Marko Herrmann
email: marko.herrmann@gmx.de

José E F Alfaya

Mauro L Lepore

Pablo E Penchaszadeh

Wolf E Arntz

Population structure

Keywords: bivalves, almeja amarilla, population dynamics, growth performance indices, Argentinean sandy beaches

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Introduction

The world"s open coastlines are dominated by almost 70 % of sandy beaches [START_REF] Bascom | Waves and Beaches, the Dynamics of theOceanSurface[END_REF][START_REF] Mclachlan | The Ecology of Sandy Shores[END_REF], at which 39 % of the global human population are living (within 100 km from the coast, CIESIN 2000). Exposed sandy beaches are one of the most dynamic environments, where sediment is being constantly moved by waves, winds, and tides [START_REF] Mclachlan | The Ecology of Sandy Shores[END_REF]. Previous investigations have demonstrated the influence of biological parameters (e.g. Defeo et al. 1992b;[START_REF] Defeo | Effects of human activities on long-term trends in sandy beach populations: the wedge clam Donax hanleyanus in Uruguay[END_REF][START_REF] Defeo | Experimental management of an exploited sandy beach bivalve population[END_REF], as well as humans (e.g. [START_REF] Defeo | Testing hypotheses on recruitment, growth, and mortality in exploited bivalves: An experimental perspective[END_REF][START_REF] Brazeiro | Effects of harvesting and density dependence on the demography of sandy beach populations: the yellow clam Mesodesma mactroides of Uruguay[END_REF], on the dynamics of populations and communities. However, physical parameters, mainly wave energy and particle size, are suggested as the main factors controlling the responses of the macrofaunal populations and communities [START_REF] Mclachlan | Adaptions of bivalves to different beach types[END_REF][START_REF] Mclachlan | Community structure on sandy beaches: patterns of richness and zonation in relation to tide range and latitude[END_REF].Rating systems for beach types (Short and Wright 1983) andbeach exposure [START_REF] Mclachlan | The definition of sandy beaches in relation to exposure: A simple rating system[END_REF]were used for a variety of previous studies investigating sandy beaches with contrasting morphodynamics (e.g. [START_REF] Souza | Zonation and seasonal variation of the intertidal macrofauna on a sandy beach of Parana State, Brazil[END_REF][START_REF] Calliari | Beach differentiation along the Rio Grande do Sul coastline (Soutern Brazil)[END_REF][START_REF] Gómez | Lifehistory of thesandhopperPseudorchestoideabrasiliensis (Amphipoda) in sandybeacheswithcontrasting morphodynamics[END_REF][START_REF] Ricciardi | Global patterns of macroinvertebrate biomass in marine intertidal communities[END_REF][START_REF] Jaramillo | Sandy beach macroinfauna from the coast of Ancud, Isla de Chiloe, Southern Chile[END_REF][START_REF] Nel | The effect of grain size on the burrowing of two Donax species[END_REF][START_REF] Veloso | Effect of morphodynamics on the spatial and temporal variation of macrofauna on three sandy beaches, Rio de Janeiro State, Brazil[END_REF][START_REF] De La Huz | The influence of sediment grain size on burrowing, growth and metabolism of Donax trunculus L. (Bivalvia: Donacidae)[END_REF]Defeo and Martínez2003;[START_REF] Laudien | Population structure, growth and production of the surf clam Donax serra (Bivalvia, Donacidae) on two Namibian sandy beaches[END_REF][START_REF] Brazeiro | Geomorphology induces life history changes in invertebrates of sandy beaches: the case of the mole crab Emerita analoga in Chile[END_REF][START_REF] Celentano | Habitat harshness and morphodynamics: life history traits of the mole crab Emerita brasiliensis in Uruguayan sandy beaches[END_REF]Delgado and Defeo 2007b;[START_REF] Schlacher | Physical impacts caused by off-read vehicles to sandy beaches: spatial quantification of car tracks on an Australian Barrier Island[END_REF]Thompson 2008, Herrmann et al. 2009b).Due to the ability to absorb wave energy, the sandy beach ecosystemis one of the most debitable types of dynamic coastlines. The wave energy is used in driving surf-zone water movement, which transports sand offshore during storms and moves it back onshore during calms. Sandy beaches are greatly influenced by the transport of sand not only driven by waves but also by wind in the backshore and dunes. Most of the global sandy beaches are naturally backed by dunes (before the human influence, e.g. constructions in coastal dunes) and interact with them by supplying or receiving immense volumes of sand. Thus, sandy beaches are extremely dynamic environments where both mass of sand and water, are always on the move [START_REF] Komar | Beach Processes and Sedimentation[END_REF][START_REF] Short | Handbook of Beach and Shoreface Morphodynamics[END_REF].

Furthermore, the sandy beach ecosystem is characterized by sediment grains, which define the pore space between sediment particles. This so-called interstitial system is important as a habitat for organisms and for the filtration of seawater.

The interstitial environment of sandy beaches is mainly controlled by the factors grain size, sorting, shape, packing, porosity, pore size and permeability [START_REF] Mclachlan | Theinsterstitial environment of sandy beaches[END_REF].The intertidal of sandy beaches is normally devoid of macroflora so that the microflora and detritus attain an important significance as a food resource for specimens living in this type of environment. The sandy beach flora is usually composed of benthic microalgae and surf-zone phytoplankton, both of which are regularly dominated by diatoms. Although surf clams are commonly the main primary consumers in soft bottom communities, and can contribute up to 95 % of the total biomass [START_REF] Mclachlan | Sand beach energetics: an ecosystem approach towards a high energy interface[END_REF][START_REF] Arntz | El Niño -Klimaexperiment der Natur: Die physikalischenUrsachenundbiologischenFolgen[END_REF][START_REF] Ieno | Spatial and temporal patterns in coastal macrobenthos of Samborombon Bay, Argentina: a case study of very low diversity[END_REF], sandy beaches are inhabited by most phyla of invertebrates as interstitial forms or as members of the macrofauna, or both.

The yellow clam Mesodesma mactroides 1 Reeve 2 , 1854 (Bivalvia: Mesodesmatidae), also known as "almeja amarilla" in Argentina and Uruguay, is a peculiar species [START_REF] Parodiz | Transgressiones oceánicas y fauna del Mar Epicontinental Argentino[END_REF][START_REF] Stuardo | Distribución de los moluscos marinos litorales en Latinoamérica[END_REF]. As one of the most common bivalves of South America [START_REF] Defeo | Development and management of artisanal fishery for yellow clam Mesodesma mactroides in Uruguay[END_REF]) it inhabits intertidal Atlantic sandy beaches from tropical (23°S Ilha Grande, Rio de Janeiro State, Brazil) to temperate latitudes (40°S Isla del Jabalí, Province of Buenos Aires, Argentina) [START_REF] Rios | Seashells of Brazil: 2[END_REF][START_REF] Fiori | Age and individual growth of Mesodesma mactroides (Bivalvia) in the southernmost range of its distribution[END_REF]. Thus, its distributional range covers about 1,800 km South Brazilian sandy beaches, 22 km in Uruguay, and 375 km in Argentina. A suspension-feeder, the yellow clam is the only Mesodesma species occurring throughout the Argentinean intertidal (Olivieret al. 1971). It co-occurs in the north of Argentina with the sympatric and numerically dominant surf clam Donax hanleyanus [START_REF] Herrmann | Population dynamics of the surf clams Donax hanleyanus and Mesodesma mactroides from open-Atlantic beaches off Argentina[END_REF]. During the 1940s and 1950s,M. mactroides was an important economic resource in Argentina.A maximum exploitation of 1,079 tonnes was reached in 1953, but the fishery was closed in 1956 following a stock crash [START_REF] Coscarón | La almeja amarilla (Mesodesma (T.) mactroidesDeshayes) de la costa de la Provincia de Buenos Aires[END_REF] from which the population never recovered.Abundances were decimated furtherby subsequent mass mortality events in March 1993 due to a red tide incident (Odebrechtet al. 1995), in the 1990s due to an unknown cause [START_REF] Fiori | Mass mortality of the yellow clam, Mesodesma mactroides (Bivalvia: Mactracea) in MonteHermoso beach, Argentina[END_REF] and most recently as a 1 M. mactroides reference specimens were deposited in the collections of the "Museum fürNaturkunde der Humboldt-Universitätzu Berlin" and the "MuseoArgentino de CienciasNaturales -Bernardino Rivadavia" underthe reference numbers ZMB/Moll.104643 and MACN-In37513, respectively. 2 In previous studies Deshayes 1854 was cited spuriously as the author who first described M. mactroides. However, this description was not published by Deshayes but by his colleague Reeve, thus the corresponding author of M. mactroides is Reeve 1854 (see Reeve 1854; Deshayes 1855) likely result of heavy metal pollution [START_REF] Thompson | Mortandad masiva de Mesodesma mactroides (Bivalvia: Mactracea) en el partido de la costa, Buenos Aires, Argentina, en septiembre[END_REF].

Illegal extractions and the impact of tourism may also be contributing factors in the failure of the species to make a recovery [START_REF] Bastida | Ⓐ indicates mass mortality of M. mactroides in South America[END_REF]. Although there are a number of recent studies on populations of M. mactroides from Uruguay (e.g., [START_REF] Defeo | Spatial and temporal structure of the yellow clam Mesodesma mactroides (Deshayes 1854) in Uruguay[END_REF]Defeo et al. 1988b;[START_REF] Defeo | Development and management of artisanal fishery for yellow clam Mesodesma mactroides in Uruguay[END_REF][START_REF] Defeo | Dinámica espacial del esfuerzo pesquero en una pesquería artesanal de la costa Atlántica Uruguaya[END_REF]Defeo et al. 1992c;[START_REF] Brazeiro | Effects of harvesting and density dependence on the demography of sandy beach populations: the yellow clam Mesodesma mactroides of Uruguay[END_REF][START_REF] Lima | Population dynamics of the yellow clam Mesodesma mactroides: recruitment variability, density-dependence and stochastic processes[END_REF], the last Argentinean study is 40 years old (Olivieret al. 1971). The aim of this study is to estimate contemporary population parameters and production and to make comparisons with data from the historical study. Additionally,the present study aims to examine the possible influence of physical and biological factors on the yellow clam growth and to compare observed population parameters with those of other members of the genus from elsewhere.

