Michael Ewald

Sandra Skowronek

Raf Aerts

Jonathan Roger Michel

Henri Lenoir

Hannes Feilhauer

Ruben Van De Kerchove

Olivier Honnay

Ben Somers

Carol Ximena Garzon-Lopez

Duccio Rocchini

Jonathan Lenoir

Carol Ximena Garzón-López

Sebastian Schmidtlein

Assessing the impact of an invasive bryophyte on plant species richness using high resolution imaging spectroscopy

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

Introduction

Invasive alien plant species can have various unwanted consequences, including adverse effects on human health [START_REF] Shackleton | The role of invasive alien species in shaping local livelihoods and human well-being: A review[END_REF] and ecosystem functioning [START_REF] Pyšek | A global assessment of invasive plant impacts on resident species, communities and ecosystems: the interaction of impact measures, invading species' traits and environment[END_REF][START_REF] Vilà | Ecological impacts of invasive alien plants: a meta-analysis of their effects on species, communities and ecosystems[END_REF]. Impacts on ecosystems include alterations of species richness and composition [START_REF] Powell | Invasive plants have scale-dependent effects on diversity by altering species-area relationships[END_REF], changes in disturbance regimes [START_REF] Pauchard | Positive feedbacks between plant invasions and fire regimes: Teline monspessulana (L.) K. Koch (Fabaceae) in central Chile[END_REF], or in ecological processes like carbon, nutrient, and water cycling [START_REF] Aerts | Invasion by the alien tree Prunus serotina alters ecosystem functions in a temperate deciduous forest[END_REF][START_REF] Liao | Altered ecosystem carbon and nitrogen cycles by plant invasion: a meta-analysis[END_REF][START_REF] Rascher | Understory Invasion by Acacia longifolia Alters the Water Balance and Carbon Gain of a Mediterranean Pine Forest[END_REF]. Once an invasive plant species has established within a new range, it is often extremely difficult to eradicate [START_REF] Rejmanek | When is eradication of exotic pest plants a realistic goal?[END_REF]. Therefore, the management of established alien species focuses on limiting the most adverse effects [START_REF] Kumschick | A conceptual framework for prioritization of invasive alien species for management according to their impact[END_REF]. This requires prioritizing management efforts towards the most harmful species and the most valuable habitats [START_REF] Gaertner | Invasive plants as drivers of regime shifts: identifying high-priority invaders that alter feedback relationships[END_REF][START_REF] Mcgeoch | Prioritizing species, pathways, and sites to achieve conservation targets for biological invasion[END_REF]. One prerequisite to prioritize these management efforts is the detection and monitoring of invasive plant species [START_REF] Latombe | A vision for global monitoring of biological invasions[END_REF].

Remote sensing techniques have high potential for monitoring invasive plants over large areas [START_REF] Rocchini | Potential of remote sensing to predict species invasions: A modelling perspective[END_REF]. Many studies have demonstrated the capability of remote sensing approaches to detect invasive plant species and to map their distribution [START_REF] Vaz | Managing plant invasions through the lens of remote sensing: a review of progress and the way forward[END_REF]. These studies covered a variety of growth forms including trees and shrubs [START_REF] Lopatin | How canopy shadow affects invasive plant species classification in high spatial resolution remote sensing[END_REF][START_REF] Somers | Invasive species mapping in Hawaiian rainforests using multi-temporal Hyperion spaceborne imaging spectroscopy[END_REF], herbs or grasses [START_REF] Müllerová | Does the data resolution/origin matter? satellite, airborne and Uav imagery to tackle plant invasions[END_REF]Skowronek et al., 2017a) and even cryptogams [START_REF] Skowronek | Transferability of species distribution models for the detection of an invasive alien bryophyte using imaging spectroscopy data[END_REF]. Most commonly, these mapping approaches use classifiers, delivering maps with information on the target species' presence and absence. Yet, remote sensing can also be used to acquire quantitative information on species abundance, such as the fractional cover (e.g. [START_REF] Falkowski | Mapping tree canopy cover in support of proactive prairie grouse conservation in western north America[END_REF][START_REF] Huang | Climate anomalies provide opportunities for largescale mapping of non-native plant abundance in desert grasslands[END_REF][START_REF] Miao | Estimation of yellow starthistle abundance through CASI-2 hyperspectral imagery using linear spectral mixture models[END_REF][START_REF] Peerbhay | Detecting bugweed (Solanum mauritianum) abundance in plantation forestry using multisource remote sensing[END_REF]. As the ecosystem impact of an invasive plant species is considerably influenced by its local abundance [START_REF] Parker | Impact: toward a framework for understanding the ecological effects of invaders[END_REF][START_REF] Bradley | Disentangling the abundance-impact relationship for invasive species[END_REF], remote sensing is offering high potential, not only to map invasive plant species, but also to indicate impact magnitudes across large areas [START_REF] Ewald | Analyzing remotely sensed structural and chemical canopy traits of a forest invaded by <Emphasis Type="Italic">Prunus serotina</Emphasis> over multiple spatial scales[END_REF][START_REF] Große-Stoltenberg | Early detection of GPP-related regime shifts after plant invasion by integrating imaging spectroscopy with airborne LiDAR[END_REF][START_REF] Hellmann | Heterogeneous environments shape invader impacts: integrating environmental, structural and functional effects by isoscapes and remote sensing[END_REF]. However, the potential to link abundance maps with potential invasion impacts has not been fully explored yet. For example, abundance information can be used to compare the invasion severity between different habitat types. Furthermore, abundance maps can be used in combination with abundance-impact relationships to display linkages between the abundance of an invader and specific ecosystem properties [START_REF] Bradley | Disentangling the abundance-impact relationship for invasive species[END_REF][START_REF] Panetta | Managing for biodiversity: impact and action thresholds for invasive plants in natural ecosystems[END_REF]. This combination can be used as spatially explicit indicator of invasion impacts on selected ecosystem properties. Here, we aim to explore this potential. The ability to retrieve plant species covers using imaging remote sensing techniques depends on the size of plant individuals or populations and the pixel-size of the images. Broadly speaking, two scenarios are possible. First, the size of the target individual or population is exceeding the pixel size of the image. This is often the case when using very high resolution remote sensing data in combination with large target individuals, such as shrubs and trees, or large monospecific stands [START_REF] Gil | Mapping invasive woody plants in Azores Protected Areas by using very high-resolution multispectral imagery[END_REF][START_REF] Guirado | Deep-learning versus OBIA for scattered shrub detection with google earth imagery: Ziziphus lotus as case study[END_REF][START_REF] Kattenborn | UAV data as alternative to field sampling to map woody invasive species based on combined Sentinel-1 and Sentinel-2 data[END_REF]. Suitable mapping approaches for this scenario include classification approaches, delivering pixel-wise presence-absence information, and object or texture-based approaches [START_REF] Bradley | Remote detection of invasive plants: a review of spectral, textural and phenological approaches[END_REF]. These approaches allow species cover to be calculated from resulting maps across larger areas [START_REF] Falkowski | Mapping tree canopy cover in support of proactive prairie grouse conservation in western north America[END_REF]. In the second scenario, the size of the target individuals or stands is smaller than the pixel size of the used image. In this specific case spectral unmixing approaches and regression techniques can be used to predict fractional cover for single pixels [START_REF] Asner | Spectral and chemical analysis of tropical forests: Scaling from leaf to canopy levels[END_REF][START_REF] Miao | Estimation of yellow starthistle abundance through CASI-2 hyperspectral imagery using linear spectral mixture models[END_REF][START_REF] Peterson | Estimating cover of an invasive grass (Bromus tectorum) using tobit regression and phenology derived from two dates of Landsat ETM+ data[END_REF].

