Jonathan Carruthers-Jones

Alice Eldridge

Patrice Guyot

Christopher Hassall

George Holmes

The Call of the Wild: investigating the potential for ecoacoustic methods in mapping wilderness areas

Keywords: wilderness, conservation, soundscape, biodiversity, urban-wild interface, participatory mapping

The critical importance of wilderness areas (WAs) for biodiversity conservation and human wellbeing is well established yet mapping criterion on which WA management policies are based take neither into account. Current WA mapping methods are framed in terms of absence of anthropogenic influence and created using visual satellite data, obviating consideration of the ecological or anthropogenic value of WAs. In this paper we suggest that taking the acoustic environment into account could address this lacuna. We report the first investigation into the potential for ecoacoustic methods to complement existing geophysical approaches. Participatory walks, including in situ questionnaires and ecoacoustic surveys were carried out at points along transects traversing urban-wilderness gradients at four study sites in the Scottish Highlands and French Pyrenees. The relationships between a suite of six acoustic indices (AIs), wilderness classifications and human subjective ratings were examined. We observed significant differences between five out of six AIs tested across wilderness classes, demonstrating significant differences in the soundscape across urban-wild gradients. Strong, significant correlations between AIs, wilderness classes and human perception of wildness were observed, although magnitude and direction of correlations varied across sites. Finally, a compound acoustic index is shown to strongly predict mapped wildness classes (up to 95% variance explained MSE 0.22); perceived wilderness and biodiversity are even more strongly predicted. Together these results demonstrate that the acoustic environment varies significantly along urban-wild gradients; AIs reveal details of Carruthers-Jones/ Call of the Wild environmental variation excluded under current methods and capture key facets of the human experience of wildness. An important next step is to ascertain the ecological and anthropogenic relevance of these differences and develop new automated acoustic analysis methods suited to mapping the environmental characteristics of WAs. Taken together, our results suggest that future management of WAs could benefit from ecoacoustic methods to take the biosphere and anthroposphere into account.

Introduction

Wilderness areas (WA) are critical to sustaining both biodiversity and human well-being [START_REF] Watson | Catastrophic declines in wilderness areas undermine global environment targets[END_REF]. WAs are considered the "last refuges" for many rare and endangered wild animals and plants [START_REF] Mittermeier | Wilderness and biodiversity conservation[END_REF] and have a significant role in ensuring the long-term persistence of biodiversity [START_REF] Soule | Rewilding and biodiversity: complementary goals for continental conservation[END_REF][START_REF] Watson | Wilderness and future conservation priorities in Australia[END_REF][START_REF] Mittermeier | Turtle hotspots: an analysis of the occurrence of tortoises and freshwater turtles in biodiversity hotspots, high-biodiversity wilderness areas, and turtle priority areas[END_REF] as well as coping with the threats of climate change and other impacts of human development. At the same time, the value of WAs for human well-being [START_REF] Barton | The wilderness expedition: An effective life course intervention to improve young people's well-being and connectedness to nature[END_REF][START_REF] Harper | Wilderness therapy, therapeutic camping and adventure education in child and youth care literature: A scoping review[END_REF] and sense of place [START_REF] Hausmann | The ecosystem service of sense of place: benefits for human well-being and biodiversity conservation[END_REF] is increasingly recognised. Urban expansion and landscape fragmentation have significantly reduced both the overall amount, size and connectivity of WAs globally [START_REF] Ellis | Anthropogenic transformation of the biomes, 1700 to 2000[END_REF] heightening the strategic importance of their systematic identification [START_REF] Kuiters | Wilderness register and indicator for Europe final report[END_REF][START_REF] Wilson | Half-earth: our planet's fight for life[END_REF][START_REF] Lin | Identifying local-scale wilderness for on-ground conservation actions within a global biodiversity hotspot[END_REF] and stimulating urgent calls for new methods to comprehensively and cost-effectively map remaining areas [START_REF] Carver | Mapping wilderness[END_REF]. In this paper we propose that the emerging field of ecoacoustics [START_REF] Et Sueur | Ecoacoustics: the ecological investigation and interpretation of environmental sound[END_REF] could serve as a useful complement to extant approaches by firstly providing a unifying framework within which anthropogenic, geophysical and ecological perspectives can be conceptually integrated and secondly providing a cost-effective, scalable method for incorporating biodiversity assessment.

Debate over the best approaches to the protection of wilderness has a long history at both global and European levels [START_REF] Thoreau | Walking". The Atlantic Monthly[END_REF]Wilderness Act, 1964;[START_REF] Soule | Should wilderness be managed. Return of the wild[END_REF][START_REF] Zunino | A perspective on wilderness in Europe[END_REF]. The International Union for the Conservation of Nature (IUCN) describes two main attributes of WAs (Protected Area designation -Category Ib): a relatively high degree of ecological naturalness (the degree to which an ecosystem has deviated from its original state due to human in influence), and the absence of human artefacts (e.g. roads, houses, train lines etc), [START_REF] Dudley | Guidelines for applying protected area management categories[END_REF]. Within Europe especially the term wildness is now argued to be less politicised than wilderness and also captures the smaller size of the remaining intact land areas on the continent [START_REF] Ward | For wilderness or wildness? Decolonising rewilding[END_REF]. Wildness quality mapping represents wildness on a relative scale capturing not just areas of high wildness but the entire continuum from urban to wild. Most examples of wildness quality maps in Europe use satellite data to create maps on a continuum from least wild (e.g. the centre of a large urban conurbation) to most wild (e.g. a remote corner of a mountainous region), [START_REF] Sanderson | The human footprint and the last of the wild: the human footprint is a global map of human influence on the land surface, which suggests that human beings are stewards of nature, whether we like it or not[END_REF]Carver et al., 2012;Muller, 2015). Maps usually comprise four key layers: perceived naturalness; absence of modern artefacts; rugged or challenging terrain; and remoteness from roads and ferries. Similar multi-criterion approaches, based on satellite data, have been developed in many regions: Australian national wilderness inventory (Leslie and Maslen, 1995), the wildness quality index for Europe [START_REF] Eea | Europe's ecological backbone: recognising the true value of our mountains[END_REF], the human footprint index at the global scale [START_REF] Sanderson | The human footprint and the last of the wild: the human footprint is a global map of human influence on the land surface, which suggests that human beings are stewards of nature, whether we like it or not[END_REF], the map of Denmark [START_REF] Müller | Where are the wilder parts of anthropogenic landscapes? A mapping case study for Denmark[END_REF], Cairngorm National Park Wildness Quality map [START_REF] Carver | Wildness Study in the Cairngorms National Park[END_REF]. This multi-criterion approach is attractive because it can be operationalized at scale using satellite data and Geographical Information Systems (GIS) to create comprehensive maps that support landscape management and decision making such as for renewable energy projects or protected area management [START_REF] Mcmorran | Scotland's wild mountains: addressing key challenges[END_REF]Ma andLong, 2019, see also Scottish Natural Heritage, 2014). However, overdependence on remotely sensed national scale datasets means that key facets of ecological and human importance are neglected. Human experiences of WAs are intrinsically situated, multisensory and subjective. The value of WAs from a human perspective cannot be mapped remotely but requires in situ assessments in order to reflect the rich, multisensorial and subjective reality of how people understand and value wild places (Ólafsdóttir et al., 2016). Current attempts to develop complementary methods to capture human-level experience repurpose everyday technologies to support terrestrial mapping: viewshed analysis, an approach adapted from computer gaming, has been explored in order to assess ground-level vistas, rather than aerial land cover [START_REF] Carver | Real-time visibility analysis and rapid viewshed calculation using a voxel-based modelling approach[END_REF][START_REF] Sang | Wild Vistas: Progress in Computational Approaches to 'Viewshed'Analysis[END_REF]; social media networks have been coopted to enable crowdsourcing of visitor perceptions of trails in the USA [START_REF] Carver | Keeping it wild: Mapping wilderness character in the United States[END_REF]See et al., 2014). Such approaches begin to capture human perspectives but focus exclusively on visual attributes of the wild landscape, and struggle to capture wider human experience.

The use of 'walking' or 'mobile' participatory methods to research people's attitudes to place has also grown markedly in recent decades as a key research tool for capturing data relating to people's experience, knowledge and attitudes to surrounding landscapes [START_REF] Macpherson | Walking methods in landscape research: moving bodies, spaces of disclosure and rapport[END_REF][START_REF] Vergunst | Ways of Walking Ethnography and Practice on Foot[END_REF]. Participatory methods have been used to capture a range of attributes, including cultural and experiential values for WAs and their potential long-term benefits [START_REF] Holden | The Roles of Cultural Values in Landscape Management: Valuing the'morethan-visual[END_REF][START_REF] Brown | Mixed methods participatory GIS: An evaluation of the validity of qualitative and quantitative mapping methods[END_REF][START_REF] Dorning | Integrating spatially explicit representations of landscape perceptions into land change research[END_REF]. Capturing stakeholder attitudes to landscape may be most accurately performed in the field, in spite of the challenges this brings [START_REF] Scott | Seeing is not everything': Exploring the landscape experiences of different publics[END_REF]. Walking research offers an intuitive and compelling means of studying human relationships with landscape and place [START_REF] De Certeau | Walking in the City[END_REF]Pink, 2007). When walking methods involve walking interviews, they have been found to generate deeper place-based narratives than sedentary research practices, particularly in terms of narrative quantity and spatial specificity to the study area [START_REF] Evans | The walking interview: Methodology, mobility and place[END_REF]. However most structured approaches to walking methodologies have focused exclusively on urban zones and a key challenge remains how this fine grained local qualitative knowledge can be implemented in a structured way so as to allow comparison between individuals and across different habitat types and landscape gradients. An outstanding methodological challenge is how to design conceptual frameworks for combining rich qualitative data that comes from these mobile methods and the quantitative data available from remote sensing which forms the bedrock of current wildness mapping approaches.

In ecological terms, the current approach to mapping wildness within Europe (using data based primarily on human influence) fails to capture key ecological characteristics, including biodiversity.

Contemporary wildness debates highlight how depleted many designated WAs are in terms of their native species as well as their overall levels of biodiversity [START_REF] Lewis | Estimating dark diversity and species pools: an empirical assessment of two methods[END_REF]Monbiot, 2013;[START_REF] Pheasant | Towards predicting wildness in the United Kingdom[END_REF][START_REF] Guetté | Naturalité»: concepts et méthodes appliqués à la conservation de la nature[END_REF]. In response, approaches to measuring the intactness of natural processes are being explored [START_REF] Dearden | Wilderness and our common future[END_REF], however, operationalising an assessment protocol for use at scale has yet to be achieved. Comprehensive, scalable methods to incorporate biodiversity assessments within WA mapping remains a significant challenge (Pettorelii et al., 2019).