Material and methods

Study area

The present study was carried out at the three south Atlantic beaches Santa Teresita (36°32'S, 56°41'W), Mar de las Pampas (37°19'S, 57°00'W) and Faro Querandí (37°29'S, 71 57°07'W) (Fig. 1), whereby we want to underline that M. mactroides was insignificant abundant at the last two mentioned location during the entire sampling period. Consequently,the population structure, growth and production of the intertidal yellow clam were studied during alternate spring tides only at Santa Teresita. According to McLachlan`s (1980) rating scale for exposure and Short and Wright`s (1983) classifications of beach types, Santa Teresita can be classified as sheltered and dissipative (for detailed physical parameters see [START_REF] Herrmann | Population dynamics of the surf clams Donax hanleyanus and Mesodesma mactroides from open-Atlantic beaches off Argentina[END_REF]. This is an open ocean beach receiving continuous wave action and is subject to semidiurnal tides, with a maximum tide range of 1.6 m; spring tides average 1.7 m and mean neap tides 0.2 m. The sea surface temperature (SST) varies between 11°C in winter and 23°C in summer.

The study site is composed of good-moderate fine sand, with a mean particle diameter of 2.26 phi [START_REF] Herrmann | Population dynamics of the surf clams Donax hanleyanus and Mesodesma mactroides from open-Atlantic beaches off Argentina[END_REF]. The well drained and oxygenated sandy beach of Santa Teresita is affected by a freshwater seepage and a southward current which brings water masses from the 290 km long and up to 220 km wide estuary of the Río de la Plata and is thus characterized by a mean salinity of 31.

Sampling and laboratory procedures

Quantitative sampling of M. mactroides was carried out at monthly intervals from December 2004 to December 2006, using a series of stations located at 4 m intervals along a transect running perpendicular to the shoreline from the spring tide high water mark to the spring tide low water mark. At each station, three replicate sand samples (40 40 cm) were excavated to 50 cm depth using a stirring corer, and sieved individually on a 1 mm mesh. To obtain monthly lengthfrequency distributions, the maximum anterior-posterior shell length (apSL)of yellow clams retained on the mesh was measured to the nearest 0.1 mm using a digital vernier calliper (Mitutoyo, model 500-161U). For biomass analysis, monthly sub samples of 35 specimens were collected and stored in 70 % ethanol.

To estimate shell-free wet mass (SFWM), soft tissues were removed and weighed after blotting on filter paper in the laboratory. Samples were subsequently dried to constant mass at 70°C, weighed again to determine shell-free dry mass (SFDM) and ignited in a muffle furnace at 550°C for 5 h in order to estimate ash-free dry mass (AFDM).

Growth, growth performance, life span and mortality

A series of 25 length-frequency distributions provided the basic information for growth estimates. A von Bertalanffy growth function (VBGF) (von Bertalanffy 1938) was fitted to the data using the electronic length-frequency analysis "ELEFAN I" routine of the FISAT II program package (Gayanilo Jr.et al. 2005

): , ( 1
)
where L t is the apSL (mm) at time t, L ∞ the asymptotic maximum apSL (mm), K(yr -1 ) is the curvature parameter, C is a constant defining the degree of seasonal oscillation (ranging from 0 indicating continuous non-oscillating growth, to 1 when growth comes to a complete halt at the "winter point" (WP)), t 0 is the theoretical age at zero length (yr) and t s is the initial point of oscillation in relation to t = 0 and the WP. The different subroutines of ELEFAN I (K-scanning) were used to identify the VBGF that best fitted the monthly size-frequency data, using the R n value as a criterion of fit.

To measure growth performance, the growth index phi prime ( ) defined as:

,

was used. This criterion was chosen because the negative correlation between K and L ∞ invalidates comparisons based on individual parameters [START_REF] Pauly | Once more on the comparison of growth in fish and invertebrates[END_REF]. Finally the overall growth performance index ,

which represents growth rate at the point of inflexion of the size-growth curve [START_REF] Pauly | Gill size and temperature as governing factors in fish growth: a generalization of the von Bertalanffy"s growth formula[END_REF], was calculated to compare growth of M. mactroides with other surf clam species.

The theoretical life span (t max ) was estimated by an inverse of the von Bertalanffy growth equation, considering maximum shell length as 95% of the asymptotic length [START_REF] Taylor | Cod growth and temperature[END_REF]:

, ( 4
)
whereL 95% represents 95% of the maximum shell length recorded during field sampling.

Total mortality (Z) was calculated for 2005 and 2006 separatelyfrom length-converted catch curves [START_REF] Pauly | Comparison of age structure and length-converted catch curves of brown trout Salmotrutta in two French rivers[END_REF] yielded by the ELEFAN II routine of the FISAT program (Gayanilo Jr.et al. 2005). Z was estimated by: ,

whereN is the number of individuals, g the regression intercept, Z(yr -1 ) is the unbiased mortality estimate, and t is the estimated age (yr) for each cohort [START_REF] Pauly | Comparison of age structure and length-converted catch curves of brown trout Salmotrutta in two French rivers[END_REF].

Biomass and production

To

, ( 7
)
whereN i and M i are the average abundance of animals (number m -2 ) and mean individual AFDM in length class i, respectively, and G i is the mass-specific growth rate:

, ( 8
)
whereb is the exponent of the size-mass relation, K, L ∞ are VBGF parameters and L i is the mean size in class i.

Mean annual biomass was estimated as , [START_REF]mactroides C Argentina Faro Querandí 37[END_REF] and annual ratios of the M. mactroides population were calculated from annual total production P and mean annual biomass .

Statistical analysis

Temporal differences of the population abundance and mean OGPwere analysed statisticallyby one-way ANOVA using the software package SPSS version 17.0(2008). Differences were considered significant at a level of α = 5 % [START_REF] Zar | Biostatistical Analysis[END_REF].

Results

No

Biomass and production

The observed relationship between length and calculated AFDM of and 0.06 g AFDM m -2 yr -1 (2006) (Fig. 5b-c). Individual production increased to its highest value at47 mmlength (0.35 g AFDM m -2 yr -1 ) and decreased thereafter (Fig. 5a). The distribution of total annual production P and abundance among the size classes are illustrated in Fig. 5b-c. Annual production ranged between 0.12 and 0.19 g AFDM m -2 yr -1 , and ratios were between 1.84 and 2.93.

Discussion and conclusion

Population structure

Comparison ofhistorical data [START_REF] Olivier | Estructura de la Comunidad, Dinámica de la Población y Biología de la Almeja Amarilla (Mesodesma mactroides) en Mar Azul[END_REF]) with present situation, both diagrammed in Fig. 3, showed considerable differences in the population structure of M. mactroides.The number of individuals wasmany times higher in 1968 compared to 2005 (up to 53 times higher in December), suggesting a much smaller reproduction successnowadays during spring time, which represents one of the two spawning seasons per year (Herrmann et al. 2009a). Not merely the number of individuals can be registered as a great difference between the historical population and the recent one, but rather the bivalves in 1968 reached a much higher apSL. Studied individuals from present work never reached the "commercial size" of 60 mm [START_REF] Coscarón | La almeja amarilla (Mesodesma (T.) mactroidesDeshayes) de la costa de la Provincia de Buenos Aires[END_REF]; indeed only three individuals with apSL of > 60 mm were recorded in the entirety of the study period. Furthermore, the analysis of Fig. 3 showed up to three cohorts in the past population structurewhereas in the current study it was found only one single cohort of M. mactroides. However, since Herrmann et al. (2009b) reported variationsin the reproductive biology of the wedge clam D. hanleyanus, depending on different beach morphodynamics, it can`t be excluded that also the population structure of M. mactroides can be diversify between beaches with contrasting morphodynamics, for which the comparison between populations of M. mactroides from Mar Azul (historically) and Santa Teresita (presently) only can point out a tendency.

Growth and growth performance

The data presented here confirm that M. mactroidesis a fast-growing species (Table1, no. 1) obtained forM. mactroidesin the current studyconforms with the values of 5.22 (Olivieret al. 1971)and 5.30 [START_REF] Fiori | Age and individual growth of Mesodesma mactroides (Bivalvia) in the southernmost range of its distribution[END_REF]) calculated from data sets of previous studies on Argentinean populations of the same species and also with the value of 5.28 for a population ofM. donacium from the Peruvian sampled after an El Niño (EN) event [START_REF] Arntz | Changes in the structure of a shallow sandybeach community in Peru during an El Niño event[END_REF]. It can be assumed that SST and food availability are key factors affecting growth and aspects of population dynamics such as production, reproduction, recruitment and mortality.

The relatively highvalues of OGP from Uruguay (Fig. 6, no. 10-13) and Peru (before and during an EN event, Fig. 6, no. [14][15], suggest a negative correlation with latitude (Fig. 7: y = 6.77-0.04x, r = 0.94, p < 0.05, N = 12). This trend provides support to the latitudinal gradient hypothesis of marine invertebrates [START_REF] Defeo | Macroecology of population dynamics and life history traits of the mole crab Emerita brasiliensis in Atlantic sandy beaches of South America[END_REF], which predicts a positive correlation between body size and latitude: populations from subtropical beaches exhibited higher growth and mortality rates and shorter life spans than those of temperate beaches.

However, this observation would be better understood if growth studies on several populations could be linked to quantitative and qualitative data regarding phytoplankton. Results compiled from several studies of two different Mesodesma species confirmed the hypothesis of [START_REF] Laudien | Population structure, growth and production of the surf clam Donax serra (Bivalvia, Donacidae) on two Namibian sandy beaches[END_REF] that OGP is habitatspecific (Fig. 6). OGPis lowest (5.22-5.81, group A) for M mactroides populating temperate regions, and highest (6.33, group B) forM. donacium of an upwelling area. Mean OGPs of the two groups were significantly different (ANOVA, F 1,14 = 21.716, p < 0.05). Climate anomalies may be detected with the help of the auximetric grid, as indicated by the lower OGPfor the upwelling surf clam M. donacium sampled in Peru during and shortly after EN (1982/1983, Fig. 6, no. 15-16) compared to data from normal upwelling years (Fig. 6, no. 14) [START_REF] Arntz | Changes in the structure of a shallow sandybeach community in Peru during an El Niño event[END_REF].

Biomass and production

Annual intertidal biomass of the M. mactroides population at Santa Teresita ranged between 0.06 and 0.07 g AFDM m -2 yr -1 and production varied between 0.12 and 0.19 g AFDM m -2 yr -1 . Unfortunately comparable information on the species from elsewhere is very scarce. To the best of our knowledge, no study of this kind has been carried out in Brazil. [START_REF] Defeo | Aspectos biocenológicos y de dinámica de la población de "almeja amarilla", Mesodesma mactroides (Deshayes, 1854) en la zona[END_REF] (1971), in which individuals of commercial size (> 60 mm) were found throughout the year. The reason may be a change in environmental conditions, which has impacted directly on food supply. Biomass data ranging from 141 to 546 g AFDM m -2 yr -1 for the similar-sized surf clamD. serra, whichoccupies a similarecological position in the Benguela upwelling system (Laudienet al. 2003) lead us to hypothesise that environmental conditions off Argentina may have been more favourable with regard to major primary production four decades ago.