Here, we used airborne imaging spectroscopy to map the fractional cover of the invasive bryophyte Campylopus introflexus-listed among the 100 worst alien plant invaders in Europe [START_REF] Essl | Alien bryophytes and lichens of europe[END_REF]. In its non-native range C. introflexus is predominately occurring in inland and coastal dunes. In early successional stages it can affect natural succession dynamics [START_REF] Ketner-Oostra | Decline of lichen-diversity in calcium-poor coastal dune vegetation since the 1970s, related to grass and moss encroachment[END_REF]. Once established, it can form dense mono-specific carpets, locally decreasing species richness within plant communities of dune ecosystems [START_REF] Biermann | Changes in a lichen-rich dry sand grassland vegetation with special reference to lichen synusiae and Campylopus introflexus[END_REF][START_REF] Ketner-Oostra | Decline of lichen-diversity in calcium-poor coastal dune vegetation since the 1970s, related to grass and moss encroachment[END_REF]. Like other mosses, C. introflexus is capable of retaining water and nutrients from precipitation, and therefore is suspected to influence biomass accumulation, soil formation, and micro-climate [START_REF] Ayres | Direct uptake of soil nitrogen by mosses[END_REF][START_REF] Essl | Little, but increasing evidence of impacts by alien bryophytes[END_REF]. Moreover, C. introflexus was reported to influence the species composition of carabid beetles and spiders in coastal dunes [START_REF] Vogels | Effects of moss-encroachment by Campylopus introflexus on soil-entomofauna of dry-dune grasslands (Violocorynephoretum)[END_REF]. Using the fractional cover derived from predicted maps as a proxy for local abundance, we aimed to evaluate the impact of C. introflexus on plant species richness within a dune habitat in a spatially explicit manner. The specific research questions were:

1. How accurately can the fractional cover of the invasive moss C. introflexus be mapped in different habitats using airborne imaging spectroscopy data?

2. Which spectral bands are the most important to predict fractional cover of C.

introflexus?

3. Which habitat type shows the highest level of invasion?

4. At which fractional cover can we expect an impact on local plant species richness?

2 Materials and Methods

Study area

The study area covers the coastal dune ecosystem in the western part of the island of Sylt (54 • 55' N, 8 • 20' E) located in northern Germany (Fig. 1). The climate is temperate oceanic with a mean annual temperature of 9

Study species

Campylopus introflexus was first observed in Europe in 1941, and has by now spread across major parts of the continent [START_REF] Hassel | The expansion of the alien mosses Orthodontium lineare and Campylopus introflexus in Britain and continental Europe[END_REF]. On the island of Römö it was first noticed in 1970 [START_REF] Frahm | Campylopus introflexus (Hedw.) Brid. neu für Dänemark[END_REF], and probably also introduced on the neighboring island of Sylt around the same time. Growing on bare sand, C. introflexus is predominately occurring in coastal dunes [START_REF] Essl | Little, but increasing evidence of impacts by alien bryophytes[END_REF].

Vegetation data

Field data were acquired between June and November 2014, across 287 plots of 3 m × 3 m surface area. Ninety of these plots were selected in situ in order to cover sites with no presence (n = 30), low (≤ 30 %, n = 31) and medium to high fractional cover (> 30 %, n = 29) of C. introflexus. Apart from the cover of C. introflexus, we also recorded the percentage cover of all co-occurring vascular plant and moss species in these 90 plots. For the 197 remaining plots, we only recorded the percentage cover of C. introflexus. These additional plots were randomly located to cover all relevant habitat types across the study area. For each of the 287 plots we recorded the center coordinates. In addition to the field data, for each plot, habitat types according to the Habitats Directive of the European union were derived from biotope maps of the year 2012 (Fig

Remote sensing data

Airborne imaging spectroscopy data were acquired on July 16 2014, between 12:21 and 13:13 local time (UTC+2), using the APEX (Airborne Prism EXperiment) sensor at a flight height of 2270 m with flight lines in a north-to-south direction. The data included images of 285 spectral bands covering a spectral range from 412 nm to 2432 nm with a pixel size of 1.8 m × 1.8 m. Imaging spectroscopy data was geometrically and atmospherically corrected using the standard procedures applied to APEX data [START_REF] Sterckx | Atmospheric correction of APEX hyperspectral data[END_REF][START_REF] Vreys | Geometric correction of APEX hyperspectral data[END_REF]. To verify atmospheric correction, ground samples were taken in September 2014 from 15 different locations at the beach and at large parking lots (serving as virtual invariant features) using a full-range field spectrometer (FieldSpec 4JR, ASD Inc., Longmont, USA). Noisy bands between 410 nm and 480 nm, and bands in the range between 1320 nm and 1450 nm and between 1670 nm and 1990 nm, being affected by atmospheric water absorption, were excluded from subsequent analyses. This resulted in a total of 233 predictor variables. To minimize the influence of shadows, we subjected the spectra to brightness normalization [START_REF] Feilhauer | Brightness-normalized partial least squares regression for hyperspectral data[END_REF]. To meet the size of the field plots, the spatial resolution of spectral images was resampled to a pixel size of 3 m × 3 m. Resampling was necessary because cover estimations were based on a surface area of 3 m × 3 m, and also because we intended to link the predicted maps with information derived from abundance-impact relationships based on plots of the same size.

For each of the 287 plots, we extracted the brightness normalized reflectance values from all pixels overlapping with the plot area. This resulted in up to four spectra originating from different neighboring pixels per plot. From these values we calculated the weighted mean for each band using the percentage overlap of each pixel with the respective plot area as weight.

Generalized partial least squares regression

To model the fractional cover of C. introflexus, we ran generalized partial least squares (gPLS) regression models using the field-derived cover values as the response variable.

Like classical partial least squares (PLS) regression, gPLS is a combined method of dimension reduction and regression that can deal with a high number of predictor variables, and also with high collinearity among the predictor variables [START_REF] Bastien | PLS generalised linear regression[END_REF][START_REF] Wold | The Collinearity Problem in Linear Regression. The Partial Least Squares (PLS) Approach to Generalized Inverses[END_REF]. For this reason, it is often used to link a response variable to hyperspectral remote sensing data, in which adjacent bands are often highly correlated and the number of predictors is exceeding the number of observations used to train the model. Similar to other dimension reduction methods like principle component analyses, in a PLS regression the information content of a dataset is translated into n latent variables, also called components, which are aligned orthogonally in the n-dimensional feature space determined by the n initial variables. In classical PLS regression, components are linked to the response variable using a least squares regression. In a gPLS, this regression is replaced by a general linear regression that can be fit to response variables ranging from 0 to 1 [START_REF] Bastien | PLS generalised linear regression[END_REF]. For the purpose of improving the prediction performance (remove noise) and for a better understanding of the model, a variable selection can be useful [START_REF] Mehmood | The diversity in the applications of partial least squares: an overview[END_REF].

