END_REF]. The act of innovating existing practices in favor of new ones is thus interlinked to the actions of motivated individuals and civil society members who collectively seek to alleviate societal challenges or emerging environmental, economic, social and institutional needs, that may remain unsolved under existing public policies and interventions, or privately-led actions.

According to this perspective, social innovation may provide approaches to (partially) tackling emerging problems in society, in addition to what may be currently provided by public institutions and private actors. However, how this is carried out, by whom, under which circumstances and with what outcomes and impacts, has not been thoroughly assessed, leaving margin for testing hypotheses on the emergence of social innovation. Additionally, while Social Innovation and its drivers have been studied in detail in the urban environment [START_REF] Seismic | Home -SEiSMiC Project[END_REF][START_REF]Social innovation in cities[END_REF], scarce information exists for understanding the emergence and impact of SI initiatives in rural areas [START_REF] Neumeier | Social innovation in rural development: identifying the key factors of success[END_REF].

What are the differences that may exist between urban and rural areas when dealing with a social innovation? Rural communities due to their remoteness form the decision-making centers, might be equally prone as urban realities to design and implement social innovations as a response to existing complex problems they might face. Frameworks for analysis and evaluation may need to account for marginalization trends in rural areas. These result from a mix of social (e.g. rural exodus), economic (e.g. lack of services and infrastructure, global market pressures) and environmental (e.g. harsh climate) features [START_REF] Price | Categorisation of Marginalised Rural Areas (MRAs). Deliverable 3.1 -Social Innovation in Marginalised Rural Areas[END_REF].

While scarce information exists on the emergence of social innovation, testing project-led initiatives may shed light into some of the factors of success. For a Social Innovation Action (SIA) to be designed and implemented, local communities and early implementers may draw from emerging needs and triggers and seek solutions that could best address them. Additionally, environmental and geographical constrains can influence successful participation [START_REF] Neumeier | Social innovation in rural development: identifying the key factors of success[END_REF], due to varying commitment to the Social Innovation process, organizational structure, and the community´s climate of acceptance and cooperation towards change. Furthermore, in remote and marginalized rural areas, these processes require the participation and alignment of several rural actors, often distant from one another, around perceived behavioral and structural changes that go beyond the individual level. These forms of collaboration and alignment of actors with different and sometimes divergent interests have been studied in other fields such as spatial design [START_REF] Goethals | Developing shared terms for spatial equality through design[END_REF], environmental planning and management [START_REF] Prabhu | Realizing Community Futures: A Practical Guide to Harnessing Natural Resources[END_REF][START_REF] Reed | Stakeholder participation for environmental management: A literature review[END_REF] or sustainable development more broadly [START_REF] Smith | Constructing grassroots innovations for sustainability[END_REF]. A deeper understanding of how those processes apply across different types of social innovation remains yet to be achieved. All these factors make crucial the definition of supportive frameworks that can assist researchers, practitioners, and local actors alike in fomenting and implementing SIAs. These operational frameworks need to ensure communities participation and involvement at the different stages of the social innovation process: from the diagnostic of the problem and challenges encountered, to the identification and design of appropriate solutions to tackle those challenges. An extensive body of literature has highlighted the benefits of actively involving stakeholders into decision making processes, spanning from the seminal work of Paulo Freire (1921) on awareness creation of oppressed community, to the characterization of the degree in which communities can and are engaged in decision making (see Arnstein's ladder of citizen participation in [START_REF] Arnstein | A Ladder Of Citizen Participation[END_REF] p.217), and finally to the recent work on participation for environmental management in tourism [START_REF] Reed | Stakeholder participation for environmental management: A literature review[END_REF][START_REF] Wondirad | Community participation in tourism development as a tool to foster sustainable land and resource use practices in a national park milieu[END_REF]. Some of the mentioned benefits are: better appraisal of the reality faced by the stakeholders and therefore better identification of adequate solutions; increased rate of adoption of identified solutions; empowerment of the social actors and increase in human capital through social learning processes; increased responsibility of the stakeholders' vis-à-vis the innovation identified and its implementation.

While SIAs can play a prominent role in supporting the emergence of Social Innovation, the triggering factors that spur action in the face of existing resources and community needs, remain unclear. In their latest formulation, the European Research programme Horizon 2020 stimulates the establishment of innovation actions as demonstration activities for exploring the technical feasibility of new or improved knowledge in terms of processes, related technology, products or services -in real world conditions [START_REF]Work programme 2014-2015[END_REF]. They constitute a new approach for hands-on practice and implementation of research, a reason why they are rising in importance. Yet, there is limited knowledge on the processes behind SIA implementation, on the researcher´s role in guiding local actors toward the establishment of an idea, on the evolution of the idea and the network from its proposal to its mature phase.

In this paper, we analyse the co-construction process which led to the initiation of seven SIAs on grassroots initiatives across Europe and the Mediterranean basin applied to forestry, agriculture, and rural development. These SIAs have been kicked off as part of the Horizon 2020 project SIMRA (Social Innovation in Marginalised Rural Areas) to boost and analyse their initial stages of development [START_REF]SIMRA Innovation Actions[END_REF].

According to the definition provided by SIMRA, SI refers to "the reconfiguration of social practices in response to societal challenges, which seeks to enhance outcomes on societal well-being and necessarily includes the engagement of civil society actors" [START_REF] Polman | Classification of Social Innovations for Marginalized Rural Areas[END_REF].

Grounded on innovation management theories and innovation entrepreneurship approaches, an operational framework co-construct SIAs was developed. This framework differs from -but complements-from rural Living Labs [START_REF] Zavratnik | Living Labs for Rural Areas: Contextualization of Living Lab Frameworks, Concepts and Practices[END_REF] insofar as these act typically as platform for catalyzing a portfolio of innovation projects, whereas our framework refers to the operationalization of individual innovation projects. This framework differs as well from Action Research as the latest focuses on creating new knowledge (research as priority), whereas the Innovation Actions rather concentrate in the testing of scientific knowledge. In particular, in this paper we focus on testing the initial phases of such operational framework, focusing on:

1. Discussing the relevance of kick-starting a SIA, providing a concrete insight for identifying common features across diverse type of actions. We will seek to respond to the following questions: who are the initial local actors? Which are the triggers and needs the local actors responded to? How has the early Social Innovation idea evolved in the later stages of the SIA process?

2. The feasibility of the SIAs: The feasibility assessment is used to evaluate whether an action identified as a SIA is actually viable and can be implemented effectively in the long term.

Feasibility will be assessed through the TELCOS assessment model, contextualized based on the specificities of each individual SIA. Our analysis will explore the feasibility challenges faced by different SIAs, in terms of technological, operational, legal, collaborative, and economic factors. These findings will help us draw general lessons for the implementation and establishment of effective SIAs in rural areas, which may be potentially applicable to different rural areas and to diverse innovations where social processes are at their core.

Methods: an operational framework to co-construct SIAs

Framework steps

An adaptive project management conceptualization [START_REF] Wysocki | Adaptive Project Framework: Managing Complexity in the Face of Uncertainty[END_REF] underlined the development of common guidelines to the SIAs [START_REF] Gorriz-Mifsud | Social Innovation Actions Guidelines[END_REF]. These guidelines envision SIAs as a process to catalyze and sustain rural grass-root initiatives as flow with key milestones in the short, medium and long terms.

In particular, the SIA co-construction framework includes: a first (i) Design phase aiming at defining the scope of action of the SIAs, by identifying the objectives that the stakeholders wish to pursue, and at operationalizing the SIA activities; an (ii) Implementation phase which carries out the planned social innovation activities, including dissemination of the activities, processes and outcomes to the wider public; an (iii) in-itinere evaluation of the initial implementation, with the possibility to redesign the strategy, a (iv) Consolidation phase during which the initiative crystallize. Lastly (v) the achievement of the targets (performance) is evaluated. In this paper we will focus on the design phase. Nonetheless, a short description of the remaining phases is also hereby provided. Along this process, the collaboration between SIA implementers and local actors might take multiple forms according to the degree of involvement, as described in. Table 1. This involvement applies also to the other phases and includes the transfer of the best scientific knowledge. In this line, academic works recommend (i) to smartly manage the social networks of rural communities, including bonding, bridging (often this is the role of the SIA implementer), and linking relationships;

(ii) a supportive internal governance system, (iii) to support local actors in their capacities to search for funding options. During the initial collaboration phase, the working groups may include the actors who directly had the idea -the innovators 2 -and their close connections -those following and supporting the leaders in the idea´s development. Working groups and SIA implementers may meet regularly to refine the first Social Innovation idea, to operationalize it (e.g. through an adapted Business Model Canvas), and to discuss whether and how the proposed activity falls within the scope of action -e.g. Once an action is selected, the working groups should check whether it complies with a set of established feasibility requirements (Feasibility assessment). In order to standardize the feasibility plan across case studies, the TELOS framework was proposed -a widely used method for feasibility study of business projects. The TELOS framework takes into consideration five feasibility areas for assessing a given project: Technical, Economic, Legal, Operational and Schedule. Due to the key role of social relationships and collaboration networks in Social Innovation initiatives, an additional component (Collaborative feasibility) was added following [START_REF] Taylor | In praise of the feasibility study[END_REF], hereafter named TELCOS (Figure 2). what was missing during the SIA implementation and is suggested to be included in future.

Consolidation phase

The consolidation phase is developed only in the medium-long term and it indicates that the SIA has achieved maturity and is self-sustained beyond the life of the project. The local actors involved work now independently from the SIA implementers who supported the action in its first steps. Additionally, the first beneficiaries of the initiative become noticeable, as those individuals that benefit directly or indirectly from the project implementation (e.g. consumers if the SIAs produce tangible products and services). Also, the first outcomes on the wellbeing of the local community start to appear. It might happen that if the initiative is relevant not only to the concrete locality where it is applied, the consolidation phase implies a replication or expansion to other areas.