Indeed, changes in the environmental conditions of the Argentinean coast are implicated in a study of historically SSTs (Bavaet al. 1999). The authors reported that EN 1992 evoked negative SST anomalies throughout the south western Atlantic. However, a detailed understanding of the collapse of the Argentinean M. mactroides population would require analysis of further historical climate and phytoplankton data.

Life span and mortality

There is some controversy concerning longevity estimates forM. mactroides. The instantaneous mortality rate (Z) for the Argentinean M. mactroideswas higher at present study (2.57-3.01 yr -1 ) compared to calculations (0.85 yr -1 ) from Olivieret al. (1971) by but fits well with mortality rates for the same species from Uruguayan beaches (Defeo et al. 1992c: 2.26-2.91 yr -1 ). [START_REF] Arntz | Changes in the structure of a shallow sandybeach community in Peru during an El Niño event[END_REF] also found similarmortality rates (Z = 2.5 yr -1 ) for the closely related M. donacium inhabiting Peruvian sandy beaches. Found differences in mortality rates indicate that the Argentinean M. mactroidespopulation this day is much more stressed than forty years ago. Factors inducing this stress are possibly engendered by ecological (e.g. mass mortality events) and environmental aspects (e.g. changing beach morphodynamics by human impacts).

Ecological aspects

The carnivorous gastropod Olivancillaria vesica auricularia, a known predator of surf clamsin Brazilian sandy beaches [START_REF] Marcus | StudiesonOlividae. BolmFac Filos CiêncUniv São Paulo[END_REF][START_REF] Gianuca | The ecology of a sandy beach in Southern Brazil[END_REF];

Rocha-Barreira de Almeida 2002), also attacksM. mactroides and D. hanleyanus at Santa Teresita (V. S. Teso, pers. communication). Predator-prey relationship tests revealed escape behaviour in both clams, when approached by the snail. Surf clams sought refuge even without direct contact, excavating themselves within 10 seconds in order to escape with the next wave (M. Herrmann, unpublished data).

From previous investigations (e.g., [START_REF] Olivier | Estructura de la Comunidad, Dinámica de la Población y Biología de la Almeja Amarilla (Mesodesma mactroides) en Mar Azul[END_REF][START_REF] Penchaszadeh | Ecología de una población de 'berberecho' (Donax hanleyanus) en Villa Gesell, Argentina[END_REF][START_REF] Bastida | Estructura poblacional de la almeja amarilla (Mesodesma mactroides) durante el verano de 1989 en la Provincia de Buenos Aires, Argentina[END_REF] with the collapse of M. donacium populations [START_REF] Penchaszadeh | Estudios sobre el mejillón (Mytilusplatensisd'Orb.) en explotación comercial del sector bonaerense[END_REF][START_REF] Tarazona | Modificaciones producidas durante "El Niño" en la infauna bentónica de áreas someras del ecosistema de afloramiento peruano[END_REF][START_REF] Arntz | Changes in the structure of a shallow sandybeach community in Peru during an El Niño event[END_REF]). Defeo and de Alava (1995) hypothesized that interspecific competition for space and food might explain the relationship between D. hanleyanus and M. mactroides. [START_REF] Olivier | Estructura de la Comunidad, Dinámica de la Población y Biología de la Almeja Amarilla (Mesodesma mactroides) en Mar Azul[END_REF] supported this by describing overlapping trophic niches of the two species and predictedthat limited food availability might soon impact onM. mactroides. However, results in the present study suggest that the two surf clam species do not compete for resources at Santa Teresita. The fact that abundances of D. hanleyanus and M. mactroides populations were previously much higher than at present [START_REF] Olivier | Estructura de la Comunidad, Dinámica de la Población y Biología de la Almeja Amarilla (Mesodesma mactroides) en Mar Azul[END_REF][START_REF] Penchaszadeh | Ecología de una población de 'berberecho' (Donax hanleyanus) en Villa Gesell, Argentina[END_REF]) makes food availabilityan even less likely limiting factor in the current scenario.

This said, thebalance between these two species appears to have been disturbed following mass mortality events affecting M. mactroides in March 1993 (Odebrechtet al. 1995),November 1995[START_REF] Fiori | Mass mortality of the yellow clam, Mesodesma mactroides (Bivalvia: Mactracea) in MonteHermoso beach, Argentina[END_REF] and September 2004 (Thompson and Sánchez de Bock 2007) (Fig. 9).D. hanleyanus, which was not affected by the mass mortality events, took over immediately as the dominant species, but failed to reach the former abundance of M. mactroides.

Similar changes in the structure of Peruvian shallow sandy beach communities have been ascribed to EN. [START_REF] Arntz | Changes in the structure of a shallow sandybeach community in Peru during an El Niño event[END_REF]

Environmental aspects

Another hypothesis explaining the recent dominance of D. hanleyanus over M. mactroides is a change in granulometry. [START_REF] Penchaszadeh | Ecología de una población de 'berberecho' (Donax hanleyanus) en Villa Gesell, Argentina[END_REF] reported that sediment characteristics are principle factors in the distribution of surf clams. [START_REF] Olivier | Estructura de la Comunidad, Dinámica de la Población y Biología de la Almeja Amarilla (Mesodesma mactroides) en Mar Azul[END_REF] and December 2006 at Santa Teresita (black histograms) compared with historical data from [START_REF] Olivier | Estructura de la Comunidad, Dinámica de la Población y Biología de la Almeja Amarilla (Mesodesma mactroides) en Mar Azul[END_REF] sampled between 1968 and 1969 at Mar Azul (grey histograms). For comparing values see Table 1.

  compare biomass of M. mactroides with values for other species, all published values retrieved were converted to AFDM (according to Breyet al. 1988), whereby a conversion factor of 0.186 (SFWM to AFDM for ethanol [70 %] preserved M. mactroides) was empirically determined. Parameters of the relationship between length and ash-free dry mass (AFDM) of M. mactroides were estimated by regression analysis , (6) whereM is the AFDM (g), L is the shell length of the size class (mm), and a and b are constants. AFDM was determined for 400 specimens of all size classes sampled between December 2004 and December 2006. Total annual productivity was calculated for the M. mactroides belt at Santa Teresita (January-December 2005 and January-December 2006) using the mass specific growth rate method (Crisp 1984; Brey 2001) from size-mass relationships, the size-frequency distribution obtained from all pooled samples and the VBGF:

  M. mactroides populations were discovered south of Santa Teresita during the sampling period. Only a few single individuals were recorded at Mar de las Pampas (August 2005: 31 ind. m -2 ) and Faro Querandí (August 2005: 19 ind. m -2 ; March, April and October 2005: 6 ind. m -2 and February 2006: 6 ind. m -2 )meaning that population parameters and production could not be estimated for these two last mentioned areas.Population structureGreatest abundance was recorded in May 2005, with 110 ± 25 ind. m -2 (mean ± SE) and absolute abundance peaked at 543 ind. m -2 (size-class 2-4 mm) in April 2005(Fig.2). Distinct cohorts were apparent in the length-frequency distributions obtained(Fig.3). In both years, a single new cohort was observed during February. New cohorts were tracked for up to eleven months in which they reached apSLs of 27 mm in 2005 and approximately 35 mm in 2006 before disappearing, after which new recruits appeared.Growth, longevity and mortalityDuring the 25 month study period 3,015 M. mactroides were collected and measured at Santa Teresita; 2,142 ind. in the first year and 873 ind. in the second year. The smallest recorded individual (apSL 2 mm) was found in winter (September 2005) and the largestspecimen(apSL64 mm) was found in summer(February 2005). The estimated VBGF of the M. mactroides population at Santa Teresita with the growth parameter K = 0.47 (yr -1 ) and L ∞ = 85 mm (apSL) are shown in Fig. 4. The corresponding index (3.53) and OGP (5.46) are compared with those of Mesodesma growth parameters, calculated from different areas. Parameter C = 0.10 indicated weak seasonal variation in growth, with lowest growth rates occurring in winter (WP = 0.7 ~ mid-August). The calculated life span (t max ) ofM. mactroides was 6.19 years and the instantaneous mortality rate (Z) was 3.01 yr -1 in 2005 and 2.57 yr -1 in 2006.

M

  . mactroidesM (AFDM) =(9.44•10 -6 )•L (mm)2.83 (N = 606, r 2 = 0.98) was used for production estimates.Annual abundance of M. mactroides(mean ± SE) was 48 ± 13 ind. m -2 in 2005 and 29 ± 3 ind. m -2 in 2006. Within the Mesodesma belt these abundance values represent an average annual biomass of 0.07 g AFDM m -2 yr -1(2005)

(

  K = 0.47; = 3.4), and fall in line with estimates from previous studies of Argentinean and Uruguayan populations (K = 0.3-0.9; = 3. Table1). However, changes in growth parameters are evidentwhen the recent values (Table1, no. 1) are compared with those ofOlivier et al. (1971) (Table1, no. 9). In 1968-70 M. mactroides showed a lower growth constant (K = 0.28 versus 0.47), indicating that yellow clam growth was slower four decades ago, but that eventual maximum length was almost even compared to the present (L t(1968-70) = 84 mm; L t(2005-06) = 85 mm). Our values are in conformity withestimated maximum shell length ofM. mactroides from previous studies in Argentina[START_REF] Olivier | Estructura de la Comunidad, Dinámica de la Población y Biología de la Almeja Amarilla (Mesodesma mactroides) en Mar Azul[END_REF] 84 mm) and Uruguay(Defeoet al. 1992a: 100 mm).WhileDefeo et al.(1992c)observed strong seasonal variation (C = 1.0) in growth rates for aM. mactroides populationin Uruguay, the variation documented by[START_REF] Fiori | Age and individual growth of Mesodesma mactroides (Bivalvia) in the southernmost range of its distribution[END_REF] was low (C = 0.45), and[START_REF] Olivier | Estructura de la Comunidad, Dinámica de la Población y Biología de la Almeja Amarilla (Mesodesma mactroides) en Mar Azul[END_REF] recorded no seasonal variations for this species from the Argentinean coast. In support of these latter findings, the present study estimated C = 0.1. Non-linear growth functions such as the VBGF are difficult to compare, and several authors (e.g.,[START_REF] Pauly | Gill size and temperature as governing factors in fish growth: a generalization of the von Bertalanffy"s growth formula[END_REF][START_REF] Munro | A simple method for comparing the growth of fishes and invertebrates[END_REF][START_REF] Moreau | Indices of overall growth performance of 100 tilapia (Cichlidae) populations[END_REF][START_REF] Laudien | Population structure, growth and production of the surf clam Donax serra (Bivalvia, Donacidae) on two Namibian sandy beaches[END_REF][START_REF] Herrmann | Population dynamics of the surf clams Donax hanleyanus and Mesodesma mactroides from open-Atlantic beaches off Argentina[END_REF]) have demonstrated the suitability of composite indices of overall growth performance (OGP) for inter-and intraspecific comparisons forvarious clam species. OGP is proportional to the maximum rate of body mass increase during a lifetime, i.e. mass increase at the inflexion point of the VBGF. Few values of maximum body mass can be found in the literature and maximal mass is proportional to L ∞ . TheOGPvalue of5.46

  estimated a biomass of 133 g AFDM m -2 (converted from SFWM values published) for the Uruguayan M. mactroides population, while four decades ago the biomass of Argentinean M. mactroides was estimated at 323 g AFDM m -2(Olivieret al. 1971, calculated from published SFWM data). Thus, the once-prominent role of this primary consumer in the ecosystem appears to be reduced. The individuals observed during the present study were small compared to the recordsof Olivieret al.