Because our response variable ranged from 0 to 1, we chose to calculate a gPLS with a logit-link function. Models were calculated using the package plsRglm 1.1.1 [START_REF] Bertrand | Partial least squares regression for generalized linear models[END_REF]) in R 3.4.4 (R Core Team, 2018). The northern end of the island was excluded from modeling because the specific vegetation community was under-represented in the dataset. We therefore also dropped data from 21 field plots located in that area. This resulted in a total of 266 plots, including 162 absence plots used in the final training dataset. All predictor variables were scaled to a mean value of 0 and a standard deviation of 1 to allow direct comparisons among estimated coefficients.

To prevent a possible underestimation of the prediction errors, model calculations were ran repeatedly using 10 different data subsets resulting from a spatially blocked subsampling procedure [START_REF] Roberts | Cross-validation strategies for data with temporal, spatial, hierarchical, or phylogenetic structure[END_REF]. Based on their coordinates, field plots were divided into ten clusters using k-means clustering. For each dataset, 20 % of the field plots were held back for independent validation, each time selecting the plots closest to the center coordinates of the respective clusters.

For the selection of relevant spectral bands, model calculations were additionally embedded in a 10-fold cross validation procedure. Bands were selected based on a stepwise backward variable selection procedure using the standardized coefficients as selection criteria. In each step we dropped 10 % of the predictor variables using the following procedure. We implemented a non-parametric bootstrap approach with 250 bootstrap iterations. For each bootstrap iteration, we recalculated gPLS regression models. Standardized coefficients for each predictor variable resulting from the 250 bootstrap iterations were then tested for significant difference from zero using a 95 % confidence interval (p < 0.05). Using the test results, we dropped predictor variables with insignificant standardized coefficient starting with those having the smallest mean absolute value.

For all models calculated in each step the optimal number of latent variables was selected based on the lowest Akaike Information Criterion (AIC) value. To prevent overfitting, we additionally limited the maximum number of latent variables to five. The procedure stopped when less than 10 % of the remaining predictor variables showed insignificant standardized coefficients. From all steps the model associated with the lowest RMSE in 10-fold cross validation was selected as final model. For each model calculated with different data subsets, we calculated prediction maps representing the fractional cover of C. introflexus within each pixel. From these maps we calculated median maps representing pixel-wise medians of predicted values.

Model evaluation

The model's predictive performances were evaluated by calculating two sets of Pearson R 2 values and root mean square error (RMSE) values based on training and independent validation data respectively. To obtain the overall predictive performance, we calculated the mean and standard error for both R 2 and RMSE values.

The relative importance of each spectral band was derived by calculating model predictions based on permuted predictor datasets. For each model, permuted datasets were created by successively randomizing the values of each predictor variable. For each permuted dataset, we re-calculated model predictions based on the original response data. Predicted values calculated from permuted datasets were then compared to the predictions from the non-permuted dataset, calculating the Pearson correlation coefficient. A high correlation coefficient indicates that a predictor (here: spectral band) is not better than a random predictor variable. In contrast, a small coefficient indicates high relevance. Accordingly, the inverse correlation coefficient was then used as variable importance of each predictor variable.

Model residuals were used to compare prediction errors for different habitat types. To get a picture from all model residuals, we ran a linear mixed-effects model (LMM) using the residuals from all models as response variable and habitat type as fixed-effect variable. The data subsets were numbered from 1 to 10 and included as random intercepts. We ran LMMs using the nlme 3.1 [START_REF] Pinheiro | nlme: Linear and nonlinear mixed effects models, R package version 3[END_REF] package in R.

Evaluating the impact of C. introflexus

To evaluate the abundance of C. introflexus in different habitat types, we calculated mean fractional covers and the total cover for each type. First, average covers were calculated from each of the 10 prediction maps for each habitat type. As a second step, we calculated the mean and standard error from these values. Total cover was derived by multiplying mean fractional cover with the total area covered by each habitat type, respectively.

To assess the impact of C. introflexus on native plant communities, we evaluated the relationship between its fractional cover and the richness of co-occurring plant species based on data acquired in 90 field plots. We used the Simpson diversity index (Simpson, 1949) for plant species richness, including all vascular plant and moss species. This abundance-impact relationship was analyzed by fitting a linear model. The Simpson index was used as response variable and the percentage cover of C. introflexus, both the first-and second-order polynomial term, was used as explanatory variable. To identify existing break points in this relationship, we ran a piecewise regression using the segmented 3.0 package in R [START_REF] Muggeo | segmented: an R package to fit regression models with brokenline relationships[END_REF][START_REF] Muggeo | Estimating regression models with unknown break-points[END_REF].

Results

Models resulted in mean R 2 values of 0.71 (standard error: ± 0.03), and 0.64 ± 0.17 as well as mean RMSE values of 0.12 ± 0.01 and 0.14 ± 0.02 based on the training (Fig. 3) and independent validation datasets, respectively (Appendix A1). The ten final models included a range of 7-183 spectral bands after variable selection. All models included bands situated in the shortwave-infrared region (SWIR), between 2150 nm and 2160 nm as well as between 2240 nm and 2250 nm (Fig. 4). The bands located around 2242 nm and 2248 nm had highest mean variable importance. Prediction errors differed between the habitat types, indicating highest prediction accuracy for crowberry heath and lowest accuracy for dune slacks (Fig. 5). Based on the median predictions C. introflexus total cover across the island of Sylt reached 2.08 km 2 . Mean cover of C. introflexus ranged from 10.2 ± 0.9 % in crowberry heath to 11.1 ± 1.0 % in grey dunes (Table 2).

The linear model relating fractional cover of C. introflexus to plot level plant species richness resulted in a R 2 value of 0.10. The predicted curve peaks at about 30 % cover

Habitat Mean cover (%)

Total coverage (km 2 ) Grey dunes 11.1 ± 1.0 0.37 Crowberry heath 10.2 ± 0.9 1.54 Dune slacks 10.6 ± 1.9 0.17 of C. introflexus (Fig. 6). The piecewise regression analysis resulted in a break point at a fractional cover of 44 %. Above this threshold increasing cover of C. introflexus was related to decreasing plant species richness within plots of 3 m × 3 m surface area. Prediction maps indicated that 7.1 %, 4.3 %, and 6.8 % of the pixels were characterized by fractional covers of 44 % or above, for grey dunes, crowberry heath and dune slacks, respectively (Fig. 7).