The need for cost-effective biodiversity assessment tools is of course not limited to WA mapping, but a requisite across all fields of conservation. Situated within the emerging discipline of Ecoacoustics [START_REF] Et Sueur | Ecoacoustics: the ecological investigation and interpretation of environmental sound[END_REF] there is increasing interest in acoustic methods for biodiversity appraisal from researchers, managers and policymakers alike. Ecoacoustics understands the acoustic environment, or soundscape (Pijanowksi et al., 2011), as a resource and therefore as a source of information about ecological status -the soundscape being structured through evolutionary processes, akin to other niche construction processes. Based on the assumption that computational analyses of acoustic recordings therefore provide a biodiversity proxy, an ecological machine listening is emerging, dubbed Rapid Acoustic Survey (Sueur et al., 2008). Over 60 computational acoustic indices have been proposed and evaluated to date [START_REF] Buxton | Efficacy of extracting indices from large-scale acoustic recordings to monitor biodiversity[END_REF] and variously been shown to map spatial heterogeneity [START_REF] Bormpoudakis | Spatial heterogeneity of ambient sound at the habitat type level: ecological implications and applications[END_REF], reflect observed changes in habitat status [START_REF] Kasten | The remote environmental assessment laboratory's acoustic library: an archive for studying soundscape ecology[END_REF], biocondition [START_REF] Eyre | BioCondition: A Condition Assessment Framework for Terrestrial Biodiversity in Queensland[END_REF], and to strongly predict species richness across a wide range of terrestrial [START_REF] Eldridge | Sounding out ecoacoustic metrics: Avian species richness is predicted by acoustic indices in temperate but not tropical habitats[END_REF][START_REF] Boelman | Multi-trophic invasion resistance in Hawaii: bioacoustics, field surveys, and airborne remote sensing[END_REF] and aquatic habitats [START_REF] Bertucci | Acoustic indices provide information on the status of coral reefs: an example from Moorea Island in the South Pacific[END_REF][START_REF] Harris | Ecoacoustic indices as proxies for biodiversity on temperate reefs[END_REF]. The increasing power and decreasing cost of hardware makes acoustic survey comparable to satellite monitoring in terms of scalability in space and time, but it has the benefit of providing high-resolution data which intimately reflect the real-time dynamics of populations in situ. Acoustic survey is a highly attractive solution for large scale ecological monitoring, especially in remote locations such as WAs, because it is non-invasive, obviates the need for expert aural identification of individual recordings, is potentially sensitive to multiple taxa and scales cost-effectively (Sueur et al., 2008).

As well as providing cost-effective monitoring methods, ecoacoustics offers a valuable conceptual framework to integrate biospheric and anthropogenic perspectives. Following [START_REF] Odum | Purism scale approach for wilderness mapping in Iceland[END_REF] classification of broad ecosystem components, elements of the soundscape are described according to their source: Geophony denotes the sounds made by abiotic processes (wind, rain etc.) in the landscape; biophony the sounds of animals; and anthrophony, the sounds of humans (Pijanowksi et al., 2011) We find the term technophony to be more useful in order to refer specifically to the noises of man-made powered machinery which are distinct in terms of their acoustic signals and resulting impact on soniferous species communication [START_REF] Gage | Visualization of temporal change in soundscape power of a Michigan lake habitat over a 4-year period[END_REF]. The soundscape is therefore a site of rich interaction between processes of the lithosphere, biosphere, hydrosphere and anthroposphere: machine listening provides a means to listen to and interpret these interactions. In terms of WA mapping, soundscape components provide descriptors for auditory correlates of existing WA criteria (e.g. distance from road) and a unified framework within which to consider facets of biodiversity and human experience which are currently absent in wildness quality mapping and excluded in decision making.

We propose a new direction for WA mapping and management by investigating the potential for ecoacoustics as both a conceptual framework and monitoring method to integrate human and ecological perspectives with current geophysical WA mapping schema. We report the first systematic investigation of the relationship between acoustic indices, wildness quality metrics and in situ human subjective perceptions of wildness and biodiversity. Our investigation is structured by the following questions: Q1) How do AIs differ along a gradient of mapped wildness categories? Q2) What is the relationship between AIs and a) wildness categories b) human subjective perceptions of wildness and biodiversity? Q3) Do AI predict a) wildness quality b) human perceptions of wildness and biodiversity?

We predict that a) overall sound levels and presence of low frequency signals will decrease with increasing wildness as we move away from roads and other human influence; If wildness is associated with higher biodiversity, then b) we would expect an increase in biophonic activity with increasing wildness. If AIs are sensitive to factors which influence human perceptions of WAs other than those captured in wildness quality metrics then we would expect c) AIs to predict human perceptions more strongly than wildness classes.

Methods

Study sites

Study transects were identified at four sites across the Scottish Highlands and French Pyrenees, each along comparable gradients from urban to wild. Existing maps of wildness were available for both countries. The four sites were Invereshie & Inshriach National Nature Reserve (I&I) on the Scottish east coast (Lat 57° 6' 45.8244'' N,Long 3° 50' 39.7644 -least wild to most wild (supplementary materials A). This existing remote sensed data on wildness provided a reference condition against which to measure other data types. These simplified Jenks maps of wildness were then used in the GIS to search for a transect that covered a viable continuum of wildness -least wild to most wild -which could be walked in 5 to 6 hours. At all sites transect gradients spanned a small village (WC 2) to a high mountain area (WC 8) and the high wild areas feature relatively intact natural areas representative of the Scottish Highlands and French Pyrenees, as defined by the local park authorities. Eight acoustic survey points and participant questionnaire points were selected along the transect at each site, matched across sites to give equivalent representations of WC and habitat. See Table 1 for the description of the subset of sites studied.

Participant recruitment and perception surveys

Stakeholder participant groups were identified using a strategic iterative snowball process [START_REF] Dougill | Learning from doing participatory rural research: lessons from the Peak District National Park[END_REF][START_REF] Reed | Who's in and why? A typology of stakeholder analysis methods for natural resource management[END_REF][START_REF] Colvin | Approaches to identifying stakeholders in environmental management: Insights from practitioners to go beyond the 'usual suspects[END_REF] which aimed to: i) avoid imposing a selective stakeholder typology, ii) develop a rounded understanding of who had an interest in the issue, and B for questionnaires). To minimise the impact of weather on participant experience, all walks were conducted on days of non-extreme weather conditions (absence of lying snow cover, high winds or heavy rain). Walks at Scottish sites were conducted between April and September 2017; walks at French site LES were conducted during June-September 2018.

Acoustic surveys

Acoustic surveys were carried out for four days at each of the eight sample points at each site sequentially July 2017; LES and POT September 2017). To avoid introduction of additional human sound sources, participatory walks and acoustic surveys were held on different days; surveys were carried out during daylight hours to match the acoustic environment experienced by walkers. Recordings were made for five minutes in every 15 minutes between 07:00 and 21:00 using eight Wildlife Acoustics Song Meter SM2+ offline digital recorders at 16 bit amplitude resolution with a 48 kHz sampling rate and a gain of +36dB, giving a total of 7168 stereo files. The Song Meter is a battery powered, offline, programmable weatherproof recorder, with two channels of omni-directional sound and a flat frequency response between 20 Hz and 20 kHz.

Recorders were fixed to trees or posts at 1.5m above ground level and orientated south to standardise for prevailing weather conditions and wind noise (see Supplementary Materials A and C

for details of acoustic survey points).

Manual screening of audio data confirmed that the left (opposite to prevailing weather) channel was consistently less distorted by wind, so the right channel was dropped from the analysis; mono recordings were pre-processed using a high pass filter at 1kHz to remove remaining artefacts whilst preserving low frequency energy associated with human influence (technophony). Equipment failure and extreme weather rendered 1275 files (17%) unusable; these sites were dropped from the recordings leaving a total of 5893 files (I&I, N=1351; BEN, N=1110; POT, N=1715; LES, N=1717). This left 5 matched sample points (from the original 8) at each of the four study sites, representing 5 different wildness classes. See Table 1.

Acoustic Indices

AIs were selected and designed based extensive literature review and our previous validation studies [START_REF] Eldridge | Sounding out ecoacoustic metrics: Avian species richness is predicted by acoustic indices in temperate but not tropical habitats[END_REF]. Six acoustic indices were selected from over 50 initially explored to characterise: a) biophonic activity as an indicator of biodiversity; b) technophonic activity as an indicator of human influence and c) overall sound energy as an indicator of absence of noise. Three ecological indices which have been demonstrably linked with biodiversity in temperate biomes were chosen as biodiversity proxies: Acoustic complexity Index (ACI) which has been reported to correlate significantly with the number of avian vocalisations in an Italian national park (Pieretti et al., 2011); Bioacoustics Index (BAI) [START_REF] Boelman | Multi-trophic invasion resistance in Hawaii: bioacoustics, field surveys, and airborne remote sensing[END_REF] which is reported to show significant association with avian species richness [START_REF] Fuller | Connecting soundscape to landscape: Which acoustic index best describes landscape configuration?[END_REF] and Acoustic Evenness Index (AEI) (Villanueva-Rivera et al., 2011) which has shown strongly predict avian species richness [START_REF] Eldridge | Sounding out ecoacoustic metrics: Avian species richness is predicted by acoustic indices in temperate but not tropical habitats[END_REF]; A novel variant of the Normalised Difference Soundscape Index NDSI [START_REF] Kasten | The remote environmental assessment laboratory's acoustic library: an archive for studying soundscape ecology[END_REF], the Relative Technophony Index (RTI) is introduced as a measure of technophony (see Supplementary Material D for details); and two standard acoustic descriptors used in machine listening tasks to track overall sound energy: Root Mean Square (RMS) and Spectral Centroid (SC, a measure of the overall distribution of sound energy across the frequency spectrum) [START_REF] Peeters | A large set of audio features for sound description (similarity andclassification) in the CUIDADO project[END_REF]. Median values for RMS and SC were used as they are more robust to outliers. See Supplementary Material D for details of all AIs. All acoustic analyses were carried out using a bespoke Python library (Guyot, 2018) which implements and extends R libraries seewave (Sueur et al., 2008) and sound ecology (Villanueva-Rivera et al., 2011).

Auditioning

In order to support interpretation of the acoustic indices, a subset of recordings were selected by taking the median value for RMS at each sample point for each site as indicative of the acoustic activity at that site. These were auditioned by JCJ, AE and PG, noting the dominant sound sources (cars, planes, people, birds, wind, rain). Recordings are available at http://tiny.cc/mdiq6y.