  [START_REF] Olivier | Estructura de la Comunidad, Dinámica de la Población y Biología de la Almeja Amarilla (Mesodesma mactroides) en Mar Azul[END_REF] estimated a maximum age of ~8 years for the Argentinean yellow clam population, using length-frequency analysis. In contrast,Defeo et al. (1988a) suggested a life span of ~3.5 years for Uruguayan Mesodesma populations, derived by the interpretation of shell growth rings. Length-frequency analysis of the current data set yields a life span of ~6 years, anintermediate estimate within the range of the aforementioned extremes.

  it is well-known that intertidal sandy beach communities of northern Argentina were inhabited by large co-existing populations of the surf clams M. mactroides and D. hanleyanus. However, the significant negative correlation between the abundances of D. hanleyanus(data taken from Herrmann 2009) and M. mactroides (Fig. 8) at Santa Teresita in 2005 suggests that the timing of abundance peaks of D. hanleyanusmay be related to periods of low M. mactroides abundance. Penchaszadeh and Olivier (1975), Dadon (2005) and Defeo and de Alava (1995) observed similar inverse correlations between D. hanleyanus and M. mactroides abundance on Argentinean and Uruguayan beaches. Furthermore, surf clam populations from the Peruvian Pacific coast also showed the same relationship, and abundances of Emerita analoga and D. marincovichi (formerly D. peruvianus) increased coincidentally

AppendixFig. 1 :

 1 Fig. 1: Map of the three study sites at the Argentinean Atlantic coast: Santa Teresita (S), Mar de las Pampas (M) and Faro Querandí (F). •indicates the southernmost distribution of M. mactroides at Isla del Jabalí. Dotted areas indicate northern and southern sandy beaches of the Province of Buenos Aires where major M. mactroides populations existed in 1953-1956 (Coscarón 1959).

Fig. 2 :

 2 Fig. 2: Population abundance of M. mactroides from December 2004 to December 2006 at Santa Teresita. * M. mactroides was not abundant during sampling in January 2006. Error bars: standard error (SE).

Fig. 3 :

 3 Fig. 3: Monthly length-frequency distribution of M. mactroides collected between December 2004

Fig. 4 :Fig. 5 :

 45 Fig. 4:The VBGF (grey lines) of M. mactroideswas estimated with the FISAT program from monthly length-frequency data (black histograms) for the period December 2004 to December 2006.

Fig. 6 :

 6 Fig. 6: Auximetric grid comparing the OGP of the Argentinean M. mactroides from the present study (❶)with several Mesodesma populations from different areas (white circles). Plot indicates two groups: temperate (A) and upwelling species (B), indicated by dashed contours. Dotted diagonal lines show equal values of OGP (numbers in dotted circles). For keys and data sources see Table1.

Fig. 7 :

 7 Fig. 7:OGPof M. mactroides from present study (❶) estimated from length-frequency data and obtained from ELEFAN,compared with several Mesodesma-populations from different latitudes.For comparing values see Table 1.

Fig. 8 :

 8 Fig. 8:M. mactroides -D. hanleyanus linear correlation of abundances (Log ind. m -2 ) from 2005 at Santa Teresita.

Fig. 9 :

 9 Fig. 9: Population trendsin the Argentinean surf clam M. mactroides: mean abundances from the

Acknowledgements

This work is part of the doctor thesis of Marko Herrmann, partly supported by "DeutscherAkademischerAustauschdienst" (DAAD) and by the University of Bremen. The authors thank Sonia Cabrera, Maria Eugenia Segade Enrique, Juan Pablo Livore, Soledad Zabalaand my wife Sandra Noir for field assistance.

Thanks are also expressed to all Argentinean colleagues from the laboratory 19 at the "Facultad de Ciencias Exactas y Naturales" (UBA) and from the laboratory 80 at the "MuseoArgentino de CienciasNaturales -Bernardino Rivadavia" (MACN).

Finally we thank Amy-Jane Beer for English editing.


Jean-François Bercher
email: jf.bercher@esiee.fr

Corinne Berland
email: c.berland@esiee.fr

J.-F Bercher

J.-F Bercher

Envelope and Phase delays correction in an EER radio architecture

come

Envelope and phase delays correction in an EER radio architecture

Introduction

Recent radiocommunication systems aiming at high data rate are based on efficient modulation schemes in which Quadrature Amplitude Modulations (QAM) are obviously preferred to frequency or phase modulation. While the 3GPP standard employs QPSK and 16QAM, higher data rate are achieved (in WLAN system for example) using Orthogonal Frequency Data Multiplexing (OFDM) modulation. Although these modulations are suitable for signal processing, the realization of the RF front end, particularly the transmitter, becomes more and more complex. In the conception of a transmitter, it is essential to achieve both efficiency and linearity. In radiocommunication standards, the quality of the transmitted signal is well defined, usually in terms of output spectrum, Adjacent Channel Power Ratio (ACPR) and Error Vector Magnitude (EVM). When the modulated signal presents an envelope variation, similar to an amplitude modulated signal, the compression effect of the Power Amplifier (PA) generates intermodulation products which directly impact the three mentioned figures of merit. This implies that with a classic class A PA, the input signal would have to present a mean power (depending on the type of modulation) lower than the input compression point: the difference is quantified in terms of back off. In order to gain in efficiency, a linearization system is often preferred to a linear amplification. In this paper, we deal with the Envelope Elimination and Restoration (EER) principle, and illustrate it in the case of a 16QAM 64 carriers OFDM modulation. The EER principle was proposed by Kahn in 1952 [START_REF] Kahn | Single-sideband transmission by envelope elimination and restoration[END_REF] and is based on the splitting of a modulated signal into two signals. The first one is a constant envelope phase modulated signal, while the second one is the envelope of the original signal. In the original EER transmitter, the splitting is realized in an analog way using a limiter and an envelope detector. The phase modulated signal is the input of an efficient PA whereas the envelope signal is sent to a switching power supply which feeds the last stage of the PA. The transmitter now evoluates toward a fully digital transmitter [START_REF] Staszewski | Alldigital PLL and transmitter for mobile phones[END_REF] as shown in Fig. 1. The digital creation of the two signals is achieved using a CORDIC processor [START_REF] Sarigeorgidis | Ultra low power cordic processor for wireless communication algorithms[END_REF]. The principle of this processor relies on the rotation of a vector (X i ,Y i ) to a new vector. In our case, the final vector is (Z i , θ ), where Z i is the magnitude of the vector and the rotation angle is our phase. VLSI implementation can profit from of a suitable iterative formulation. The phase signal is sent to the PA amplifier using a modulated PLL. This solution is preferred to a I/Q RF modulator in order to reduce transmitted spurious emissions. To limit distortion, the envelope signal is usually amplified with a class S modulator using a Σ ∆ modulator rather than a classic PWM. The final amplification is realized by a high efficiency amplifier. As the input signal is a constant envelope signal, switched class of power amplifiers are preferred due to their high efficiency. As demonstrated in [START_REF] Raab | Idealized operation of the class e tuned power amplifier[END_REF], the class E power amplifier is very suitable for this application since the expression of the output voltage is directly proportionnal to its power supply. This property is mandatory as we intend to reinject the envelope variation through the modulation of its power supply. However the main drawback of this architecture is its sensitivity to delay mismatch between the two signals.

The delays introduced by the two paths can be mismatched due to pipeline differences in the paths and the delay in the anti-alias filter (amplitude path), as well as small contributions from other analog delays [START_REF] Sander | Polar modulator for multi-mode cell phones[END_REF]. Furthermore, in a production environment, delays should be matched to variations in process including supply voltage, frequency, output power and temperature [START_REF] Morgan | Highly integrated transceiver enables highvolume production of GSM/EDGE handsets[END_REF]. This usually requires careful factory calibration procedures. The mismatch deteriorates both the EVM and the output spectrum of the transmitted signal [4; 24] and has to be corrected. A linear interpolation was suggested in [START_REF] Jau | Linear interpolation scheme for compensation of path delay difference in an envelope elimination and restoration transmitter[END_REF] to compensate for the mismatch, while a group delay equalizer was proposed in [START_REF] Mártires | Differential delay equalization in a Kahn EER transmitter[END_REF]. In these two cases, it still remains to identify the delay mismatch and track its possible variations. In this article, we propose a scheme and an associated algorithm that covers the whole calibration problem: identification, correction and tracking.

In the first part, we demonstrate the sensitivity of EER applied to a 16QAM 64 carriers OFDM modulation and bring forward the maximum tolerable delay mismatch for this modulation. The second part presents an efficient algorithm which corrects this default. We then focus on the implementation of the algorithm and on the importance of the interpolation filter used to resynchronize the signals. In the final part, an analysis of the behaviour and performances of the algorithm is provided. Simulation results performed on HPADS are presented and show the performances achieved with this solution in terms of output spectrum, EVM and ACPR.