Evaluation phase

This phase is often relevant for the innovators to justify the achievements towards donors and funders, and to advocate for necessary inputs to sustain the running of the initiative. Depending on the expected period of the objectives, the outputs, outcome or impacts can be identified. The availability of different sources of disaggregated pre-SIA and post-SIA data (e.g. municipal statistics, initial vs final surveys) contribute to build appropriate indicators. This phase is frequently overlooked as per its high resource consumption, and its usual disconnection from programs supporting only the initial steps.

Framework application

Social Innovation study sites

The areas in which the seven SIAs were initiated testing the operational framework are located in France, Lebanon, Italy, Norway, Spain, and United Kingdom (Figure 3). These sites encompass different types of rural areas as well as different types of marginalization. A brief description of the context of each area is hereby provided. Having a clearly defined context become crucial as it represents the conditions that enable or constrain the Social Innovation, amplifying and echoing communities´ needs and triggering factors [START_REF] Secco | Set of methods to assess SI implications at different levels. Instructions for WP5 & 6[END_REF].

Gúdar-Javalambre County, Spain (ES1):

The southernmost county of Aragon is considered one of the hotspots of biodiversity in the Mediterranean [START_REF] Médail | Biodiversity Hotspots in the Mediterranean Basin: Setting Global Conservation Priorities[END_REF] but is simultaneously one of the most depopulated areas in Spain. With 8,328 inhabitants and a density of 3.5 inhab/km 2 , it belongs to the so-called "triangle of desert". Its landscape covers a mosaic of pine and evergreen oak forests, agroforestry, cereal crops and pastures. Yet, mountain agriculture in this area is harsh due to climate (irregular precipitations) and accessibility (steep terrains). Farm economy concentrates around black truffle production and trade, given its predominantly calcareous soil -hence leaving other productions as secondary and less interesting for landowners. The area is a typical location for second homes and rural tourism, mainly from the nearby region of Valencia. [START_REF] Iedom | Rapport Annuel 2019[END_REF]. The local economy is constrained its ultra-peripherality that engenders additional costs in relation to food provisioning, very narrow markets due to the size of the territory, difficulties to position itself on export markets, weak cash flow of rural enterprises.

Solsonè s County

Data collection and analysis

Starting in 2016, seven teams used the SIA operational framework to foster grassroot initiatives in each study area. The heterogeneity of study areas and their respective challenges implied a diverse set of activities implemented. These comprised i.a. literature reviews, stakeholder analysis, organization facilitation and observation of internal meetings with local actors, organization of workshops -for brain-storming, discussion forums and dissemination-, survey development, softcoaching, or rural hackathons. These activities are in line with action-orientated research, which combines research and practical work in a process that aims to improve strategies, practices and knowledge [START_REF] Kidd | Reflecting on Participatory, Action-Oriented Research Methods in Community Psychology: Progress, Problems, and Paths Forward[END_REF]. The internal meetings were organized to discuss about the adequacy and operationalization of the activities the local actors were willing to implement within the SIA process, and their corresponding feasibility assessment. The feasibility assessment was formalized in a report [START_REF] Marini Govigli | Insights from the Feasibility Assessment of Innovation Actions[END_REF].

The SIAs activities were systematically recorded in a "Social Innovation Action Diary". The SIA Diary was conceptualized as a quasi-ethnographic tool of participant observation, where the SIA implementers registered the progress of the initiative, as well as complementary explanatory observations. Therefore, the SIA implementers not only facilitate and coach local actors, but also "immersed themselves within their chosen empirical setting" for a period during which "the researcher's experience, in terms of his or her participation and/or observation at the research site, is used to generate a narrative-based interpretation of the events that took place" [START_REF] Dey | Methodological issues: The use of critical ethnography as an active research methodology[END_REF]. The SIA Diaries served then as repository of the steps done, troubles faced, debates, agreements taken, or crises arisen during the SIA process.

Based on the diaries and the feasibility reports, a content analysis has been performed contrasting the operational framework presented in Section 2. In addition, the suitability of the proposed initiatives was analyzed in regard to the Social Innovation definition, as well as to the Social Innovation evaluation framework, following [START_REF] Secco | Set of methods to assess SI implications at different levels. Instructions for WP5 & 6[END_REF]. This analysis allowed identifying the following dimensions for each SIA:

• Triggers, as "punctual events or accumulations of unmet needs causing a response and a change in practices by local communities" [START_REF] Secco | Set of methods to assess SI implications at different levels. Instructions for WP5 & 6[END_REF];

• Needs, as "the demands of vulnerable groups in society" [START_REF] Secco | Set of methods to assess SI implications at different levels. Instructions for WP5 & 6[END_REF];

• The structure of the core group of Local Actors involved in the SIA;

• The structure of the network the SIA plans to build up;

•

The interventions planned (i.e. activities) to conduct and the expected outcomes.

Results and discussion

By testing the Social Innovation Action co-construction framework, it emerges that the implementation of the suggested process allowed to select purposeful actions for each of the study areas. These were: Nonetheless, the process leading to these actions differed across the study areas. Figure 4 shows an adaptation of the co-construction framework to one of the seven study areas (ES1). The remaining six figures are provided in Appendix A. In the next sections we discuss patterns and differences across three main initial stages of the design phase of SIAs: (i) the scoping phase and the first contacts with the key local actors in the area, (ii) the definition of a social innovation idea to develop during the SIAs, and (iii) the feasibility assessment of the idea and further redesigning processes.

Scoping

Meetings with key local actors were held in the different regions. However, for some cases the chosen SIA had to be revised and modified to better adapt to the local context and territorial dynamics (Table 3). These meetings enabled the different SIA implementers to identify main contextual factors, individual and collective needs for each SIA. Additionally, they served to (i) define the scope of the action, (ii) define the core group of local actors involved, (iii) list the additional actors to be engaged in the SIA and (iv) start envisioning potential activities to conduct over the course of the years. These initial contacts were mainly held with local authorities (ES1, UK), Local Action Group3 (IT, ES1), operational groups 4 (GUA) and/or local actors (NO, LE, ES1, ES2) interested in pursuing a given idea on the territory. This shows how the local social fabric is fundamental for revealing and/or identifying unmet territorial needs. [START_REF] Neumeier | Social innovation in rural development: identifying the key factors of success[END_REF] refers to this first stage of social innovation as "Problematisation", which is triggered by "an initial impetus for which an actor or small group of actors decide to change their behaviour and attitudes". The idea and identification of a problem may be done by this initial core group, by an external influence (SIA implementers), or by a combination of both. The outcomes of these exercises of envisioning activities and practices that could potentially enhance the community's well-being has also been identified in the literature as way of revealing the unmet territorial needs [START_REF] Spijker | Knitting green spaces with the threads of social innovation in Groningen and London[END_REF]. This highlights that, often for social innovations, the development of activities is not necessarily preceded by context analysis and formal strategic design. Literature findings also shows that policymakers may have a strategic role in strengthening social entrepreneur initiatives [START_REF] Stephan | Institutions and social entrepreneurship: The role of institutional voids, institutional support, and institutional configurations[END_REF], therefore creating alliances with them early on in the process may be beneficial for long-term sustainability of the social innovation initiatives.

Social Innovation embryos are often processes known only through proximal networks to the innovators. In order to disclose them, a search through networks on Rural Development stakeholders (ES2) or a brainstorming workshop (ES1, UK) were conducted.

The role adopted by the SIA implementers was very similar across all seven SIAs, mainly facilitating the process of designing and implementing the actions (low degree of involvement, Table 1). In terms of the pre-existing relationships with the local actors, however, there were significant differences. They may be differentiated in three types; The strength of the linkages to the territory, the role adopted by the SIA implementers, and the type of pre-existing relationships all play a definitive role during the first steps of the design phase.

In particular, our results show how the SIAs with strong connections to the territory and wellestablished working relationships with the local actors had smoother process for selecting the Social Innovation idea to implement. Literature on action research acknowledges that abiding respect for people's knowledge, and understanding and addressing the issues that communities confront is crucial for achieving success [START_REF] Brydon-Miller | Why action research?[END_REF].

The degree of participation of the public sector (local councils, counties and LAGs), private (social business, cooperatives, commerce trade and cooperative trade associations), and non-profit sectors (local and community foundations, local development agencies) in the SIAs varied significantly. In two cases (ES2, UK) the action was moved forward by a single motivated individual, who was seeking support in moving forward the initiative. In one case (LE) the aim was increasing cooperation of individuals already members of a group (an existing cooperative), while in the other cases (ES1, NO, IT), social networks were newly created, fostered by the H2020 project. This diversity of composition and ways of constructing the social networks reflects the wide diversity that we can find in other well-documented cases of social innovation worldwide [START_REF] Spijker | Knitting green spaces with the threads of social innovation in Groningen and London[END_REF][START_REF] Calzada | Critical social innovation in the smart city era for a city-regional European horizon[END_REF][START_REF] Simra | Database: Social innovations in marginalised rural areas[END_REF].

Lastly, we observed how the needs that each SIA aimed to meet were located at the core of the three sustainable development pillars. Namely the environmental domain (e.g. improving the ecological resilience of the landscape, preserving ecosystem services and biodiversity), the social domain (supporting institutional void in rural areas and for marginalized sectors of the society, strengthen community involvement), or economic one (boost business formation led by marginalized communities). This further aligns with the very wide body of literature that supports that social innovation dynamics materialize in a wide range of sociocultural and socio-political contexts, in all their dimensions [START_REF]The international handbook on social innovation: collective action, social learning and transdisciplinary research[END_REF].

