Olivier Roux
email: olivier.roux@ird.fr

Régis Céréghino

Pascal J Solano

Alain Dejean

Caterpillars and Fungal Pathogens: Two Co-Occurring Parasites of an Ant-Piant Mutualism

Abstrad

ln mutualisms, each interacting species obtains resources from its partner that it would obtain less efficiently if alone, and so derives a net fitness benefit. ln exchange for shelter (domatia) and food, mutualistic plant-ants protect their host myrmecophytes from herbivores, encroaching vines and fungal pathogens. Although selective filters enable myrmecophytes to host those a nt species most favorable to their fitness, sorne insects can by-pass these filters, exploiting the rewards supplied whilst providing nothing in return. This is the case in French Guiana for Cecropia obtusa (Cecropiaceae) as Pseudocabima guianalis caterpillars (Lepidoptera, Pyralidae) can colonize saplings before the installation of their mutualistic Azteca ants. The caterpillars shelter in the domatia and feed on food bodies (FBs) whose production increases as a result. They delay colonization by ants by weaving a silk shield above the youngest trichilium, where the FBs are produced, blocking access to them. This probable temporal priority effect also allows female moths to lay new eggs on trees that already shelter caterpillars, and so to occupy the niche longer and exploit Cecropia resources before colonization by ants. However, once incipient a nt colonies are able to develop, they prevent further colonization by the caterpillars. Although no higher herbivory rates were noted, these caterpillars are ineffective in protecting their host trees from a pathogenic fungus, Fusarium moniliforme (Deuteromycetes), that develops on the trichilium in the absence of mutualistic ants. Therefore, the Cecropia treelets can be parasitized by two often overlooked species: the caterpillars that shelter in the domatia and feed on FBs, delaying colonization by mutualistic ants, and the fungal pathogen that develops on old trichilia. The cost of greater FB production plus the presence of the pathogenic fungus likely affect tree growth.

Introduction

Mutualisms are interspecific interactions involving two or more species where each partner obtains resources that it would obtain less efficiently if alone, and so derives a net fitness benefit [START_REF] Bronstein | Our current understanding of mutualism[END_REF][START_REF] Bronstein | Coevolutionary dynamics and the conservation of mutualisms[END_REF][START_REF] Herre | The evolution of mutualisms: exploring the paths between conflict and cooperation[END_REF]. These mutualistic partnerships are transmitted from one generation to the next in one of two ways. ln vertical transmission, hosts transmit symbiont offspring directly to their own offspring [START_REF] Zientz | Insights into the microbial world associated with ants[END_REF], [START_REF] Klein | Mealybug-carrying by swarrning queens of a southeast asian bamboo-inhabiting ant[END_REF], while in horizontal transmission the partners need ta renew their association after each reproductive event [START_REF] Herre | The evolution of mutualisms: exploring the paths between conflict and cooperation[END_REF], [START_REF] Wilkinson | Horizontally acquired mutualisms, an unsolved problem in ecology[END_REF]. Biotic pollination, seed dispersal by animais, ant-plant associations and interactions between rhizobia or mycorrhiza and plant roots are transmitted horizontally and can be mutualistic [START_REF] Zientz | Insights into the microbial world associated with ants[END_REF], [6--9].

Myrmecophytes (orant-plants) are involved in mutualisms with a limited number of so-called plant-ants that they shelter in domatia (i.e. hollow branches or thorns and leaf pouches) and usually provide with food through extra-floral nectaries (EFNs) and/or food bodies (FBs). ln retum, plant-ants protect their host plant from herbivores, competitors, encroaching vines and fungal pathogens [1 0]. Because the transmission of ant-plant mutualisms is horizontal, myrmecophytes have evolved severa! types of selective fllters enabling them ta host those ant species most favorable ta their fitness [START_REF] Davidson | The evolutionary ecology of symbiotic aniplant relationships[END_REF], [START_REF] Ward | Systematic studies on Pseudomyrmex acacia-ants (Hymenoptera: Forrnicidae: Pseudomyrmecinae)[END_REF]. Hostplant selection by founding ant queens, for example, seems ta be driven by chemical compounds [13--16]; however ta enter into the domatia of certain myrmecophyte species, founding ant queens must be the right size or be able ta recognize and ta gnaw an entrance hale into the prostomata or thinner area, generally devoid ofvessels [START_REF]The ecology and evolution of ant-plant interactions[END_REF], [START_REF] Davidson | Cecropia and its biotic defenses[END_REF][START_REF] Grangierj | Mechanisms driving the specificity of a myrmecophyte-ant association[END_REF][START_REF] Brouat | Plant Jock and ant key: pairwise coevolution of an exclusion fùter in an ant-plant mutualism[END_REF].

Nevertheless, these mechanisms do not keep the mutualism between myrmecophytes and plant-ants free from conflict, competition and/or exploitation by other ants or by non-ant species. ln this context of competition for resources, the abilities of species are generally unequal, leading ta hierarchically-organized systems with dominant and subordinate species. Ta survive, weaker competitors must develop colonization strategies, be resistant ta perturbations, manage with fewer resources or have good longevity with the aim of conserving their access ta the "niche" [20--22]. One alternative way ta obtain an advantage over a better competitor is ta be the frrst ta obtain access ta resources and ta monopolize them. Often this advantage allows poor competitors ta persist longer in habitats than they would otherwise [START_REF] Palmer | Burning bridges: priority effects and the persistence of a competitively subordinate acacia-ant in Laikipia, Kenya[END_REF], [START_REF] Wilbur | Priority effects in experimental pond communities: responses of Hyla to B'!fo and Rima[END_REF]. This phenomenon is known as "temporal priority" and has been documented in many taxa such as mycorrhizal fungi [START_REF] Kennedy | Priority effects determine the outcome of ectomycorrhizal competition between two R/Uzopogon species colonizing Pinus muricata seedlings[END_REF][START_REF] Kennedy | Root tip competition among ectomycorrhizal fungi: are priority effects a rule or an exception?[END_REF][START_REF] Mack | Balancing multiple mutualists: asynnnetric interactions among plants, arbuscular mycorrhizal fungi, and fungal endophytes[END_REF], plants [START_REF] Bertness | Competition and facilitation in marsh plants[END_REF], [START_REF] Ladd | Priority effects produced by plant litter result in nonadditive competitive effects[END_REF], amphibians [START_REF] Wilbur | Priority effects in experimental pond communities: responses of Hyla to B'!fo and Rima[END_REF], [START_REF] Lawler | Temporal overlap, competition, and priority effects in larval anurans[END_REF] and insects [START_REF] Palmer | Burning bridges: priority effects and the persistence of a competitively subordinate acacia-ant in Laikipia, Kenya[END_REF], [START_REF] Hodge | Effects of temporal priority on interspecific interactions and community development[END_REF].

It is well known that mutualistic plant-ant species compete for their host-plant [START_REF] Vasconcelos | Ant colonization of Maieta guianensis seedlings, an Amazon ant-plant[END_REF], [START_REF] Davidson | Competition among ants for myrmecophytes and the significance of plant trichomes[END_REF][START_REF] Yu | Experimental studies of species-specificity in Cecropia-ant relationships[END_REF][START_REF] Fonseca | Ontogenetic succession in Amazonian ant trees[END_REF]. Moreover, mutualistic ants are not the only ones competing for this resource. lndeed, parasites of ant-myrmecophyte mutualisms-mostly ant species-are able to colonize the myrmecophytes, but do not provide them with protection [START_REF] Yu | Parasites of mutualisms[END_REF], [9]. This was fust shawn for Pseudmnyrmex nigropiiosur that colonizes myrmecophytic Aca&ia and consumes their EF'Ns and FBs, but exhibits no defensive behavior [START_REF] Janzen | Pseudomyrmex nigropilosa: a parasite of a mutualism[END_REF]. Sorne non-ant insects are also able to colonize and parasitize myrmecophytes, benefiting from the shelter and food provided by the plant in different ways; for example, the larvae of the clerid beetle Phyllobamu.s sp. parasitize myrmecophytic Pip" trees, feeding on bath the FBs and on mutualistic ants. The fitness of the host trees is reduced due ta a greater investment in FB production and a decrease in biotic protection by the guest ants, with a subsequent increase in herbivory [START_REF] Letoumeau | Code of ant-plant mutualism broken by parasite[END_REF]. Also, females of the chrysomelid beetle CrMlomera sp. open an entrance hole in the prostomata of Curopia trees in the same manner as mutualistic A,eteca ants. They then lay eggs in the domatia and the larva.e feed on young leaves [3 7].

These insects, often less numerous and less aggressive than ants, must find weaknesses in the mutualism to be able to colonize their host tree and exploit its resources. Using chernical mirnicry or camouflage to counter ant aggressiveness is one solution for getting past ant defenses [START_REF] Maschwitz | Lycaenids parasitizing symbiotic plant-ant partnerships[END_REF], [START_REF] Lenoir | Hefetz A (200 1) Chentical ecology and social paxasitism in ants[END_REF]; however, being the fll'St to arrive to colonize new treelets might also represent a serious advantage because the biotic defense provided by mutualistic plant-ants is not yet in place and nearly leaves the plant without indirect deferues.

The myrmecophyte CemJjlia obtura (Cecropiacae), the focal species of this study, is mutualistically associated with several A,eteca species (Dolichoderinae) whose founding queens and workers recognize the zone where the prostomata is situated and so establish colonies in the internodal domatia [START_REF] Longino | Geographie variation and community structure in an antplant mutualism: Al;teca and Cecropia in Costa Rica[END_REF], [START_REF] Longinojt | Azteca ants in Cecropia trees: taxonomy, colony structure and behavior[END_REF]. In addition to shelter, the plant provides the ,Azteea colonies with food in the form of g1ycogen-rich Müllerian bodies produced by the trichilia, pads of dense trichomes situated at the base of the leaf petiole, and lipid-rich pearl bodies produced beneath young leaves. Mutualistic A,eteca workers generally protect their host trees from defoliating insects, encroaching vines and fungal pathogens [START_REF] Davidson | Cecropia and its biotic defenses[END_REF], [START_REF] Longino | A taxonontic review of the genus Azteca (Hymenoptera: Fornticidae) in Costa Rica and a global revision of the aurita group[END_REF][START_REF] Dejean | Predation and aggressiveness in host plant protection: a generalization using ants from the genus A;:,teca[END_REF][43], but this is not the case for saplings [START_REF] Val | Does ontogeny cause changes in the defensive strategies of the myrrnecophyte Cecropia peltata?[END_REF], [START_REF] Faveri | The Azteca-Cecropia Association: axe ants always necessary for their host plants?[END_REF].

Because the transmission of the A;;.teca--Cecropia mutualism is horizontal, the size of the internodes plus the production rate of the food bodies do not permit A,eteca colonies to develop before the saplings reach ca. 1 rn in height [START_REF] Val | Does ontogeny cause changes in the defensive strategies of the myrrnecophyte Cecropia peltata?[END_REF], [START_REF] Coley | Costs and benefits by tannins in a neotropical tree[END_REF]. Before Ctcropio. saplings reach this minimwn size, herbivores and parasites may use this absence of mutualistic plant-ants to opportunistically tak.e over the Cecropia's resources; whereas, after this period of rime, potential invaders must overcome the biotic defenses conferred by plantants [START_REF] Val | Does ontogeny cause changes in the defensive strategies of the myrrnecophyte Cecropia peltata?[END_REF].