Impact of delay mismatches on an OFDM modulation

Using a 16QAM 64 carriers OFDM modulation is an interesting case study for the validation of this kind of architecture because of its high Peak to Average Power Ratio (PAPR). In fact, for an OFDM modulation, the PAPR is equal to the number of carriers, which corresponds to 10 log(64) = 18dB in our simulation case. A few results have already been presented on the study of the whole transmitter with this modulation [START_REF] Baudoin | Influence of time and processing mismatches between phase and envelope signals in linearization systems using envelope elimination and restoration, application to hiperlan2[END_REF] and this paper indicates that the critical specification is the synchronization of signals. Fig. 2 and Fig. 3 present the impact of delay mismatch in the range of ± T 3 , with T the time symbol, on the output spectrum and on the EVM. Table 1 gives the values of ACPR for these delays. The symbol rate is 20MHz. It is important to notice that the EVM calculation for OFDM modulations must be realized on each carrier separately and then averaged. Fig. 4 shows that the effect of the desynchronization on the first subcarrier for delays T /27 and T /5 acts as additional noise. As a guideline, we take specifications from 802.11.a standard: the rms EVM is specified at a maximum of 11.22% and the output spectrum has to remain under -40dBc above a 30MHz frequency offset from the carrier measured in a 1MHz resolution bandwidth. When analysing the EVM results, we can observe that a desynchronization of T /3 produces an EVM value of 14% and when considering the output spectrum, in 1MHz bandwidth, the value at 30MHz from the carrier is about -7.5dBc. In fact, the value of -40dBc is only obtained for a delay of T /27 which gives an EVM value of about 0.6% This demonstrates that the synchronization of signals impacts so strongly on the output spectrum that it is mandatory to implement a correction algorithm.

The correction algorithm

This paper is an extended version of [START_REF] Bercher | Envelope/phase delays correction in an EER radio architecture[END_REF] where we presented a preliminary version of the algorithm. In this version, the algorithm is presented in more details and implementation issues are discussed. More specifically, we discuss the role, performances and implementation of the interpolation step and provide a detailed analysis of the algorithm performances.

Envelope and phase alignment

According to the central limit theorem, for complex modulation schemes such as the OFDM, when the number of subcarriers is large, the emitted signal can be approximated as a gaussian distributed complex random variable. For a narrowband stationary signal written as

ξ (t) = x(t) cos(ω 0 t) + y(t) sin(ω 0 t) (1)
it is well known that R xx (τ) = R xy (τ) and that R xy (τ) = -R yx (τ), where R xx and R xy are respectively the autocorrelation and intercorrelation functions. The processes x(t) and y(t) are always uncorrelated at the same instant, that is R xy (0) = 0. If ξ (t) is normal, then x(t) and y(t) are independent at the same instant. The complex gaussian process ξ (t) is completely characterized by its mean and autocorrelation function

R ξ ξ (τ) = R xx (τ) cos(ω 0 τ) + R xy (τ) sin(ω 0 τ). (2)
Let us denote S ξ ξ ( f ) and S xx ( f ) the spectra of ξ (t) and x(t), that is the Fourier transforms of the autocorrelation functions R ξ ξ (τ) and R xx (τ). When S ξ ξ ( f ) is symmetric with central frequency f 0 , the in-phase and quadrature components x(t) and y(t) are uncorrelated, that is R xy (τ) = 0. The baseband spectrum S xx ( f ) is generally proportional to the square of the transfer function of the emission filter, which shall be shaped as a square-root Nyquist filter. Consequently, the autocorrelation function R xx (τ) = R yy (τ), which is the inverse Fourier transform of the baseband spectrum, is the impulse response h of a (full) Nyquist filter. The process ξ (t) can also be written as

ξ (t) = ρ(t) cos (ω 0 t -φ (t)) (3)
where ρ(t) is the envelope and φ (t) is the phase process, with

x(t) = ρ(t) cos(φ (t)), y(t) = ρ(t) sin(φ (t)).
In the case of a complex gaussian process, it is well known that the envelope and phase are independent at the same instant and respectively distributed according to the Rayleigh and uniform distributions. The case of delayed envelope and phase is less known. In fact, it appears that for gaussian processes, the envelope ρ(t) and phase φ (t -∆ ) are also Rayleigh and uniform distributed, and are always independent (see [START_REF] Bercher | Monte-carlo estimation of time mismatch effect in an OFDM EER architecture[END_REF]) with no reference to the correlation coefficient, whatever the delay between envelope and phase components. In consequence, the output do not convey any information on the time alignment or mismatch between the envelope and phase components. As a result, it is not possible to correct the relative delay between the envelope and phase components from the sole observation of the system output. Calibration then needs to rely on a feedback loop and involve a direct comparison between the initial "aligned" signal and the observed signal.

In a calibration step without data transmission, we can also modify the modulation scheme and design non-gaussian sequences with some dependence between envelope and phase. In such a situation, a 'contrast' based on the output properties may be devised in order to align the components. However, we will focus here on the feedback solution that preserves the data and modulation technique.

The compensation algorithm

Let us denote z(t) the output of the system. Due to the delays ∆ 1 and ∆ 2 that affect the envelope and phase components, we have

z(t) = ρ(t -∆ 1 ) cos (ω 0 t -φ (t -∆ 2 )) = ρ(t -∆ 1 ) cos (φ (t -∆ 2 )) cos (ω 0 t) +ρ(t -∆ 1 ) sin (φ (t -∆ 2 )) sin (ω 0 t).
We propose here to correct the delays using an adaptive precompensation. The synchronization algorithm relies on the idea of introducing two advances µ 1 and µ 2 in order to prec- ompensate the delays, as illustrated in Fig. 5. In such a case, the output becomes

z(t) = ρ(t + µ 1 -∆ 1 ) cos (ω 0 t -φ (t + µ 2 -∆ 2 )) (4)
and we will adjust the advances µ 1 and µ 2 in order to min- imize a statistical distance between z(t) and ξ (t). A natural criterion is the minimization of the quadratic distance

J(µ 1 , µ 2 ;t) = E |ξ (t) -z(t)| 2 , (5)
where E [•] is the statistical expectation operator. This expression can also be rewritten as

J(µ 1 , µ 2 ;t) = E e x (t) 2 cos(ω 0 t) 2 + E e y (t) 2 sin(ω 0 t) 2 +E[e x (t)e y (t)]sin(2ω 0 t)
where e x (t) and e y (t) are the errors for the in-phase and quadrature components respectively,

e x (t) = (ρ(t) cos φ (t) -ρ(t 1 ) cos φ (t 2 )) (6a) e y (t) = (ρ(t) sin φ (t) -ρ(t 1 ) sin φ (t 2 )) (6b)
and where, in order to simplify the expressions, we noted

t 1 = t + µ 1 -∆ 1 and t 2 = t + µ 2 -∆ 2 . (7)
After time averaging, this simply reduces to

J(µ 1 , µ 2 ) = 1 2 J x (µ 1 , µ 2 ) + J y (µ 1 , µ 2 ) (8)
with

J x (µ 1 , µ 2 ) = E e x (t) 2 and J y (µ 1 , µ 2 ) = E e y (t) 2 . (9)
The global criterion equals the sum of two elementary criteria on the in-phase and quadrature components. We can readily obtain the same criterion (up to a factor) using the demodulated, baseband, version of the signal.

In practice, we indeed work in baseband with digital signals.

After sampling, we compare the digital input signal to the sampled baseband output. In the following, we will keep T for the symbol period and note T s the sampling period. Consequently, we will note the different discrete time indexes as follow

   t(n) = nT s t 1 (n) = nT s + µ 1 -∆ 1 t 2 (n) = nT s + µ 2 -∆ 2 . ( 10
)
The output sampling clock does not need to be synchronous to the input: it may be a divided version of the input clock, and any propagation delay will be absorbed in the correction procedure.

Since it is simpler to generate delayed signals than advanced signals, we introduce a small processing delay D in Fig. 5.

ρ(t+µ 1 -∆ 1 ) φ(t+µ 2 -∆ 2 ) RF Transmitter ρ(t+µ 1 ) φ(t+µ 2 ) ρ(t)cos(φ(t)) ρ(t+µ 1 -∆ 1 )cos(φ(t+µ 2 -∆ 2 )) + - ρ(n) φ(n) µ 1 ,µ 2 z -D ρ(t)sin(φ(t)) ρ(t+µ 1 -∆ 1 )sin(φ(t+µ 2 -∆ 2 ))
Fig. 5 Principle of delays correction.

Let us consider the criterion J x (µ 1 , µ 2 ) in ( 9) on the in-phase component. Developing and taking into account the independence between ρ(t 1 ) and φ (t 2 ) lead to

J x (µ 1 , µ 2 ) = 2C x (0, 0) -2C x (µ 1 -∆ 1 , µ 2 -∆ 2 ) (11)
where

C x (τ 1 , τ 2 ) = E [ρ(t) cos(φ (t))ρ(t -τ 1 ) cos(φ (t -τ 2 ))] (12) with τ = 1 = µ 1 -∆ 1 and τ 2 = µ 2 -∆ 2 , is a kind of 'correla- tion function'. The term C x (0, 0) simply reduces to C x (0, 0) = E ρ(t) 2 E cos(φ (t)) 2 = R xx (0), the variance of the in-phase x(t) component. It is important to note that C x (τ, τ), ob- tained with τ 1 = τ 2 = τ is nothing else but the correlation function R xx (τ).
Since we know that the correlation function is proportional to the shaping filter h, it appears that the behaviour of the criterion J x (µ 1 , µ 2 ) is closely related to the shaping filter. Regarding the quadrature component and criterion J y (µ 1 , µ 2 ), the same conclusions and formulas found in [START_REF] Jau | Linear interpolation scheme for compensation of path delay difference in an envelope elimination and restoration transmitter[END_REF]12) are easily obtained by substituting x by y and cos by sin. In the case of an OFDM modulation, the criterion J(τ 1 , τ 2 ) was evaluated numerically by Monte-Carlo simulations with a square root Nyquist filter (square root raised cosine with 0.5 roll-off). This is illustrated in Fig. 6 for delays (advances) between -4T and 4T . We can recognize here the general shape of the (inverted) impulse response of a raised cosine, somewhat distorded and modulated. The criterion does not only present a global minimum at τ 1 = 0, τ 2 = 0, but also several other minima. Derivation of a closed-form formula for criteria (9) involving ( 12) is a challenging if not impossible task. However, in Fig. 7, where the criterion for delays less than 1.5T is shown, we clearly observe that any descent algorithm will avoid local minima for delays ∆ 1 , ∆ 2 ≤ T , with initial conditions set to zero.