Defining

The definition of the main social innovation around which the local actors decide to work on to alleviate the needs of the local society is a key milestone of the SIA process. While all actions eventually identified an idea to be developed, the original idea never stopped evolving at any point during the designing phase. As an example, in the ES1 case, the topic of the SIAs changed three times during the process of designing (from broad land stewardship-type of initiative, to land banking for wildfire risk prevention), and was additionally refined during the implementation of the idea. (Fig. 4).

Redefining an idea repeatedly during a design thinking approach is widely recognized as a key feature of creativity, and a likely approach to achieve concrete results [START_REF] Dorst | Creativity in the Design Process: Co-Evolution of Problem-Solution[END_REF][START_REF] Christiaans | Cognitive models in industrial design engineering: A protocol study[END_REF].

In the seven SIAs, the idea changed and got adapted in three main situations: (i) when important barriers were identified by the actors involved (ES1, ES2), (ii) when the public reacted negatively to the proposed idea (ES1), and (iii) when eventual institutional, economic, technological and collaborative barriers either emerged during the design process or were existing but not eventually acknowledged properly (ES1, NO, IT). Existing literature confirms these findings. Considering the needs of others and confirmation from early users of the benefits of the proposed ideas emerge as key element for defining an idea to work upon [START_REF] Ross | Bottom-up grassroots innovation in transport: motivations, barriers and enablers[END_REF]. Additionally, appropriate technical, financial and governmental policies can also hinder the implementation of a given action, therefore if additional barriers emerge local actors might be forced to modify appropriately the scope of the action.

This shows how the process of defining a SIAs is rather complex and recursive, requiring several contacts with all interested Local Actors, and might be subjected to a constant evolution through time, especially in the first phases of the process. This is a limitation of the current framework, which was designed to define an idea and proceed to its implementation, without fully acknowledging the possible evolution of the concept including at later phases of the process.

Feasibility assessment

Six out of seven SIAs performed a feasibility assessment during the designing phase. This allowed to refine the actions, and to identify certain challenges, which would need to be tackled during the implementation phase (Table 4). The feasibility of the proposed actions shows some recurring elements across the analyzed SIAs.

• Technical feasibility -In three out of six SIAs, local actors had the expertise required to perform the tasks listed in each initiative. In two cases, technical support emerged as key factor for strengthening the initiative (ES1, UK). Additional skills or expertise was to be sought from the SIA implementers involved (i.e. soft skills) or through the involvement of external consultant and experts (e.g. professional forestry guidance, communication experts, investment and management skills, marketing skills, IT skills, and networking and facilitating skills). This shows the key role of the SIA implementers in supporting local actors with specific skills.

• Economic feasibility -All SIA implementers and Local Actors recognized that SIAs require continuous financial support to their operational and maintenance costs in the medium and long-term. Currently, this aid takes the form of national and international grants (e.g. H2020 as in the case of SIMRA, Erasmus+, LAG funds, EARDF funds), which partially covered the planned expenditure of the Actions. Such funds are crucial for SIAs. Their principal benefits are social (e.g. improving demography, increasing social resilience, fostering gender balance, empowering youth) and environmental (e.g. improving the fire resilience of the landscape), rather than economic in nature.

• Legal feasibility -The legal frameworks at local and national levels, through laws and monetary, information and coordination policy instruments supported the implementation of the SIAs activities in all cases. Nonetheless, in three cases ad hoc legal forms could benefit the long-term sustainability of the SIAs through mainly fiscal incentives, and tax exemptions (ES1, LE, IT).

• Collaborative feasibility -The majority of the SIA implementers had a positive perception of the local actors' participation in the SIAs activities. This reflects existing, long-term collaborations (e.g. public-private partnerships, volunteers-public partnerships), and awareness of local actors for the need of close collaboration as a prerequisite to the successful undertaking of planned activities. Some critical issues emerged in those cases where institutional support declined or was ceased (ES1), discouraging the network of members of the Innovation actions.

• Operational feasibility -Some of the issues which can hinder future implementation of the Innovation Actions were highlighted, including: lack of active participation of the stakeholders during the implementation of the initiative (ES1, NO, IT), funding barriers (UK), ad-hoc training of staff (UK), and uncertainties over the future prospects of the initiative (IT);

• Schedule feasibility -With respect to timing, all activities planned appeared to be feasible for implementation within the project timeframe and as part of its legacy. It is however likely that in some cases their outcomes will be clearer over a longer time period.

The feasibility assessment appeared to be an efficient assessment tool and motor for future action to be taken. Providing a structured and systematic framework to analyses the current situation and collect the relevant information, enabled both the social innovators and the SIA implementers to reflect upon some of the main challenges faced by the local actors. Indeed, social innovation can be hindered or enabled by a variety of factors that are underpinned in the different dimensions of this feasibility framework. For example among factors identified by [START_REF] Živojinović | Social Innovation to Sustain Rural Communities: Overcoming Institutional Challenges in Serbia[END_REF] in the Serbian context are: poor law enforcement, lack of trust, norms and values. [START_REF] Camps | Exploring how social capital facilitates innovation: The role of innovation enablers[END_REF] highlight the role of social capital and its components (structural, cognitive and relational) on innovation capabilities at the firm level; [START_REF] Camps | Exploring how social capital facilitates innovation: The role of innovation enablers[END_REF] highlight the importance of well targeted and flexible funding as well as cooperation and collective efforts in the development of social innovation in the forestry sector.

Having a clearer picture of the potential challenges also enabled the SIA implementers to reflect upon what they could 'offer' the local actors both in term of their own skills as social innovation practitioner (facilitation, mentoring, planning skills…), as well as in terms of the boundaries and limits of the project (time, amount of funding, type of action funded). Moreover, because the feasibility assessment comprised six dimensions, it allowed to conduct a holistic approach to the social innovation initiatives regardless of their nature or the domain to which they applied (rural development, agriculture or forestry). [START_REF] Moulaert | Towards social holism: social innovation, holistic research methodology and pragmatic collective action[END_REF] recommend holism as a research methodology and pragmatic collective action arguing that it enables the researcher to account for relationships between the part and the whole, here the whole being the overall social innovation and the parts being its different dimensions (technical, economic, legal, collaborative, operational, schedule). Indeed, by focusing on understanding how results could be achieved under those different dimensions, the feasibility assessment helped identify interdependencies between different aspects of the social innovation project ideas. As an example, understanding existing actors and networks involved in the GUA case highlighted the necessity to reinforce the collaborative dimension of the initiative before being able to move onto other dimensions such as the technical dimension. In the UK case, reinforcement of technical capacities such as monitoring and communication will in turn increase the social innovators' capacity to secure future funding.

Recommendations and Conclusions

Conceptualizing frameworks can assist practitioners in identifying bottlenecks for implementing SIA projects and processes. They can also allow standardizing processes across study areas, which might differ geographically, economically, or in terms of the initiative taken forward.

However, SIAs are also strongly dependent on local dynamics and networks. Communities' customs and norms, formal and unofficial relationships are thus crucial for the success of the SIA, but can be hardly subject to theory at the start of a SIA process.

Additionally, the needs and triggers supporting SIAs are not static but may change over time, as such, ideas initially proposed will change over time, adapting to emerging constraints or new opportunities. Rigid approaches are thus not appropriate when designing and implementing SIAs, but rather support to local actors in developing Social Innovation initiative may need to be continuous and flexible. Additionally social innovations require to be considered through the entire cycle of the spiral model: from ideas, to prototyping and piloting, to implementation, and upscaling.

It is the cycle from ideas to production and markets (e.g. new entrepreneur opportunities), and towards innovative policy and governance mechanisms within each scale and across the scales (c.f. [START_REF] Weiss | How to support innovation in the forest sector: Summary and conclusions[END_REF]). This implies that attention has been given to new social networks, partnerships, horizontal and vertical networks [START_REF] Nijnik | Can social innovation make a difference to forest-dependent communities?[END_REF].

The role of the SIA implementer is also key and twofold. On one hand, SIA implementers ought to facilitate local actors in designing a context-appropriate initiative; on the other hand, SIA implementers may observe, helping to conceptualize, synthetize and disseminate the information they gather from the development of the SIA process. These multifaceted positions imply that the role of the external implementers should be clarified at the beginning of the SIA processes, in order to minimize potential for conflict, and maximize support and cohesion with the local actor group.

Preliminary information on SIA multiple drivers as well as perceptions on the possible challenges that the initiative may face allow to zoom into the role of the SIA in the community and for the involved partners. In particular, in some cases it raised awareness of local actors on the challenges that could obstacle the feasibility of their action in the future. The facilitation process led by the SIA implementers together with local actors provides some hints at how a rural acceleration service of social innovations might look like. Indeed, the process of establishing and guiding social innovation actions can be compared to the arrangement and organization of start-ups incubators and accelerators. While the existing literature is mainly oriented at business development (e.g. [START_REF]Global Best Practices Report on Incubation and Acceleration Key Trends and Analysis of the 2015 Incubation and Acceleration Ecosystem[END_REF][START_REF] Mrkajic | Business Incubators in Developing Countries : Guidelines for Incubator Managers[END_REF]), mentoring and aid programs specifically designed for supporting SIA may overlap with start-ups incubator services, where by focusing primarily on social innovation early ideas, local actors are guided into the development stage and are supported on the establishment of their core activities.