Statistical Analyses

To explore how each AI differs along a wildness gradient (Q1), Wilcoxon signed-rank tests were carried out to test for differences in AI values between pairs of WCs across days. To investigate the relationship between AIs and WCs and human perceptions of wildness and biodiversity (Q2), twotailed Spearman's rank correlation tests were carried out between each of the six AIs and respective wildness measures and human perceptual judgements. All analyses were carried out for all sites combined as well as sites individually.

Previous ecoacoustic research has demonstrated that compound metrics are more powerful than any single AI in predicting biodiversity metrics such as species richness and/or abundance; [START_REF] Eldridge | Sounding out ecoacoustic metrics: Avian species richness is predicted by acoustic indices in temperate but not tropical habitats[END_REF][START_REF] Towsey | The use of acoustic indices to determine avian species richness in audio-recordings of the environment[END_REF], therefore to test whether acoustic analyses predict either WC or human perceptions of wildness (Q3), multivariate random forest regression models [START_REF] Breiman | Random forests[END_REF] were built using all 6 AIs as predictors and either WC or human perception of wildness or biodiversity as response. Multivariate random forest regression creates a model based on multiple decision trees to describe a response variable based on one or more predictors, then merges those trees to obtain a more accurate prediction; they are tolerant of deviations from parametric assumptions and skew in the data. The total percentage variance explained and mean squared error (MSE) of the model provide an indication of the predictive strength and accuracy. The relative contribution of predictors was assessed using Variable Importance (VIMP): the difference between prediction error when a given predictor variable is noised up by randomly permuting its values, compared to prediction error under the observed values.

Results

Do AIs differ along urban-wild gradients?

AIs plotted by WC for all sites combined reveal a large degree of scatter for individual AI values (Fig 2 .) suggesting a wide range of variation within and between wildness classes across sites. SC

shows the strongest increasing trend overall. RMS and BAI show the strongest decreasing trend. Comparisons among sites (Fig. 3.) show significant differences between WCs in five out of six AIs demonstrating strong variation in acoustic environment across the urban-wild gradients studied. In all but one case (AEI at BEN) there are significant differences between extremes of wilderness gradients (WC2 and WC8), but none vary as a simple monotonic function of WC and considerable variation is observed in the patterns of significant difference between sites, suggesting that there are variations in the soundscape beyond those reflected in current wildness quality maps.

The clearest trend is observed in the SC which tends to increase, reflecting an overall reduction in low frequency energy as we move along urban-wild gradients; RMS similarly tends to decrease, reflecting an overall reduction in amplitude of all sound signals. Within this general trend, median values for some survey points are significantly above (POT 3) and below (LES 5 and 6,BEN 6) values at the ends of the gradient in urban and most wild sites, others show markedly larger variance (BEN 6). RTI largely mirrors SC, showing significant decreases from peri-urban (W2) to remote sites (W8) across locations. BEN is the exception here where there is a significant increase with increasing wildness. The same sites LES 5 and 6 and POT 3 show marked deviations from otherwise almost linear trends. Biophonic activity, as indicated by the ecological indices (BAI, AEI and ACI) tends to decrease from urban to wild sites with significantly greater values between WC2 and WC8 at each site except BEN which shows an increase. The clearest trend is visible at I&I.

What is the relationship between AIs and a) wildness categories b) human subjective perceptions of wildness and biodiversity?

Correlation analyses (Fig. 4) suggest that WCs largely reflect human perceptions of wildness and biodiversity, with strong positive correlations in all sites tested. AIs show predominantly moderate, significant correlations with WC, but these vary in magnitude and direction across sites (Fig. 4 top rows).

In line with analyses of AI against wildness class (Fig. 3), acoustic features SC and RMS show the strongest and most consistent relationships. SC shows a moderate, positive relationship with WC and distance from road at sites I&I, LES and POT; relationships with human perceptions of wildness and biodiversity are similar at I&I and LES, however BEN shows no relationship between SC and human perceptions of wildness, but a moderate positive relationship with biodiversity. RMS shows a moderate (I&I, LES) to strong (POT) negative relationship with WC and distance from road.

The relationship between WC and ecological indices ACI, AEI and BAI are significant but vary in magnitude and direction across sites, suggesting variation in levels of biodiversity along the urbanwild gradient between sites. This pattern of relationships seen with WC is the same for distance from road and human perceptions, except at BEN where fewer significant relationships are observed.

Finally, RTI shows significant relationships, but contrary to our predictions does not show positive relationships with WC or distance from road: moderate negative correlations are observed with all four measures in I&I, small positive relationships in BEN and no significant correlations observed in LES (WC) or POT (WC or Road). Crosses denote non-significant correlations (95% confidence intervals). Note that no human data is available for POT.

Do AIs predict mapped and perceived wildness?

Multivariate regression models show that the 6 AIs tested strongly predict wildness class at each site with low error (Fig. 5). Variance explained for all sites combined is lower than for any individual site apart from BEN, suggesting that variation between sites is stronger than that along the urban-wild gradient. Variable importance varies between sites ( variance, in line with prediction that sound levels will decrease and dominant frequency will increase along urban-wild gradients. 1.15) and are surprisingly poor predictors of distance from road (Table 3).

Discussion

We investigated the relationships between AIs, currently designated wildness classes, and human subjective judgements of wildness and biodiversity. A range of statistical analyses were used to investigate how sound levels, frequency content and soundscape components varied along urbanwild gradients at four different sites. Our results demonstrate that i) The soundscape varies significantly over wildness class; ii) There are significant variations in soundscape across wild classes between study sites; iii) biophonic activity does not necessarily increase with increasing wildness; iv) AIs predict human perceptions of wildness more strongly that current wildness classes.

i) The soundscape varies significantly over wildness class

The simple acoustic features investigated are inline with the first prediction, that overall sound levels and presence of low frequency signals will decrease with increasing wildness: the increase in SC at all sites (Fig. 3) suggests that peri-urban site are dominated by lower frequency components than wilder locations; decrease in RMS across the same gradient suggests that sites generally get quieter as wildness class increases. These results demonstrate that the soundscape varies with human influence and that acoustic metrics recapitulate existing components of wildness mapping.

The RTI was introduced as a measure of the relative dominance of low frequency energy and an explicit proxy for distance from human influence. We predicted that RTI would decrease as we move from urban to wild spaces, mirroring SC, as traffic noise decreases with increasing distance to the road. This trend is observed at I&I (Fig 3,4) but is far from consistent across sites. Reviewing the sound recordings reveals that there are high levels of car noise at WC2 and that WC8 is relatively quiet without plane or wind noise, which may explain the clear predicted trend found at this site. Conversely the opposite trend is evident at BEN; a review of the sound recordings reveals this is driven two factors: Firstly, the low value of RTI at WC2 and WC3 is not due to absence of traffic, rather that the close proximity of cars, and increased noise from wet roads generating high frequency energy (up to 8kHz) and therefore lower values of RTI. At the wilder locations WC6 & WC8 jet fighter activity (low frequency) is clearly audible in the sound recordings at the higher, more exposed locations leading to higher values for RTI. Derived from the NDSI [START_REF] Kasten | The remote environmental assessment laboratory's acoustic library: an archive for studying soundscape ecology[END_REF], this band-limited index is based on the assumption that the sound of human industry (technophony) contains predominantly low frequency components. This is true at landscape scales or where sample sites may be surrounded by vegetation cover which attenuates high frequency components of signals; in urban settings where traffic is close proximity to sample points these assumptions of band-limited sound signals break down and so new approaches to acoustic monitoring may be needed across urban-wild gradients.

ii) There are significant variations in soundscape across urban-wild gradients and between sites. Within these overall trends there are significant differences in soundscape along gradients and differences in the magnitude and direction of correlations, suggesting there is wider environmental variation than that captured in current wildness quality mapping methods. The ecological relevance of these variations requires further study. Auditioning the recordings reveals that the anomalous trends in BEN are due primarily to gusting winds in the WC6 and 8 which generates broadband energy, resulting in the large variance in both SC and RMS at WC6. The anomalous patterns observed at LES 5 and 6 (high RMS, low SC) are due to river and running water near the site which generates acoustic energy in the low frequency band.

The need for high quality biophysical naturalness metrics linked to land use and management, as well as broader ecological approaches to measuring the intensity and biophysical impact of anthropogenic activities, have been cited as a key future challenges for the mapping of wildness (Leslie, 2016). The considerable variation observed in the patterns of significant difference between sites (Fig. 3) suggests that AIs are sensitive to differences between recorder locations not captured by current wildness class mapping schema. Systematic interpretation of these differences requires more detailed ecological data sets as a baseline.

iii) Biophonic activity does not necessarily increase with increasing wildness Values for ecological indices BAI, ACI & AEI do not show either strong positive correlations, or significant increases across gradients as predicted under the assumption that biodiversity increases along urban-wild gradients. This could be due to absence of biophonic activity or inadequacy of the indices. Auditioning of the recordings for site BEN suggest that the trend from other sites is confounded here by high wind noise (gusts) and rain (drops) creating acoustic energy within a range that is commonly associated with bird vocalisation, especially at sites WC6 and WC8. BAI fell from WC2 to WC8 at all sites except BEN suggesting lower levels of biophonic activity at the wild end of the urban to wild gradients measured. Auditioning for site I&I revealed much higher levels of biophonic activity (abundance and species richness) at WC2 compared with all other sites at I&I, and WC8 was effectively silent.

It is proposed that this pattern is driven by a number of factors. Firstly, as a general trend all high wildness sites were also higher in terms of altitude than low wildness sites. Biodiversity is known to fall with altitude and the resultant lower temperatures, as is evidenced by the importance to avian richness of summer temperature [START_REF] Lennon | Bird diversity and environmental gradients in Britain: a test of the species-energy hypothesis[END_REF]Marzluff et al. 2012). Other studies have shown that avian richness is higher in urban areas with diverse habitats than in upland areas [START_REF] Rosenfeld | Assessing the ecological significance of linkage and connectivity for avian populations in urban areas[END_REF]; and more widely that biodiversity can be higher in urban gardens than in semi-natural landscapes [START_REF] Thompson | Urban domestic gardens (I): putting small-scale plant diversity in context[END_REF]. Furthermore, the high wildness sites of the type found at WC8 in I&I and BEN for example, are recognised as being ecologically impoverished compared to their original post-glacial state [START_REF] Hobbs | Woodland restoration in Scotland: ecology, history, culture, economics, politics and change[END_REF][START_REF] Fisher | Review of Status and Conservation of Wild Land in Europe[END_REF].