Discussion

Mapping the fractional cover of C. introflexus

We demonstrated that imaging spectroscopy can be used to map the fractional cover of an invasive alien bryophyte. Similar results for mapping invasive alien plant species cover have been found by previous studies, however in different contexts. For example, Peterson ( 2005) used multi-seasonal Landsat 7 images to map the cover of a non-native grass species using regression approaches. [START_REF] Guo | Hyperspectral measurement of seasonal variation in the coverage and mipacts of an invasive grass in an experimental setting[END_REF], predicted the fractional cover of an invasive grass species using hyperspectral data derived from field spectroscopy. Similarly, spectral mixture analyses can be used to predict the fractional cover of selected target species [START_REF] Andrew | The role of environmental context in mapping invasive plants with hyperspectral image data[END_REF][START_REF] Somers | Endmember variability in spectral mixture analysis: a review[END_REF]. For example, Asner and (2008) predicted the cover of several non-native tree species in a rainforest from imaging spectroscopy data using a spatial unmixing approach. In line with these previous studies, our results underline the high potential of remote sensing to map fractional cover of invasive alien plant species. Future remote sensing approaches should focus on mapping quantitative data, such as fractional cover, as they provide more information than qualitative (presence-absence) data.

Martin

Similar to presence-absence mapping one major requirement for the success of mapping species cover is that the spectral signal of the target species differs from that of the background vegetation [START_REF] Bradley | Remote detection of invasive plants: a review of spectral, textural and phenological approaches[END_REF]. In our study, plots with a high C. introflexus cover were spectrally different from those without presence (Appendix A2). Reflectance spectra of C. introflexus were characterized by higher reflectance in the short wave infrared (SWIR) and lower reflectance in the near infrared (NIR), compared to the dominating vascular plant species of the studied habitat types. This finding is in contrast to [START_REF] Bubier | Spectral reflectance measurements of boreal wetland and forest mosses[END_REF] who found lower canopy reflectance for mosses compared to vascular plant species in both the NIR and SWIR. In the case of C. introflexus, high SWIR reflectance is probably due to the high proportion of windblown sand that intermingles with the moss carpets. Although reflectance differed between C. introflexus and the co-occurring vegetation across large parts of the spectrum, variable importance indicated that only distinct sections were important for predicting fractional cover. The high importance of spectral bands in the SWIR for the prediction of C. introflexus cover can be attributed to differences in leaf structure or water content [START_REF] Jacquemoud | PROSPECT+SAIL models: A review of use for vegetation characterization[END_REF][START_REF] Kokaly | Characterizing canopy biochemistry from imaging spectroscopy and its application to ecosystem studies[END_REF]. Leaf water content can differ substantially between vascular plant species and mosses such as C. introflexus. Furthermore, in contrast to vascular plants, mosses are characterized by only a very thin cuticle and thus differ in leaf structure (Glime, 2017). Therefore, it is likely that selected spectral bands in the SWIR allow a discrimination of different C. introflexus covers from co-occurring vascular plant species. Furthermore, spectral bands in the SWIR may have helped to discriminate C. introflexus from bare soil. The high importance of spectral bands located in NIR in some of the calculated models (Fig. 4) can be attributed to differences in canopy structure. In contrast to most of the co-occurring species, C. introflexus carpets are characterized by a very homogeneous and dense canopy, allowing only little within-canopy scattering of incoming radiation [START_REF] Ollinger | Sources of variability in canopy reflectance and the convergent properties of plants[END_REF]. The NIR was also recognized to be important for the discrimination of different moss species [START_REF] Bubier | Spectral reflectance measurements of boreal wetland and forest mosses[END_REF]. In contrast to structural properties, differences in pigment concentration can be excluded to be important for predicting C. introflexus cover, as characteristic bands in the visible region of the spectrum did not play an important role in the models [START_REF] Ustin | Retrieval of foliar information about plant pigment systems from high resolution spectroscopy[END_REF].

It is important to note that [START_REF] Skowronek | Transferability of species distribution models for the detection of an invasive alien bryophyte using imaging spectroscopy data[END_REF], using the same hyperspectral images, found different bands to be important for predicting the presence and absence of C. introflexus. This previous study located important bands in the 1700-1750 nm region and around 2000 nm. This suggests that discriminating abundances of one species may require different spectral information than discriminating between presences and absences. Similar to [START_REF] Skowronek | Transferability of species distribution models for the detection of an invasive alien bryophyte using imaging spectroscopy data[END_REF], we identified distinct narrow bands in the SWIR to be important for mapping C. introflexus. This suggests that hyperspectral data is necessary for mapping C. introflexus, as SWIR bands from multispectral sensors usually cover wavelength ranges that are too broad (e.g. Sentinel 2, Landsat 8). Prediction accuracies differed between the habitat types, indicating less-accurate cover predictions for dune slacks and grey dunes in comparison to crowberry heath. Lower accuracies for dune slacks can probably be attributed to the small sample size (Table 1). For grey dunes, lower accuracies can presumably be attributed to the spectral variability that was highest in this habitat type (Appendix A3). In contrast to the two evergreen heath habitats, grey dunes were characterized by a higher cover of grasses and herbs, that show stronger phenological variation. We assume that the high spectral variability observed for grey dunes was caused by the variable proportion of senescent leaves we observed in some of the plots. Furthermore, grey dune plots were more variable concerning bare soil coverage. Previous studies showed that the mapping accuracy of species is influenced by the spectral variability of the studied habitat [START_REF] Andrew | The role of environmental context in mapping invasive plants with hyperspectral image data[END_REF][START_REF] Somers | Endmember variability in spectral mixture analysis: a review[END_REF]. In addition, grey dunes and C. introflexus canopies seem to be spectrally less different compared to the other habitat types (Appendix A3). This may be due the higher influence of other native moss species on the canopy reflectance of grey dunes or due to the higher bare soil coverage compared to other habitat types. When mapping plant species, high spectral similarity between the target species and its background vegetation may influence prediction accuracies negatively [START_REF] Andrew | The role of environmental context in mapping invasive plants with hyperspectral image data[END_REF]. Apart from found differences in prediction accuracies, our results indicate that imaging spectroscopy is suited for cover predictions in different habitat types and hence also for comparing the potential impact of invasive plant species in these habitat types.

As C. introflexus does not exhibit strong phenological variation, it is likely that the results from our study can be transferred to other similar study areas. A previous study by [START_REF] Skowronek | Transferability of species distribution models for the detection of an invasive alien bryophyte using imaging spectroscopy data[END_REF] showed that model transfer is promising when predicting presence-absence of this species. However, as for other moss species, spectral characteristics may strongly vary with water content and thus with local weather conditions [START_REF] Neta | Development of new spectral reflectance indices for the detection of lichens and mosses moisture content in the Hudson Bay Lowlands, Canada[END_REF].

Evaluating the impact of C. introflexus

This study demonstrates that remote sensing data can be used to evaluate the level of invasion in different habitat types. Our results indicate that C. introflexus is most abundant in grey dunes, in agreement with the findings of [START_REF] Klinck | The alien invasive moss Campylopus introflexus in the Danish coastal dune system[END_REF]. As the local abundance is one of the determining factors for the impact magnitude of an invasive alien species, cover maps can be used as baseline information for the potential impact of C. introflexus. Such information covering different habitat types can be helpful to prioritize limited management resources.