We noted that recently-perturbed areas are rapidly occupied by thousands of C. obtusa saplings, permitting sorne caterpillars to live in the domatia and feed on the FBs, and that caterpillar presence was associated with the development of a fungus on the old trichilia. To broaden our understanding of the biological interactions and coexistence of these caterpillars within the A,eteca-CemJjlia mutualism, we conducted a correlation study where we posed the following questions. (1) Do caterpillars mainly or even exclusively feed on the FBs and, if so, does this activity increase FB production as is known for mutualistic plant-ants [START_REF] Letoumeau | Code of ant-plant mutualism broken by parasite[END_REF]? [START_REF] Bronstein | Coevolutionary dynamics and the conservation of mutualisms[END_REF] Is caterpillar presence associated with the greater herbivory of Cecropia saplings? (3) Does this presence favor fungal development on the trichilia with deleterious consequences for the plant? (4) Can A;;.teca ants prevent colonization by caterpillars, or, inversely can caterpillars delay or even prevent colonization by ants thus allowing them to exploit Cecropia saplings longer?

Materlals and Methods

Ethics Statement

This study was conducted according to relevant national and international guidelines.

Study sites

We conducted this study between 2000 and 2009 in French Guiana near the Petit Saut dam (5°03'39n N-53°02'36"W) and near the Montagne rks singes (5°04'19.6"N-52°41'42.5"W). We selected and tagged C. obtura that were ca. In these areas, C. obtura is mostly associated with A;;.teca afjari and A. o'IJO.ticeps whose colonies exploit Müllerian bodies (Fig. lA), tend hemipterans in the host tree domatia and prey on insects landing on the leaves [START_REF] Dejean | Predation and aggressiveness in host plant protection: a generalization using ants from the genus A;:,teca[END_REF].

The caterpillars' diet

To assess if the caterpillars feed exclusively on the FBs, in 2001 and 2002, we selected 83 saplings on which we had found at least one caterpillar. Preliminary studies conducted during a 24-hour period permitted us to leam that FBs are produced during the afternoon, between 15:00 and dusk (see also [START_REF] Belin-Depoux | La fonction myrmécophile de Cecropia obtusa Trecul (Cecropiaceae) en Guyane française[END_REF][START_REF] Lapierre | Wasps robbing food from ants: a frequent behavior?[END_REF][START_REF] Mcclure | Territorial aggressiveness on the axboreal ant Azteca alfori by Camponotus hlondus in French Guiana due to behavioural constraints[END_REF][START_REF] Dejeana | A temporary social parasite of tropical plant-ants improves the fitness of a myrrnecophyte[END_REF]), and that caterpillars are active during FB production. During a more comprehensive survey, we observed their behavior for 5 minutes per sapling in the morning between 8:00 and 11 :00 as weil as during FB production in the aftemoon between 15:00 and 18:00. Observations were also made at night between 21:00 and 5:00 to ensure that the caterpillars were not active noctumally.

Food body production, herbivory, ant and caterpillar presence and tree growth

To quantify FB production, in 2003 we selected 30 saplings, eacb bearing at !east five leaves: 10 sheltered A. a!fari, 10 sheltered caterpillars and the remaining 10 were unoccupied. Because FB production increases when they are removed by ants or experimentally [1 7], we conducted experiments where bath ants and caterpillars were prevented access to the upper zone of the trees where most of the FB production occurs. We did this by placing a ring of Tanglefoot® around the trunk under the lowermost leaf to isola te the top of each sapling, and th us prevent the ants and caterpillars from having access to the leaves and to the FBs. We then plugged the entrance hales situated in this upper, isolated section with a spot of Tanglefoot®. The ants and caterpillars could, nevertheless, move freely in and out of the domatia using the lower entrance hales giving them access to the lower part of the trunk and to aider leaves with inactive trichilia. We placed aluminum foi! shelters around the isolated trichilia to protect them from flying insects and to gather the FBs that dropped off [START_REF] Lapierre | Wasps robbing food from ants: a frequent behavior?[END_REF], [START_REF] Bérenger | Special relationships of certain predatory Heteroptera Reduviidae with plants. First known case of a phytophagous Harpactorinae[END_REF]. For each sapling and during 20 days, at ca. 19:00, we removed the FBs produced that day and that had fallen from the !east mature trichilia onto the aluminum foi!, and counted them. We compared the results using a repeated measures ANOVA followed by a Newman-Keuls' post-hoc test for multiple comparisons (GraphPad Prism 4.03 software).

We also tested, in 2003-2004, if the presence of ants and caterpillars affected the presence of defoliating insects by scoring the amount of herbivory on the oldest leaf on 90 saplings (30 sheltered A. a!fari, 30 sheltered caterpillars, and the 30 others were unoccupied), each bearing at !east five leaves. We chose the oldest leaves because they provide an idea of the history of the defoliation over the preceding ca. 18 months which corresponds to the lifespan of C. obtusa leaves [START_REF] Dejean | Predation and aggressiveness in host plant protection: a generalization using ants from the genus A;:,teca[END_REF]. We evaluated the percentage of foliar surface eaten by insects (FSE) using the following scale: (!) leafintact; (2) slightly attacked: O%<FSE:=;25%; (3) somewhat attacked: 25%<FSEo=;50%; (4) very attacked: 50%<FSEo=;75%; and (5) extremely attacked: FSE>75%. We compared the results using the Kruskal-Wallis test.

To verifY how caterpillars can delay or even prevent ant colonization and vice versa, every 8 months betweenjune 2006 and June 2008, we noted which ant species or if caterpillars sheltered in the domatia and fed on the FBs on the 64 C. obtusa situated near the Montagne des singes. An additional survey was conducted injuly 2009. lnjuly 2008, we measured the height of the trees that had sheltered (!) Azteca colonies during the experimental period (A. ovaticeps: N = 1 0; A. a!fari: N = 22), (2) neither ants nor caterpillars (N = 1 0), or (3) caterpillars during the en tire experimental period or that had been replaced by an Azteca colony only during the last part of the experimental period (N = 14). Ali of these trees are the same age as they developed just after the dirt raad was built near the Montagne des singes and have a similar exposure to the sun and to rain. We compared the results using an ANOVA and Newman-Keuls' post-hoc test.

Fungal infestation of the trichilia

Between 2003 and 2005, we recorded the number of trees whose trichilia had a fungal infestation out of 610 C. obtusa saplings sheltering an Azteca colony (N = 349), caterpillars (N = 83), or not occupied (N = 178). We scored the presence versus absence of fungal infestation on the trees as "!" and "0", respectively, and compared the results using the Kruskal-Wallis test and Dunn's post-hoc test.

To analyze how ant or caterpillar presence affected fungal development, we took samples of 20 trichilia with and 20 without developed mycelium from C. obtusa in ali cases (i.e. trichilia taken from trees sheltering ants, caterpillars or unoccupied) and cultivated the mycelium in aseptic conditions in Sabouraud's nutritive substrate (N = 120 trichilia). We analyzed these samples under a microscope to verifY how the mycelium develops on the trichilia. We frrst fixed the sampled trichilia with FAA (formalin, acetic acid, alcohol), and then embedded them in paraffin. We stained tissue sections with basic fuchsin light -green or toluidine blue contrasted with sodium molybdate.

Voucher specimens of the adult maths obtained after the metamorphosis of the caterpillars were identified as Pseudocabima guianalis (Lepidoptera, Pyralidae, Phyticinae) and were deposited at the Systematic Entomology Laboratory of the United States Department of Agriculture (USDA), Beltsville, Maryland. Fungal samples were identified as Fusarium moniliforme (Deuteromycetes) and were deposited at the Laboratoire de biologie et taxonomie des microchampignons, Muséum National d'Histoire Naturelle, Paris, France.

Results

The caterpillars

Observations made on the 83 saplings sheltering caterpillars permitted us to note that the frrst instar caterpillars lived under a silk shelter that they built between the stipules developing around the terminal bud, the trunk and the youngest leaf. They only left this shelter between 15:00 and 18:00 to feed on the FBs produced daily during that rime period by the youngest trichilia, or sometimes by the other tricbilia. As the shoot grows, the stipules, which normally drop off, are trapped by the silk (Fig.

Fungal presence

Fusarium moniliforme was present on 323 of the 610 C. obtusa saplings (53.0%), sometimes completely covering the trichilia (Fig. 1 C). The percentage of infested individuals was significantly lower among saplings sheltering an Azteca colony than those sheltering P. guianalis caterpillars or not occupied, while the difference between the latter two cases was not significant (Fig. 2).

Normally, FBs are sub-spherical and homogeneous with reserve cells bordered by a cell wall and a thin cuticle. The penetration of the F. moniliforme mycelium into an FB occurs once it is already formed, but not necessarily completely developed. ln

Q) Q) 30 イ ョ セ @ co Q) ë ;: 20 Q) セ @ 10 Q)
Il.. FB cells show no reaction either in the cell wall or in the cytoplasm. Once inside the FB, the mycelium progressively invades ail of the cells, down to the base of the FB (Fig. lE).

The first cells of the inner triclûlium react to the presence of the mycelium when the FB is highly invaded. They seem to contain more tannin and are more elongated.

We noted a proliferation of the mycelium in in vitro cultures in Petri dishes with bath healthy and infected trichilia, betraying the presence of spores in ail cases (i.e. trichilia taken from trees sheltering ants, caterpillars or unoccupied). colonies. Among the others, 178 (29.2%) were totally unoccupied, while the remaining 83 (13.6%) sheltered three to six P. guimullis caterpillars at different larval stages (Fig. Sic). For the 64 C. obtusa surveyed during 3 years near the MonltJgne lhs singes, at the start of the survey the peroentage of C. obtusa sheltering P. gu.ianoJis caterpillars was by far superior (39.1% or 25 trees out of 64; Fig. 3), illustrating that there are variations between areas. When present, caterpillars were also more numerous with some trees sheltering up to 12 caterpillars. Saplings were also associated with the two Curopio-ant species typical of the area, A. a!fari and A. ovatictps, as weil as, unexpectedly, the tire ant Solenopsis saevis.rima (tree N"7). Also, six trees were unoccupied at the start of the survey, and three of the 25 saplings bearing caterpillars were also occupied by A. aljari (trees N" 24, 43 and 52). This dual hosting was also observed later in the swvey for two additional trees (trees N" 14 and 41 ), but after a few months, ail five trees were occupied only by kteca colorùes.

Ant and caterpillar occupancy of

Note that, in the end, tree N ° 52 was colonized by A. ッ カ 。 エ ゥ エ Z セ ウ N @ Over the course of the different surveys, unoccupied trees were colonized by caterpillars (four cases) or directly by Azteca ants (trees N° 19 and 25). Although S. saevissima workers exploited the FBs and were aggressive towards flying insects landing on their host tree foliage, tree N"7 was colonized in the end by caterpillars (Fig. 3). While the two Az/tca species occupied more and more trees over time, the number of trees sheltering P. guianalis caterpillars lirst increased and then decreased. They were replaced by A. aljari Food body production, herbivory and tree growth FB production was significantly higher for saplings sheltering kteca ants than for those sheltering caterpillars and for the latter compared to unoccupied saplings (Fig. 4).