Gradient algorithm

Since we do not have a closed-form for the criterion nor a direct explicit solution for its global minimizer, we need to exhibit the solution using a descent algorithm. We simply use a gradient algorithm that consists in iterating the following formulas:

     µ 1 (n + 1) = µ 1 (n) -γ 1 (n) ∂ J(µ 1 ,µ 2 ) ∂ µ 1 µ 1 =µ 1 (n) µ 2 (n + 1) = µ 2 (n) -γ 2 (n) ∂ J(µ 1 ,µ 2 ) ∂ µ 2 µ 2 =µ 2 (n) (13)
where γ 1 (n) and γ 2 (n) are two adaptation steps that possibly depend on the iteration index n. The gradients are given by

∂ J(µ 1 , µ 2 ) ∂ • = 1 2 ∂ J x (µ 1 , µ 2 ) ∂ • + ∂ J y (µ 1 , µ 2 ) ∂ • (14)
and we readily obtain

∂ J x (µ 1 , µ 2 ) ∂ µ 1 = -E dρ(u) du u=t 1 (n) cos φ (t 2 (n))e x (t(n)) (15a) ∂ J y (µ 1 , µ 2 ) ∂ µ 1 = -E dρ(u) du u=t 1 (n) sin φ (t 2 (n))e y (t(n)) (15b)
and

∂ J x (µ 1 , µ 2 ) ∂ µ 2 = -E d cos φ (u) du u=t 2 (n) ρ(t 1 (n))e x (t(n)) (16a) ∂ J y (µ 1 , µ 2 ) ∂ µ 2 = -E d sin φ (u) du u=t 2 (n) ρ(t 1 (n))e y (t(n)) (16b)
with e x and e y defined by (6a,6b). The update equations are obtained using these gradients in [START_REF] Johansson | On the design of adjustable fractional delay FIR filters[END_REF]. However, we do not have the analytical expressions of the statistical expectations involved in these formulas. Therefore, we have to resort to using a stochastic approximation of these theoretical recursions. A popular solution in adaptive filtering is the Least Mean Squares (LMS) algorithm that simply consists in omitting the statistical expectation. The LMS then involves the instantaneous gradient rather than the (correct) statistical average. Furthermore, the equations are updated at each new sample. This gives

µ 1 (n + 1) = µ 1 (n) + γ 1 (n) dρ(u) du t 1 (n) × cos (φ (t 2 (n))) e x (t) + sinφ (t 2 (n))e y (t(n)) (17)
µ 2 (n + 1) = µ 2 (n) + γ 2 (n)ρ(t 1 (n))× d cos φ (u) du t 2 (n) e x (t) + d sin φ (u) du t 2 (n) e y (t(n)) . ( 18
)
Practical implementation of formulas ( 17) and ( 18) requires -computation of the errors e x (t(n)) and e y (t(n)) defined in (6a) and (6b) which results from the comparison of the system input and output, -computation of the derivatives that can be simply approximated by finite differences of ρ(t 1 (n)) and cos (φ (t 2 (n))).

Of course, the algorithm can be simplified by considering a sole error component rather than two. For instance, one can simply put e y = 0 in the previous equations. In practice, simulations show that the gain associated with the second component is extremely small.

For formulas [START_REF] Mártires | Differential delay equalization in a Kahn EER transmitter[END_REF] and [START_REF] Morgan | Highly integrated transceiver enables highvolume production of GSM/EDGE handsets[END_REF], the computational load is about 8 real multiplications per iteration. However, since the derivatives must be computed at time t 1 (n) and t 2 (n), that is at the output of the corrected system, ρ(t) and cos (φ (t)) must be separately accessible. This implies a quadrature demodulation before the feedback loop. Furthermore, with this approach, we need to adjust two advances µ 1 , µ 2 and apply them to the input signal. Adopting very high sampling frequencies in order to get the required precision may not be an efficient solution. Digital interpolation is a more effective solution as it keeps reasonable sampling frequencies and save consumption.

Impact of the interpolator

Contrary to what we indicated in [START_REF] Bercher | Envelope/phase delays correction in an EER radio architecture[END_REF], the interpolation procedure has a significant impact on the performances of the algorithm in terms of the EVM and output spectrum.

According to the Shannon-Nyquist sampling theorem, we know [29] that any band-limited signal x(t) can be recovered exactly from its samples x(m) = x(mT s ) taken at the sampling frequency 1/T s by the formula

x(t) = ∑ m x(m)sinc(π(t -mT s )/T s ), (19)
where sinc is the cardinal sine. This indicates that, in principle, the samples convey enough information to reconstruct the original signal at any desired time. In particular, it is possible to reconstruct x(t -τ), for any τ, and therefore new shifted samples x(kT s -τ) from the original samples x(kT s ), according to

x(kT s -τ) = ∑ m x(mT s )sinc(π(kT s -τ -mT s )/T s ). ( 20
)
The above expression is in the form of a digital convolution and can be implemented as a filtering operation. However, because the underlying filter has an infinite (sinc) impulse response, and is non causal, practical implementation introduces truncation and delay. Another possibility is to use a convenient approximation of the ideal interpolator mentioned above. The MMSE FIR interpolator [START_REF] Oetken | A new approach for the design of digital interpolating filters[END_REF] is the minimum mean square error approximation of the ideal filter with finite impulse response. Although optimum implementations of these interpolators exist [START_REF] Crochiere | Optimum fir digital filter implementations for decimation, interpolation, and narrow-band filtering[END_REF], the coefficients shall be pre-computed and tabulated for each possible fractional delay and these types of structures should be reserved for the interpolation with fixed delays.

Another class of interpolators relying on polynomial approximation can be used instead. Indeed, the Weierstrass approximation theorem states that every continuous function defined on an interval can be uniformly approximated as closely as desired by a polynomial function. We can then use a polynomial to approximate the value of the function, given a series of samples, at the desired delay. Such interpolators are especially interesting for our application since they can be described by FIR filters. They can be implemented very efficiently in hardware, and their coefficients can be computed in real time rather than taken from a table. An efficient structure was devised by Farrow [8; 15] and improvements to the structure can be found in [25; 7; 5]. In this structure, the delay is directly adjustable without modification so that it is suitable for our adaptive synchronization problem.

A related problem is Sample Rate Conversion (SRC) which is often considered in digital front ends [10; 9]. In SRC, a digital signal has to be converted into another digital signal but with a different sampling frequency. Caution must be exercised to avoid aliasing in the operation. In this situation, polynomial interpolators are usually disqualified because they do not provide enough anti-aliasing. In the last decade, following [START_REF] Vesma | Optimization and efficient implementation of FIR filters with adjustable fractional delay[END_REF], many solutions have been developed for the synthesis of adjustable fractional delay filters with larger bands and better anti-aliasing capabilities [26; 13; 12; 30; 32]. It is worth mentioning that adjustable fractional delay filters can also be obtained using programmable allpass Infinite Impulse Response (IIR) filters [16; 31]. However, such filters are more sensitive to quantization, transients may occur when changing coefficients, and synthesis is complicated by the stability issues.

In our application, interpolation operates directly on the digital input signal, without any rate change. Aliasing can still occur due to the sampling operation of the output, which is needed for our feedback loop. However, the system specifications, in particular the power limitation in adjacent channels, severely constrain the design and limit the images of the original band-limited spectrum. Furthermore, since the over-sampling ratio T /T s is typically greater than 5, an antialiasing filter can be easily designed.

The over-sampling ratio being high is an important factor since it allows the use of very low-order interpolators. Indeed, the frequency response of the corresponding filters is almost flat in magnitude and linear in phase in the band of interest. This is illustrated in section 4.1. Then, in section 4.2, we examine and compare the performances of the different interpolators in terms of interpolation error and EVM at the output of the transmitter.

The interpolators and their frequency responses

In this section, we choose to compare four interpolators [1, Chapter 25]: Linear, Bessel, third and fifth order Lagrange interpolators. We first give the expressions of these interpolators and then compare their frequency responses. The first interpolator, the forward linear interpolator, is the simplest, and is given by:

x(m, τ) = x(m) + τ (x(m + 1) -x(m)) (21)
where x(m, τ) represents the interpolated value of the input value at the time (m + τ)T s .

The second interpolator studied is the third order Lagrange interpolator:

x(m, τ) = x(m) + τ (x(m) -x(m -1)) + τ(τ+1) 2 (x(m + 1) -2x(m) + x(m -1)). ( 22
)
This interpolation uses three successive points. The first part of the expression is similar to the linear interpolator and the second part is a correction term calculated using the point before and after the central one.

The third interpolator we looked at is the Bessel central difference interpolator, fourth order, described as follow:

x(m, τ) = x(m) + τ (x(m + 1) -x(m)) + τ(τ-1) 4 (x(m + 2) -x(m + 1) -x(m) + x(m -1)).
This interpolation uses four successive points and is also similar to the linear interpolation with an additional correction term. Compared to the third order Lagrange formulation, the correction term is not symmetrical. The last interpolator is the fifth order Lagrange interpolator:

x(m, τ) = (τ 2 -1)(τ-2)τ 24 x(m -2) -(τ 2 -4)(τ-1)τ 6 x(m -1) + (τ 2 -1)(τ 2 -4) 4 x(m) -(τ 2 -4)(τ+1)τ 6 x(m + 1) + (τ 2 -1)(τ+2)τ 24 x(m + 2). (23)
These different interpolators can be clearly viewed as FIR filters, where impulse responses can be deduced from the above equations. Therefore, it is certainly interesting to compare their frequency responses to the frequency response of a pure delay. This comparison is shown in Fig. 8 for the magnitude and in Fig. 9 for the phase. With the exception of the linear interpolator, it appears that the interpolators have interesting performances for normalized frequencies below 0.2 (over-sampling ratio greater than 5). While the two Lagrange interpolators show the flattest magnitude, the Bessel interpolator exhibits a better phase linearity. This information will be completed by other measures of performances, namely interpolation error and EVM at the output of the transmitter.