In our specific case, average fractional cover for C. introflexus was in the same order of magnitude for the three habitat types. Cover values derived from prediction maps were slightly higher than observed in the 160 randomly placed field plots (mean fractional cover of 9 %, 5 % and 6 % for crowberry heath, grey dunes and dune slacks, respectively). This may be because the discrete field sampling missed relevant spatial variation in C. introflexus fractional cover. Furthermore, differences may be caused by uncertainties of the used mapping approach. Potential difficulties may arise from its unreliability to detect low cover occurrences at the sub-pixel level [START_REF] Bradley | Remote detection of invasive plants: a review of spectral, textural and phenological approaches[END_REF]. For example Skowronek et al. (2017b) found that the detection of C. introflexus becomes unreliable when the species is covering less than one third of the pixel size. We therefore do not expect that this approach can be used to differentiate between low cover and absence. However, it has high potential to discriminate between low, medium and high abundance areas, and thus can indicate the magnitude of potential invasion effects.

To highlight invasion impacts concerning specific ecosystem properties, cover maps can be combined with abundance-impact relationships. Here we studied the relationship between C. introflexus cover and the species richness of co-occurring plant species. Our results suggest a cover threshold of 44 % above which increasing cover of C. introflexus is related to decreasing plant species richness. In combination with a cover map, this threshold can be used to highlight high impact areas in a spatially explicit manner. Previous studies found variable abundance-impact relationships strongly depending on the target species and studied variables (e.g. [START_REF] Bradley | Disentangling the abundance-impact relationship for invasive species[END_REF][START_REF] Fried | Comparing an exotic shrub's impact with that of a native life form analogue: Baccharis halimifolia vs Tamarix gallica in Mediterranean salt marsh communities[END_REF][START_REF] Gooden | Impact threshold for an alien plant invader, Lantana camara L., on native plant communities[END_REF][START_REF] Mcalpine | Ecological impacts of ground cover weeds in New Zealand lowland forests[END_REF][START_REF] Große-Stoltenberg | Early detection of GPP-related regime shifts after plant invasion by integrating imaging spectroscopy with airborne LiDAR[END_REF]. It is important to note that thresholds derived from abundance-impact relationships should be applied with caution because they do not necessarily imply a causal link between invasive species abundance and a specific ecosystem property. Still, abundance or cover thresholds are meaningful, particularly to prioritize the control of established invaders in valuable habitats [START_REF] Panetta | Managing for biodiversity: impact and action thresholds for invasive plants in natural ecosystems[END_REF]. Regarding our case study it should also be considered that invasive alien plants may affect some native species or functional groups even at lower abundance levels, than indicated by the threshold [START_REF] Panetta | Managing for biodiversity: impact and action thresholds for invasive plants in natural ecosystems[END_REF]. For example, interactions of C. introflexus with other mosses and lichens are not specifically displayed in the threshold defined in this study and should be subject to further investigations. Here, particularly long term effects on natural succession dynamics are not fully understood [START_REF] Hasse | Campylopus introflexus invasion in a dune grassland: succession, disturbance and relevance of existing plant invader concepts[END_REF]. Cover maps created from remote sensing time series could help study these dynamics at the landscape level.

Conclusion

In this study, we successfully mapped the fractional cover of an invasive alien moss C. introflexus using imaging spectroscopy. Important spectral bands for the prediction of covers were situated in the SWIR, suggesting that variation in cover was related to differences in leaf water content and leaf structure between C. introflexus and the co-occurring native vegetation. Resulting maps can be used as baseline information to study potential impact on ecosystem functioning, because the local abundance of invasive alien species is one of the determinants to explain their impact magnitude. We showed that cover maps can be used to evaluate the level of invasion in different habitat types. Furthermore, cover maps in combination with thresholds derived from cover-impact relationships might be useful to highlight areas of particular concern. Both outputs represent valuable information to prioritize management of established invasive alien species.

Figure 1

 1 Figure 1 Location of the study area (a), outline of the island of Sylt (b) and habitat maps including plot locations for the northern (c) and southern part of the island (d).

Figure 2

 2 Figure 2 Habitat types included in this study; a: grey dunes, b: crowberry dunes, c: dune slacks (photos by S. Skowronek).

Figure 3

 3 Figure 3 Median predicted values retrieved from generalized partial least squares regression models vs. observed values for the fractional cover of C. introflexus in 3 m × 3 m plots from 10 training datasets. Error bars represent ranges between the 25 % quantile and the 75 % quantile of model predictions.

Figure 4

 4 Figure 4 Relative importances of spectral bands included in 10 gPLS models using different training datasets.

Figure 5

 5 Figure 5 Prediction error of C. introflexus fractional cover for different habitat types. Displayed values represent absolute deviations of predicted covers from observed covers calculated using a linear mixed effects model including all deviations resulting from repeated gPLS model calculations.

Figure 6

 6 Figure 6Relationship between C. introflexus cover and plant species richness given by the Simpson diversity index observed for 90 field plots (3 m × 3 m) and predicted using a linear model.

Figure 7

 7 Figure 7 Map examples showing (a) an RGB image (b) the predicted fractional cover of C. introflexus (c) a map of habitat types and (d) pixels with fractional covers higher than 0.44 (in red) indicating potential negative impact on plant species richness.

Table 1

 1 Habitat types included in this study and total coverage across the Island of Sylt. # Training plots: number of plots included in the dataset used for model calibration. The remaining plots covered other habitat types.

	Used Acronym Description	Area (km 2 ) # Training plots
	Grey dunes	Fixed coastal dunes with herbaceous vegetation	3.3	44
	Crowberry dunes Decalcified fixed dunes with Empetrum nigrum	15.1	167
	Dune slacks	Humid dune slacks	1.6	13

Table 2

 2 Fractional covers of C. introflexus for different habitat types based on predicted maps. Mean cover represents mean pixel values for each habitat type averaged for all calculated maps. The total coverage is giving the total area covered by C. introflexus.

Acknowledgements

This study is part of the project DIARS (Detection of invasive plant species and assessment of their impact on ecosystem properties through remote sensing) funded by the ERA-Net BiodivERsA, with the national funders: ANR (Agence Nationale de la Recherche); BelSPO (Belgian Federal Science Policy Office); and DFG (Deutsche Forschungsgemeinschaft). Michael Ewald was funded through the DFG research grant SCHM 2153/9-1. Furthermore this study was supported by the INPANT project (SR/01/ 321) within the STEREOIII program funded by the BelSPO. Data processing and the calculation of prediction maps was supported by the computational resource bwUni-Cluster funded by the Ministry of Science, Research and the Arts and the Universities of the State of Baden-Württemberg, Germany, within the program bwHPC. The authors would like to thank the nature conservation authority of Northern Friesland and private land owners for providing maps and granting permission to access the study area.


Sami Mekki
email: mekki@enst.fr

Jean-Luc Danger
email: danger@enst.fr

Benoit Miscopein
email: benoit.miscopein@orange-ftgroup.com

Joseph J Boutros
email: boutros@tamu.edu

Chi-squared Distribution Approximation for Probabilistic Energy Equalizer Implementation in Impulse-Radio UWB Receiver

This paper addresses the hardware implementation feasibility assessment of a probabilistic energy equalizer for UWB impulse radio (IR) transmission. The receiver is based on energy detection and targets data rates greater than 100Mbps. The aim of this study is to bypass certain complex mathematical functions like the chi-squared (χ 2 ) distribution and approximate it by functions easier to implement. Fixed point data types simulations are achieved and compared to the results obtained from the reference floating point non-approximated function. Simulation results, under IEEE 802.15.3a channel models , confirm our choice which allows an easy hardware implementation for these complex functions.