We did not note significant differences in the peroentage of foliar surface eaten by defoliating insects between the C. obtusa sheltering an Af;teca colony, caterpillars, or not occupied by either ants or caterpillars (Kruskal-Wallis test, H 2 90 = 1.813; P>0.05).

Nevertheless, caterpillar presence a.ffected trec growth as those sheltering kteca colonies during this experimental period were significandy taller at the end of the survey than those sheltering caterpillars or those that were unoccupied (Fig. 5). The differences were not significant between trees sheltering colonies of the two kteca species, or between trees sheltering caterpillars or that were unoccupied.

Discussion

Al! in ail, these results constitute a new step in understanding the nature of the parasitism of the Azteca-CuropitJ mutualism by nonant insects. Like Coelomera chrysomelid beetles [3 7] and mutualistic ants, P. guianalis caterpillars at their third larval stage recognize the prostomata and gnaw an entrance hale to shelter in the host tree domatia. The difference with the damage caused by Coelomera is that it is direct as they feed on young leaves [3 7], whereas P. guimullis caterpillars, like mutualistic ants, feed on the FBs produced by the plant. They are indirectly detrimental ta their host tree because they allow Fusarium to develop on the triclûlium. Moreover, although it is possible that female maths select the most productive trees for their offspring [START_REF] Derridj | Chenticals on the leaf surface, information about the plant available to insects[END_REF], the caterpillars seems to induce an ゥ ョ 」 イ ・ 。 セ セ ・ @ in FB production, as do kteca workers and clerid beede (Phyllobaenus sp.) larvae on Piper [START_REF] Letoumeau | Code of ant-plant mutualism broken by parasite[END_REF]. However, unlike sorne mutualistic ants [START_REF] Heil | Food body production in Macaranga triloha (Euphorbiaceae): a plant investment in andherbivore deferree via symbiotic ant partners[END_REF], P. guimullis caterpillars do not provide any services in return for this increase in FB production. This type of increase in FB production is usually stimulated when mutualistic ants remove the FBs because the space thus made available reduces the pressure on the trichilia and favors the production of the next group of FBs [START_REF] Davidson | Cecropia and its biotic defenses[END_REF], [START_REF] Folgarait | Responses of Cecropia to experimental removal of Mullerian bodies[END_REF]. In the absence of mutualistic ants, FB production remains law, but is Iùgh enough to be attractive to founding ant queens [START_REF] Davidson | Cecropia and its biotic defenses[END_REF], [START_REF] Folgarait | Responses of Cecropia to experimental removal of Mullerian bodies[END_REF]. Because our experimental design did not allow Az,teca ants, caterpillars or other insects access to the trichilia, FB production should have been reduced to the same level as on unoccupied trees. Y et, this was not the case, suggesting tbat a factor other than FB removal plays a role, such as the plant obtaining nutrients from its ant or caterpillar inhabitants [START_REF] Sagers | Carbon and nitrogen isotopes trace nutrient exchange in an ant-plant mutualism[END_REF].

When parasitic ants are present, the establishment of mutualistic species is durably prevented, and plant fitness is lessened due to increased herbivory [START_REF] Yu | Parasites of mutualisms[END_REF]. The presence of non-ant parasites does not imply the exclusion of mutualistic ants, but host tree leaves can suffer herbivory if these insects feed on the plant [3 7] or on plantants that are therefore unable to protect their host trees [START_REF] Letoumeau | Code of ant-plant mutualism broken by parasite[END_REF].

However, here, the presence of P. guianalis caterpillars was not associated with greater herbivory compared to trees sheltering Azteca colonies or unoccupied trees because saplings rely on secondary anti-defoliator compounds and structures for their protection [START_REF] Jolivet | Ants and plants: an example of coevolution[END_REF], [START_REF] Val | Does ontogeny cause changes in the defensive strategies of the myrrnecophyte Cecropia peltata?[END_REF], [START_REF] Faveri | The Azteca-Cecropia Association: axe ants always necessary for their host plants?[END_REF].

Even though the saplings did not suffer greater herbivory, the protective mutualism is very disrupted as, when present, Az,teca colonies significandy limit the development of the Fusarium mycelium. Indeed, we show that in the absence of mutualistic ants, Fusarium developed on the trichilia of both unoccupied trees and trees sheltering caterpillars. Moreover, this fungus is known to produce growth-inhibiting mycotoxines that are also responsible for necrosis in plants [START_REF] Cole | Toxin from Fusorium moniliforme: effects on plants and animais[END_REF], [START_REF] Nelson | Taxonomy and biology of Fusorium moniliforme[END_REF] and insects [START_REF] Teetor-Baxsch | Entomogenous Fusorium species[END_REF], [START_REF] Gupta | Isolation of beauvericin as an insect toxin from Fusarium semitectum and Fusarium monili.forme vax. suhglulinllru[END_REF]. Consequently, likely due to the presence of this pathogen plus the cost of producing FBs, the growth rate of the trees that sheltered caterpillars during the survey conducted at the Montagne des singes was affected if compared to those that sheltered Azt«a colonies during the same period (Fig. 5). Therefore, mutualistic Azt,eca likely control the extent of the fungal infection in the same way that, by defending myrmecophytic Piper from stem-boring insects, PJWJQk ants reduce fungal infections [START_REF] Letoumeau | Ants, stem-borers and fungal pathogens: experimental tests of a fitness advantage in Piper ant-plants[END_REF]. On the other band, when deprived of their mutualistic Crematogaster ants, myrmecophytic Macaranga suffer from both shoot borers and pathogenic fungi [START_REF] Heil | Reduced chitinase activities in ant plants of the genus Macaranga[END_REF]. lndeed, ants' antifungal activity is well known [START_REF] Jolivet | Ants and plants: an example of coevolution[END_REF], [START_REF] Letoumeau | Ants, stem-borers and fungal pathogens: experimental tests of a fitness advantage in Piper ant-plants[END_REF], [START_REF] Heil | On benefits of indirect deferree: short-and long-term studies of antiherbivore protection via mutualistic ants[END_REF] and can be due to chernicals produced by the venom, the metapleural or the mandibular glands [63--66] or results from the activity of symbiotic bacteria [6 7]. On the plant side, it bas been noted that sorne myrmecophyte species have lost their intrinsic physiological defenses against fungal infection [START_REF] Letoumeau | Ants, stem-borers and fungal pathogens: experimental tests of a fitness advantage in Piper ant-plants[END_REF], [START_REF] Heil | Low chitinase activity in Acacia myrrnecophytes: a potential trade-off between biotic and chentical deferrees?[END_REF] . The spores of Fusarium can be disseminated by bath wind and insects, particularly Lepidoptera larvae that are resistant [START_REF] Teetor-Baxsch | Entomogenous Fusorium species[END_REF] , explaining why the P. guianalis caterpillars were not infected by Fusarium, while the host plant trichilia were. Because we did not note a difference in the amount of herbivory between ant-inhabited and ant-free CtcropitJ, one can hypothesize that Fusarium might be the main selective driving force in the present situation. In that case, the earlier the C. obtusa treelets shelter mutualistic Azteca colonies, the more they will grow due to the antifugal activity of the ants (particularly by suppressing spore germination [START_REF] Stow | Chemical and genetic defenses against disease in insect societies[END_REF]). Later, as the trees grow and their ability to synthesize secondary antiherbivore compounds lessens, Azteca workers, that belong to larger and larger colonies, will take over and provide their host trees with biotic protection from herbivorous insects [START_REF] Davidson | The evolutionary ecology of symbiotic aniplant relationships[END_REF], [START_REF] Davidson | Cecropia and its biotic defenses[END_REF], [START_REF] Dejean | Predation and aggressiveness in host plant protection: a generalization using ants from the genus A;:,teca[END_REF]. We cannot exclude that the occupancy by caterpillars could be favored by the plant's characteristics or rrùcro-environmental conditions rather than by the competitive and colonizing abilities of the insects. Nevertheless, because C. obtusa is a pioneer species that develops in large numbers in recently cleared areas, sorne insect species with a high rate of dispersal can be the flrst to reach the resources provided by the trees (see [START_REF] Bérenger | Special relationships of certain predatory Heteroptera Reduviidae with plants. First known case of a phytophagous Harpactorinae[END_REF] for Reduvidae feeding on C. obtura FBs before the installation of AQ.!ca colonies). It is thus likely that a temporal priority enabled P. guian4lis caterpillars to install themselves on certain trees prior to the arrivai of the plant's mutualistic Azteca ants with which they are involved in competitive exclusion (see also [START_REF] Palmer | Burning bridges: priority effects and the persistence of a competitively subordinate acacia-ant in Laikipia, Kenya[END_REF], [START_REF] Wilbur | Priority effects in experimental pond communities: responses of Hyla to B'!fo and Rima[END_REF] for a temporal priority concerning ant-myrmecophyte mutualisms).

Caterpillars

When caterpillars do successfully colonize a tree, severa! overlapping generations can be observed-the youngest sheltering under the stipules developing around the terrrùnal bud, the trunk and the youngest leaf, and the oldest in the host tree domatia. During the hours of FB production, both young and old caterpillars share the FBs on the trichilia. Smaller caterpillars likely benefit from the silk woven above the trichilia by larger mates as protection from competing ants, predators and/ or parasitoids which seem repelled, and so do not walk on it {pers. obs.). The overlap between different generations of caterpillars plus the fact that certain trees can be occupied during severa! years imply that female P. guian4lis moths lay eggs on trees already sheltering caterpillars.

When an incipient Azttca colony successfully colonizes their host tree, large caterpillars seem to deny the first workers access to the most productive trichilia by weaving silk above them. They, thus, indirecdy slow dawn colony growth as these foraging workers only have access to the lower, less-productive trichilia often already covered by F. monili:forme. Consequendy, the development of the colonies depends mosdy on the ants attending hemipterans in the internodes of the host trees (if any). So, although caterpillars can delay Azteca colonization, the Cecropia trees are fmally exclusively occupied by Azteca ants. Furthermore, once Azteca colonizes a Cecropia tree, the workers exploit the FBs on the upper, mostproductive trichilia and patrol the foliage, rendering the situation irreversible by preventing colonization by caterpillars. lndeed, the Azteca workers, that are able to capture insects the size of a female P. guianalis math [START_REF] Dejean | Predation and aggressiveness in host plant protection: a generalization using ants from the genus A;:,teca[END_REF], probably destroy any insect eggs that have been successfully laid on their host plant's foliage (see [START_REF] Djiéto-Lordon | Symbiotic mutualism with a community of opportunistic ants: protection, competition, and patterns of tree occupancy by ants associated with the myrrnecophyte Barteria nigritana (Passifloraceae)[END_REF] and references therein).

ln conclusion, P. guianalis caterpillars are able to "break the code" [START_REF] Letoumeau | Code of ant-plant mutualism broken by parasite[END_REF] of the Azteca-Cecropia mutualism by recognizing the prostomata and exploiting the resources Cecropia normally supplies to mutualistic Azteca; they even induce greater FB production. Although no higher herbivory rates were noted, these caterpillars are ineffective in keeping a fungus from developing on the trichilia of their host trees, something that mutualistic Azteca ants can do. By denying mutualistic ants access to FBs and young leaves, P. guianalis caterpillars become a more formidable competitor of mutualistic ants and so are parasites of both Cecropia saplings and the Azteca-Cecropia mutualism.