Comparison of performances

In order to evaluate interpolation errors, we compare an original signal to a reconstructed one. This comparison is realized on the envelope of the OFDM modulated signal previously introduced. Starting with an original signal with 300 samples per time symbol, we subsample it by a factor of 100. This gives an over-sampling ratio of 3. The signal is then reconstructed by interpolating the missing 200 values between x(m -1) and x(m + 1), and the interpolated values at a delay ∆ from the sample, are compared to the value of the original Let us note T s the sampling period and call E o the original signal, E i the interpolated signal. The quadratic error (rms) is expressed as: When the interpolators are used between x(m) and x(m + 1), their rms quadratic errors are similar with a maximum error in the middle of the two samples. The Bessel and 5th order Lagrange interpolators achieve similar results with a maximum error of about 0.4%, while the two other interpolators present less than 0.6%. However, a problem arises when the interpolator is used in the range of x(m -1) and x(m). The linear interpolator is obviously not up to par with a 3.5% of error followed by the Bessel with 2.5%. The asymmetry can degrade the quadratic error performances. During iterations, positive as well as negative values of τ may indeed appear. The best solution remains certainly the two Lagrange formulas which are quasi symmetrical and more appropriate to our problem. The order of the interpolator can also make a difference in terms of the complexity of the implementation. Using HPADS, the interpolators can be validated with the study of the EVM (computed for the OFDM after FFT demodulation) and the output spectrum. The simulation is realized differently than with the previous quadratic error evaluation. Here the signal sampled at T /3 is delayed or advanced by τ taken between ±1 and resynchronized using the interpolator. This is a complementary analysis to the previous analysis and is better suited to standard transmitter analysis. Fig. 11 shows the EVM at the output of the transmitter and presents the same profile as the error quadratic curves in Fig. 10. The main problem is the relatively high values of the EVM.

Err(∆ ) = 100 ∑ k (E o (kT s -∆ ) -E i (kT s -∆ )) 2 ∑ k (E 2 o (kT s -∆ ) (24)
Values reach a maximum of 2% for the 5th order Lagrange interpolator and 3.5% for the 3rd order Lagrange interpolator. The latter is not acceptable for our application. We then need to sample the signal to a higher rate. Fig. 12 presents the EVM values when changing the sampling period of the signal from T /3 to T /9 (odd values are here preferred to facilitate the demodulation of the OFDM signal in simulation). The x axis of the curve is normalized to symbol duration. The difference between samplings at T /3 and T /9 is dramatic, going from 2% to 0.1%. The sampling rate not only has an impact on the EVM value, it also affects the output spectrum. Ts=T/9

Ts=T/7

Fig. 12 EVM values for the 5th order Lagrange interpolator with different sampling periods.

Implementation using the Farrow structure

A polynomial interpolator can efficiently be implemented using a Farrow structure [START_REF] Farrow | A continuous variable digital delay element[END_REF] or its extensions [27; 13; 5]. We only describe here the original Farrow structure but other possibilities will deserve further investigation. The Farrow structure relies on the parallelization of N FIR filters which can be expressed as

y i (m) = M-1 ∑ k=-M h i (k)x (m -k). ( 25
)
The output of these filters are combined so that the output signal can be expressed as

y (m, τ) = N-1 ∑ i=0 τ i y i (m) = N-1 ∑ i=0 τ i M-1 ∑ k=-M h i (k) x (m -k) (26)
For instance, rewriting the equations of the 5th order Lagrange interpolator [START_REF] Staszewski | Alldigital PLL and transmitter for mobile phones[END_REF] gives

y 0 (m) = 5 4 x (m) , y 1 (m) = -2 24 (m + 2) + 4 6 x (m + 1) -4 6 x (m -1) -2 24 (m -2) y 2 (m) = -1 24 (m + 2) + 4 6 x (m + 1) -5 4 x (m) + 4 6 x (m -1) -1 24 (m -2) y 3 (m) = 2 24 (m + 2) -1 6 x (m + 1) + 1 6 x (m -1) -2 24 (m -2) y 4 (m) = 1 24 (m + 2) -1 6 x (m + 1) + 1 4 x (m) -1 6 x (m -1) + 1 24 (m -2)
and can be implemented on HPADS as presented in Fig. 13.

For the 5th order Lagrange filter, this implementation uses 5 multiplications, 20 additions and 15 arithmetic divisions. As for the 3rd order Lagrange interpolator, the implementation leads to only 2 multiplications, 5 additions and simpler divisions that only requires to divide by 2. This can certainly be an argument for the choice of the interpolator, not only in terms of performances but also in terms of simplicity of implementation.

Fig. [START_REF] Johansson | On the design of adjustable fractional delay FIR filters[END_REF] Farrow structure for the 5th order Lagrange interpolator.

On the interpolation noise

If the interpolator were perfect, the algorithm, in the absence of observation noise, would find the exact solution. However, as illustrated in Figs. 10, 11 and 12, the system is characterized by an internal "self-noise" associated with the interpolator. Indeed, for equally spaced points t 0 ,t 1 , . . .t n , the remainder of the Lagrange interpolation polynomial of degree n for estimating

x(t 0 + τT ) is [28] n i (t 0 + τT ) = T n+1 ω(τ) x (n+1) (ξ ) (n + 1)! (27)
with ω(τ) = τ(τ -1) . . . (τn), and where ξ is an unknown point in the interval [t 0 ,t n ]. Thus, we can evaluate the variance

σ 2 i (τ) = E n i (t 0 + τT ) 2 of this interpolation noise n i (t) and obtain E n i (t 0 + τT ) 2 = (-1) n+1 T 2(n+1) (n + 1)! |ω(τ)| 2 R (2(n+1)) xx (0) (28) using the relation E |x (n+1) (t)| 2 = (-1) n+1 R (2(n+1)) xx (0).
As a result, the variance of the interpolation noise is nonstationary since it depends on the interpolation point t 0 +τT . With this formula, it also appears that the higher the sampling rate and the higher the interpolation order, the lower the variance. To illustrate this, Fig. 14 compares the theoretical formula [START_REF] Volkov | Numerical Methods[END_REF] to simulation results (see also the results in Fig. 10).

Analysis of the algorithm and results

Typical results for delays 0.18T and 0.47T for envelope and phase respectively, are shown in Fig. 15. The measured EVM after convergence is only 0.7% compared to 14% without correction. The resulting spectrum is reported in Fig. 16. It shows very interesting performances: the spectrum is improved by 30dB compared to the uncorrected case. The remaining noise floor at -50dBc in 200kHz bandwidth corresponds to the interpolation errors. Surprisingly, it appears that the spectrum obtained with the values at the output of the correction algorithm is slightly better than the spectum obtained with the true values of delays. This is commented in §5.2. However interesting these results are, it is important to examine the role of the interpolator, the impact of the values of the adaptation step and to study figures of merit such as the settling time (convergence speed), the bias and variance of results as well as the overall EVM. We evaluated these different points by Monte-Carlo experiments for a varying adaptation step γ and for all delays less than T . In the study, we took the same adaptation γ = γ 1 = γ 2 step for both adap- tations. Furthermore, the signal power was normalized to one in order to be independent of signal scales.

To analyse these different characteristics, we use a simplified 'toy' model that is simpler to analyse than our original problem. Let z(t) = x(t -∆ ) be an observed, delayed version of an original signal x(t). In order to identify and correct the delay, we adopt the following recursion

µ(n + 1) = µ(n) + γ dx(u) du u=t-∆ +µ(n) [x(t) -x(t -∆ + µ(n))]
(29) with x(t -∆ + µ(n)) the interpolated value at time (t -∆ + µ(n)).

Settling time

The convergence speed is an important issue for the practical use of such algorithm. With ∆ µ(n) = (µ(n) -∆ ) small, we have

x(t) -x(t + ∆ µ(n)) ≃ -∆ µ(n) ẋ(t) + n i (t), (30)
where ẋ(t) is the derivative of process x(t) and where n i (t) represents the interpolation noise. Then,

∆ µ(n + 1) = ∆ µ(n) -γ∆ µ(n) ẋ(t + ∆ µ(n)) ẋ(t) + γ ẋ(t + ∆ µ(n))n i (t) (31)
Therefore, the mean trajectory is

E [∆ µ(n + 1)] = E [∆ µ(n)] 1 -γR ẋ ẋ(∆ µ(n)) , (32)
with E [ ẋn i ] = 0. Considering that R ẋ ẋ = -Rxx , and since we supposed ∆ µ(n) small, so that Rxx (∆ µ(n)) ≃ Rxx (0), the previous equation can be solved recursively and we get

E [∆ µ(n + 1)] ≃ 1 + γ Rxx (0) n E [∆ µ(0)] , (33)
with Rxx (0) < 0. This means that µ(n) converges exponentially to ∆ , with a time constant t c = -1/ log 1 + γ Rxx (0) ≃ 1/ γ Rxx (0) . This is illustrated in Fig. 17, where are presented the settling times for the algorithm measured as the rise from 0 to 95% of the final value, t s = 3t c . This figure shows that the convergence speed is clearly independent of the interpolator and the simulation results are in line with the above development.

We also examined the variation of the convergence speed with respect to the values ∆ 1 , ∆ 2 . The results reported in Fig. 18 clearly indicates that the convergence speed is independent of the final values. Envelope delay Phase delay Fig. [START_REF] Morgan | Highly integrated transceiver enables highvolume production of GSM/EDGE handsets[END_REF] Settling time for all possible delays (less that T ), with γ = 0.2.

Bias

A careful examination of the results shows that both envelope and phase are often affected by a small, but undeniable, bias. In addition, we have observed in practice that the quadratic error associated with the biased estimates is lower than the error obtained with the true values. Similarly, it appears that the spectrum with identified values is slightly better than the spectrum associated with the true values. This can be explained as follow. The error e(t) is

e(t) = x(t)-x(t -∆ + µ(n)) = x(t)-x(t -∆ + µ(n)) +n i (t), (34)
so that its variance is

σ 2 e = E e(t) 2 = E x(t) -x(t -∆ + µ(n)) + E n i (t) 2 , ( 35
) that is σ 2 e = E e(t) 2 = 2 (R xx (0) -R xx (µ(n) -∆ )) + σ i (µ(n)), (36)
where σ i (µ(n)) = E n i (t) 2 is the variance of the interpolation noise that depends on the value of the advance µ(n), see the discussion in §4.4. The purpose of the algorithm is to find a value µ minimizing σ 2 e . The first term clearly decreases when µ(n) → ∆ while the second term may increase. Therefore, the procedure will find the best values that minimize the sum of the two terms, realizing a trade-off between bias and variance of the interpolation noise. The solution is theoretically given by

Ṙxx (µ(n) -∆ ) = - dσ 2 i (µ(n)) dµ(n) . (37)
Fig. 19 presents the bias measured for the envelope and phase component with respect to the adaptation parameter. We find that the bias still exist and do not depend on the adaptation parameter γ as it was indicated by the analysis. The bias is a function of the interpolator type and order.