I. Introduction

Ultra-wideband impulse radio (IR) is considered as a promising candidate for indoor communications and wireless sensor networks, as described in [START_REF] Yang | Ultra-wideband communications: an idea whose time has come[END_REF]. Both transmitter and receiver are facing difficulties of implementation due to the technological limits. Coherent IR UWB reception based on Rake receiver is limited in number of implementable Rake fingers [START_REF] Choi | Performance of ultra-wideband communications with suboptimal receivers in multipath channels[END_REF]. An alternative is given by the transmitter reference (TR) method [START_REF] Hoctor | Delay-hopped transmittedreference rf communications[END_REF], however the electronic architecture is more complex as it needs analog delay lines and mixers. Non-coherent energy detection receiver is far less complex as a few components like shottky diodes and capacitors suffice. Nevertheless transmitting impulses at high data rate causes inter-symbol interference [START_REF] Lottici | Inter-symbol interference mitigation in high-data-rate uwb systems[END_REF], [START_REF] Zhang | Interference mitigation for coexistence of heterogeneous ultra-wideband systems[END_REF], [START_REF] Şahin | Inter-symbol interfrence in high data rate uwb communications using energy detector receivers[END_REF] which could seriously decrease the performances of a simple energy detector.

In [START_REF] Mekki | Probabilistic equalizer for ultra-wideband energy detection[END_REF], it was proposed a probabilistic energy equalizer that handles the interferences among symbols and inside the symbols in an M -array pulse position modulation (M-PPM). However, equalization function is mathematically complex and leads to hardly feasible implementations. The problem comes mainly from the energy expression, which follows a chi-squared (χ 2 ) distribution [START_REF] Proakis | Digital Communications[END_REF]. Several papers have looked for an approximation of the chi-squared distribution [START_REF] Severo | Normal approximation to the chisquared and non-central F probability functions[END_REF], [START_REF]A normal approximation for the chi-square distribution[END_REF]. Nevertheless the proposed approximations remain complex for digital implementations. In this paper, the approximation defined in [START_REF] Zhang | Approximate and asymptotic distributions of chi-squared-type mixtures with applications[END_REF] is studied for a fixed point data types implementation. The chisquared approximation is implemented and tested within the probabilistic energy detector equalizer proposed in [START_REF] Mekki | Probabilistic equalizer for ultra-wideband energy detection[END_REF]. The equalizer is followed by a channel decoder which allows joint channel decoding in the same iterative loop.

This article is organized as follows:

The system model is defined in Section II where the conditional distribution of the received energy is established for equalization purpose. Section III reviews the energy equalization principle. In Section IV, the conditioned energy distribution defined in Section II is approximated by a simple function for hardware implementation. Results with the approximated distribution are compared to the chisquared distribution in Section V. In the same section the hardware implementation results in fixed point precision data types are also depicted and compared to the theoretical results in floating point precision. Final remarks, conclusion and discussion about forthcoming work in the field are given in Section (VI).

II. System Model

We consider an UWB communication system based on impulse radio (IR) transmission with a signal bandwidth W . An M -ary Pulse Position Modulation (M-PPM) is assumed for symbol modulation, where each symbol is defined by the position of a pulse over a time symbol T s . However, transmitting pulses at high data rate over dispersive channels [START_REF] Foerster | Channel modeling sub-committee report final[END_REF] leads to inter-symbol interference (ISI) or inter-slot interference (IStI). It means that the channel impulse response of a transmitted pulse may provoke collisions on the following slots or the following symbols. The resulting received signal in the time interval [(n -1)T s , nT s [ is defined as

y n (t) = s n (t) + z n (t) = ∞ k=0 x n-k (t) + z n (t) (1)
where z n (t) is an additive white Gaussian noise with mean zero and variance σ 2 and x n-k (t) being the channel response of the (n -k) th transmitted symbol defined by:

x n-k (t) = p(t -A n-k T slot ) ⊗ h(t) (2)
where h(t) is the channel impulse response, ⊗ denotes the convolution product, p(t) is the pulse shape, T slot is the time slot duration for an M-PPM modulation and A n,k takes value in {0, 1, . . . , M -1} according to transmitted symbol. Without loss of generality, the pulse is considered as being a Dirac δ(t) function, thus

x n-k (t) = h(t -A n-k T slot ).
This assumption does not affect our logical thinking.

It should be noted that the receiver can handle only a finite number of interfering symbols. Let K be the number of interfering symbols considered by the receiver. So, it assumes that the UWB received pulse can not exceed P interfering slots linked to K by P = (K -1)M + 1. The actual value of the number of interfering symbols might be larger than the supposed value K. Thus, s n (t) turns to

s n (t) = K-1 0 x n-k (t) (3)
The radio frequency (RF) reception architecture is assumed to be an energy detector receiver so that the energy per time slot m in n th symbol is given by

E n,m = nTs+(m)T slot nTs+(m-1)T slot (s n (t) + z n (t)) 2 dt (4)
where T s = M T slot . For digital processing concerns, (4) must be discretized into a sum of finite samples. In [START_REF] Urkowitz | Energy detection of unknown deterministic signals[END_REF], it was shown that the energy of a signal, of duration T slot , with a bandwidth W can be approximated by the sum of 2W T slot samples. This leads to

E n,m = 2L =1 (s n,m + z n,m ) 2 (5)
where 2L = 2W T slot is the number of degrees of freedom (DoF) over the interval T slot , and s n,m and z n,m are respectively the th sample of s n (t) and z n (t) in m th slot of n th symbol.

The received energy per slot defined by

E n,m = 2L =1 X 2 , where X = s n,m + z n,m , follows a central chi- squared [8] distribution with 2L DoF if (s n,m ) 2 = 0, given by p(E n,m |0) = 1 σ 2L 2 L Γ(L) E L-1 n,m e -En,m 2σ 2 (6)
where Γ(z) is the gamma function [START_REF] Abramowitz | Handbook of Mathematical Functions with Formulas, Graphs,and Mathematical Tables[END_REF]. Furthermore, if (s n,m ) 2 = 0, the energy per slot follows a non-central chi-squared distribution [START_REF] Proakis | Digital Communications[END_REF] with 2L DoF given by

p(E n,m |B n,m ) = 1 2σ 2 E n,m B n,m L-1 2 e -(En,m+Bn,m) 2σ 2 I L-1 B n,m E n,m σ 2 (7)
where the non-centrality parameter

B n,m ∆ = 2L =1 (s n,m ) 2 = (s n,m ) 2 (8)
is the resulting energy on the slot m of the n th symbol without considering the additive noise. I L-1 (u) is the (L-1) th -order modified Bessel function of the first kind [START_REF] Abramowitz | Handbook of Mathematical Functions with Formulas, Graphs,and Mathematical Tables[END_REF].