While the process of co-constructing allows to mobilize interested local actors to join the Social Innovation idea, broadening and strengthening the scope of the initiative; additional training and mentoring services (such as the hereby proposed feasibility assessment) allows local actors to proactively face issues that can emerge in the early establishment phases of the SIAs. All these tools contribute to strengthen the Social Innovation initiative into a much robust one. Yet, much remains to be done to link social innovations with desired policy outcomes [START_REF] Koontz | What Do We Know and Need to Know About the Environmental Outcomes of Collaborative Management?[END_REF]. Traditional accelerator units support businesses by providing connections to investors. Similarly, a social innovation acceleration service supports local initiatives by expanding their scope, increasing citizen awareness and, potentially, leading to additional private (e.g. fundraising, crowdfunding campaigns) or public (grants from organizations) expenditure. Nevertheless, at this stage the information available from the seven SIAs is not enough to validate whether the process undertaken in the SIAs will obtain fruitful outcomes. Further work needs to be undertaken to closely follow the implementation and the consolidation phases of the SIAs, and to attempt to assess the overall impact of this SIA accelerator program.

Author

Figure 1 .

 1 Figure 1. Conceptual flows of the SIA process through the design, implementation and consolidation phases. SIA flow: the process through which a Social Innovation idea is developed into a concrete initiative (consolidation); Agent flow: the evolution and enlargement of the actors´ network along the SIA process; Idea flow: the evolution of the Social Innovation idea from broad to narrow (i.e. operationalization). Source: own elaboration.

  does the SI idea fit the Social Innovation definition? (Defining). In order to analyse the relevance of the proposed idea vis-à-vis the challenge to tackle, tailored exercises based on techniques such as the Drivers-Pressures-Status-Impacts-Response (DPSIR), Cognitive Mapping, or the Theory of Change, help in discussing with the local actors how the suggested interventions/activities are actually aligned with the problem on focus.

1 A

 1 SIA implementer refers to the person belonging to a research team in charge of organizing the SIA whose role varied between facilitation, research, and mentoring.

Figure 2 .

 2 Figure 2. The TELCOS framework with the six feasibility areas.

Figure 3 .

 3 Figure 3. Country and location of the seven Social Innovation Actions. Source: Own elaboration based on Google Earth.

1 .) 3 . 7 .

 137 Fostering the development of a land banking initiative aiming at reverting the abandonment of agricultural and forestry parcels with a specific emphasis on wildfire prevention (ES1) 2. Setting the foundation for a pilot project about forests, social networks and human health, (ES2Integrating immigrants, mainly refugees or work migrants, in the rural area by motivating them to join locals in hiking activities (NO) 4. Social inclusion of women from disadvantaged backgrounds as to provide them businesstraining and marketing tools to sustain the development of small entrepreneurship initiatives (UK) 5. Create and support new networks of local young people interested in building socially innovative and sustainable businesses (IT) 6. Transforming traditional food recipes into standardized and scientifically monitored products, to support local women in developing profitable businesses and gain community respect (LE). Supporting the development of a shared vision of multifunctional forests and enhancing related human and social capital (GUA).

  (i) Pre-existing relationship; When one or several members of the research teams had a well-established working relationship, preceding the starting of the project (IT, NO, LE, GUA), (ii) Common contacts; When none of the members of the research teams and local actors had had a working relationship before, but they had common contacts (ES1, UK) (iii) No linkage; When research teams and local actors only met each other during the scoping phase (ES2).

Figure 4 .

 4 Figure 4. Adaptation of the SIA conceptual framework to the action implemented in Gúdar-Javalambre County, Spain (ES1) in the period 2016-2018 (design phase only), highlighting the flow of the action, the flow of the agents' group, and the evolution of the Social Innovation idea.

  feasibility assessment was done on a SIA that had to be almost fully re-designed in other to move it forward, which finally constituted a radical change in the SIA. As such assessment does not correspond to the current SIA taking place in the Solsonès County (Spain), we decided to not present it here.

Figure A4 .

 A4 Figure A4. SIA conceptual framework for IT.

Figure A5 .

 A5 Figure A5. SIA conceptual framework for LE.

Figure A6 .

 A6 Figure A6. SIA conceptual framework for GUA.

  in return, will help get future funding. There is some expertise within the local partner Local Action Group, which is valorised and being used. Where the expertise is lacking within the Local Action Group, it is provided by the collaboration between the volunteer and the public sector is feasible, as they depend on each other if they want to fulfil the scope of the activities (integration). Working alongside other charities, NGOs and council will enhance the success of the programmes. The key players continue to grow in number and change, again, depending on availability of funding. There is good collaboration amongst the key actors making their involvement feasible. The main local partner, Local Action Group, is already a publicprivate partnership including a large variety of local entities (23 actors). Consequently, this makes it easier to collaborate with all of them at once, as well as the possibility to take advantage of the network and contacts that the Local Action Group has in the territory.

Table 1 .

 1 Possible relationships developed between local actors and the SIA implementer.

Working path Explanation of the SIA implementer role Degree of involvement of SIA implementers

		Providing tips and practical tools to
	Facilitation	the local actors enabling the working	Low
		group to work more efficiently
	Mentoring	Actively guidance to local actors in the development of the action	Low-Medium
	Collaborative management	Direct collaboration in the management of the action	Medium-High
	Direct involvement	Members, partners or beneficiaries of the action	High
	2.1.1. Design phase
	The design phase is the first step of a SIA. It is the moment during which key local actors and
	the SIA implementer 1 meet to discuss about the challenge to tackle, to brainstorm alternatives and to
	identify a suitable SIA. The SIA target is defined through a collaborative and participatory local
	process (Scoping) in relation to the individual and collective needs of the actors and the communities
	involved as well as the contextual factors (the social, environmental, legislative and cultural
	conditions that might enable or constrain social innovations). Hence, a participatory gap analysis
	contributes to discern the current and desired scenario, and consequently the opportunities for action.

  , Spain (ES2): The Solsonès county (Lleida, Spain) is located in Central Catalonia and has two well differentiated parts, the north, covered by high mountains, and the south, with flatlands devoted to agriculture, mainly dryland crops. The county has a population of 13,392 inhabitants (2018), out of which over 9000 live in Solsona, the capital. The county population density is 2.3 inhabitants/km2 . The Solsonès has a Mediterranean-continental climate, with very hot summers and cold winters. Most of the county is covered by oak forests and pine forests. Gudbrandsdalen makes up half of the county of Oppland and has altogether 71,036 inhabitants, with a population density of 4.6 inhabitants/km 2 . The county host seven national parks and offer a unique mountain farm environment with a rich cultural heritage. The Guadeloupe is an outermost island region of Europe, located in the French West Indies. With an area of 1,628 km 2 , Guadeloupe is one of the 35 Hotspots for Biodiversity in the World[START_REF] Myers | Biodiversity hotspots for conservation priorities[END_REF]. It has a population of 382,704 inhabitants. The economy is mainly a tertiary economy based on services provided by the private and public sectors (85.6% of the added value) while industries, construction and agriculture contribute to 8.0%, 4.6% and 1.7% to the added value respectively[START_REF] Iedom | Rapport Annuel 2019[END_REF]. Major constraints to the economy are the ultra-peripherality in relation to European markets, narrow local markets, difficulties to access export markets and weak cash flow of rural enterprises. When it comes to forestry and agroforestry, the Regional Rural Development program highlights the lack of research and development infrastructure of those sectors, remoteness from the European decision centers adding to the marginalization of the territory and those sectors. With a mainly tertiary economy based on services provided by the private and the public sectors, 23.1% of the population is unemployed, this proportion being of 47% amongst the 15-29 age class population

	Gudbrandsdalen, Norway (NO): continues to be followed by worrisome trends in youth outmigration, including highly educated
	professionals.
	Bekaa Bekaa Region, Lebanon (LE): Deir el Ahmar, North Bekaa, is situated 75 km away from
	Beirut. The region has 11,000 inhabitants and a population density of 155.7 inhab/km 2 . The area has
	an important pilgrimage site (Our Lady of Bechouat) which attracts visitors all year long, and a florid

valley region faces challenges in the northernmost municipalities related to depopulation, elderly population, unemployment and a high proportion of inhabitants depending on welfare services and benefits. This challenge the sustainability of the region. Immigration from refugees and work migrants has been over the latter 10 to 15 years an important vector in sustaining or curbing the population decrease. At the same time, this has created integration challenges in civil society in these parts of the Gudbrandsdalen.

Lancashire-Cumbria, England (UK): Lancashire is a county in North West England. The county has a population of 1,449,300, with a density of 470 inhabitants/km 2 . Apart from the coastal towns, Lancashire is largely rural with the land devoted to vegetable crops and a large nature reserve in the northwest corner of the county, straddling the border to the South with Cumbria. Cumbria is a county with a population of half a million in an area of 6,678 km², one of the most sparsely populated counties in the United Kingdom, with 73.4 people per km 2 . Cumbria is the third largest county in England by area and is predominantly rural. However, the region has struggled with a high rural share of deprivation and dependence on welfare services and social benefits, under the themes of an aging population, unemployment, low income, qualifications/skills, housing and access to services and transportation.

Valbelluna, Italy (IT):

Valbelluna is located in northeastern Italy, in the Veneto Region, along the Piave River and includes the Dolomiti Bellunesi National Park. It has a population of 142,803 inhabitants, with a density of 106.2 inhabitants/km 2 . The larger towns, supported by manufacturing poles, concentrate two thirds of the population and have witnessed population growth (4.57% over a ten year period), while the more marginal and rural municipalities located in the foothills have witnessed population loss (-3.51%). Significantly, a large share of employment is taken up by large multinational companies (e.g. eyewear), a smaller share in tourism and services while the primary sector accounts for less than 1% of productivity. Following the 2008 crisis in manufacturing, unemployment and outmigration rose quite dramatically. More recently, while overall unemployment has started to decrease, youth unemployment has decreased much less sharply and agricultural sector. Nonetheless, change in the climate, water scarcity, and limited marketing opportunities can pose serious treats to the agriculture production in the village and its future sustainability. The area understudy can be considered marginal due to population aging, migration to the big cities, youth unemployment, influx of Syrian refugees and limited access to main services such as public transport.