A second key factor is the deployment period for the recorders which for practical reasons did not coincide with peak annual avian activity. This was compounded by unseasonably bad weather at both the Pyrenees study sites. Auditioning revealed that bird vocalisation across these two sites was much lower than would be expected based on expert knowledge of the sites.

The results for the AIs designed to capture biophonic activity were also confounded by the high frequency components of noise from cars, wind and rain as outlined above. AIs such as BAI for example were originally designed for monitoring use in more remote, tropical areas low in technophony. In the current study the higher frequencies resulting from technophony are also detected by the sound recorders at the low wildness peri-urban sites (WC2 & WC3). iv) AIs predict human perceptions of biodiversity more strongly than wildness classes AIs strongly predicted wildness class, but human perceptions of wildness and biodiversity are even more strongly predicted. As discussed above, further work is needed to validate the use of acoustic methods for biodiversity monitoring in wilderness mountain areas, but these results suggest that acoustic methods are sensitive to the same factors which influence human experiences of wilderness, which current mapping methods are insensitive to.

Recommendations

The complexity and dynamically evolving nature of the relationship between humans and their landscape requires us to change our perspectives and seek new ways of understanding these complex spaces that are also better suited and more robust for use in planning, nature conservation and policy making [START_REF] Hennig | Applying Integrated Nature Conservation Management: Using Visitor Management and Monitoring to Handle Conflicts Between Winter Recreation and Grouse Species in Berchtesgaden National Park[END_REF]. Our results stimulate further work in the application of ecoacoustic methods in wildness mapping methodologies in order to better reflect ecological and human processes and values. The next important steps are, firstly the development of new AIs better suited to assessing the components of soundscape relevant to measuring wilderness across urban-wild interfaces; secondly validation of these acoustic methods with baseline data from biodiversity and local habitat assessment in order to establish the ecological significance of the variations across sites observed here. This will require repeated local spatial replications as well as replications across different biomes. The transects selected for this study spanned a continuum of low to high wildness across a relatively short distance of around 10km. Whilst this was necessary, allowing human participants to walk the transects in a single day, it is felt that future iterations of the methodology would benefit from using a protocol, with short, medium and long transects across a more diverse range of habitats/ecotones. At some study sites the issue of scale could be further explored by using a nested protocol so the long transect would contain a short and a medium transect. Using longer transects would allow placement of a greater number of recorders as an array inside larger areas of a given wildness class and thus provide better understanding of the characteristics of these areas, how acoustic events relate to local environmental data as well as highlighting any possible edge effects.

More specifically the use of precisely positioned and configured arrays of sound recorders inside a particular wildness class would also add greater resolution to the analyses, capturing the 'near field' ecoacoustic events which constitute the detail of the localised variation in the perceptual experience of a wilderness soundscape [START_REF] Farina | Acoustic codes from a rural sanctuary: How ecoacoustic events operate across a landscape scale[END_REF]. Combining this multi-scalar approach with longer term deployment of the recorders over the period of a full year is also recommended to capture seasonal variation in acoustic events and would also reduce sensitivity to extreme weather or other ephemeral events.

New acoustic analyses for wilderness mapping are also needed to deal with geophonies associated with extreme weather and water in highland wilderness areas. Our results show that simple acoustic features (RMS and SC) are more strongly correlated with changes in the soundscape across urbanwild gradients than ecological indices. This is in line with previous work in which these simple descriptors were also stronger predictors of avian species richness [START_REF] Eldridge | Sounding out ecoacoustic metrics: Avian species richness is predicted by acoustic indices in temperate but not tropical habitats[END_REF]. Results also suggest that technophony (cars passing) contains high frequency components and that geophonic components (wind, rain and rivers) are similarly broad spectrum, rendering band-limited indices unsuitable.

Two distinct approaches to automated acoustic analyses have been deployed to date: ecoacoustic indices such as those tested here which have been designed by hand, based on assumptions about the statistical structures of particular soundscape components (biophonies and technophonies). In contrast, automated identification of individual species calls [START_REF] Stowell | Automatic large-scale classification of bird sounds is strongly improved by unsupervised feature learning[END_REF] has tended to use supervised clustering or neural network style models trained with pre-selected canonical examples of specific species calls. Both approaches may be problematic for WA assessment due to wide variation of acoustic signatures across different biomes of the same wildness designation. An alternative approach, which warrants further investigation, is to repurpose the internal representations of neural networks to serve as machine-generated acoustic indices. When neural networks are trained, they create internal models of the data they are trained on -or "latent spaces"; these are efficient data representations which could be used for clustering or classification of similar acoustic events in an unsupervised manner. In the context of wilderness mapping, this has the advantage in providing a means to machine-generate acoustic features without recourse to spurious assumptions or the need for extensive training data. Moreover, once trained on a known urban-wild gradient, the model could be used for monitoring or mapping extant or newly designated wilderness areas in similar ecotones.

Conclusion

The critical ecological and societal importance of WAs is well recognised yet the metrics and maps which subserve current wilderness management policies take neither directly into account. We report the first investigation into the potential for ecoacoustics to provide a cost effective, scalable method for biodiversity assessment and framework for integrating anthropogenic and ecological perspectives within existing geophysical frameworks. Our results demonstrate that a small suite of AIs strongly predict wilderness gradients, but also reveal considerable environmental variation within areas of equal wildness as designation under current metrics. The potential ecological relevance of this variation requires further investigation which we propose is best addressed by adding full habitat and bird surveys to the data collection methods reported here, as well as detailed transcription of soundscape components from site recordings. We further demonstrate that AIs predict human perceptions of biodiversity and wildness, suggesting that acoustic methods capture important facets of the human experience of wildness. We argue that in recognising the acoustic environment as the nexus of atmospheric, biospheric and anthropogenic processes, ecoacoustics also provides a framework within which to integrate ecological and anthropogenic perspectives on wilderness, answering calls for new approaches to conceptualising and measuring wild spaces as the site of complex and dynamic human-environment relations (Leslie, 2016;[START_REF] Hennig | Applying Integrated Nature Conservation Management: Using Visitor Management and Monitoring to Handle Conflicts Between Winter Recreation and Grouse Species in Berchtesgaden National Park[END_REF]. We recommend that ecoacoustics should be incorporated into future WA mapping and management and suggest new research directions to develop and validate acoustic methods suited to the unique conditions of wilderness areas, across a range of biomes.

  '' W), Beinn Eighe National Nature Reserve (BEN) on the Scottish west coast (Lat 57° 36' 8.6004'' N, Long -5° 19' 0.624'' W) (Fig 1) Lesponne, Haute Pyrenees (LES) Southern France (Lat 42° 58' 51.0528'' N, Long0° 8' 44.214'' E), and Pouey Trenous, in the centre of the Pyrenees National Park (POT), Southern France (Lat 42° 50' 6.2016'' N, Long -0° 9' 35.9244'' W). Transects were identified through a combination of desk-based GIS analysis to identify the optimum gradients, supported by discussions with local experts from Scottish Natural Heritage (SNH), Centre for Mountain studies and Pyrenees National Park respectively. SNH developed a version of their wildness quality map for Scotland for use in the definition of Wild Land Areas (SNH 2014) which used a statistical method known as "Jenks" classification, to reclassify all pixels on the map with a similar value for wildness into eight classes, least to most wild. This simplified Jenks version of the wildness map was made available by SNH for this project. The authors used an identical statistical process to reclassify the map of hautenaturalité of the Haute-Pyrenees, produced by IUCN France, into eight Jenks wildness classes (WC)

Figure 1 .

 1 Figure 1. Example of transect walk along gradient of Jenks wildness classes for Scottish site Beinn Eighe National Nature Reserve (BEN) on the Scottish west coast (Lat 57° 36' 8.6004'' N, Long -5° 19' 0.624'' W) showing human perception and acoustic survey points. See Supplementary Material A for corresponding maps of other sites.

  iii) ensure no social groups were excluded. Participants were recruited through local press, social media, organisational contacts, and member groups such as mountain clubs. Human perception data was collected in situ along the same experimental transects from a total of 73 participants (BEN n=11, I&I n=31, LES n=31) (see Carruthers-Jones et al., 2019 for details). No human data was collected at POT because extreme weather made the paths impassable during the planned human survey period. Participants were briefed and guided along the transects in groups of 8 or less. At each sample point (see for example Fig 1), participants rated their immediate surrounding landscape in terms of wildness and biodiversity on a scale of 1(least) -7 (most) (see Supplementary Material

Fig 2

 2 Fig 2 Scatter plots of AIs (rows) across WCs for all sites combined with linear model fitted (in black).

Fig 3 Fig 4

 34 Fig 3 Tukey's box and whisker plots for AIs values (rows) across days for each WC for each study site (columns). Horizontal lines represent medians; the box represents the interquartile range; whiskers represent min and max values within 1.5 IQR. Non-significant differences (p < 0.05) between sites are denoted by bars ns. Individual AI values shown as points.

Table 1 .

 1 Descriptions of habitat at each of the five wildness classes studied

	Jenks Wild Class	Description of Scottish sites	Description of French sites
	2	Least wild, urban site	Least wild, urban site
	3	Lowland, Plantation native woodland	Lowland, woodland edge
	5	Middle mountain, open mountain heath/	Middle mountain, grazing
		moorland
	6	Upland, natural native woodland site	Upland grazing pasture surrounded by
			native woods
	8	Most wild, mountain, upland scrub	Most wild, mountain, ancient woodland

Table 2

 2

	road	Pwild	Pbio	ACI	AEI	RMS	SC	RTI	BAI
									1
	WC
									0.8
	road								0.6
	Pwild							0.4
			Pbio						0.2
				ACI					0
					AEI				0.2
						RMS			0.4
							SC		0.6
									0.8
								RTI
									1

), however RMS and SC are in the top three most important variable at all sites, together explaining over 50% of the

Table 2 .

 2 Relative variable importance for AIs as predictors of wildness classes at each site and all sites combined

	Cumulative Percentage Variance Explained	0 25 50 75 100					1.15 MSE 0.22 MSE 0.31 MSE 0.80 MSE 1.20 MSE	site	All I&I BEN LES POT
		RMS	SC	AEI	RTI	BAI	ACI
				AI
			Relative Variable Importance
	AI		ALL	I&I	BEN	LES	POT
	RMS	1.00	0.96	1.00	0.88	0.89
	SC	0.66	1.00	0.62	1.00	1.00
	AEI	0.56	0.41	0.74	0.95	0.04
	RTI	0.50	0.37	0.58	0.27	0.29
	BAI	0.48	0.54	0.50	0.23	0.36
	ACI	0.43	0.11	0.61	0.45	0.09

Figure

5

. Cumulative percentage variance explained by multivariate random forest regression models with 6 AIs as predictors and wildness class as response for all sites combined (76.21% MSE 1.15) each site (I&I 92.44% MSE 0.31, BEN 72.37% 1.2 MSE, LES 84.56% 0.8 MSE, POT 95.81% 0.22 MSE). Out of bag error rates for the six AI model are detailed at the right of each curve. AIs were added to each model in the order of variable importance for all sites combined (x axis).