  1.15 rn to 1.30 rn tall (N = 61 0) between 2000 and 2006 in a cleared 1.5 ha zone situated near the dam, and on plants growing along a straight, recently-opened dirt raad near the Montagne rks singes (N = 64) between 2006 and 2009.

Figure 1 .

 1 Figure 1. Trlchlllum of Cecropia. A-Azteca affari worloer.; removing food bodies (arrow) from a Cecropia obtusa trichilium. 8-Pseudocabima guianalis caterpillars occupying the upper part of a C obtusa. They gnawed an entrance hole into the prostomata (p) in order to enter into the hollow internodes. Secreted strands of silk caver the trichilium and keep the witherecl stipules (DS) from falling. C-Trichilium of C obtusa infested by Fusarium moniliforme. D-Penetration by a filament of F. moniliforme (arrows) inside a food body (FB). E-F. moniliforme spreading throughout a food body (FB and black arrows). The cells of the trichilium at the base of the FB seem to react to the presence of fungus (white arrow).

  lB; Fig. S 1 a). From their third instar, ca. 1.5-cm-long caterpillars, like ants, gnawed the prostomata in arder to shelter in the last internode. They wove a silk shield above the upper part of the tree trunk and the youngest trichilium (Fig. lB, Fig. Si b-d), and left the domatia only to feed on the FBs under the shelter of strands of silk. Pupation occurred inside the domatia. Larval and pupal development took about 30 days. Just-emerging maths leave the trunk by flying out through the stomata that are widened by the caterpillars when they are in their last larval stage.

  Figure ID, a filament of F. moniliforme can be seen entering into an FB. The bordering cells are in necrosis, as are the frrst reserve cells. The trichomes around the FB react strongly to the presence of the fungus, saturating their cell walls with lignin (Fig. lE); whereas the

Figure 2 .

 2 Figure 2. Trlchlllum lnfested by Fusarium monilifomle. Percentages of Cecropia obtusa saplings whose trichilia were attacked by Fusarium moniliforme in three situations: saplings sheltering an Azteca colony, saplings sheltering Pseudocabima guianalis caterpillars, and unoccupied saplings (N = number of saplings in each case). Statistical comparisons; Kruskai-Wallis test; H 3 610 = 338.1; P<0.001; Dunn's multiple comparison test, different letters indicate significant differences at P<0.001.

  or A. ovaJi.ceps colonies on 15 and 10 trees, respectively. At the end of the experiment-so 3 years after the beginning of the survey-six trees still sheltered caterpillars. During this en tire lapse of time, the trees occupied by A . a!fari or A. ovatictps were never colonized by caterpillars. In J uly 2009, four CecropiJJ trees still sheltered caterpillars.

Figure 3 .

 3 Figure 3. Host successional patterns for Ceaopm treelets. Host successional patterns for each Cecropia sapling monitored during the 3year survey on the dirt road near the Montagne des singes. Dashes correspond to trees shelterlng both caterplllars and A. a/far/. Trees were

Figure 5 .

 5 Figure 5. Helght of CemJpiatreelets. Mean height of trees (±SE) that sheltered colonies of one of the two Azteca species, Pseudocobima guionolis caterpillars, or that sheltered neither Azteca nor caterpillars during the experimental period. Statistical comparisons (normality and equal variance tests passed); ANOVA: F 2

  Comparison of the mean food body production per leaf and per day (±SE) by the youngest tricl1ilia on Cecropia obtusa saplings during 20 successive days in three situations: saplings sheltering an Azteco alfori colony, saplings sheltering Pseudocabima guionolis caterpillars, and unoccupied saplings (10 individuals in each case). Statistical comparisons; repeated measures ANOVA: F 2 30 =64.81; P<0.001; Newman-Keuls' post-hoc test: different letters indicate significant differences at P<0.001.

			1
	Azteca	Caterpillars	Unoccupied

Figure 4. Food body production.

Acknowledgments

We are grateful to the Laboratoire Environnement de Petit Saut for its logistical help, A. Yockey-Dejean for proof-reading the manuscript, M. A.

Solis (USDA, Beltsville, Maryland, USA) and E. Bury (Muséwn National d'Histoire Naturelle, Paris, France) for identifYing the P. guianalis and F. moniliforme, respectively. We would also like to thank anonyrnous referees for their helpful comments.

Supporting Information


Huaqiang Shu

Laurent Ros

Eric Pierre Simon

† H Shu

Third-order Complex Amplitudes Tracking Loop for Slow Flat Fading Channel On-Line Estimation *

Keywords: Channel estimation, Rayleigh fading, Jakes' spectrum, Random Walk model (RW), Phase-locked loop (PLL), Kalman filter (KF)

This paper deals with channel estimation in tracking mode over a flat Rayleigh fading channel with Jakes' Doppler Spectrum. Many estimation algorithms exploit the time-domain correlation of the channel by employing a Kalman filter based on a first-order (or sometimes second-order) approximation model of the time-varying channel. However, the nature of the approximation model itself degrades the estimation performance for slow to moderate varying channel scenarios. Furthermore, the Kalman-based algorithms exhibit a certain complexity. Hence, a different model and approach has been investigated in this work to tackle all of these issues. A novel PLL-structured thirdorder tracking loop estimator with a low complexity is proposed. The connection between a steady-state Kalman filter based on a random walk approximation model and the proposed estimator is first established. Then, a sub-optimal mean-squared-error (MSE) is given in a closed-form expression as a function of the tracking loop parameters. The parameters that minimize this sub-optimal MSE are also given in a closed-form expression. The asymptotic MSE and Bit-Error-Ratio (BER) simulation results demonstrate that the proposed estimator outperforms the first and second order Kalman-based filters reported in literature. The robustness of the proposed estimator is also verified by a mismatch simulation.

I. INTRODUCTION

Channel estimation is a fundamental task for a wireless communication receiver. This paper deals with channel path Complex Amplitude (CA) estimators in tracking mode. The statistical channel model assumed in this paper to describe the wireless channel is the Rayleigh fading channel with Jakes' Doppler spectrum model [START_REF] Jakes | Microwave Mobile Communications[END_REF] (also called the Clarke Model [START_REF] Clarke | A statistical theory of mobile radio reception[END_REF]). It is widely accepted in the literature for frequency-flat correlated fading channels.

Many channel path CA tracking algorithms use a Kalman filter (KF). KF-based algorithms exhibit a certain complexity and the design of a KF requires to dispose of a linear recursive state-space representation of the channel. However, the exact Clarke model does not admit such a representation. An approximation often employed in the literature consists of approaching the fading process as Auto-Regressive (AR) [START_REF] Baddour | Autoregressive modeling for fading channel simulation[END_REF], in the perspective to design a KF [START_REF] Komninakis | Multi-input multi-output fading channel tracking and equalization using Kalman estimation[END_REF]- [START_REF] Ros | Second-order modeling for Rayleigh flat fading channel estimation with Kalman Filter[END_REF]. The larger the order of the model, the better the approximation of the actual fading statistics, but also the larger the complexity. So, despite its complexity, KF-based algorithms do not ensure optimal performance if the structure or the tuning of the approximation model are not well suited, as developed hereafter. A widely used channel approximation model results from a first-order Auto-Regressive model (AR1) as recommended by [START_REF] Wang | On veryfying the first-order Markovian assumption for a Rayleigh fading channel model[END_REF], combined with a Correlation Matching (CM) criterion to fix the AR1-coefficient (equal then to the standard Bessel AR1-coefficient, J 0 (2πf d T ), for a given normalized Doppler frequency f d T ). The KF channel estimator resulting from this choice, called AR1 CM -KF in this paper, was used in several papers concerning various systems such as in Multiple-Input-Multiple-Output (MIMO) systems [START_REF] Komninakis | Multi-input multi-output fading channel tracking and equalization using Kalman estimation[END_REF], [START_REF] Liu | Space-time coding and Kalman filtering for time-selective fading channels[END_REF], or in Orthogonal Frequency Division Multiplexing (OFDM) systems [START_REF] Chen | Kalman-filter channel estimator for OFDM systems in time and frequency-selective fading environment[END_REF], [START_REF] Al-Naffouri | An EM-Based Forward-Backward Kalman Filter for the Estimation of Time-Variant Channels in OFDM[END_REF], [START_REF] Hijazi | Joint data QR-detection and Kalman estimation for OFDM time-varying Rayleigh channel complex gains[END_REF], [START_REF] Simon | Joint Carrier Frequency Offset and Channel Estimation for OFDM Systems via the EM Algorithm in the Presence of Very High Mobility[END_REF]. The AR1 CM -KF seems to be convenient for the very high mobility case, leading to quasi-optimal channel estimation performance compared to lower bounds, as seen, for example, in [START_REF] Hijazi | Joint data QR-detection and Kalman estimation for OFDM time-varying Rayleigh channel complex gains[END_REF]- [START_REF] Tugnait | Doubly selective channel estimation using exponential basis models and subblock tracking[END_REF] (in these works the AR1 CM -KF is actually used to track the Basis Extension Model coefficients of the high speed channel). But for most conventional Doppler speeds whereby the channel variation within one symbol duration can be neglected (i.e. f d T ≤ 10 -2 , as in [START_REF] Komninakis | Multi-input multi-output fading channel tracking and equalization using Kalman estimation[END_REF]- [START_REF] Ros | Second-order modeling for Rayleigh flat fading channel estimation with Kalman Filter[END_REF], [START_REF] Ros | Paths complex gain tracking algorithms for ofdm receiver in slowly-varying channels[END_REF]), the AR1 CM -KF estimator usually exploited in the literature is far from being effective [START_REF] Barbieri | On the ARMA approximation for fading channels described by the Clarke model with applications to Kalman-based receivers[END_REF]. This poor performance has been recently explained analytically in [START_REF] Ghandour-Haidar | On the use of first-order autoregressive modeling for Rayleigh flat fading channel estimation with Kalman filter[END_REF], mainly because the CM criterion is shown to be inappropriate to tune the AR1-coefficient in slow or moderate fading scenario (since the choice of J 0 (2πf d T ) ≈ 1 -1 4 (2πf d T ) 2 for the AR1 coefficient is too close to the value 1 to ensure a good tradeoff between tracking ability and noise mitigation). A better tuning of the AR1-coefficient can focus on minimizing the estimation variance in output of the KF as proposed in [START_REF] Barbieri | On the ARMA approximation for fading channels described by the Clarke model with applications to Kalman-based receivers[END_REF] (with analytic MSE performance for a given Doppler and Signalto-Noise Ratio (SNR) scenario in [START_REF] Ghandour-Haidar | On the use of first-order autoregressive modeling for Rayleigh flat fading channel estimation with Kalman filter[END_REF]), i.e. using a minimum asymptotic variance (MAV) criterion without imposing the CM constraint. The resulting estimator is called AR1 M AV -KF in this paper.