Variance

Variations after convergence are also an important issue. These variations can be modeled as an additive noise on the estimates. With our toy model, we obtained expression [START_REF] Yli-Kaakinen | An algorithm for the optimization of adjustable fractional-delay all-pass filters[END_REF] for the trajectory of the algorithm. This expression is only valid, on average, if ∆ µ(n + 1) has a zero mean. Thus, taking now ∆ µ(n + 1) = µ(n + 1)μ, where μ is the value at convergence and rearranging the terms, we get

∆ µ(n + 1) = ∆ µ(n) 1 -γ ẋ(t + ∆ µ(n)) ẋ(t) + γ ẋ(t + ∆ µ(n))n i (t). = A n ∆ µ(n) + B n n i (t) (38)
Taking the square and the expectation and making the, clearly false, hypothesis that A n and ∆ µ(n), B n and n i (t) are uncorrelated, we obtain

σ 2 µ (n + 1) = E A 2 n σ 2 µ (n) + E B 2 n σ 2 i ( 39
)
with σ 2 µ (n) the variance of µ at step n. We also find E A 2 n ≃ 1 + 2γ Rxx (0) + γ 2 E ẋ(t) 4 (with µ -∆ ≃ 0), and E B 2 n = -γ 2 Rxx (0). At convergence, we finally obtain

σ 2 µ = E B 2 n σ 2 i 1 -E [A 2 n ] = γσ 2 i 2 + γE [ ẋ(t) 4 ] / Rxx (0) . (40)
This expression shows that the variance of the estimates increases linearly with γ and that it is only associated to the 'self-noise' in the system. Despite using approximations in the derivation, the simulation results are once more in line with this formula. Fig. 20 gives the variance of the estimates of the envelope and phase delays as a function of γ. The linear dependence can be clearly seen.

EVM -Quadratic error

The EVM is a crucial parameter in digital communications. As we will observe, it sums up some of the other properties. In the case of a single carrier modulation, the EVM is defined by

E |Z e -Z o | 2 /E |Z o | 2
where Z e = ρ e e jφ e and Z o = ρ o e jφ o stand for the complex envelope of the emit- ted signal and the original signal respectively. Let us insist on the fact that the situation is more complicated for multicarrier modulations. In our setting, the expression becomes

EVM = E ρ 2 o + ρ 2 e -2ρ o ρ e cos (φ o -φ e ) E [ρ 2 o ] . ( 41
) Therefore, if ∆ φ = φ o -φ e is small, cos ∆ φ ≃ 1 -∆ φ 2 /2, the expression becomes EVM = E (ρ o -ρ e ) 2 + ρ o ρ e ∆ φ 2 E [ρ 2 o ] ≃ E ∆ ρ 2 + ρ 2 o ∆ φ 2 E [ρ 2 o ] (42) or EVM ≃ E ∆ ρ 2 E [ρ 2 o ] + E ∆ φ 2 (43)
assuming that ρ o and ∆ φ are independent. As a result, the EVM is a function of the quantities we previously studied.

We already established that the estimates are biased and that their variances, which are proportional to interpolation noise, increase linearly with γ. Therefore, the EVM behaves as

EVM ≃ γ 2 σ 2 ρ E [ρ 2 o ] + σ 2 φ + B 2 ρ E [ρ 2 o ] + B 2 φ , (44)
where σ 2 ρ and σ 2 φ are the variances of interpolation noise for the envelope and phase, and B ρ and B φ the bias terms. Since √ 1 + αx ≃ 1 + αx/2, the EVM rms may also show the same behaviour. This is the case in Fig. 21 where we clearly see a linear slope and an offset value. It also shows that the Lagrange and Bessel interpolators have the best performances. It is also interesting to look at the EVM with respect to all possible delays and with a fixed γ. Fig. 22 presents these results for γ = 0.05. We can observe a 'modulation' which is related to the variable variance of the interpolation noise with respect to the delay (see Fig. 14). Here, as in previous examples, we took T /T s = 5 and indeed notice error minima at interpolation nodes. Envelope delay Phase delay Fig. 22 EVM at output of the algorithm with γ = 0.05, for all possible delays. The 'modulation' is related to the non-stationary interpolation noise that vanishes at interpolation nodes.

Summary of the results

In the previous developments we have examined the behaviour of our algorithm according to the choice of interpolators and adaptation steps. Using specific examples, we have demonstrated a vast improvement can be achieved in terms of EVM and output spectrum, as shown in figures 15 and 16. Furthermore, using an approximate 'toy' model of the algorithm, we have shown theoretically and checked numerically, that -the settling time decreases exponentially with the adaptation step (Fig. 17), -a small estimation bias exists that does not depend on the adaptation parameter but rather of interpolation order (Fig. 19). The presence of this bias is explained as a way to reduce the whole quadratic error, -the variance (40) behaves approximately as a linear function of the adaptation parameter (Fig. 20), -the EVM (44) increases linearly with the adaptation step γ (Fig. 21).

As a consequence, a trade-off has to be made between the two important parameters that are the settling time and the EVM. The choice of the order of the interpolation is also important with respect to performances and implementation costs. The main values for the bias and variance can be found in Table 2 for different interpolators and several values of the adaptation step γ. The results for the settling time and the EVM are reported in Table 3 for the same conditions. In order to ensure independence in signal scales, the signal power was normalized to one.

Conclusion

In this paper, we proposed and analysed a new synchronization algorithm tailored for an EER architecture. Performances are indeed severely degraded in case of delay mismatch between envelope and phase paths. The proposed algorithm is based on an adaptive LMS structure. We studied the influence of the adaptation on the quadratic error, EVM and output spectrum. The algorithm uses an interpolation procedure that is characterized in terms of performances and in terms of implementation. We also examined the influence of the sampling rate. These studies demonstrated the strong interest of such a mandatory correction algorithm and the enhancement of transmitter performances. Regarding the algorithm, a few points can be further investigated: adopting different adaptation steps for the two recursions, decrasing the number of adaptation steps and alternating the two recursions. Higher system view point has to be taken into account in order to define the best use of the procedure: full tracking of delays or training periods. A for the implementation, modified and optimized Farrow structures should be considered.

This solution has drawbacks. For instance, it requires I/Q demodulation and uses a Cordic processor to split the envelope and phase of the PA output signal (derivative computation). We are currently working on these issues to improve our system.

Other concerns are the behaviour of the system with further mismatches such as a complex gain in the feedback loop, non-linear distortions and quantization noise. Preliminary results show that the algorithm is robust against the gain and noise mismatch, in the sense that the optimum solution remains unchanged. Evidently further mismatches imply a degradation of performances so future work will include attempts to also correct these mismatches with the same feedback loop.

Fig. 1

 1 Fig. 1 Polar transmitter using a CORDIC processor

Fig. 2 Fig. 3

 23 Fig. 2 Impact of the delay mismatch on output spectrum (taken in 200kHz bandwidth)

Fig. 4

 4 Fig.[START_REF] Bercher | Monte-carlo estimation of time mismatch effect in an OFDM EER architecture[END_REF] Impact of the delay mismatch on the constellation for T/27 and T/5

Fig. 6 Fig. 7

 67 Fig.[START_REF] Crochiere | Optimum fir digital filter implementations for decimation, interpolation, and narrow-band filtering[END_REF] Shape of the criterion for the envelope and phase delays ∈ [-4T, 4T ], with T the symbol period. The shape of the criterion is related to the impulse response of the shaping filter and presents local minima.

Fig. 8 Fig. 9

 89 Fig. 8 Magnitudes of the frequency responses of the four interpolators.

Fig. 10

 10 Fig.[START_REF] Hentschel | The Digital Front-End -Bridge Between RF-and Baseband-Processing[END_REF] Comparison of the normalized quadratic error (rms) between the four interpolators.

Fig. 11

 11 Fig. 11 Comparison of EVM values for the different interpolators.

Fig. 14

 14 Fig.[START_REF] Kahn | Single-sideband transmission by envelope elimination and restoration[END_REF] Standard deviation of the interpolation noise for a Lagrange 5 th order interpolator. Theoretical formula (28) (plain line) is compared to simulation results (dashed line).

Fig. 15 Fig. 16

 1516 Fig.[START_REF] Laakso | Splitting the unit delay[END_REF] Convergence of µ 1 and µ 2 to the true delays ∆ 1 = 0.47T and ∆ 2 = 0.18T .

Fig. 17

 17 Fig.[START_REF] Mártires | Differential delay equalization in a Kahn EER transmitter[END_REF] Settling times for the different interpolators, as a function of the adaptation step, for ∆ 1 = 0.36T and ∆ 2 = 0.12T .

Fig. 19

 19 Fig. 19 Bias µ -∆ for the different interpolators, as a function of the adaptation parameter γ. Phase (indicated by -• -) and envelope bias are almost constant with respect to γ.

Fig. 20

 20 Fig. 20 Variance σ 2 µ of delays estimates, ∆ 1 = 0.36T, ∆ 2 = 0.12T , for the different interpolators, as a function of the adaptation parameter γ. Phase (indicated by -• -) and envelope variances increase linearly with γ.

Fig. 21

 21 Fig.[START_REF] Sander | Polar modulator for multi-mode cell phones[END_REF] EVM rms (in %) at the output of the algorithm, for delays ∆ 1 = 0.36T, ∆ 2 = 0.12T as a function of the adaptation parameter γ.

Table 1

 1 ACPR obtained for different delay mismatches. It is evaluated for the full 64 carriers 20MHz OFDM modulation, with a channel spacing of 25MHz.

	Delay	T /27 T /13	T /5	T /3
	ACPR (dB)	-37	-29.7 -20.7 -15.7

Table 3

 3 Settling times of the algorithm and EVM rms (in %) for the output, for different interpolators and several values of the adaptation step γ.

		Settling time		EVM rms %
	γ		3rd Lag	Bessel	5th Lag
	0.05	228	1.045	0.89	0.815
	0.1	127	1.07	0.9	0.84
	0.5	45	1.24	1.05	1.009

Table 2

 2 Bias (µ -∆ ) and variance σ 2 µ of the estimates of delays, for different interpolators and several values of the adaptation step γ.

	γ 0.05 ρ 0.05 φ 0.1 ρ 0.1 φ 0.5 ρ 0.5 φ	3rd Lag -15.5 -19 -15 -28 -14 -17	Bias ×10 3 Bessel 8 2 8 4 8 4	5th Lag -9 -9 -8 -8 -7 -9	3rd Lag 6.2 5.8 11.5 4.5 56.5 13	Variance ×10 5 Bessel 10 6.59 21 4.6 89 11.5	5th Lag 5.8 5.75 11.5 4.5 56 11.5

Acknowledgements

The authors are indebted to the anonymous referees for their careful reading of the manuscript as well as their valuable comments and suggestions that helped improve the presentation of this paper. Thanks are extended to S. Trudel, P. J. Duriez and L. Garbell for their invaluable and friendly proofreading of the manuscript.