Finally, a comparison of models built for all components shows that for all sites combined, compound AIs predict human subjective judgements of biodiversity (82.22%; MSE 0.25) and wildness (77.29%; MSE 0.64) even more strongly than mapped wildness classes (76.21%; MSE

Carruthers-Jones/ Call of the Wild

Acknowledgements

This work was carried out with support from European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 642935 (ENHANCE).

Computational analysis of the sound recordings was undertaken on ARC3, part of the High Performance Computing facilities at the University of Leeds, UK.

The authors would like to thank Scottish Natural Heritage, The Centre for Ecology and Hydrology and the Pyrenees National Park for their assistance.

Author Contributions JCJ, AE, CH and GH designed the study; AE supervised the acoustic surveys; JCJ identified the transects and carried out human and acoustic surveys; PG, AE, JCJ and CH conducted the analyses and interpreted the results; JCJ and AE prepared and finalised the manuscript.


Nhat-Quang Quang Nhan

Philippe Rostaing

Karine Amis Cavalec

Ludovic Collin

Emanuel Radoi

Karine Amis

Optimized max-dmin precoder assuming maximum squared Euclidean weight-mapping and turbo detection

Keywords: Linear MIMO precoder, turbo detection, itera tive receiver

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

I. INTRODUCTION

Deploying multiple antennas at both transmitter and receiver of a wireless link (MIMO) is one of the key technology to tackle the challenges of higher data rate and increasing data traffic that the modern radio-cellular networks have to face up [START_REF] Li | MIMO techniques in Wi MAX and LTE: a feature overview[END_REF]. In time domain duplex (TDD) closed-loop schemes, the channel state information (CSI) is readily avail able at the transmitter through a feedback link, which enables to design a precoder that adapts to the channel conditions by exploiting the multiple antennas at the transmitter. Addition ally, the turbo detection, which applies the well-known turbo principle [START_REF] Berrou | Near optimum error correcting coding and decoding: Turbo-codes[END_REF], shows a great advantage to combat intersymbol interference. It was firstly introduced in [START_REF] Picart | Turbo-detection: a new approach to combat channel frequency selectivity[END_REF] and later applied in many MIMO systems [START_REF] Haykin | Turbo-MIMO for wireless communications[END_REF]. A turbo detection scheme contains a soft demapper block and a soft FEC decoder block, which exchange extrinsic information to improve the detection and decoding processes at each iteration and, finally, enhance error-rate performance at the convergence. In this paper, we consider the concatenation of the MIMO precoder with a binary recursive systematic convolutional code assuming turbo detection at the receiver. We assume perfect CSI at both the transmitter and the receiver.

We start from the two forms of the max-dmin precoder [START_REF] Collin | Optimal minimum distance-based precoder for MIMO spatial multiplexing systems[END_REF], respectively referred to as F 7"1 or Facta and we aim at optimizing the switching threshold from one to another. The max-dmin precoder was initially designed to maximize the minimum Euclidean distance between two received sequences assuming a Gray-mapping and no outer FEC code. For both F 7"1 and Facta, the received constellation is fixed and is only scaled by a positive factor depending on a channel parameter 1 explained hereinafter in the paper, which enables a MIMO mapper design that maps directly the encoded binary sequence onto the received constellation [START_REF] Nhan | Optimized MIMO symbol mapping to improve the turbo cliff region of iterative precoded MIMO detection[END_REF]. In [START_REF] Nhan | Optimized MIMO symbol mapping to improve the turbo cliff region of iterative precoded MIMO detection[END_REF], we individually optimized the MSEW-mapping for both F 7"1 and Facta. Taking into account the outer FEC and a turbo detection, we analyzed the ending point of the EXIT chart for lower error-floor pur pose and observed that beyond a given value of the parameter 1, both precoders achieve the same ending point, making the switching threshold useless. The resulting MSEW-mapped max-dmin precoder outperforms all Gray-mapped max-dmin precoders [START_REF] Li | MIMO techniques in Wi MAX and LTE: a feature overview[END_REF]. In this paper, to further enhance the error-rate performance, we define a switching threshold between MSEW mapped F 7"1 and Facta so as to maximize the minimum Euclidean distance between two received sequences whose binary mappings differ by exactly one bit. We refer to the new precoder as max-fmin precoder. EXIT charts as well as error rate simulations carried over random MIMO channels support the analysis.

The remainder of this paper is organized as follows. Section II briefly introduces the system model along with max-dmin MIMO precoder. In Section III, the optimal MSEW mappings at the received constellations of max-dmin precoder are firstly recalled. Subsequently, the new max-fmin precoder is proposed. Also in this section, EXIT chart analysis is introduced for verification. Simulated error rates are presented in Section IV to validate the analysis. Section V concludes the paper and gives some perspectives.

II. SYSTEM MODEL AND PRELIMINARIES

A. System model

Let us consider a MIMO system with nR receive, nT trans mit antennas and b independent data streams to be transmitted. A binary recursive-systematic convolutional (RSC) code is used to encode the information binary sequence. The FEC codeword is interleaved before entering a MIMO symbol map per. In the MIMO symbol mapper, interleaved FEC-encoded bits are grouped into packets and each packet is mapped onto a b-dimensional symbol vector s. The vector s is then precoded with a precoding matrix F and transmitted through the MIMO channel. At the receiver side, after MIMO detection, a soft symbol demapper iteratively exchanges extrinsic information with a BCJR soft decoder [START_REF] Bah | Optimal decoding of linear codes for minimizing symbol error rate[END_REF]. The detection output, denoted by y, reads

y = GHFs + G1], ( 1
)
where F is the nT x b precoding matrix with the power constraint IIFII} = 1, II.IIF is the Frobenius norm, G is the b x nR postcoding matrix, H is the nR x nT channel matrix, and 1] is the nR x 1 additive white circularly-symmetric complex gaussian noise vector. Let E[.] and (.) t stand for the expectation and the conjugate transpose respectively and let In R be the identity matrix of size nR-Then we assume E[1]1]t] = O"�In R and E[sst] = O";Ib. The system model is shown in Fig. 1, where L1, L� and L},; respectively stand for the a priori, the a posteriori and the extrinsic log likelihood ratios (LLRs) of the soft demapper, while the equivalent notations for the BCJR soft decoder are L�, L� and L�.

According to [START_REF] Collin | Optimal minimum distance-based precoder for MIMO spatial multiplexing systems[END_REF], the following channel transformation is applied to simplify the system model. Let us define F d and F v such that F = F v x F d. Matrices F v and G are unitary and chosen so as to transform the MIMO channel into a virtual channel with b independent parallel sub-channels, i.e. the columns of F v and G t respectively are the b most significant right-singular and left-singular vectors of the singular value decomposition of H. Note that F d denotes the new precoding matrix. It also satisfies the power constraint IIFdll } = 1. The equivalent model is written as

(2)
where 1]v is the bx 1 virtual noise vector with E[1]v 1] t ] = O"�Ib.

Hv = (� 1 � 2 ) = P (CO�I Si �,), (3)
where p and 1 respectively represent the channel gain and angle. As 0"1 ;::: 0"2 > 0, we have 0 < 1 :s; 7r / 4. Therefore, any random MIMO channel can be simply characterized by the pair (P,1) thanks to the virtual transformation. Let us define the instantaneous received SNR as

2 2 SNR = ;21IHII} = "2 p 2 . (4) r, (1 "1
With this definition of SNR, a channel is only characterized by its angle 1.

In most MIMO coded schemes, a binary to M-ary sym bol conversion is inserted between the FEC code and the MIMO precoder. Traditionally, the interleaved FEC codeword is mapped onto mono-dimensional M -ary modulated symbols. The mapped symbols are then converted into a b-dimensional symbol vector s. In contrast to the conventional mapping, in this paper, we apply the work in [START_REF] Nhan | Optimized MIMO symbol mapping to improve the turbo cliff region of iterative precoded MIMO detection[END_REF] by considering a direct mapping of the interleaved FEC codeword into s. Note that elements of s are M -ary modulated symbols. We denote by Q the set of all symbol vectors s. Then, the size of Q is M b .

The process at soft demapper of the well-known MIMO turbo detection scheme, which is widely applied in many research papers [START_REF] Haykin | Turbo-MIMO for wireless communications[END_REF], is not presented here due to space limitation. 6 -1 ' ,

• if 0 :s; 1 :s; 1 0 (6) where 'IjJ = arctan � n -�. Apparently, in the case I :s; 10, i.e.

Fd = Facta = v'2 0 o ) ( 1 ei% ) sin1j; -1 ei% '
F d = F T l' the precoder only spreads power on the first sub channel. In the case I > 10, i.e. F d = Facta, the precoder spreads power on both sub-channels. Fig 3 shows the received dmin normalized by p of F rl and F acta. We can see that, in order to keep the high value of the received normalized dmin, the max-dmin precoder uses 10 as a threshold to switch between F rl and Facta. Note that 10 is SNR-independent and designed for uncoded system. The study of this threshold for the turbo detection, under assumption of optimized mapping at the MIMO symbol mapper, will be presented in Section III-B.

III. ANALYSIS

A. Influence of mapping at the received constellation From (3) and the precoding matrix ( 5) or [START_REF] Nhan | Optimized MIMO symbol mapping to improve the turbo cliff region of iterative precoded MIMO detection[END_REF], it comes that HvF d = aA, where A is a fixed matrix and a is a scalar, which depends on I and p. Therefore, the received constellation (of F rl or Facta) is unchanged and just scaled by a scalar factor a for each channel. Consequently, at the MIMO symbol mapper, grouping and mapping the encoded binary bits into 8 is equivalent to map the encoded bits onto the received constellation symbol, which is defined by x = HvF d8.