On the other hand, [START_REF] Ros | Second-order modeling for Rayleigh flat fading channel estimation with Kalman Filter[END_REF] analytically shows that the MSE performance of a KF can still be improved by switching from the AR1 model to an integrated random walk (RW) model (also called integrated Brownian model) for the approximation model. Such a model was a second-order approximation model that better takes into account the fact that the exact channel CA continues in a given direction during several symbols for low f d T and then exhibits a strong trend behaviour. The Kalman estimator based on this special second-order model is called RW2-KF in this paper.

So far, all our discussion dealt with KF-based algorithms, but we now wish to obtain simpler adaptive algorithms. This can be done by deriving steady-state versions of the KF based on a random-walk (RW) approximation model, yielding a recursive linear filter with constant coefficients. In general, such algorithms converge slower than the KF, but can reach the same asymptotic performance in tracking mode. To start with, we should bear in mind that the LMS algorithm, which is the most popular adaptive algorithm, can be viewed as a steadystate version of a KF based on a first-order RW approximation model [START_REF] Lindbom | Tracking of time-varying mobile radio channels .1. the wiener lms algorithm[END_REF]. The second-order LMS was first proposed by [START_REF] Gazor | Prediction in LMS-type adaptive algorithms for smoothly time varying environments[END_REF] in a channel estimation context. It was derived as the steadystate version of a KF based on a second-order integrated RW approximation model (RW2-KF). However, the author does not specify how to tune the two constant coefficients of the model. [START_REF] Xue | Second-order LMS based wireless channel tracking: implementation under imperfect carrier synchronization[END_REF] proposed a method to obtain optimal coefficients, and [START_REF] Ros | Paths complex gain tracking algorithms for ofdm receiver in slowly-varying channels[END_REF], [START_REF] Ros | Complex amplitudes tracking loop for multipath slow fading channel estimation in OFDM systems[END_REF] presented the CA tracking algorithm and its optimization from a second-order phase-locked-loop (PLL) point of view. Indeed, as it has been already shown by Driessen [START_REF] Driessen | DPLL bit synchronizer with rapid acquisition using adaptive Kalman filtering techniques[END_REF] and Christiansen [START_REF] Christiansen | Modeling of PRML timing loop as a Kalman filter[END_REF] in the case of the phase estimation problem, a proportional-integral (second-order) PLL has the same structure as the KF when considering the second-order integrated RW model, and thereafter the closed-form Kalman gain expression is given in [START_REF] Patapoutian | On phase-locked loops and Kalman filters[END_REF]. The algorithm of [START_REF] Ros | Paths complex gain tracking algorithms for ofdm receiver in slowly-varying channels[END_REF], [START_REF] Ros | Complex amplitudes tracking loop for multipath slow fading channel estimation in OFDM systems[END_REF] is called the second-order complex amplitude tracking loop (RW2-CATL).

In this paper, we propose and study a low-complexity flat fading channel estimator based on a third-order integrated RW model (RW3). The contributions of this paper are multi-fold with the purpose to solve the following questions: Why could a PLL-structured estimator asymptotically work like a traditional one (e.g. a KF) ? What is the relationship between them ? How much can a well-chosen third-order CATL outperform, in terms of asymptotic MSE performance, the more complex KF based only on first-or second-order models (e.g. AR1 CM -KF, AR1 M AV -KF, or RW2-KF) ? How to tune properly and in a simple way the coefficients of such a third-order CATL, assuming Rayleigh-Jakes channel and a given scenario of f d T and SNR? What is then the closed form expression of the MSE of such a channel estimator ? How does a distorted foreknowledge of Doppler frequency or noise power information influence the estimator performance ?

Section II gives the system model. In section III, we propose and analyze a third-order Complex-Amplitude-Tracking Loop, called RW3-CATL, for the time-varying channel estimation. Section IV describes the proposed method to correctly tune the loop coefficients, and section V validates our model and assumptions by means of MSE and BER simulations.

II. MODEL AND ESTIMATION OBJECTIVE

We consider the estimation of a flat Rayleigh fading channel in a digital modulation system. The discrete-time observation is:

r (n) = α (n) • x (n) + v (n) , ( 1
)
where n is the symbol time index;

x (n) = a (n) + jb (n) with a (n) , b (n) ∈ ℜ is the transmitted phase modulated (M -PSK) or
quadrature amplitude modulation (M -QAM) symbol, the sequence of transmitted symbol is assumed to be zero-mean and stationary with normalized variance : E x (n) 2 = σ 2 x = 1; v (n) is a zero-mean additive white circular complex Gaussian noise with variance σ 2 v ; and α (n) is a zero-mean circular Gaussian channel Complex Amplitude with variance σ 2 α . Note that this model can be applied to more advanced systems such as OFDM system, where α would then represent the channel gain to be estimated at one pilot frequency as in [START_REF] Chen | Kalman-filter channel estimator for OFDM systems in time and frequency-selective fading environment[END_REF], and x (n) could then be a known (or pilot) symbol in the channel estimation perspective (data-aided scenario).

The normalized Doppler frequency of this channel is f d T , where f d is the Doppler frequency and T is the symbol period. A Jakes' Doppler spectrum is assumed for this channel:

Γ α (f ) =    σ 2 α πf d 1-f f d 2 , if |f | < f d 0, if |f | ≥ f d . (2)
The autocorrelation coefficient of the stationary CA α is then defined for lag q by: R α

[q] = E α (n) • α (n-q) * = σ 2 α J 0 (2πf d T • q), (3
) where J 0 is the zeroth-order Bessel function of the first kind. Given the observation model (1) and the Doppler spectrum statistical constraint (2) for the dynamic evolution of the CA, we look for an on-line unbiased estimation α(n

) of α (n) . The MSE σ 2 ǫ def = E ǫ (n) 2 of the estimation error ǫ (n) def = α (n) - α(n) will be investigated.

III. COMPLEX AMPLITUDE TRACKING LOOP

A. From steady-state KF to PLL-structured CATL 1) RW3 model and RW3-KF: similar to the method presented in [START_REF] Gazor | Prediction in LMS-type adaptive algorithms for smoothly time varying environments[END_REF], in the slow to moderate fading scenario, we can firstly depict a steady-state KF [START_REF] Kay | Fundamentals of Statistical Signal Processing: Estimation Theory[END_REF], but based here on a RW3 model, denoted as RW3-KF. The model can be formulated in discrete time update equations as:

α(n) =α (n-1) + δ (n-1) + 1 2 ξ (n-1) , (4)
δ (n) =δ (n-1) + ξ (n-1) , (5) ξ (n) =ξ (n-1) + u (n) , (6)
where u (n) is a zero mean circular complex Gaussian with variance σ 2 u . The equation ( 4) is the discrete version of the Taylor series expansion of a continuous signal. So in this approximation model, the approximate process of α (n) , denoted α(n) , is updated by a time increment of δ (n-1) + 1 2 ξ (n-1) every symbol period with δ (n) and ξ (n) respectively approximate the first-and second-order derivative of the continuous signal. The observation model is given by (1), which could be rewritten in separating the parameter α (n) from the transmitted signal as:

y (n) = α (n) + w (n) , (7)
with

y (n) = r (n) x (n) and w (n) = v (n)
x (n) . Note that w (n) remains a zero-mean additive white circular complex noise with variance

σ 2 w = K mod • σ 2 v where K mod = E 1 x (n)
2 is a constant factor, known for a given modulation scheme. For a special case of constant-energy modulation, e.g. M -PSK, we would have σ 2 w = σ 2 v and K mod = 1. Then we reform [START_REF] Chen | Kalman-filter channel estimator for OFDM systems in time and frequency-selective fading environment[END_REF] and ( 4)∼(6) in matrix form as:

y (n) =Sa (n) + w (n) , (8)
a (n) =Ma (n-1) + u (n) , (9)
with the selection vector S = 1 0 0 , the state vector a (n)

def = α(n) δ (n) ξ (n)
T , the state noise vector

u (n) = 0 0 u (n)
T and the 3 × 3 evolution matrix M = 1 11 2 ; 0 1 1; 0 0 1 . The observation equation ( 8) and the state evolution equation ( 9) compose the statespace model of the KF. The corresponding two-stage KF equations are then written as:

Time Update Equations â(n|n-1) = Mâ (n-1|n-1) , (10)
P (n|n-1) = MP (n-1|n-1) M T + U, (11)
Measurement Update Equations

K (n) = P (n|n-1) S T SP (n|n-1) S T + σ 2 w , (12)
â(n|n) = â(n|n-1) + K (n) (y (n) -Sâ (n|n-1) ), (13)
P (n|n) = (I -K (n) S)P (n|n-1) , (14)
with â(n|n-1) def = α(n|n-1) δ(n|n-1) ξ(n|n-1)

T the prediction of state vector, â(n|n) def = α(n|n) δ(n|n) ξ(n|n)

T the estimation of state vector, and the 3 × 3 state noise matrix is defined as U = 0 0 0; 0 0 0; 0 0 σ 2 u , K (n) is the Kalman gain vector, P (n|n-1) , P (n|n) are respectively the covariance matrices (both 3 × 3) of the prediction error and the estimation error. Define also the error signal as:

v ǫ(n) = y (n) -Sâ (n|n-1) . ( 15
)
Note that the computation of the error signal requires the knowledge of x (n) since y (n) is the equalized version of the received signal r (n) . Two different scenarios can then be considered : either treat x (n) as pilot symbols, or use the decisions instead. In the decision-directed scenario,x (n) is replaced by the a priori decision x(n|n-1) to compute

y (n) = r (n)
x(n|n-1) , where x(n|n-1) is decided from the previous estimation of the channel. The value of x(n|n-1) depends on the applied modulation schemes, e.g. in QPSK modulation,

x(n|n-1) = √ 2 2 •sgn{Re(α * (n|n-1) •r (n) )} + j √ 2 2 •sgn{Im(α * (n|n-1) •r (n) )
}, with sgn{•} the sign function. In this work, we concentrate on the performance of the channel estimator 1 . So the theoretical analysis is derived assuming symbols are known (pilot-aided scenario) or perfectly decided, and the effect of decision error (in the decision-directed scenario) will be observed in the simulation part.

2) Time-domain equations of the steady-state RW3-KF:

Since the linear model ((9)(8)) is observable and controllable [START_REF] Chui | Kalman Filtering with real-time applications[END_REF], an asymptotic regime is quickly reached [START_REF] Kay | Fundamentals of Statistical Signal Processing: Estimation Theory[END_REF]. In other words, K (n) converges to a constant when n is large enough, i.e.