Guadeloupe, France (GUA):

Table 2 .

 2 The six SIAs and the main features of their networks.

	447
	Acronym	Location	Scoping meetings	Local Actors' working group	Working group size	Type of network	Needs/opportunities
	ES1	Gúdar-Javalambre County, Spain	Meeting with local authorities, brainstorming workshop with the local community	Public, non-profit	10	External relation among collective actors	Resilient landscape facing wildfires Loss of traditional agricultural varieties Harsh farming and low amount of farmers
			Skype meeting with local				Arrival of external expertise and ideas to
	ES2	Solsones County, Spain	agency for rural development, meeting with LAG, Meeting with potential initiative, E-mail	Non-profit	1	One person (innovator) weaving new networks	the area Insufficient number of social initiatives regarding health, culture, environment, and integration in the county.
			brainstorming
	NO	Gudbrandsdalen, Norway	Meeting with national trekking association, and NGOs	Public-non-profit	6	External relation among collective actors	Integration of settled refugees Retaining local population in the area, Increase community involvement
							Support for all women from deprived
							backgrounds compounded with living in
	UK	Lancashire-Cumbria, UK	Meeting with local authorities, brainstorming workshop	Private (social enterprise)	1	One person (social entrepreneur)	deprived rural areas by setting up their own businesses Refugee women support to build
							confidence, ability to earn and socially
							integrate
	IT	Valbelluna, Italy	Meeting with local authorities	Public-private-non-profit	12	External relation among collective actors	Lack of support for new young-lead businesses in cultural, environmental and social businesses
	LE	Bekaa Region, Lebanon	Meeting with potential social innovators, Internal meeting to identify SI	Private	12	Internal relation among cooperative members	Overcome cultural, social, political, technical, economic challenges.

Table 4 .

 4 Results from the TELCOS feasibility assessment of the six Social Innovation Actions.

	Darker colors indicate stronger feasibility concerns of the different dimensions. An extended table is
			provided in Appendix B.
	Feasibility dimension	ES1	NO	UK	IT	LE	GUA

A Local Action Group (LAG) is a non profit-making composition made up of public and private organisations from rural villages having a broad representation from different socio-economic sectors, and often funded through the European Agricultural Fund for Rural Development.

An Operational Group is a group of people from a diverse combination of practical and scientific backgrounds who work on concrete, practical solutions to a problem or innovative opportunity and whose project is funded by the EU Rural Development policy. (https://bit.ly/332MDna )

Funding: This project has received funding from the European Union's Horizon 2020 Research and Innovation Programme under Grant Agreement No 677622. Acknowledgments: ETIFOR thanks the participants in SIparte, University of Padova, iPitt, Matteo Aguanno and David Rech. SEEDS-Int is grateful to the Jana al Ayadi Cooperative, Mrs Angele Ishak, and Ms Lory Boutchakdjian. The Eastern Norway Research Institute /Inland Norway University of Applied Sciences thanks DNT Gudbrandsdalen and Vågå municipal refugee services. EFI is indebted to Mijares Vivo, the Irrigation

Communities of Olba and Albentosa, the municipalities of Olba and Albentosa, LAG AGUJAMA and the Gúdar-Javalambre County. FORECO thanks Mireia Pecurul, Jaume Subirà, Begoña López, the Arrels Primary School students, and the owners of "Finca la Salada". Lancaster University Management School thanks Jane Binnion, her team, and the women at the Growing Club. The James Hutton Institute would like to thank the SYAPROVAG and its partners in the VALAB Operational Group: the National Parc of Guadeloupe, INRA, GDAEcoBio, ARECA, Fleur de Carbone, EPLEFPA Guadeloupe, Rézo 129 and the Chamber of Agriculture of Guadeloupe.

Conflicts of Interest:

The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results. The timescale is feasible.

The timescale of the activities proposed is feasible.

The timescale of the activities proposed is feasible. The SIA is an on-going programme with a planned annual rolling programme.

The timescale of the activities proposed is feasible.

The timescale of the activities proposed is feasible.

The timescale of the proposed activities is feasible.


Wenmeng Xiong
email: wenmeng.xiong@bjut.edu.cn

Changchun Bao

Maoshen Jia
email: jiamaoshen@bjut.edu.cn

José Picheral
email: jose.picheral@centralesupelec.fr

Study of MVDR beamforming with spatially distributed source: Theoretical analysis and efficient microphone array geometry optimization method

Keywords: Microphone Array, Minimum Variance Distortionless Response, Beamforming, Coherent Distributed Source, Geometry Optimization, White Noise Gain

In this paper, the minimum variance distortionless response (MVDR) beamforming technique is studied in the presence of a spatially coherently distributed (CD) source. In the first part, we propose the CD-MVDR beamforming in which the steering vector of the CD source model is used instead of the conventional point source model. We derive a theoretical expression of the white noise gain of CD-MVDR beamforming, which is inversely proportional to the square of the difference between the angular dispersion of the actual source and that of the CD-MVDR beamforming model. In the second part, based on the performance analyses, we propose an efficient optimization method for the microphone array geometry to reduce the impact of the CD source angular dispersion on the performance of the conventional MVDR beamforming. Simulation results validate our proposed theoretical expressions and show that with the proposed geometry, the conventional MVDR beamforming tends Article Title to be significantly more robust to the angular dispersion of the CD source for both white noise gain (WNG) and directivity factor (DF).

1 Introduction s a famous signal enhancing technique, beamforming [START_REF] Benesty | Microphone array signal processing[END_REF] has been widely used in speech enhancement, acoustic imaging, wireless communications, radar processing, etc. The beamforming techniques can be divided into two categories: fixed beamforming and adaptive beamforming. The fixed beamforming algorithms, including delay-and-sum beamforming, are independent of the received signals at the microphone array, while the adaptive beamforming algorithms are more flexible, the coefficients of the Beamformers can be changed according to the received signals, to preserve the signal of interest (SOI) while reducing the noise and the interference. Among the adaptive beamformings, the MVDR beamforming [START_REF] Benesty | Microphone array signal processing[END_REF] is prevail for its convenient implementation. The MVDR beamforming has two different ways to make use of the signals received on the microphone array, leading to two objective functions: the one generally employed by the conventional MVDR filter minimizes the power of the output signals of the microphone array; the other one generally employed by the MVDR filter minimizes the power of the residual noise of the microphone array with the reconstructed interference plus noise covariance matrix [START_REF] Gu | Robust adaptive beamforming based on interference covariance matrix sparse reconstruction[END_REF]. Indeed, the latter one can achieve a better output signal to interference plus noise ratio (SINR) when the input signal to noise ratio (SNR) is high.

In addition, plenty of robust beamforming techniques have been proposed to improve the performance with point source model [18; 19; 25; 29]. However, in many practical applications, the source can no longer be considered a point source and angular dispersion should be taken into consideration. According to the statistical characteristics of source, the models of the spatially distributed source can be classified into two types: the incoherently distributed (ID) sources and the coherently distributed (CD) ones [START_REF] Valaee | Parametric localization of distributed sources[END_REF]. An ID source means that signals coming from different points of the same distributed source are assumed uncorrelated. The acoustic sources caused by the vibration of the front wheels and the rearview mirrors on a moving car can be considered examples of ID sources. A CD source means that the signal components from the same source are delayed and scaled replicas of one point in the source. The trumpet on a big loudspeaker can be considered an example of a CD source. The localization of the distributed source has been widely studied in the literature [START_REF] Brooks | A deconvolution approach for the mapping of acoustic sources (damas) determined from phased microphone arrays[END_REF] [START_REF] Xiong | Sparsity-based localization of spatially coherent distributed sources[END_REF], but few works have been publicly reported on the enhancement techniques of a target distributed source. In [START_REF] Wang | On robust and high directive beamforming with small-spacing microphone arrays for scattered sources[END_REF] and [START_REF] Zheng | Sparse array beamforming design for coherently distributed sources[END_REF] the proposed beamforming for a distributed source can achieve an optimal value of SINR due to the Rayleigh quotient. However, the covariance matrix of the received signals emitted by the target source is required for the proposed method yet difficult to be obtained in practice. Thus, in our work, instead of utilizing the covariance matrix of the received signals, we assume that the steering vector of the distributed source is known a priori in the linear constraint in beamforming. We try to optimize the microphone array geometry to compensate for the performance degradation caused by the inaccuracy of the steering vector model.