The selected mapping depends on a key parameter denoted by £ and defined as [START_REF] Bah | Optimal decoding of linear codes for minimizing symbol error rate[END_REF] where sand s' represent any pair of symbol vectors with associated binary conversions differing by exactly one bit. The minimum value of £ over the received constellation is denoted by £min. At high SNR, we deduce that the fault detections, which may lead to errors in the iterative process of the turbo detection, mostly occur between the pair of received constellation symbols separated by £min. Indeed, it is shown in [START_REF] Nhan | Optimized MIMO symbol mapping to improve the turbo cliff region of iterative precoded MIMO detection[END_REF] that maximizing £min results in a significant error-rate performance improvement for the max-dmin precoded MIMO system assuming turbo detection at the receiver. The MSEW mapping proposed in [START_REF] Tan | Analysis and design of symbol mappers for iteratively decoded bicm[END_REF] consists in first maximizing £min and then minimizing the number of pairs (s, s') such that £(s, s') = £min.

Let us recall the optimal MSEW mappings for F rl and F acta that we obtained in [START_REF] Nhan | Optimized MIMO symbol mapping to improve the turbo cliff region of iterative precoded MIMO detection[END_REF]. By denotin � 16 possible values of 8 by 80 = [so sO ] T , 81 Then, the optimal mappings for F rl and F octa are presented in TABLE I. Note that, in this table, the Gray-direct mapping stands for the conventional mapping that Gray-maps every 2 encoded bits into one 4-QAM symbol before converting the modulated symbols to the b-dimensional symbol vector 8 (b = 2) using parallel to symbol converter. In this subsection, we propose an optimization for max-dmin precoder in turbo detection scheme. The precoder optimization relies on the received constellations. We first remind that considering F rl or F octa, the received constellation is fixed and scaled by a positive factor a which depends always on T Assuming no outer FEC code and a Gray-mapping, the max-dmin precoder defined in [START_REF] Collin | Optimal minimum distance-based precoder for MIMO spatial multiplexing systems[END_REF] switches between F rl and F octa depending on the threshold /0 chosen so as to maximize the minimum Euclidean distance. In [START_REF] Nhan | Optimized MIMO symbol mapping to improve the turbo cliff region of iterative precoded MIMO detection[END_REF], considering the outer FEC code and the turbo detection at the receiver, we studied the optimization of the switching threshold so as to lower the error-floor, i.e. to maximize Iftoor (see Fig. 6). We showed that the MSEW mapping involved no switch in that case as beyond a given value of / the criterion maximizing Ifloor leads to the selection of F rl only.

= [so SI ] , 82 = [so S2 ] T , 83 = [so S3 ] T , 84 = [SI so r 85 = [SI SIr 86 = [SI S2 ] T , 87 = [SI S3 ] T , 88 = [S2 so ] , 89 = [S2 SI ] , 810 = [S2 S2 ] T , 811 = [S2 S3 ] T , 812 = [S3 SO ] T , 813 = [S3 SI ] T , 814 = [S3 S2 ] T and 815 = [S3 S3 ] T , where So = (-1 -i)/v'2, SI = (-l+i)/v'2, S2 = (l-i)/v'2, S3 = (l+i)/v'
In this paper, we consider the maximization of £min as the criterion to define the threshold /0 assuming the MSEW mapping. In Fig. 4, we plotted the value £min as a function of T We observe that /1 = 30.7° is the best threshold. Beneath /1 F rl outperforms F octa and beyond /1, the trend is reversed. We define a new precoder from the newfound switching threshold /1: F d = F rl when / :s; /1 and F d = F octa otherwise. We refer to this new precoder as max-£min precoder.

In Fig. 5, the probability density functions in terms of / of different MIMO configurations are plotted for the case b = 2 data streams, where nd = InT -nRI. It is shown that the max-£min precoder uses on average more F rl form compared to the max-dmin precoder. In addition, we observe from Fig. 5 that the percentages of 17.28° :s; / :s; 30.7° are 44.88%, 59.16% and 57.78% for nd = 0, nd = 1 and nd = 2 respectively. In this interval (17.28° :s; / :s; 30.7°), max-dmin uses F octa while the max-£min precoder uses FrI .

We conclude this section by mentioning that we plotted £min as a function of / in the Gray-direct mapping case for both F rl and F octa' We observed that both curves intersect at a value of / roughly equal to /0. This study confirms the choice of /0 as the switching threshold for the Gray-direct mapping. The figure is not given in the paper due to space limitation.

C. Validation through EXIT chart

In this subsection, we use EXIT chart [START_REF] Hagenauer | The exit chart-introduction to extrinsic information transfer in iterative processing[END_REF] to analyze the evolution of the MI between the bit-interleaved coded binary sequence c and its LLRs at input and output of the demapper when max-dmin and max-£min are used. The extrinsic MI at output of demapper is a function of the a priori knowledge Ii and the SNR. We define I� = TI (Ii , SNR) . Similarly, I� = T 2 (1�). For details about the mathematical expressions of TI and T 2 , the reader can refer to [START_REF] Hagenauer | The exit chart-introduction to extrinsic information transfer in iterative processing[END_REF], [START_REF] Ten-Brink | Designing iterative decoding schemes with the extrinsic information transfer chart[END_REF]. The mutual information is averaged over 100 trials. Recall that the signal-

2 2
to-noise ratio is defined by SNR = 0"211HII} = ;2 p2.

(117 7J

To demonstrate the difference between max-£min and max-dmin precoders, let us randomly pick up a channel in the interval 17.28° :s; / :s; 30.7° to plot the EXIT chart. Without loss of generality, we select / = 22°. We remind that, with this channel, max-dmin uses F octa form while max-£min uses F rl form. Fig. 6 shows the EXIT chart of the iterative receiver over the selected channel (/ = 22°) at SNR = 8 dB. The dashed line represents the EXIT function of the BCJR decoder of the (13, 15)octa RSC code, which is independent from SNR. Firstly, it is observed that using Gray-like mapping results in an early crossing between the two EXIT functions and leads to worse error-rate performance. This, again, confirms the advantage of MSEW compared to the conventional Gray like mapping. Secondly, by considering MSEW mapping, we observe that the tunnel of the EXIT chart is wider in the case of max-£min precoder (F rl + MSEW) than in the case of max-dmin (F oct a + MSEW). This predicts a better error-rate performance at the turbo-cliff region. The ending point (Iftoor) of the EXIT function of max-£min is also higher than the one of max-dmin, which predicts a better performance at the error floor by using max-£min.

IV. SIMULATION RESULTS

Monte-Carlo simulation has been carried out over random MIMO channels, i.e. each element of H is distributed as Hi,j rv CN(O, 1). Thus, on average, each channel element has unit energy, i.e. E[IHij 12] = 1. Note that, with the definition of instantaneous SNR in (4), the error-rate performance does not depend on p2. Therefore, the system performance for different values of / is obtained by taking the average of the randomly generated channels. The half-rate (13, 15) octa rRSC code is used as FEC encoder. The frame length is set to 800 uncoded bits and interleaved by a random interleaver. Fig. 7 shows the average bit-error-rate (BER, in solid lines) and frame-error-rate (FER, in dashed lines) performances of the considered system over random 2 x 2 MIMO channels. Firstly, the error-rate performance in case max-dmin precoder is used with Gray-like mapping is plotted to compare with the case max-dmin used with MSEW mapping. As observed, in terms of FER, the MSEW-mapped max-dmin achieves a gain of roughly l.5 dB at FER = 10-2 compared to the Gray-like mapped max-dmin. The gain is even more significant at high SNR due to slope changes of the FER curves. In terms of BER, at high SNR, the gain is roughly 0.8 dB at BER= 10-4 . At low SNR, BER performance of Gray-direct is better than that of MSEW, which is in accordance with the EXIT chart analysis, since the corresponding EXIT function of Gray-direct mapping begins higher and, therefore, it avoids intersection at the bottleneck. This also makes the gap between the FER and BER of Gray-direct mapping is larger than that of MSEW mapping as observed in Fig. 7. However, in terms of FER performance, MSEW still achieves a gain, even at low SNR. This is due to the early crossing of the EXIT function of the Gray-direct mapping and the EXIT function of decoder (see Fig. 6). The results above show the advantage of MSEW mappings for max-dmin precoder over random channels. Secondly, we observe from Fig. 7 that the performance gains are even more significant when max-£min is used. Precisely, in terms of BER, max-£min precoder achieves the gains of more than 1.5 dB and roughly 0.75 dB compared to Gray like mapped and MSEW-mapped max-dmin respectively. As demonstrated in Fig. 6, with MSEW mapping, the correspond ing EXIT function of max-£min precoder is higher than the one of max-dmin precoder at both bottleneck and Ifloor regions. This results in the performance gains of max-£min compared to MSEW-mapped max-dmin at any value of the considered SNR range, as demonstrated in Fig. 7.

It is essential to emphasize that since the max-£min aims to maximize £min, the performance gain of max-£min compared max-dmin is even more significant at high SNR, where the errors mostly occur due to wrong detection between pairs of symbols, whose binary patterns differ by exactly one bit. Indeed, as shown in Fig. 7, at FER = 10-6, while the FER curve of max-£min continues to go down, the FER curve of MSEW-mapped max-dmin begins to saturate.

V. CONCLUSION

In this paper we focused on the design of precoders assum ing an outer FEC code and a turbo detection at the receiver. We considered the two forms of the max-dmin precoder [START_REF] Collin | Optimal minimum distance-based precoder for MIMO spatial multiplexing systems[END_REF] with the MSEW mapping defined in [START_REF] Nhan | Optimized MIMO symbol mapping to improve the turbo cliff region of iterative precoded MIMO detection[END_REF]. We optimized the switching threshold so as to maximize the minimum Euclidean distance between two received sequences whose associated binary sequences differ by one position. We referred to the optimized precoder as max-£min. Simulations (EXIT chart and BER performance) showed the improvement achieved by max-£min compared to max-dmin. Future works will consider the optimization of the generic expression of F d according to the maximization of £min.

Fig. 1 :

 1 Fig. 1: System model.

Fig. 2 :

 2 Fig. 2: Equivalent channel transformation.

B.

  The max-dmin linear precoderWe focus on a commonly used precoder named max-dmin[START_REF] Collin | Optimal minimum distance-based precoder for MIMO spatial multiplexing systems[END_REF], in which, F d was designed to maximize the minimum Euclidean distance, which is denoted by dmin = min Ilxm -xell where x = HvF dS, between the m# received constellation symbols. In case b = 2 and 4-QAM modulation, the optimal solution de ends on 1 and by defining the threshold 10 = arctan £:ft (�17 28 °) Fd reads 2 2+

FdFig. 3 :

 3 Fig. 3: The received normalized dmin versus T • if 10 < I :s; 7r / 4

1. 4 Fig. 4 :Fig. 5 :

 445 Fig. 4: Normalized £min versus I with MSEW mappings.

  2 and [.j T stands for vector transposition, the bit-interleaved encoded bits of c are grouped into length-4 packets and mapped directly onto 8.