III. Energy Equalization Principle

To benefit from the iterative process of a communication system, we consider a probabilistic equalizer that can be embedded into the iterative loop of a channel decoder based on SISO (Soft-Input/Soft-Output) decoding. Thus, the considered equalizer takes the accumulated energy per slot (i.e. E n,m ) and per symbol (i.e. E n = (E n,1 , E n,2 , . . . , E n,M )) as reference, in order to retrieve the transmitted symbol x n . So the detector computes a conditioned probability p(E n |x n ) regarding the interfering symbols on x n . It has been shown in [START_REF] Mekki | Probabilistic equalizer for ultra-wideband energy detection[END_REF] that the equalization is performed by computing

Encoder Pulse Generator Mapper Channel Filter codeword SISO Decoder Data Equalizer Table Mem read Decoded Tslot (.) 2 H c {d i } {p n-k (t)} z(t) E n,m {B j } p(E n,m |B n,m ) σ 2 p(E n |x n ) s(t) π(x k ) bits { d i }
p(E n |x n ) = xn-1 . . . xn-K+1 M m=1 p(E n,m |B n,m ) K-1 k=1 π(x n-k ) (9)
where π(x n-k ) is the a priori probability provided by the SISO decoder and p(E n,m |B n,m ) is defined in Section II.

It was also established that the set of all the possible values that B n,m could take, has a finite cardinal. Figure 1 resumes the transmission and the receiver design under consideration.

In order to reduce the complexity and make the equalizer feasible, we investigate the implementation in finite precision. Moreover the probability given by Equation ( 9) needs some mathematical simplifications and approximations of the probability density function (pdf) p(E n,m |B n,m ), corresponding either to the central (6) or non-central chisquared [START_REF] Mekki | Probabilistic equalizer for ultra-wideband energy detection[END_REF] distribution. This will be investigated in the following section.

IV. Chi-Squared Distribution Approximation for Hardware Implementation

The chi-squared distribution defined by ( 6) and ( 7) is a three variable function (E n,m , B n,m and σ 2 ). Thus, building a look-up table according to these parameters would occupy a great memory. For instance, if the energy distribution is coded with 7 bits and if E n,m , B n,m and σ 2 are respectively 14-bit, 6-bit and 6-bit long, the space memory allocated to this look-up table would occupy 448Mbits (or 56Mbytes). This corresponds to a costly silicon area in a FPGA or ASIC technology and thus incompatible with low-cost constraints.

An approximation for the chi-squared distribution is thus necessary. In the literature, there are some proposals for the calculation of the non-central chi-squared distribution [START_REF] Ross | Algorithm for calculating the noncentral chisquare distribution[END_REF] and the use of the normal approximation to the chi-squared distribution [START_REF] Severo | Normal approximation to the chisquared and non-central F probability functions[END_REF], [START_REF]A normal approximation for the chi-square distribution[END_REF], [START_REF] Zhang | Approximate and asymptotic distributions of chi-squared-type mixtures with applications[END_REF], but those approximations require high bit precision and are therefore too complex for digital design. An intuitive approximation can be found by considering the Remark in [START_REF] Zhang | Approximate and asymptotic distributions of chi-squared-type mixtures with applications[END_REF] which stands that when a variable Σ is used to approximate a variable Ω, it is equivalent to match the mean and variance of Σ and Ω. It is notably shown in [START_REF] Zhang | Approximate and asymptotic distributions of chi-squared-type mixtures with applications[END_REF], that a chi-squared distribution can be approximated by a Gaussian distribution. However the smaller the number of DoF 2L, the larger the approximation error. Due to the large bandwidth W in UWB-IR, the number DoF could be big enough [START_REF] Saadane | On the estimation of the degrees of freedom of in-door UWB channel[END_REF] to consider the Gaussian distribution as an approximation to the chi-squared density. For instance 2L is around 30 for W = 3 GHz and T slot = 5 ns. According to the previous Remark, the Gaussian approximation has the same mean and variance as the non-central chi-squared distribution, i.e. E n,m ∼ N (m χ 2 , σ 2 χ 2 ), given by [START_REF] Proakis | Digital Communications[END_REF]:

m χ 2 = 2Lσ 2 + B n,m (10) σ 2 χ 2 = 4Lσ 4 + 4σ 2 B n,m (11)
This can be extended to the central chi-squared distribution by considering B n,m = 0. Using these results and the aforementioned assumptions, we obtain the approximation for the energy distribution (noticed p) per slot, ∀ B n,m ≥ 0 and 2L >> 2 as It is noticed that the error tends to zero as σ 2 decreases (Figure 2(b)). According to [START_REF] Mekki | Probabilistic equalizer for ultra-wideband energy detection[END_REF], the energy equalizer operates at σ 2 < 1; i.e. σ 2 = 1 corresponds to SN R = -3dB for a pulse energy equals to unity in coded system. So, the maximum error, considered between the chi-squared and Gaussian distributions, is = 5 × 10 -3 as shown in Figure 2(a). We denote the normal function by

p(E n,m |B n,m ) ≈ p(E n,m |B n,m ) = exp - (En,m-m χ 2 ) 2 2σ 2 χ 2 2πσ 2 χ 2 (12)
φ(t) = 1 √ 2π e -t 2 /2 (13)
Using ( 10), ( 11) and ( 13), equation ( 12) can be rewritten as follows

p(E n,m |B n,m ) = 1 σ 2 χ 2 φ   E n,m -m χ 2 σ 2 χ 2   (14)
As the energy distribution is simply deduced from the normal function φ(t), the digital implementation can only use two look-up tables. The first one contains the values of the normal function φ(t) , ∀t ≥ 0. The second one contains the values of the ratio 1/ √ x , ∀x > 0. The input/output precision of the look-up tables will be analyzed in the simulation section according to the hardware constraints.

V. Simulation Results

The BER computation of the energy equalizer with the approximated Gaussian distribution has been performed via simulations in both floating point precision and fixed point precision data types. Before considering simulations in fixed point precision, we compare the performance of the receiver with the Gaussian approximation [START_REF] Foerster | Channel modeling sub-committee report final[END_REF] and with the exact calculation of the chi-squared distribution in floating point precision. The block fading multipath channel is generated randomly according to IEEE 802.15.3a UWB channel models CM1 and CM2 [START_REF] Foerster | Channel modeling sub-committee report final[END_REF]. Channel estimation is out of the scope of this paper. The channel state information (CSI) is assumed perfectly known at the receiver side. Nevertheless, channel parameters can be approached by the mean of the expectation-maximization (EM) algorithm as studied in [START_REF] Mekki | Em channel estimation in a low-cost uwb receiver based on energy detection[END_REF].