The optimization of array geometry can be divided into two categories. The first category refers to choosing a subset of active microphones according to different criteria from a large scale of candidate microphones in a wireless acoustic sensor network (WASN), for the sake of improving the information transform efficiency and energy saving. In [START_REF] Zhang | Microphone subset selection for mvdr beamformer based noise reduction[END_REF] the best subset of sensors is determined by minimizing the transmission cost while constraining the output noise power. In [START_REF] Prabhakar | Sparsity-promoting sensor selection for non-linear measurement models[END_REF] the Camér-Rao Bound (CRB) is used as an optimization goal and the microphone subset selection problem is formulated as a sparsity based optimization problem. Similar work has been extended to correlated noise scenarios in [START_REF] Liu | Sensor selection for estimation with correlated measurement noise[END_REF]. In [START_REF] Bertrand | Efficient sensor subset selection and link failure response for linear mmse signal estimation in wireless sensor networks[END_REF] and [START_REF] Szurley | Energy aware greedy subset selection for speech enhancement in wireless acoustic sensor networks[END_REF] the microphone subset selection criteria are the utility of each sensor and the SNR, respectively. The criteria are solved by greedy methods. The second category refers to the design of the geometry of the microphone array directly. For example, the V shape is optimum for ambiguity-free array [START_REF] Gazzah | Optimum ambiguity-free directional and omnidirectional planar antenna arrays for doa estimation[END_REF]. The spiral array provides performance advantages over a wide frequency spectrum owing to the absence of inter-element spacing redundancy [START_REF] Aldeman | A hybrid spiral microphone array design for performance and portability[END_REF]. Beamforming with the famous differential microphone arrays (DMAs) can achieve a frequency invariant and super directive performance with a small inter-element spacing [START_REF] Benesty | Study and design of differential microphone arrays[END_REF]. Recently, to make a compromise between the WNG and DF, a novel class of microphone arrays have been proposed [START_REF] Cohen | Differential kronecker product beamforming[END_REF] to decompose the steering vector of the source as a Kronecker product of two sub-steering vectors, one sub-steering vector corresponds to a microphone array with a smaller aperture and the other sub-steering vector corresponds to a microphone array with a larger inner space than the original one. The novel class of microphone array have been extended to circular [START_REF] Zhao | Co-prime circular microphone arrays and their application to direction of arrival estimation of speech sources[END_REF], cuboid [START_REF] Wang | Design of kronecker product beamformers with cuboid microphone arrays[END_REF], and other geometries. Besides the Kronecker product method, one can also design optimal nonuniform linear DMAs with a specified target directivity pattern [START_REF] Chen | On the design and implementation of linear differential microphone arrays[END_REF].

In our work, we first propose CD-MVDR beamforming by extending the conventional MVDR beamforming to the CD source scenario, and analyse theoretically the performance of CD-MVDR beamforming: with Taylor expansions, the WNG is found to be inversely proportional to the square of the difference between the angular dispersion of the actual source and that in the CD-MVDR beamforming model. As a second step, we propose a criterion to minimize the impact of the angular dispersion on the WNG across the whole angle sector of interest of the direction of arrival (DOA) of the SOI with a minimum inter-element space constraint. Although such a criterion is effective in theory, it is difficult to implement in practice due to the high computational burden. Therefore, an iterative algorithm is proposed to efficiently optimize the array geometry, which locates the microphones one by one on the edge of the constraint region given by the located microphones. With the proposed locally-optimal geometry, both the WNG and the DF become more robust to the angular spread of the source.

This paper is organized as follows: the wide band CD source model and the conventional MVDR beamforming are briefly reviewed in section 2. The performance of CD-MVDR beamforming in the presence of the CD source is studied in section 3 and some simulation results are shown to validate the theoretical expressions. In section 4, we propose an array geometry optimization algorithm based on the results of the previous section. Finally, conclusions are given in section 5.

2 Signal model and MVDR beamforming Let us consider a far-field wide band CD [START_REF] Valaee | Parametric localization of distributed sources[END_REF] source with angular spread ∆ in an anechoic environment impinging on a microphone array from the DOA θ 0 at the speed of sound, i.e., c = 340m/s. The microphone array is composed of M sensors and assumed to be planar. With respect to the origin in the polar coordinate, the position of the m th microphone is given by the distance r m and the angular ϕ m (see Figure 1). The signal received at the m th microphone at moment t is given by:

y m (t) = π -π s(t + r m cos(θ 0 + φ -ϕ m )/c)g(φ)dφ + n m (t), (1)
where s(t) is the SOI, n m (t) is the noise observed at the m th microphone, g(φ) is introduced to describe the angular distribution of the CD source π -π g(φ)dφ = 1. With the short time Fourier transform (STFT), assuming that the window length of the STFT is long enough, the wide band signal in frequency domain ỹm (ω) can be given as: ỹm (ω) = e jωrm cos(θ0+φ-ϕm)/c g(φ)dφs(ω) + ñm (ω),

where ỹm (ω), s(ω) and ñm (ω) are the STFT of y m (t), s(t) and n m (t), respectively, j is the imaginary unit with j = √ -1.

(2) can also be expressed in a vector form as:

y(ω) = c(θ 0 , g, ω)s(ω) + n(ω), (3)
where

y(ω) = [ỹ 1 (ω), . . . , ỹM (ω)] T ∈ C M ×1 , n(ω) = [ñ 1 (ω), . . . , ñM (ω)] T ∈ C M ×1 , c(θ 0 , g, ω) = [c 1 (θ 0 , g, ω), . . . , c M (θ 0 , g, ω)] T ∈ C M ×1
, is the steering vector for a CD source which can be given by:

c(θ 0 , g, ω) = π -π a(θ 0 + φ, ω)g(φ)dφ, (4)
where a(θ 0 , ω) is the steering vector for a point source, which can be given by:

a(θ 0 , ω) = [e jωr1 cos(θ0-ϕ1)/c , ..., e jωr M cos(θ0-ϕ M )/c ] T , (5)
where the superscript T is the transpose operator. The steering vector in (4) can be seen as a set of point sources impinging from θ 0 + φ with the steering vector in [START_REF] Bertrand | Efficient sensor subset selection and link failure response for linear mmse signal estimation in wireless sensor networks[END_REF] and weighted by the distribution function g(φ).

Assuming that the source and the additive noise are uncorrelated, with (3), the correlation matrix of the signals received at the microphone array is given by:

R y E[y(ω)y H (ω)] = σ 2 s c(θ 0 , g, ω)c H (θ 0 , g, ω) + R n (ω), (6)
where E[•] denotes the mathematical expectation, the superscript H denotes the conjugate-transpose operator,

σ 2 s E[|S(ω)| 2 ] is the power of the SOI, R n (ω) E[n(ω)n H (ω)
] is the correlation matrix of the additive noise n(w) whose rank is M in general. Notice that in the case of spatially white noise, R n (ω) = σ 2 n I M , where σ 2 n is the noise power, and

I M is a M × M diagonal matrix; while in the case of spherically isotropic noise, R n (ω) = σ 2 n Γ n (ω), where [Γ n (ω)] ij = sinc[ωd ij /c] = sin(ωd ij /c)/(ωd ij /c), [Γ n (ω)] ij is the (i, j)th element of the matrix Γ n (ω)
, and d ij is the distance between the i th and the j th microphones. Note that our work focuses on the CD source since no explicit expression exists for the steering vector in (4) for the ID source.

MVDR beamforming and its performance

The main idea of the conventional MVDR beamforming is given by minimizing the output power of the microphone array while preserving the target source. Assuming that a point source impinges from θ 0 , MVDR beamforming is formulated as:

argmin h(ω) h H (ω)R y h(ω) subject to h H (ω)a(θ 0 , ω) = 1, (7)
where h(ω) can be considered as a finite impulse filter applied to each microphone output at angular frequency w.

In this paper, we investigate the performance of MVDR beamforming in the same sub-frequency band. Therefore, in the following, we omit the ω in the definitions for simplicity. The solution of ( 7) can be given as:

h = R -1 y a 0 a 0 R -1 y a 0 , (8)
where a 0 is a simplification of a(θ 0 , ω).

The two important optimization goals of beamforming are given as follows: WNG: The white noise gain is defined as the ratio between the output SNR and the input SNR of the microphone array in the scenario of spatially white noise, which also describes the robustness of the Beamformer:

G W (h) = σ 2 n h H R s h σ 2 s h H R n h . ( 9
)
In the case of one target source, the WNG can be given as 1/h H h. DF: The directivity factor is defined as the ratio between the output SNR and the input SNR of the microphone array in the scenario of a diffuse sound field, which also indicates the beampattern gain at the look direction to the average beampattern gain at other directions:

G D (h) = 1 h H Γ n (ω)h . ( 10
)

CD-MVDR beamforming

In this section, we first extend the conventional MVDR beamforming for the CD source scenario as CD-MVDR beamforming, and then study its performance.

CD-MVDR beamforming

The CD-MVDR beamforming is designed to enhance the signal of a CD source located at θ B with a generalized steering vector c(θ B , g, ω). In this case the beamforming h(ω) is the solution of:

min h(ω) h H (ω)R y h(ω) subject to h H (ω)c(θ B , ω) = 1. (11)
In the following, the steering vector model c(θ B , g, ω) in the CD-MVDR beamforming is shortened as c B . As a consequence, the CD-MVDR beamforming filter is given as:

h = R -1 y c B c B R -1 y c B . ( 12
)

CD-MVDR beamforming performance

To further investigate the performance of CD-MVDR beamforming, we first assume that the angular spread of the CD source is not too large (≤ 10 • ) in the far-field, and the distribution shape is symmetrical. Thus,

π -π g(φ)dφ = 1, π -π φg(φ)dφ = 0.
Considering the second order Taylor approximation in the angular spread, the steering vector for the CD source in (4) can be approximated by:

c 0 ≈ π -π (a 0 + ȧ0 φ + ä0 φ 2 )g(φ)dφ = a 0 + 1 2 ä0 δ 2 0 , (13)
where the steering vector c(θ 0 , g, ω) of the actual source is simplified as c 0 , a 0 is the steering vectors of the point source with the same DOA of c 0 , ȧ0 = ∂a(θ) ∂θ | θ0 , ä0 = ∂ 2 a(θ)

∂θ 2
| θ0 , δ 2 0 = φ 2 g(φ)dφ is the angular dispersion of the distributed source. The first order term in φ is canceled due to the spatial symmetry of the distribution function of the source. Note that if the distribution function of one source is simple such as Uniform, Gaussian, or Raised Cosine, its angular dispersion can be expressed explicitly as: Uniform :

δ 2 0 = ∆ 2 0 /12, Gaussian : δ 2 0 = ∆ 2 0 ,
Raised cosine :

δ 2 0 = π 2 -6 12π 2 ∆ 2 0 , (14)
where ∆ 0 is a parameter of the angular distribution function of the actual distributed source. For Uniform and Raised cosine distribution, ∆ 0 is the length of the support of the source; for Gaussian distribution, ∆ 0 is the standard deviation of the Gaussian function.