+Fig. 6 :

 6 Fig. 6: EXIT chart for the iterative receiver used with different precoders and the associated mappings, 'Y = 22°, SNR = 8 dB, RSC(13,15).

5 ;Fig. 7 :

 57 Fig. 7: Average BER (solid lines) and FER (dashed lines) performance of the precoded turbo detection over random 2 x 2 MIMO channels.

TABLE I :

 I The

	Gray-direct F rt locta MSEW F q Facta	[0 1 23456789 10 11 12 13 14 15] [7 2 1 11 13 48 14 12 61 015 5309] [2 5709 12 10 15 11 14 8 13 43 16 ]

optimized binary representation in the constellation map of precoder max-dmin for different mappings (each decimal value represents for 4 binary bits, e.g. (8)10 = (lOOOh) [6]. Mapping Mode lso ... Si ... S15 j B. Optimized max-£min precoder

ACKNOWLEDG MENT

The authors would like to acknowledge Dr. Robert G. Maunder, University of Southampton, UK, for the helpful discussion about EXIT chart analysis.


Massinissa Lalam
email: massinissa.lalam@enst-bretagne.fr

Karine Amis Cavalec

Dominique Leroux

Dongning Feng
email: d.feng@student.unsw.edu.au

Jinhong Yuan

K Amis

An improved iterative decoding algorithm for block turbo codes

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

I. INTRODUCTION

Concatenated block coding technique was introduced by P. Elias in 1954 [START_REF] Elias | Error-free coding[END_REF]. In 1994, R. Pyndiah introduced binary (BCH codes [START_REF] Bose | On a class of error correcting binary group fields[END_REF]) and non-binary (RS codes [START_REF] Reed | Polynomial codes over certain finite fields[END_REF]) block turbo codes (BTCs) with a soft-input soft-output (SISO) iterative decoding algorithm [START_REF] Pyndiah | Near optimum decoding of product codes[END_REF] which offers a good trade-off between performance and complexity. Since BTC can achieve high data rates with a low decoding complexity, it has found widespread applications (LMDS, UMTS, IEEE 802. [START_REF] Lalam | On the use of reed-solomon codes in space-time coding[END_REF]) and has become a key ingredient in the telecommunication system design process.

Soft iterative decoding of BTC requires the generation of a large set of codewords to compute reliable soft output for each bit. Various approaches have been developed in the literature in order to create a codeword list for a linear binary block code, denoted C(n, k), where n is the length of the code and k its dimension. However, we can distinguish them into three major classes.

The first class of algorithm uses encoding to decode the received sequence. In the OSD [START_REF] Fossorier | Soft-decision decoding of linear block codes based on ordered statistics[END_REF]-BMA [START_REF] Martin | On softinput soft-output decoding using "box and match" techniques[END_REF] or in [START_REF] Gazelle | Reliability-based code-search algorithms for maximum-likelihood decoding of block codes[END_REF] or [START_REF] Martin | Soft-input softoutput list-based decoding algorithm[END_REF], the principle is to reorder the soft values of the received sequence according to their reliability. Error patterns of increasing Hamming weight are applied at the k most reliable independent positions (MRIPs) and encoding is performed on the thresholded resulting sequences. All produced codewords are stored in the list of candidates. However, a new code generator must be evaluated for the encoding process due to the reordering, which increases the decoder complexity.

The second class of algorithm uses an algebraic decoder to construct a list of candidates. In the GMD [START_REF] Forney | Generalized minimum distance decoding[END_REF], the Chase [START_REF] Chase | A class of algorithms for decoding block codes with channel measurement information[END_REF] or in [START_REF] Kaneko | An efficient maximum-likelihood-decoding algorithm for linear block codes with algebraic decoder[END_REF], the principle is to locate the least reliable independent positions (LRIPs) in the received sequence. After thresholding, error patterns are applied at these LRIPs and algebraic Peterson [START_REF] Peterson | Error-correcting codes[END_REF] or Berlekamp-Massey (BM) [START_REF] Berlekamp | Algebraic Coding Theory[END_REF] algorithm is performed. If the result is a codeword, then it is stored in the list of candidates.

The third class of algorithm uses both encoding and decoding to decode the received sequence as in [START_REF] Hu | An efficient hybrid decoding algorithm for reedsolomon codes based on bit reliability[END_REF].

The techniques described above process at the binary level. Generally, these algorithms are also applied for non binary block codes, but to their binary representation, e.g the Chase-2 algorithm for RS codes in [START_REF] Aitsab | Performance of reed-solomon block turbo code[END_REF]. This approximation leads to suboptimal performance.

The conventional SISO iterative decoding algorithm using the Chase algorithm is considered very efficient for BTCs with high code rates such as Hamming-BTCs (Hamming codes as component codes). However, for an high error correction capability (typically t ≥ 2) its performance is far from the capacity limit and new algorithms have to be found.

In this paper, we propose to use a sliding encoding-window (SEW) algorithm [START_REF] Lalam | On the use of reed-solomon codes in space-time coding[END_REF] to decode binary or non-binary BTC with systematic cyclic codes as its component codes (BCH or RS codes). For this type of codes, the SEW algorithm can easily generate a list of codewords, which are close to the received sequence in terms of the Hamming distance. By combining this list of codewords with the one generated by a classical algebraic-based decoder, such as the Chase algorithm, we can form an enlarged set of test patterns. Based on the new test patterns, the SISO iterative decoding algorithm can produce a better log-likelihood ratio (LLR) of each decoded digit. This eventually improves the bit error probability in the iterative decoding of the BTC.

This paper is organized as follows. Part II briefly introduces the transmission scheme. Part III describes the SEW algorithm and its combination with the Chase or the BM algebraic decoding. Part The parameters of the BTC C = C 1 ⊗ C 2 are the products of the component code parameters such as the code length n = n 1 × n 2 , the information length k = k 1 × k 2 and the Hamming distance d = d 1 × d 2 . Thus, a BTC has a larger minimum Hamming distance. All of the rows of the matrix c are codewords of C 2 and all of its columns are codewords of C 1 [START_REF] Macwilliams | The Theory of Error Correcting Codes[END_REF]. This is a very important property of BTCs since in the iterative decoding process, both information bits and parity check bits can exploit the extrinsic information from the received sequences.

Symbols can be either q-ary symbols (RS case) or bits (BCH case). For non binary codes, each q-ary symbol can be represented by an unique set of m bits, with q =2 m .

Let c be the binary representation of one binary or nonbinary codeword of C and c =2 c -1 be the modulated representation of c using a BPSK. BPSK symbols are sent through a Gaussian channel and the received vector r is given by:

r = c + n ( 1
)
where n is the additive white Gaussian noise (AWGN). The optimum decoded sequence can be obtained by using the maximum-likelihood (ML) decoding algorithm. It chooses a codeword ĉ from the code set C, which has a smallest Euclidean distance from the received vector, r:

ĉ =argmin c∈C r -c 2 =argmin c∈C r -2c +1 2 (2)
Unfortunately, it is infeasible to search all the codewords in the ML decoding due to an extremely large number of codewords up to 2 mk for a large value of k.

III. DECODING ALGORITHM

A. Chase Algorithm

The Chase algorithm [START_REF] Chase | A class of algorithms for decoding block codes with channel measurement information[END_REF] is a sub-optimum decoding algorithm, which belongs to the argument list decoding algorithm. Instead of reviewing all the codewords of a code, the Chase algorithm searches the codewords within a sphere of radius d -1 about the received word r due to the fact that the ML codeword ĉ is located in the sphere of radius d -1 with a very high probability at a high SNR. Thus, all possible error patterns of binary weight less than or equal to d -1 are considered.

However, to decrease the complexity, we use the second form of the Chase algorithm applied at the binary level. Let y be the hard decision of r. We search the n d LRIPs of r (lowest absolute values) and then make all possible commutations (0 or 1)a tt h e s en d positions in y.T h e2 n d produced sequences are decoded using a classical algebraic decoder. Only valid codewords are added to the code subset C(n d ).

The decoded codeword ĉ is chosen according to [START_REF] Bose | On a class of error correcting binary group fields[END_REF], where we use the subset C(n d ) instead of the whole codeword set C.

For non-binary codes, we use the same approach as in [START_REF] Aitsab | Performance of reed-solomon block turbo code[END_REF] where the algorithm is still applied at the binary level on the binary representation of a codeword.

In order to apply the Chase algorithm to the SISO iterative decoding, it is desirable that the code set C(n d ) should contain the decoded codeword ĉ and its concurrent (competitor) codewords for all the binary positions. However, for nonbinary codes and high error correction capability binary codes (typically t ≥ 2), it appears that the code subset generated by the Chase algorithm is too small to find the concurrent codewords for all bits. In this case, the SISO iterative decoder has to use a coefficient to approximate the soft decoder output. This will lead to a performance degradation. To deal with this problem, we consider a sliding encoding-window (SEW) algorithm [START_REF] Lalam | On the use of reed-solomon codes in space-time coding[END_REF] for decoding the systematic cyclic codes. In particular, the SEW algorithm can generate a large set of codewords used by the decoder to find the concurrent codewords in order to compute more reliable soft-decision outputs. This can eventually improve the iterative decoding performance.

B. Sliding encoding-window algorithm

We recall that in systematic codes the information part is visible in the codeword, generally at the beginning or the end of the codeword. With a cyclic q-ary code, any cyclic permutation over q-ary symbols of a codeword is still a codeword.

The SEW algorithm exploits both the systematic and cyclic properties of a code, but processes at the binary level, to easily generate candidates, which are in the neighbourhood of the received sequence. The SEW algorithm consists of two successive steps: the encoding phase (EP) and the sliding phase (SP) repeated until all bits are covered with codeword diversity. We describe the algorithm for q-ary RS codes, but its extension to binary codes (m =1) is straightforward.

Let W be the block of the km soft values extracted from r corresponding to the information part, n e ≤ km be the number of the LRIPs inside W and W be the hard decision of W. By permuting the bits (0 or 1)a tt h en e LRIPs of W and encoding the result, we produce 2 ne codewords "close" to y: at least km-n e bits are identical. Codewords generated during this encoding phase (EP) are stored in the set C ne .

As we use systematic codes, we introduce codeword diversity only over the (n -k)m bits of the redundancy part. To create diversity over all bits, we exploit the q-ary cyclic property of the RS code. We cyclically shift W of ∆ q-ary symbols (km bits) over r. After this sliding phase (SP), we perform the EP on the new window. We call W the sliding encoding-window and ∆ the SEW step.