A. Chi-squared versus Gaussian approximation simulations in double precision

We consider an UWB-IR system as defined in Figure 1. Transmission is ensured by a 4-PPM modulation at 100 M bit/s. Thus we get 2 bits per transmitted symbol. We have implemented a duo-binary turbo code as it is defined in the standards [START_REF]Digital Video Broadcasting (DVB); Interaction channel for Satellite Distribution Systems[END_REF], [START_REF]Digital video broadcasting (DVB); interaction channel for satellite distribution systems; guidelines for the use of en 301 790[END_REF]. This channel coder is chosen because it is suited to QPSK (quadratic phase shift keying) and 4-PPM modulations. The encoded data, at the input of the encoder, are 864-bit long blocks. The turbo encoder rate is 1/2 and 10 iterations of the SISO decoder are performed at the receiver side. The equalizer is jointly implemented into the iterative loop of the decoder to benefit from the iterative process of the decoder. The efficiency of the energy equalizer will not be treated in this paper, the reader should refer to [START_REF] Mekki | Probabilistic equalizer for ultra-wideband energy detection[END_REF] for more details concerning the equalizer performances. The receiver assumes that there are only two interfering symbols, i.e. K = 2 and P = 5, but the real number of interfering symbols could be more. The CSI is assumed over P time slots duration and not otherwise. In our case, for a data rate of 100Mbps, the time slot duration is 5 ns, so the receiver has a perfect CSI only over 25ns. This duration is sufficient for channel models as CM1 and CM2, although their respective maximum excess delay are 80 ns and 115 ns as it is studied in [START_REF] Mekki | Probabilistic equalizer for ultra-wideband energy detection[END_REF]. It is noticed that the results with Gaussian approximation match the chi-squared performances in floating point precision.

B. Simulations in fixed point precision

Simulation in fixed point precision is subject to hardware constraints. The duo-binary turbo coder hardware implementation is out of the scope of this paper. The digital design of the channel coder is furnished by TurboConcept for an optimum efficiency [START_REF]TC1000-xX DVB-RCS Turbo Decoder v2.1[END_REF]. The energy detector of UWB platform is a logarithmic one [START_REF]AD8318 1 MHz to 8 GHz, 70 dB Logarithmic Detector/Controller[END_REF]. To guarantee the scalar value of the energy E n,m for equalization, a look-up table of the function 10 x is required. Computer simulations in fixed point precision are achieved by means of the SystemC class sc fix [START_REF]SystemC User's Guide[END_REF]. The Gaussian approximation for energy equalization are computed through the look-up tables of the functions:

φ(x) = e -x 2 /2 √ 2π , g(x) = 1/ √
x and h(x) = 10 x . Figure 4 shows the Gaussian approximation computation architecture for the chi-squared distribution. According to the class sc fix of SystemC, a signed or an unsigned object are defined by two parameters: the total word length noted as wl, i.e. the total number of bits used in the type, and the integer word length noted as iwl,

ROM mean & variance ROM ROM ROM σ 2 χ 2 log(E n,m ) log(2Lσ 2 ) 10 x Chi-squared 1 √ x φ(x) p(E n,m |B n,m ) B n,m B n,m m χ 2 2Lσ 2 En,m-m χ 2 q σ 2 χ 2 E n,m
Fig. 4. Gaussian approximation architecture to the chi-squared distribution.

i.e. the number of bits that are on the left of the binary point (.) in a fixed point number. The remaining bits stand for the fractional part of the object. Hence each object is represented by a pair of parameters noted < wl, iwl >.

Simulations have been carried out with different parameter sizes. Table I shows the word sizes of the parameters considered for the fixed point simulations. We notice, that the quantization error of the energy distribution is about 1/2 7 , which is bigger that the considered maximum error = 5 × 10 -3 . Then the quantization error will mask the error brought by the approximation of the chi-squared distribution.

Table II

φ(x) = e -x 2 /2 √ 2π < 8, 2 > < 18, 0 > 4.5 g(x) = 1/ √ x < 14, 7 > < 10, 4 > 160 h(x) = 10 x < 6, 2 > < 14, 7 > 0.875
We notice that the total memory occupied by the lookup tables is lower than the chi-squared look-up table considered previously with respect to E n,m , B n,m and σ 2 . Although the simplifications defined in table II have led to reduce the amount of occupied memory, the inverse square root defined by g(x) could be determined iteratively by applying the Newton-Raphson method [START_REF] Muller | Arithmétique des ordinateurs[END_REF]. The inverse square root of a parameter a is obtained by the conver-gence of the suite (see Appendix I for details):

u n+1 = 1 2 u n 3 -au 2 n ( 15
)
where

lim x→+∞ u n = 1 √ a (16)
The choice of the initial value u 0 is very critical for the convergence of ( 15) in a minimum of iterations. Due to the difficulties to get an optimal generic initial value u 0 for a minimum iteration in hardware implementation, in what follows Table II is implemented instead of Newton-Raphson method. Simulations according to table I and II under the same conditions as for double precision lead to the results depicted in Figure 5. Results in fixed point precision data types are close to those obtained in double precision with chi-squared distribution. These results are promising for a hardware implementation with the given parameters size.

VI. Conclusion

One purpose of this paper is to approximate the chisquared distribution by a gaussian distribution with mean and variance as the chi-squared to facilitate the hardware implementation in IR-UWB energy detection system. It is shown that the error is marginal and offers the way to hardware simplification. By using different parameter quantizations, it has been proven that the Gaussian approximation can largely reduce the necessary memory and the whole complexity of the probabilistic energy equalizer, with very limited performance decrease. Further studies will be investigated for memory and processing optimization. For instance computing of probabilities could be done in logarithmic domain, or a polynomial approximation could be applied to replace some look-up tables. More research should be investigated on the initial value of Newton-Raphson method for digital design of the inverse square root, as it was studied in floating point precision [START_REF] Lomont | Fast inverse square root[END_REF].

Appendix I Newton-Raphson Method

Newton-Raphson method, also known as Newton's method, consists on computing the root that leads to f (x) = 0 by the mean of iterative process, where f (x) is a differentiable function defined on the interval [a, b] with values in the real numbers R. The following suite

u n+1 = u n - f (u n ) f (u n ) (17)
converges to the root. The process is initialized with an arbitrary value u 0 that is important for a fast convergence (see [START_REF] Muller | Arithmétique des ordinateurs[END_REF], [START_REF] Jordan | Calculus of Finite Differences[END_REF] for convergence proof and for more details). This procedure can be unstable near a horizontal asymptote or a local extremum. However, with a good initial choice of the root's position, the algorithm can be applied iteratively Newton-Raphson application to g(x) = 1/ √ x: The function g(x) is defined on the interval ]0, +∞[ with value on ]0, +∞[. Let α ∈]0, +∞[ such as y = 1/ √ α, we define the function f (x) by

f (x) = α + 1 x 2 (18)
where f (x) is differentiable on the interval ]0, +∞[ with value on R * + . It is noticed that 1/ √ α is a root of f (x) = 0. We note u n an approximation of the current root. A better approximation u n+1 can be derived from [START_REF] Mekki | Em channel estimation in a low-cost uwb receiver based on energy detection[END_REF] as

u n+1 = 1 2 u n 3 -αu 2 n (19)

Fig. 1 .

 1 Fig. 1. Transmitter and receiver design.

Figure 2 ( 5 Fig. 2 .

 252 Figure 2(a) shows the error measured by |p(E n,m |B n,m )p(E n,m |B n,m )| for E n,m ≥ 0, B n,m > 0 and σ 2 = 1.

Fig. 3 .

 3 Fig.3. Chi-squared vs Gaussian approximation in float precision for CM1 and CM2 using duo-binary turbo code at rate 1/2 with K = 2.

Fig. 5 .

 5 Fig. 5. chi-squared float precision versus the Gaussian approximation in fixed point precision for K = 2.

  lists the Input/Output size look-up table necessary for density computation.