Similarly, the second order approximation of the steering vector in CD-MVDR beamforming [START_REF] Gazzah | Optimum ambiguity-free directional and omnidirectional planar antenna arrays for doa estimation[END_REF] can be given as:

c B ≈ a B + 1 2 äB δ 2 B . (15)
In the case that the a-priori information of the central DOA is well estimated, that is, θ 0 = θ B and thus a 0 = a B , the relationship between ( 13) and ( 15) can be given as:

c 0 = c B + 1 2 ä0 (δ 2 0 -δ 2 B ). ( 16
)
We focus on the study of the CD-MVDR beamforming performance when the angular spread is roughly known. We aim to express the performance with respect to the angular dispersion error δ 2 0 -δ 2 B where δ 2 0 is the dispersion of the actual source and δ 2

B is the dispersion used by the CD-MVDR beamforming. One can notice that the case δ 2 B = 0 addresses the performance study of the standard MVDR beamforming in presence of a CD source.

In addition, with the Woodbury formula [START_REF] Horn | Matrix analysis[END_REF], the inverse of the correlation matrix R y in (8) can be given by:

R -1 y = R -1 n - R -1 n c 0 c H 0 R -1 n 1 + c H 0 R -1 n c 0 . (17)
Introducing ( 17) and ( 16) into [START_REF] Prabhakar | Sparsity-promoting sensor selection for non-linear measurement models[END_REF], and keeping the second order Taylor approximation in δ, the MVDR Beamformer can be given by:

h ≈ h 0 + ∆h, ( 18
)
where

h 0 = R -1 n c B /c H B R -1 n c
B is the term depending on the central DOA of the source, the array geometry and the noise on the microphone array; ∆h

= (δ 2 0 -δ 2 B ) • R -1 n c B Real{ä B R -1 n c B } -1 2 R -1 n äB c H B R -1 n c B -1 2 R -1 n c B äH B R -1 n c B /c H B R -1 n c B is the term due to (δ 2 0 -δ 2 B
). Note that when the conventional MVDR beamforming is used, that is, the angular spread in the steering vector c B in CD-MVDR beamforming equals to 0, c B shrinks to a B , and (δ 2 0 -δ 2 B ) shrinks to δ 2 0 . In this case, ∆h depends explicitly on the angular dispersion of the actual source.

Similarly, introducing (18) into ( 9), the approximation of WNG as an explicit function of the angular spread can be deduced as:

GW (h, δ 2 0 , δ 2 B ) ≈ M 2 M σ 2 n + (δ 2 0 -δ 2 B ) 2 • 0.25M 2 äB 2 + 0.25M c H B äB 2 -M Real{ä H B c B } 2 + 0.5M Real{(ä H B c B ) 2 } . ( 19
)
From ( 19) we can see that the WNG is inversely proportional to the square of the angular dispersion difference between the actual source and CD-MVDR [START_REF] Mohammadzadeh | Robust adaptive beamforming with improved interferences suppression and a new steering vector estimation based on spatial power spectrum[END_REF] proves that the WNG can be used to correct the angular dispersion in the CD-MVDR beamforming model such that:

δ2 B = argmax δ 2 B G W (h, δ 2 0 , δ 2 B ), (20)
where G W (h, δ 2 0 , δ 2 B ) here is obtained in (9).

Numerical simulation results

In this subsection, we validate our theoretical results on CD-MVDR beamforming in [START_REF] Mohammadzadeh | Robust adaptive beamforming with improved interferences suppression and a new steering vector estimation based on spatial power spectrum[END_REF] and [START_REF] Panayotov | Librispeech: An asr corpus based on public domain audio books[END_REF]. A female speaker reading a book as the source [START_REF] Panayotov | Librispeech: An asr corpus based on public domain audio books[END_REF] impinges on the array from θ 0 = 45 • , a uniformly distributed source propagation model is generated according to (3). The total snapshots number is 2 17 , the sampling rate of the source is 16kHz, and the window length of the STFT is 256. A uniform linear array with M = 13 microphones is used. We work in the sub-frequency band of 1000Hz, d = 0.2λ = 0.068m, where λ is the wavelength of the source. In all the simulations, the signal to noise ratio (SNR) is set to 10 dB. 100 Monte Carlo experiments are executed.

In Figure 2, we plot the WNG of CD-MVDR beamforming as a function of the angular spread ∆ B in the CD-MVDR beamforming steering vector model for a uniform CD source with ∆ 0 = 5 • (Figure 2(a)) and a Gaussian CD source with ∆ 0 = 1.5 • (Figure 2(b)), respectively. The angular distribution shapes in the CD-MVDR beamforming models are identical to the actual sources. We can see that our theoretical results obtained with [START_REF] Mohammadzadeh | Robust adaptive beamforming with improved interferences suppression and a new steering vector estimation based on spatial power spectrum[END_REF] are generally consistent with the simulation results. The WNG reaches a maximum value when the angular dispersion of the actual source and that in the CD-MVDR beamforming model approaches, and then decreases when the gap between the actual angular dispersion and that in the model increases.

In Figure 3, we plot the actual and estimated angular dispersion as a function of the angular spread, for both uniform CD source and Gaussian CD source. The blue lines represent the actual angular dispersions obtained directly with [START_REF] Horn | Matrix analysis[END_REF], while the red stars represent the estimated angular dispersions obtained with [START_REF] Panayotov | Librispeech: An asr corpus based on public domain audio books[END_REF]. The distribution function in the CD-MVDR beamforming model is set to be uniform. We can see that by maximizing [START_REF] Panayotov | Librispeech: An asr corpus based on public domain audio books[END_REF] the angular dispersion can be well estimated, even if the distribution function of the actual source and that of the model in CD-MVDR beamforming are mismatched (see Figure 3(b)).

Array geometry optimization algorithm

In the previous section, we can see that CD-MVDR beamforming requires a-priori information on the angular dispersion of the CD source. The performance of CD-MVDR beamforming degrades if the angular dispersion in its model is mismatched with that of the actual source. In this section, we optimize the microphone array geometry to improve the performance of the conventional MVDR beamforming in the CD source scenario, where the a priori information of the angular dispersion of the CD source is dispensable. For simplification, the microphone array geometry optimizations are given with the WNG. We can see in the simulations in section 4.1 that the proposed geometries can also improve the DF.

In the following, we seek to maximize the WNG by minimizing the influence of the angular spread on the WNG. Firstly, we define:

α m = ωr m cos(θ 0 -ϕ m )/c, β m = ωr m sin(θ 0 -ϕ m )/c, where m = 1, ..., M.
Further, when the conventional MVDR beamforming is used, expression (19) remains valid with δ 2 B = 0 and c B becomes equal to a B . We note the terms concerning the angular dispersion in GW in [START_REF] Mohammadzadeh | Robust adaptive beamforming with improved interferences suppression and a new steering vector estimation based on spatial power spectrum[END_REF] as F and they are given as:

F = 0.25M 2 äH B äB +0.25M a H B äB 2 -M Real{ä H B a B } 2 +0.5M Real{(ä H B a B ) 2 }. ( 21
)
To explore the intrinsic structure of F, we introduce α m and β m in [START_REF] Simmer | Adaptive microphone arrays for noise suppression in the frequency domain[END_REF]. With Euler's formula, F can be given as:

F = ( δω c ) 4 - 1 4 M ( β 2 m ) 2 + 1 4 M 2 β 4 m + 1 4 M 2 α 2 m - 1 4 M ( α m ) 2 .
(22) According to Cauchy inequality, it can be easily seen that F ≥ 0, and the global minimum of F can be obtained when α 1 = ... = α m = ... = α M , and β 1 = ... = β m = ... = β M . However, it is not possible to put all the microphones in the same place in practice. It is generally necessary to impose a constraint on the minimum distance between the microphones, for the sake of eliminating a coherence too high [START_REF] Simmer | Adaptive microphone arrays for noise suppression in the frequency domain[END_REF], alleviating poor noise sensitivity at low frequencies [START_REF] Derkx | Theoretical analysis of a first-order azimuth-steerable superdirective microphone array[END_REF], as well as reducing the mutual coupling [START_REF] Liu | Super nested arrays: Linear sparse arrays with reduced mutual coupling-part i: Fundamentals[END_REF][16], etc. Therefore, in this paper, we also give a minimum distance between two neighbor microphones as a constraint for the optimization of the microphone geometry. Intuitively, a more compact array with a smaller aperture is more robust to the angular spread for the MVDR beamforming.