K (n) = K (n+1) = K (∞) def = k 1 k 2 k 3 T . ( 16
)
By combining ( 10) and ( 13), we obtain the measurement update equations of α with the steady-state RW3-KF:

α(n|n) = α(n-1|n-1) + δ(n-1|n-1) + 1 2 ξ(n-1|n-1) + k 1 v ǫ(n) , (17) δ
(n|n) = δ(n-1|n-1) + ξ(n-1|n-1) + k 2 v ǫ(n) (18) ξ(n|n) = ξ(n-1|n-1) + k 3 v ǫ(n) , (19)
with v ǫ(n) defined by [START_REF] Tugnait | Doubly selective channel estimation using exponential basis models and subblock tracking[END_REF].

3) The steady-state RW3-KF as an RW3-CATL: Note that k 2 and k 3 are embedded in the derivatives of the CA in [START_REF] Lindbom | Tracking of time-varying mobile radio channels .1. the wiener lms algorithm[END_REF], this makes it difficult to control directly the estimator through the use of the gains. It is however interesting to note that if we study these equations in Z-domain, the gains can be separated from the variables, and this allows us to analyse the estimator as a PLL-structured tracking loop with a PII filter (or Proportional-double-Integral filter). The expression of (17)(18) [START_REF] Xue | Second-order LMS based wireless channel tracking: implementation under imperfect carrier synchronization[END_REF] in Z-domain are:

α(z)(1 -z -1 ) = δ(z)z -1 + 1 2 ξ(z)z -1 + k 1 v ǫ (z), (20)
δ(z)(1 -z -1 ) = ξ(z)z -1 + k 2 v ǫ (z), (21) ξ
(z)(1 -z -1 ) = k 3 v ǫ (z). (22)
Substituting ( 21) [START_REF] Christiansen | Modeling of PRML timing loop as a Kalman filter[END_REF] in [START_REF] Ros | Complex amplitudes tracking loop for multipath slow fading channel estimation in OFDM systems[END_REF], we have:

α(z)(1-z -1 ) = k 1 + (k 2 + 1 2 k 3 )z -1 1 -z -1 + k 3 z -2 (1 -z -1 ) 2 v ǫ (z), (23)
The equation [START_REF] Patapoutian | On phase-locked loops and Kalman filters[END_REF] shows the final estimate α as a filtered version of the error signal v ǫ , as in a PLL. Let us define:

µ 1 = k 1 , (24)
µ 2 = k 2 + 1 2 k 3 , (25)
µ 3 = k 3 , (26)
and

v Lag1 (z) = v ǫ (z) 1 -z -1 , ( 27
) v Lag2 (z) = v ǫ (z) (1 -z -1 ) 2 , (28)
(23) can then be rewritten as:

α(z)(1 -z -1 ) = µ 1 v ǫ (z) + µ 2 v Lag1 (z)z -1 + µ 3 v Lag2 (z)z -2 , ( 29
) or equivalently in discrete-time domain:

α(n|n) = α(n-1|n-1) + µ 1 v ǫ(n) + µ 2 v Lag1(n-1) + µ 3 v Lag2(n-2) .
(30) From ( 13), we have:

α(n|n) = α(n|n-1) + k 1 v ǫ(n) . ( 31
)
Fig. 1: Equivalent structure of the RW3-CATL By combining ( 30), ( 31) and ( 24), we get the prediction equation:

α(n+1|n) = α(n|n) + µ 2 v Lag1(n) + µ 3 v Lag2(n-1) . ( 32
)
These equations, derived from the steady-state RW3-KF, can be slightly rearranged to resemble the most traditional form of a PLL-like structure, as shown in the next subsection.

4) Final time-domain equations of the RW3-CATL:

We can now easily sum up the equations of the proposed third-order Complex Amplitude Tracking Loop (CATL), as:

Error signal: v ǫ(n) = y (n) -α(n|n-1) , ( 33
)
Loop Filter:

v Lag1(n) = v Lag1(n-1) + v ǫ(n) , (34) v Lag2(n) = v Lag2(n-1) + v Lag1(n) , (35) v c(n) = µ 1 v ǫ(n) + µ 2 v Lag1(n) + µ 3 v Lag2(n-1) , (36)
Numerically Controlled Generator:

α(n+1|n) = α(n|n-1) + v c(n) , (37)
Final estimate:

α(n|n) = α(n|n-1) + µ 1 v ǫ(n) . (38)
Here (33) is from ( 15), ( 38) is obtained from (31) using ( 24), (34) and ( 35) are respectively a result of ( 27) and ( 28), and finally (36) and (37) are derived from (32) using (38). The structure of our PLL-like estimator based on the discrete-time equations (33)∼(37) is shown in Fig. 1. As in a digital PLL, the RW3-CATL is composed of an error detector, a loop filter and a numerically controlled generator.

The error detector compares firstly the received signal with a reference signal equal to the previous prediction of the parameter, α(n|n-1) . It delivers the error signal v ǫ(n) to the proportional-double-integral filter

F P II (z) = µ 1 + µ2 1-z -1 + µ3z -1
(1-z -1 ) 2 which is controlled by the three filter coefficients µ 1 , µ 2 and µ 3 . As the steady-state Kalman gains are real positive (this can be proved by solving the Ricatti equations), the loop filter coefficients µ 1 , µ 2 , µ 3 are also real positive values. The 34) and ( 35) are respectively the first-order and the second-order digital integrations (or accumulations) of the error signal v ǫ(n) . The loop filter output v c(n) is then used as a command by the numerically controlled generator to generate the next prediction α(n+1|n) from the previous one α(n|n-1) , according to the integration process (37). This structure is similar to the one presented in [1]2 , which is deduced from a standard third-order DPLL [START_REF] Lindsey | A survey of digital phase-locked loops[END_REF]. We will also demonstrate the equivalence between the RW3-CATL and the standard third-order DPLL in the following analysis. However, unlike the conventional PLL, the final output of the CATL is not the prediction (or a priori estimate) α(n|n-1) but the final (or a posteriori) estimate of the complex amplitude α(n|n) , according to equation (38), as in the KF principle. Thus an additional correction branch is added, represented by the dashed line in Fig. 1. The RW3-CATL is a time-invariant filter, it hence does not need to update its coefficients. On the contrary, the RW3-KF has to update its coefficients (the Kalman gain and the error variances) every symbol period.

signals v Lag1(n) , v Lag2(n) defined in (
Note that, thanks to the second integration in v Lag2(n) , this digital third-order loop does not exhibit acceleration-dependent steady-state error in the case of second-order variations of the CAs. In other words, the RW3-CATL characterizes the variation of the channel parameter by taking into account its slope and its curvature, while second-order loops only consider the slope.

B. General properties 1) Closed-loop transfer function of RW3-CATL:

By combining [START_REF] Chen | Kalman-filter channel estimator for OFDM systems in time and frequency-selective fading environment[END_REF] and [START_REF] Tugnait | Doubly selective channel estimation using exponential basis models and subblock tracking[END_REF], we have:

v ǫ(n) = α (n) -α(n|n-1) + w (n) . ( 39
)
The error signal is thus a combination of the prediction error (α (n) -α(n|n-1) ) and the channel noise. By combining (38) and (39), we obtain the error signal -estimation error relation:

v ǫ(n) = 1 1 -µ 1 • (α (n) -α(n|n) ) + 1 1 -µ 1 • w (n) . (40)
Transform (40) to the Z-domain:

v ǫ (z) = 1 1 -µ 1 • (α(z) -α(z)) + 1 1 -µ 1 • w(z). ( 41
)
By combining (27)(28)(29), we get:

α(z)(1 -z -1 ) = [µ 1 + µ 2 • z -1 1 -z -1 + µ 3 • z -2 (1 -z -1 ) 2 ] • v ǫ (z). (42)
Then substituting (41) in (42) leads to:

α(z) = L(z) • α(z) + L(z) • w(z), (43)
where L(z) is the Z-domain transfer function of the 3rd-order CATL defined by (44) with

F (z) = µ 1 + µ2•z -1 1-z -1 + µ3•z -2 (1-z -1 ) 2 . L(z) = F (z) (1 -µ 1 )(1 -z -1 ) + F (z) = (µ 1 -µ 2 + µ 3 )(1 -z -1 ) 2 + (µ 2 -2µ 3 )(1 -z -1 ) + µ 3 (1 -µ 1 )(1 -z -1 ) 3 + [(µ 1 -µ 2 + µ 3 )(1 -z -1 ) 2 + (µ 2 -2µ 3 )(1 -z -1 ) + µ 3 ] (44) L(z) = (m + 2)ζω n T • (1 -z -1 ) 2 + (1 + 2mζ 2 )(ω n T ) 2 • (1 -z -1 ) + mζ(ω n T ) 3 (1 -z -1 ) 3 + (m + 2)ζω n T • (1 -z -1 ) 2 + (1 + 2mζ 2 )(ω n T ) 2 • (1 -z -1 ) + mζ(ω n T ) 3 (45)
In order to be able to compare with a classic analog thirdorder PLL, L(z) can be rewritten in a more interpretable form3 in (45) as a function of the natural pulsation ω n = 2πf n with f n the natural frequency, the damping factor ζ and the capacitance ratio m, where:

(m + 2) • ζω n T = µ 1 -µ 2 + µ 3 1 -µ 1 , (46)
(1 + 2mζ 2 ) • (ω n T ) 2 = µ 2 -2µ 3 1 -µ 1 , ( 47
) mζ • (ω n T ) 3 = µ 3 1 -µ 1 . ( 48
)
The capacitance ratio m is an additional factor for thirdorder PLL used to adjust the step response character [START_REF] Hu | Principle of Automatic Control (4th edition)[END_REF]. By comparing (44) and ( 45), (µ 1 , µ 2 , µ 3 ) can be expressed by (ω n , ζ, m) as:

µ 1 = (m + 2)ζω n T + (1 + 2mζ 2 )(ω n T ) 2 + mζ(ω n T ) 3 1 + (m + 2)ζω n T + (1 + 2mζ 2 )(ω n T ) 2 + mζ(ω n T ) 3 , ( 49
)
µ 2 = (1 + 2mζ 2 )(ω n T ) 2 + 2mζ(ω n T ) 3 1 + (m + 2)ζω n T + (1 + 2mζ 2 )(ω n T ) 2 + mζ(ω n T ) 3 , ( 50
)
µ 3 = mζ(ω n T ) 3 1 + (m + 2)ζω n T + (1 + 2mζ 2 )(ω n T ) 2 + mζ(ω n T ) 3 .
(51) 2) Stability: The condition of stability of the causal rational system L(z) is obtained when all the roots of the denominator polynomial are inside the unit circle. In view of the complexity of the third-order transfer function, we resort to a simplified Jury-Marden method [START_REF] Antoniou | Digital Signal Processing[END_REF].