Due to the systematic coding, the obtained codewords must be cyclically shifted in the reverse order so that their kmlength information part matches the position of W in r.W e store shifted codewords in C ne .

We repeat the EP and SP until all bits are covered with diversity. Obviously, the minimum step size is ∆=1, which means that every time, we slide the encoding window by one symbol position for a total of n EPs to process. Considering that the main objective of the SEW algorithm is to generate the codeword diversity for all n coded symbols and that each EP can potentially provide a codeword diversity over n -k symbols, it is natural to set the step size of the sliding window operation ∆=n -k. This is the step size which can be used to generate the codeword diversity for all coded symbols continuously with least overlapping. In this case, we need ⌈ n n-k ⌉ EPs to perform, where ⌈.⌉ is the upper integer part. We can show that 2 ne ≤ card(C(n e )) ≤⌈ n n-k ⌉2 ne . It is interesting to note that a small step size ∆ can provide better decoding performance but it suffers from a higher computational complexity. Therefore, it is possible to choose ∆ between 1, ⌈ n n-k ⌉ to achieve a good trade-off between the complexity and performance. Unless we precise a value of ∆, we will use ∆=n -k in the following.

The decoded codeword ĉ is chosen according to [START_REF] Bose | On a class of error correcting binary group fields[END_REF], where we use the subset C(n e ) instead of the entire code set C.

The SEW algorithm is different from conventional encoderbased decoding algorithms in the way that no permutation or reordering of the received bits according to the reliability is required. Therefore, there is no need to recompute either the generator or parity check matrix, which is carried out by the Gaussian elimination method by checking the linear independence of the most reliable set of bits. Note that the complexity of encoding is much lower than that of the Gaussian elimination operation, one gets the performance enhancement at a slightly increased complexity.

C. Combination with decoding-based algorithm

The SEW algorithm always generates a codeword, but does not guarantee finding the most probable. In fact, if the thresholded input contains less than t errors, any algebraic decoding will produce the most probable codeword. Therefore, we combine the SEW algorithm with at least one algebraic decoding, to ensure that if the input is successfully algebraically decoded, the proposed decoder produces the most probable codeword. In the following, we consider the Chase and the BM algorithms.

Depending on the algebraic decoding-based algorithm chosen, we combine the code set generated by the SEW C(n e ) with either the code set of the Chase C(n d ) or the decoded codeword if the BM succeeds. We denote C(n e ) + this enlarged codeword set.

D. SISO decoder with combined decoding algorithm

The superior performance of BTC is due to the SISO iterative decoder. Decoding is performed either column-wise first and then row-wise or vice versa. For each bit of the selected codeword ĉ, its LLR is computed according to:

Γ(ĉ i )=ln Pr(ĉ i =1|r) Pr(ĉ i =0|r) (3)
We use the subset C(n e ) + instead of the entire set C to compute [START_REF] Reed | Polynomial codes over certain finite fields[END_REF]. Following the same approach as in [START_REF] Lalam | Sliding encoding-window for reed-solomon code decoding[END_REF], the LLR can be approximated by:

Γ(ĉ i ) ≈< c i,1 -c i,0 , r > (4)
where <.,.>is the scalar product and:

c i,j =argmax c∈C(ne) + |ci=j < 2c -1, r > (5)
From ( 4) we can estimate the extrinsic information w i of the i th bit:

Γ(ĉ i ) ≈ < c i,1 -c i,0 , r > ≈ r i + j =i (c i,1 j -c i,0 j )r j ≈ r i + w i (6)
The extrinsic information from the horizontal (vertical) codes can be used to update the received vector r for decoding the vertical (horizontal) codes.

r(m)=r + α(m)w(m) (7)
where m is the index of the iteration, w(m) is the extrinsic information computed from the previous decoding step and α(m) is a weighting factor [START_REF] Pyndiah | Near optimum decoding of product codes[END_REF]. The decoding procedure described above is then generalized by cascading the elementary decoders as illustrated in Fig 1.

A parameter β is used for the computation of w i , when either c i,1 or c i,0 is not available in C(n e ) + for the i th position [START_REF] Pyndiah | Near optimum decoding of product codes[END_REF].

IV. NON-BINARY BTCS

A. Codeword diversity

The SEW algorithm naturally creates codeword diversity. For SEW (n e =8) and the code RS(15, 9, 7), Fig. 2 represents an average histogram of all codewords of C ne sorted by their Hamming distance from the transmitted codeword. Distances are computed at the binary level over 1000 decodings. Fig. 2. SEW(ne =8) codeword diversity at the binary level for RS [START_REF] Aitsab | Performance of reed-solomon block turbo code[END_REF][START_REF] Forney | Generalized minimum distance decoding[END_REF][START_REF] Gazelle | Reliability-based code-search algorithms for maximum-likelihood decoding of block codes[END_REF] As the signal to noise ratio increases, the SEW algorithm generates more codewords in the neighbourhood of the sent codeword. For the code RS(15, 9, 7),t h eS E W( n e =8 ) performs 768 encoding operations, which can generate 767 and 766 different codewords at E b /N 0 =0dBand 6d B , respectively. However, in the radius of Hamming distance of 15 from the transmitted codeword, there are more codewords (325)a t E b /N 0 =6dBthan that (163 codewords) at E b /N 0 =0dB. The enlarged set of the codewords in the close neighbourhood of the correct codeword can help to improve the log-likelihood ratio of the decoder output.

This figure shows that the SEW algorithm provides sufficient codeword diversity to be used in a SISO context in order to compute soft output reliability. Due to this codeword diversity, the choice of the β parameter, used when no concurrent codewords are found for a given position [START_REF] Pyndiah | Near optimum decoding of product codes[END_REF], is less determining.

B. Simulation results

We use 4 iterations in the simulations and the parameters of the BTC [START_REF] Pyndiah | Near optimum decoding of product codes[END_REF] defined for each half-iteration are:

• α = {0.0, 0.25, 0.25, 0.3, 0.35, 0.4, 0.45, 0.5} • β = {0.25, 0.5, 0.75, 1.0, 1.25, 1.5, 1.75, 2.0} For the BTC RS(15, 9, 7) 2 the decoding algorithms are : Chase(n d =4 ), BM + SEW(n e =8 )a n dB M+S E W( n e = 12). Fig. 3 presents the bit error rate (BER) on a Gaussian channel. We observe a gain of 0.75 dB and 1.5d B at a BER of 10 -4 with the addition of the SEW (n e =8 )a n dt h eS E W (n e =12) algorithm respectively.

For the BTC RS(31, 27, 5) 2 ,Fig. 4sho wstheBERandthe word error rate (WER) performance of the code with various decoding algorithms on a Gaussian channel: Chase

(n d =5 ), B M+S E W ( n e =8), Chase(n d =5)+S E W ( n e =8), BM + SEW(n e =8, ∆=1)a n dB M+S E W( n e =12).
We can observe that while the BM + SEW (n e =8) achieves the same BER performance as the Chase (n d =8) algorithm, the BM + SEW (n e =8 ) performs 0.15 dB better than the Chase (n d =5 ) algorithm in WER. In addition, we see that the BM + SEW (n e =8, ∆=1) and Chase (n d =5)+S E W (n d =8) have the same performance and outperform the Chase (n d =5 ) algorithm by 0.2 and 0.25 dB in BER and WER, respectively. Finally, we note that the BM + SEW (n e =12)is 0.5 and 0.6d Bbetter in BER and WER respectively than the classical Chase (n d =5 ). However, its complexity is greater than all the previous algorithms, but the SEW process can be parallelized as the 8 EPs performed for the RS(31, 27, 5) are independent from each other.

V. B INARY BTCS

The BTC BCH(31, 16, 7) 2 has been investigated over an AWGN channel with BPSK modulation by using Monte-Carlo simulation. The number of LRIPs of Chase algorithm is set as n d =5 . The number of LRIPs of the SEW algorithm is set as n e =6to maintain a reasonable complexity. The number of the iterations of the SISO iterative decoder is set to 4. The BER performance comparison of the combined and the conventional algorithms is shown in Fig. 5.

It is obvious that the considered algorithm outperforms the conventional Chase-Pyndiah decoding algorithm for the BTC. In Fig 5, we also show the effect of step size ∆ of the sliding operation on the code error performance. We can see that reducing the step size ∆ from 15 to 8 can slightly improve the

5 ≈ 0. 3 ≈ 1

 531 Fig. 1. SISO Decoder

Fig. 3 .

 3 Fig. 3. BER vs Eb/N0 -BTC-RS(15, 9, 7) 2 on a Gaussian channel -4 iterations

Fig. 4 .Fig. 5 .

 45 Fig. 4. BER vs Eb/N0 and WER vs Eb/N0 -BTC-RS(31, 27, 5) 2 on a Gaussian channel -4 iterations

  C 1 (n 1 ,k 1 ,d 1 ) and C 2 (n 2 ,k 2 ,d 2 ) respectively as its component codes, where n i is the codeword length, k i is the information length and d i is the minimum Hamming distance of the code, i =1 , 2. The information symbols are put in a k 1 × k 2 symbol matrix. The k 2 columns are encoded with the code C 1 (n 1 ,k 1 ,d 1 ), then the n 1 obtained rows are encoded with the code C 2 (n 2 ,k 2 ,d 2 ).

	IV presents simulation results with non-binary
	RS-based BTCs on Gaussian channel, while Part V deals with
	binary BCH-based BTCs. Part VI concludes the paper.
	II. TRANSMISSION SCHEME
	Information symbols are coded with a BTC scheme at the
	transmitter side before modulation. A BTC can be constructed
	by concatenating two or more short block codes with a block
	interleaver to become a long powerful code with a large
	minimum Hamming distance. Let us consider two linear block
	codes,

VI. CONCLUSION

We considered an improved iterative decoding algorithm for turbo product codes. The algorithm exploits the code properties of the cyclic codes to generate an enlarged set of test patterns close to the received word, which is combined with the result of an algebraic decoding based algorithm, such as Chase or BM. The addition of the SEW algorithm can improve the reliability of extrinsic information hence to improve the bit and the word error performance of BTCs.

We show that the proposed algorithm can improve the BER of BTCs up to 1.5d B . When combined with the Chase algorithm, the SEW algorithm produces an enlarged codeword set at the expense of a slightly increased complexity. Compared to the other encoder-based decoding algorithms, the proposed algorithm has the advantage that there is no requirement to recompute the generator or parity-check matrix by using Gaussian elimination operations.