In addition, we can see that F is a function of θ 0 , since α m and β m are both function of θ 0 . Therefore, minimizing F will lead to an optimal geometry for a fixed θ 0 . If the DOA of the source is changed, the performance of the MVDR beamforming will degrade as well. In this paper, we propose a criterion which optimizes the geometry globally across the whole angle sector of the possible DOA:

F g = 2π 0 Fdθ. ( 23
)
After tedious calculations, the global criterion F g can be given by:

F g = 1 4 π 3 M 2 ( x 2 2 + y 2 2 ) - 1 4 π 3 M x 2 + y 2 + 1 4 π 5 M 2 3 4 x 4 4 + 3 4 y 4 4 + 3 2 x y 2 2 - 1 4 π 5 M 3 4 x 4 2 + 3 4 y 4 2 + x y 2 + 1 2 x 2 2 y 2 2 , (24)
where

x = [x 1 , ...x m , ...x M ], y = [y 1 , ...y m , ...y M ],
x m and y m are the Cartesian coordinates of the m th microphone, respectively, • p is the operator of pnorm, • is the mean value operator, is the element wise product operator. Therefore, the optimization of the microphone array geometry means finding the x and y satisfying: where d is the minimum inter element space constraint that we impose a-priori. The criterion above is not convex and can be solved by greedy search across the whole region of interest for all the microphones with an extremely high computational burden. To overcome this problem, here we propose to solve [START_REF] Wang | Parameter-free landweber iteration method for robust adaptive beamforming[END_REF] with an iterative method which can find at least one local solution of the optimization problem: Intuitively, from [START_REF] Szurley | Energy aware greedy subset selection for speech enhancement in wireless acoustic sensor networks[END_REF] we can see that all the microphones have the trend to be located as close as possible. Without loss of generality, we assume that the first microphone is placed at the origin and the second on the x axis at (d, 0). Defining that C(m) is the circle centered at the m th microphone with a radius d, the m th microphone will be located on the edge of the union of C(1),C(2),...,C(m -1), at the same time the criterion of minimizing F g is satisfied for the first m microphones. The iterative optimization algorithm is formulated as in table 1.

{x opt , y opt } = arg min x,y F g 2 2 s.t. (x m , y m ) -(x n , y n ) 2 ≥ d, where m, n ∈ [1, ..., M ] , m = n, (25)

Numerical simulation results

In this subsection, we present the locally-optimal geometry obtained with the algorithm in [START_REF] Wang | Parameter-free landweber iteration method for robust adaptive beamforming[END_REF] with various numbers of microphones and show the advantages of the proposed geometry by comparing the performance of the conventional MVDR beamforming using various array geometries.

In Figure 4, the locally-optimal geometries are plotted for various values of the microphone number M . The red dots represent the microphones and the black dashed lines represent the circles centered at the microphones with radius d = 0.2λ; where d is the minimum inter-element space constraint. We can see that the locally-optimal geometry is fan-shaped, and Figure 4(e) and 4(h) show that the locally-optimal geometry has the trend to be concentric.

In Figure 5, the WNG and DF as a two-dimensional function of the angular spread and the DOA of the actual source are illustrated. We can see that in Figure 5(a) the WNG is robust to the DOA of the source, as the proposed locally-optimal geometry is obtained by minimizing the impact of the angular spread of the CD source on the WNG of MVDR beamforming. On the contrary, in Figure 5(b) the DF is less robust to the DOA than the WNG. Similarly, Figure 5(c) and Figure 5(d) show the WNG and DF with spiral geometry, where the aperture of the spiral geometry is approximately equivalent to the proposed locally-optimal geometry. We can see the advantage of the proposed geometry for both the WNG and DF.

In Figure 6, we compare the WNG and the DF of the MVDR beamforming with a Fermat spiral array (Figure 6 the proposed array. All the microphone arrays compose of M = 13 microphones and have the same minimum inter-element space constraint. The spherically isotropic noise is generated according to the method in [START_REF] Habets | Generating sensor signals in isotropic noise fields[END_REF]. In Figure 6(c) we can see that the performance degradation with the proposed geometry is significantly less than the others, while the performance degradation with the spiral geometry is the largest, which can be explained by the fact that under the same minimum inter-element space constraint, the proposed geometry has the minimum array aperture as well as the minimum value of F in [START_REF] Szurley | Energy aware greedy subset selection for speech enhancement in wireless acoustic sensor networks[END_REF]. Moreover, in Figure 6(d) we can see that, even though our proposed criterion for the optimization of the array is based on the WNG, with the locally-optimal geometry the DF degrades significantly less than the others.

In Figure 7 and Figure 8, we compare the proposed method with the modeldriven microphone subset selection (MSS) method proposed in [START_REF] Zhang | Microphone subset selection for mvdr beamformer based noise reduction[END_REF]. As one of the state-of-the-art methods, the MSS method selecting the most informative subset of microphones from a WASN for MVDR beamforming. Its main idea is as follows:

argmin h,p∈{0,1} M diag(p)c 1
.t.h H R n h ≤ β α , h H a = 1, (26)
where p = [p 1 , p 2 , ..., p M ] ∈ {0, 1} M , and p i = 1 means that the i th microphone is selected, c = [c 1 , c 2 , ..., c M ] T denote the pairwise transmission cost between each microphone and the fusion center (FC) of the WASN, β denotes the output noise power after beamforming when all microphones are used (i.e., p = 1 M ), α ∈ (0, 1] is an adaptive factor to control the output noise power compared to β. With this configuration, the number of microphones is always equal to 13. We can see from Figure 7 and Figure 8 that the advantages of the proposed method lie in the following aspects: 1) The MSS method concerns choosing informative microphones from a large number of candidate microphones, while the proposed method only needs a small number of microphones. 2) For the (h) Spherically isotropic noise, α = 0.3, θ 0 = 135 • , FC (0, 0) Fig. 7: Array geometries obtained by the model-driven MSS method in [START_REF] Zhang | Microphone subset selection for mvdr beamformer based noise reduction[END_REF] in different noise scenarios (inter element space d = 0.2λ) MSS method, the parameter α is introduced to control the output power of the and can thus impact the array geometry (see the subfigures in Figure 7); and the position of the FC can also impact the array geometry (see Figure 7(e) and Figure 7(f)). The proposed method does not have such parameters.

3) The array geometry obtained by the MSS method is sensitive to the DOA of the SOI in some scenarios, see Figure 7(g) and Figure 7(h) for example, while the proposed criterion in our paper is robust to the DOA of the SOI. 4) Figure 8 illustrates that if the FC is placed at the origin, the proposed method performs approximately as well as the MSS method in the spatially white noise scenario and outperforms slightly the MSS method in the spherically isotropic noise scenario. However, if the FC is placed with a bias with respect to the origin, for example at (d, 0), with the MSS method the robustness of beamforming to the angular distribution of the SOI decreases evidently.

Conclusions

In this paper, we have first extended the conventional MVDR beamforming into the CD source scenario as the CD-MVDR beamforming; and we have found that the WNG of CD-MVDR beamforming is inversely proportional to the square of the angular dispersion difference between the actual source and the CD-MVDR beamforming model. In the second part, we have proposed an efficient optimization method of microphone array geometry based on the minimization of the WNG of the conventional MVDR beamforming. We have found that the locally-optimal geometry is fan-shaped and has the trend to be concentric with one microphone at the center. Numerical results have shown that the proposed locally-optimal geometry is more robust than the MVDR beamforming in the CD source scenario for both the WNG and the DF.

2. 1

 1 The CD source model

Fig. 1 :

 1 Fig. 1: Illustration of planar microphone array and spatially distributed source.

Fig. 2 :Fig. 3 :

 23 WNG vs. angular spread model in CD-MVDR beamforming (Uniform CD source with ∆ 0 = 5 • and Gaussian CD source with ∆ 0 = 1.5 • .) Angular dispersion vs. angular spread of actual CD source beamforming model. Besides,

Table 1 : 2 s

 12 Iterative method for array optimization Parameters: M,d Initialization: x 1 = y 1 = 0, x 2 = d, y 2 = 0 x 3 = ... = x M = y 3 = ... = y M = 0 For m = 3 : M {xm, ym} = arg min xm,ym Fg(x 1 , ..., xm, y 1 , ..., ym) 2 .t. {xm, ym} ∈ edge{C(1) ∪ ... ∪ C(m -1)} end

  (a)), an arbitrary array (Figure 6(b)), and

Fig. 4 :

 4 Locally-optimal geometry with different number of microphones (interelement space constraint d = 0.2λ)

Fig. 5 :

 5 WNG and DF vs. angular spread and DOA (uniformly distributed source, (a) and (b): the locally-optimal geometry with M = 13, (c) and (d): the spiral geometry with M = 13)

Fig. 6 :

 6 WNG and DF vs. angular spread (θ 0 = 45 • , uniformly distributed source, M = 13) s

Figure 7

 7 illustrates the array geometry examples obtained by the MSS method in different noise scenarios and with different values of α. The FC is placed at the origin or at (d, 0). The red dots represent the candidate microphones and the blue circles represent the selected microphones. α = 0.15 in spatially white noise scenario (Figure7(b) and 7(c)) and α = 0.25 in spherically isotropic noise scenario (Figure7(e) and 7(f)) are considered in Figure8to compare the SNR gain performance with our proposed geometry.

  Spatially white noise, α = 0.07, θ 0 = 45 • , FC (0, 0) Spatially white noise, α = 0.15, θ 0 = 45 • , FC (0, 0) Spatially white noise, α = 0.15, θ 0 = 45 • , FC (d, 0) Spatially white noise, α = 0.2, θ 0 = 45 • , FC (0, 0) Spherically isotropic noise, α = 0.25, θ 0 = 45 • , FC (0, 0) Spherically isotropic noise, α = 0.25, θ 0 = 45 • , FC (d, 0) Spherically isotropic noise, α = 0.3, θ 0 = 45 • , FC (0, 0)

Fig. 8 :

 8 WNG and DF vs. angular spread (θ 0 = 45 • , uniformly distributed source, spatially white noise, M = 13)

Fundings

This work was supported by the National Natural Science Foundation of China (Grant No.61831019, No.61971015, and No.62101013).

Data availability statement

The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.

Declaration

The authors declare that they have no conflict of interest.