The third-order denominator polynomial of L(z) in ( 44) is

D(z) = a 0 z 3 +a 1 z 2 +a 2 z+a 3 , with a 0 = 1, a 1 = µ 1 +µ 2 -3, a 2 = 3 -2µ 1 -µ 2 + µ 3 , a 3 = µ 1 -1,
the criterion of stability are given by:

• D(1) > 0; • D(-1) < 0; • a 0 > |a 3 |, |c 0 | > |c 2 |, with c 0 = a 0 a 3 a 3 a 0 , c 2 = a 2 a 3 a 1 a 0 .
After some manipulations, we obtain the condition of stability of the RW3-CATL, i.e., L(z) is stable if and only if:

0 < µ 1 < 2, ( 52
)
0 < µ 3 < µ 1 µ 2 , ( 53
)
4µ 1 + 2µ 2 + µ 3 < 8. (54)
We can rewrite L(z) in the frequency-domain, by setting z = e pT , with p = jω = j2πf . Assuming slow reaction of the loop during one symbol time T (i.e. f n T ≪ 1), the digital loop transfer function is close (approximation z -1 ≈ 1 -pT ) to the usual third-order low-pass transfer function in analog PLL ( [START_REF] Shan | Design and Implementation of Bandwidth Adaptable Third-Order All Digital Phase-Locked Loops[END_REF], eq.( 2)(4)):

L(e pT ) ≈ (m + 2)ζω n • p 2 + (1 + 2mζ 2 )ω 2 n • p + mζω 3 n p 3 + (m + 2)ζω n • p 2 + (1 + 2mζ 2 )ω 2 n • p + mζω 3 n .
(55) Fig. 2 shows the magnitude-frequency graph of the RW3-CATL transfer function and the third-order analog PLL transfer function, respectively given by ( 45) and (55) with different parameters. We can see that in the low-frequency domain (f T ≪ 1) and for loops with slow reaction (f n T ≪ 1), the two transfer functions match very well, and then the analog version gives a good approximation of the RW3-CATL transfer function.

C. Asymptotic mean squared error analysis

From (43) we know that the estimation error is zero-mean, thus the RW3-CATL is an unbiased estimator. By using the definition ǫ(z) = α(z) -α(z), (43) can be re-written as:

ǫ(z) = (1 -L(z)) • α(z) -L(z) • w (n) . ( 56
)
The variance of estimation error is therefore divided into two parts. One comes from the variation of the parameter α and the other comes from the loop noise w:

σ 2 ǫ = E{ǫ (n) • ǫ * (n) } = σ 2 ǫα + σ 2 ǫw . ( 57
)
The component σ 2 ǫα (dynamic error variance) results from the high-pass filtering (1 -L(z)) of the input CAs α (n) , so we have: where Γ α (f ) is the power spectral density (PSD) of α given by [START_REF] Jakes | Microwave Mobile Communications[END_REF]. And the component σ 2 ǫw (static error variance) results from the low-pass filtering L(z) of the input loop noise w (n) :

σ 2 ǫα = + 1 2T -1 2T Γ α (f ) • |1 -L(e j2πf T )| 2 df , (58)
σ 2 ǫw = + 1 2T -1 2T Γ w (f ) • |L(e j2πf T )| 2 df . ( 59
)
1) Static error variance σ 2 ǫw : Since the noise is assumed white, the PSD of noise Γ w (f ) = σ 2 w T is constant all over the system bandwidth. Thus (59) can be rewritten as:

σ 2 ǫw = σ 2 w • T + 1 2T -1 2T |L(e j2πf T )| 2 df B L , ( 60
)
where B L is the so-called equivalent noise bandwidth (doublesided normalized). B L can be derived (i.e. to evaluate the two-sided complex integral) by using the method presented in [START_REF] Winkelstein | Closed form evaluation of symmetric two-sided complex integrals[END_REF]. For a third-order system, B L is a sixth-degree algebraic expression of ω n T (see appendix A). But with the condition f n T ≪ 1 in our case, the higher order terms than ω n T are negligible, so that B L can be finally approximated as:

BL ≈ 2πfnT • (2m 3 ζ 4 + 12m 2 ζ 4 + 8mζ 4 + 6mζ 2 + 4ζ 2 + 1) 4m 2 ζ 3 + 8mζ 3 + 4ζ B . ( 61
)
2) Dynamic error variance σ 2 ǫα : The expression for σ 2 ǫα is given by (58) in an integral form. To obtain an analytical expression of σ 2 ǫα , the general assumptions f d ≤ f n ≪ 1/T is helpful which allows us to do some approximations. By using this assumption, we could approach the term |1 -L(e j2πf T )| 2 by two asymptotes (see appendix B for derivation):

|1 -L(e j2πf T )| 2 ≈ f 6 (mζ) 2 f 6 n if f ≪ f n , 1 if f ≫ f n . ( 62
)
and for the special point f = f n , we have:

|1 -L(e j2πfnT )| 2 ≈ f 6 n (mζ) 2 f 6 n + [m 2 (4ζ 2 -1) + 4].f 6 n .
(63) The two straight asymptotes of the log magnitude in (62) are also evidently shown in Fig. 2. Note that the case of f ≫ f n needs not to be taken into account for the integral computation (58), because for the Rayleigh-Jakes model, the spectrum of α, Γ α , has a bounded support, i.e. |f | ≤ f d , and for a good tracking of α, we assume f n greater or equal to f d . So we use the low frequency asymptote for our approximation, yielding:

|1 -L(e j2πf T )| 2 ≈ f 6 (mζ) 2 f 6 n , if f ≤ f n . ( 64
)
And to obtain an acceptable approximation around f n , we impose that the function |1 -L(e j2πf T )| 2 crosses the low frequency asymptote at f = f n , yielding the following constraint (see (63)):

m 2 (4ζ 2 -1) + 4 = 0. (65)
We can see in Fig. 2 that the linear approximation (64) is quite good, especially when the constraint (65) is applied. Thus the dynamic error variance σ 2 ǫα becomes4 :

σ 2 ǫα ≈ 1 (mζ) 2 +f d -f d Γ α (f ) • f f n 6 • df . ( 66
)
Further more, with the hypothesis that f ≤ f n , a variable change cos(θ) = (f /f d ) permits us to compute an exact analytical solution of the integral (66) as:

σ 2 ǫα ≈ 5 16 • 1 (mζ) 2 • f d f n 6 • σ 2 α . (67)
IV. COMPUTATION OF THE RW3-CATL PARAMETERS

The MSE of the RW3-CATL σ 2 ǫ (57) is minimized for a set of optimal parameters (m, ζ, f n ) obtained through a threedimension optimization. A closed-form analytical expression for this problem can be obtained if we impose the constraint (65), leading to a sub-optimal solution. This constraint minimization is solved with the method of Lagrange multipliers. In section V, we show that the sub-optimal solution yields a performance very close to that of the optimal solution.

By combining (60) and (67), we have now the closed-form expression of the global MSE of the RW3-CATL:

σ 2 ǫ (m, ζ, f n ) = 5 16 • 1 (mζ) 2 • f d f n 6 • σ 2 α + σ 2 w • B L . ( 68
)
For the optimization of (68) over m, ζ and f n with constraint (65), the auxiliary function to be minimized is given by:

J = 5 16 •σ 2 α • 1 (mζ) 2 • f d f n 6 +σ 2 w •B L +λ• m 2 (4ζ 2 -1) + 4 , ( 69
)
where λ is the Lagrange multiplier. The detailed computation is given in appendix C, yielding the following sub-optimal parameter values. m is the root of: m 11 +2m 10 -16m 9 -12m 8 +112m 7 -176m 6 -512m5 +... , is given by:

( f n f d )(Jakes) = 5 64 • 1 πf d T • σ 2 α σ 2 w • Q 1 7 , (71)
with

Q = 1 m 3 ζ 4 D m + ζ 3 D ζ , ( 72
)
where D m given by ( 85) and D ζ given by ( 86) are functions of m and ζ, as defined in Appendix C. Note that the sub-optimal f n varies as the 7th root of SNR.

Then the sub-optimal MSE can be calculated by:

σ 2 ǫ (Jakes) = C • (σ 2 α ) 1 7 • (σ 2 w • f d T ) 6 7 , ( 73
)
with

C = 2 (mζ) 2 • ( 1 Q ) 6 7 + BQ 1 7 • 10π 6 1 7 . ( 74
)
V. SIMULATION

In this section, the performance of the RW3-CATL in terms of MSE and BER is assessed through simulations, and is compared to that of reference algorithms based on KF. For all our simulations the channel autocorrelation function is assumed to be given by the widely accepted Jakes' model, as stated in Section II. Except for Fig. 9, all the results are given in Data-aided mode (with then known pilot symbols).

A. Validation of the theoretical analysis

In the previous section, a method to solve the minimization has been provided, yielding sub-optimal parameters. Now, it remains to check that the MSE obtained with these suboptimal parameters is close to that obtained with the optimal parameters. We recall that the optimal solution is obtained without taking into account the constraint whereas the suboptimal solution is obtained with the constraint (65). Note that the optimal solution can be found only by means of numerical optimization. The optimal solution is obtained as follows.

First, we define a domain for m and ζ corresponding to typical practical values for these parameters: {0 < m 20, 0.05 < ζ < 0.5}. For each point of this domain, we calculate by means of a one-dimension numerical optimization the f n value that minimizes the MSE, and we keep then the value of this minimum MSE. Since f n depends on SNR and f d T , this numerical computation procedure can be done for various SNR and f d T . As a result, Fig. 3 shows the MSE as a function of m, ζ, computed for SNR = 0 dB and f d T = 10 -3 . It is noteworthy that there exists a valley-belt in which the lowest MSE values are located. To obtain the set of optimal parameters, it remains to find the global minimum by means of a numerical search. The global minimum is shown in Fig. 3 by a star point. The sub-optimal parameters are also plotted (triangle point). We recall that the sub-optimal parameter values are m = 3.19, ζ = 0.39, and for this SNR scenario f n /f d = 2.0 (f n is computed with (71)). Note that the sub-optimal point is exactly located on the cross point of the constraint line and the valley-bottom line. The MSE value for the sub-optimal parameters is very closed to that for the optimal parameters, which validates our sub-optimal solution. Fig. 4 compares the simulated and theoretical MSE versus f n for f d T = 10 -3 , and SNR = 0, 20, 40 dB. The suboptimal loop parameters (m = 3.19, ζ = 0.39) are considered (see section IV). The theoretical dynamic and static error variances (dashed lines) σ 2 ǫα and σ 2 ǫw are obtained by numerical integration of (58) and (59), respectively. The approximated error variances (square points) computed by the approximated formulae ((60) with (61) and (67)) are also plotted. It is observed that the approximated MSEs match very well the theoretical MSEs. On the other hand, we can also observe that the component σ 2 ǫα is the main contribution of σ 2 ǫ for small f n , whereas the component σ 2 ǫw dominates when f n increases. This is understood from (67) and (61) since σ 2 ǫα is inversely proportional to f6 n , while σ 2 ǫw is proportional to f n . Simulated MSEs have also been plotted. The simulated dynamic error variance σ 2 ǫα was obtained by forcing the noise w (n) to zero, whereas the simulated static error variance Fig. 3: MSE (57) versus (m, ζ) computed by numerical integration of ( 58) and ( 59) with SNR = 0 dB, the constraint line is given by (65) ǫw was obtained by maintaining the CA to a constant. We can observe that all the theoretical curves are very close to the simulated ones too, which validates our theoretical analysis and the approximations. Therefore, the abscissa of the minimum of the simulated MSE σ 2 ǫ also matches very well with the (theoretical closed-form (92)) optimal natural frequency (such that f n /f d (Jakes) = 2, 3.9, 7.3 respectively for SNR = 0, 20, 40 dB).

B. Comparison with Kalman estimators in literature

Fig. 5 compares the asymptotic MSE (i.e. in tracking mode) of the RW3-CATL with that of the AR1 CM -KF [START_REF] Komninakis | Multi-input multi-output fading channel tracking and equalization using Kalman estimation[END_REF]- [START_REF] Al-Naffouri | An EM-Based Forward-Backward Kalman Filter for the Estimation of Time-Variant Channels in OFDM[END_REF], the AR1 M AV -KF [START_REF] Barbieri | On the ARMA approximation for fading channels described by the Clarke model with applications to Kalman-based receivers[END_REF] [10] and the RW2-KF [START_REF] Ros | Second-order modeling for Rayleigh flat fading channel estimation with Kalman Filter[END_REF] by means of Monte-Carlo simulations 6 for f d T = 10 -4 and f d T = 10 -3 . Note that our proposed RW3-CATL algorithm assumes the on-line Bayesian Cramer-Rao bound (BCRB) as reference [START_REF] Hijazi | Bayesian Cramer-Rao Bounds for complex gain parameters estimation of slowly varying Rayleigh channel in OFDM systems[END_REF]. It is observed that the asymptotic MSE performance of the AR1 CM -KF is very poor. This result corroborates the works cited in the introduction, which point out that the AR1 CM -KF is convenient for high mobility (f d T >> 10 -2 ), but exhibits poor performance at f d T ≤ 10 -2 as proved by [START_REF] Ghandour-Haidar | On the use of first-order autoregressive modeling for Rayleigh flat fading channel estimation with Kalman filter[END_REF]. As expected, the RW2-KF performs better than AR1 CM -KF and AR1 M AV -KF. Finally, the asymptotic MSE of the RW3-CATL with the loop parameters properly chosen (see section IV) is the closest to the BCRB (which could be concluded from the MSE expressions of the 4 estimators). This result shows that it is preferable to use a well-chosen thirdorder algorithm based on simple CATL to a KF when the later is based only on first-or second-order models. According to (73), the theoretical asymptotic MSE of the RW3-CATL is proportional to the 6 7 power of the noise variance σ 2 w (note that SNR = σ 2 α σ 2 w with here σ 2 α = 1 and σ 2 w < 1), versus to the 4 5 and 2 3 power for the RW2-KF [START_REF] Ros | Second-order modeling for Rayleigh flat fading channel estimation with Kalman Filter[END_REF] and AR1 M AV -KF [10] respectively, which is validated by Fig. 5.

Fig. 6 shows the MSE of different systems versus f d T . The gain in performance of the RW3-CATL is greater for small values of f d T . When f d T increases, the MSEs of the AR1 M AV -KF, RW2-KF and RW3-CATL systems seem to converge to the MSE of the AR1 CM -KF. This is again understood from (73) that the theoretical asymptotic MSE of the RW3-CATL is proportional to the 6 7 power of the f d T .

C. Mismatched design

According to the analysis in section IV, we know that the knowledge of the SNR and f d is required to design the RW3-CATL parameter f n . And (68) shows that f d and the SNR are the two key factors that directly impact the estimation MSE. In this section, we thus depict the sensitivity to imperfect knowledge of SNR and

f ′ d -f d f d = ±10%, ±20%, ±50%
) and SNR fixed at 20 dB, the corresponding theoretical values are also attached. We also find that, the RW3-CATL can sustain a certain f d error, for example, within ±20%, there is no evident mismatch between the simulation MSE and the optimal MSE. Besides, an underestimated f d will cause more severe degradation than a same level overestimated f d , as shown by the 1.5f d (+50%) and 0.5f d (-50%) lines in Fig. 8.

D. BER performance

A BER simulation is carried out to evaluate the actual performance of the RW3-CATL estimator. The transmitted symbols are QPSK modulated. The data frame is composed of 200 continuous pilot symbols and then 1800 unknown symbols. In this context, the channel estimation is in halfblind mode (alternatively by pilots and decisions). Note that the a priori decision x(n|n-1) is used to compute the error signal [START_REF] Tugnait | Doubly selective channel estimation using exponential basis models and subblock tracking[END_REF], but the final decision is computed as:

x(n) = √ 2 2 • sgn{Re( α * (n) • r (n) )} + j √ 2 2 • sgn{Im(α * (n) • r (n) )}.
Fig. 9 shows that the BER of RW3-CATL remains close to the perfect line (BER with perfect acknowledge of channel) for all SNRs, while the BER of other estimators are further away from the perfect line as SNR increases. We notice also that the classical Kalman estimator based on AR1-model leads to poor BER performance due to the mismatch of AR-1 model, and that this BER can be dramatically reduced in using the integrated-RW-model-based estimators (RW2-KF and RW3-CATL). The third-order estimator performs even much better than the second-order one.

VI. CONCLUSION

In this paper, a channel path complex amplitude estimator over slow to moderate flat fading channels has been proposed. The proposed estimator is based on a third-order tracking loop, which is proved equivalent to a steady-state KF based on a same order integrated random walk (RW) model. The connection between a steady-state KF based on a RW model and the proposed PLL-like estimator is established. This explains the fact that the RW3-CATL can reach in tracking mode the same asymptotic performance as that of a steady-state RW3-KF, even though the former converges slower than the KF. The complete theoretical MSE analysis has been provided. A closed-form formula of the asymptotic MSE as a function of Doppler frequency and SNR is given. We have demonstrated that, by fixing the capacitance ratio to 3.19, the damping factor to 0.39, and by computing the natural frequency with a given expression depending on the Doppler frequency and SNR, it is possible to achieve near-optimal performance in terms of asymptotic MSE. Simulation results (MSE and BER) show that, with these well-chosen parameters, the proposed algorithm outperforms the KF of the literature (based on firstor second-order models), as long as the mobility is moderate (i.e. f d T < 10 -2 ), which is a very common scenario. The mismatch simulation shows the robustness of the RW3-CATL in harsh environment test, where the mobility (in terms of f d ) or the background noise power (in terms of SNR) information is distorted. In addition, our proposed algorithm is a computationally less demanding technique than these KFbased algorithms, since it does not require to compute the coefficients at each time period. The simple case of a flat fading channel was considered in this article, but the results can be applied or generalized to more complex systems, such as wireless OFDM systems.

APPENDIX A EQUIVALENT NOISE BANDWIDTH OF RW3-CATL

Using the result of [START_REF] Winkelstein | Closed form evaluation of symmetric two-sided complex integrals[END_REF], the two-sided complex integral in the form of (60) could be evaluated by the solution of a closed matrix equation. The matrices are composed by the coefficients of numerator and denominator of the integrand. The transfer function of a third-order system:

L(z) = b 0 z 3 + b 1 z 2 + b 2 z + b 3 a 0 z 3 + a 1 z 2 + a 2 z + a 3 , (75)
then the corresponding matrix equation is given by:

    a 0 a 1 a 2 a 3 a 1 a 0 + a 2 a 1 + a 3 a 0 a 2 a 3 a 0 a 1 a 3 0 0 a 0         a 0 B L M 1 M 2 M 3     =     b 2 0 + b 2 1 + b 2 2 + b 2 3 2(b 0 b 1 + b 1 b 2 + b 2 b 3 ) 2(b 0 b 2 + b 1 b 3 ) 2b 0 b 3     . ( 76
)
In our case, we have from (44

) that b 0 = µ 1 , b 1 = -2µ 1 + µ 2 + µ 3 , b 2 = µ 1 -µ 2 , b 3 = 0, a 0 = 1, a 1 = µ 1 + µ 2 + µ 3 -3, a 2 = 3 -2µ 1 -µ 2 , a 3 = µ 1 -1.
Combining with (49), ( 50), (51) leads to (77), where: Under the general assumption f n ≪ 1/T , the squared modulus of the high pass-filter 1 -L can be written from (55) as:

A = m 3 ζ 3 , B = 8m 3 ζ 4 + 4m 2 ζ 2 , C = 20m 3 ζ 5 + 5m 3 ζ 3 + 30m 2 ζ 3 + 5mζ, D = 16m 3 ζ 6 + 22m 3 ζ 4 + 68m 2 ζ 4 + 16m 2 ζ 2 + 34mζ 2 + 2, E = 24m 3 ζ 5 + 4m 3 ζ 3 + 48m 2 ζ 5 + 56m 2 ζ 3 + 64mζ 3 + 14mζ + 12ζ, F = 8m 3 ζ 4 + 48m 2 ζ 4 + 32mζ 4 + 24mζ 2 + 16ζ 2 + 4, G = 16m 3 ζ 6 + 22m 3 ζ 4 + 68m 2 ζ 4 + 18m 2 ζ 2 + 34mζ 2 + 2, H = 24m
|1 -L(e j2πf T )| 2 (for |f | ≪ 1/T ) = f 6 / m 2 ζ 2 f 6 n + [(m + 2) 2 ζ 2 -2(1 + 2mζ 2 )]f 2 n f 4 + • • • [(1 + 2mζ 2 ) 2 -2(m 2 + 2m)ζ 2 ]f 4 n f 2 + f 6 . ( 78
)
The red curves in Fig. 2 shows the magnitude of |1 -L(e j2πf T )| 2 as a function of f for different values of m and ζ and for f n T = 10 -3 . Note that with the form in (78), the integral (60) is too tedious to derive. In order to obtain an For that purpose, let us consider the asymptotic behaviour of the log magnitude as a function of frequency. At low frequencies, i.e. f ≪ f n , we get f 2 n f 4 ≪ f 6 n and f 6 ≪ f 6 n , yielding:

m 2 ζ 2 f 6 n + [(m + 2) 2 ζ 2 -2(1 + 2mζ 2 )]f 2 n f 4 + ... [(1 + 2mζ 2 ) 2 -2(m 2 + 2m)ζ 2 ]f 4 n f 2 + f 6 ≈ (mζ) 2 f 6 n . ( 79
)
At high frequencies, i.e. f ≫ f n , we get:

m 2 ζ 2 f 6 n + [(m + 2) 2 ζ 2 -2(1 + 2mζ 2 )]f 2 n f 4 + ... [(1 + 2mζ 2 ) 2 -2(m 2 + 2m)ζ 2 ]f 4 n f 2 + f 6 ≈ f 6 . ( 80
)
By combining (79) and (80), we obtain thus (62). Then, by using f = f n , (63) is found directly from (78).

APPENDIX C MINIMIZATION OF ASYMPTOTIC MSE WITH LAGRANGE MULTIPLIERS METHOD

We apply the method of Lagrange multipliers to minimize (68) with constraint (65). Given the auxiliary function, the problem reduces to solve the following system of equations: Since m and ζ are real positive parameters, from (84) we have:

                                 ∂J ∂f n = 2πσ 2 w T B -
ζ = (m 2 -4) 2m , ( 87
)
which indicates that m > 2. Replace all the ζ in (81), the system of equations becomes: .

                       C 1 B ′ - 6C 2 f 7 n • m 2 -4 4 = 0, (88)
C 1 D ′ m f n - 2C 2 f 6 n • m(m 2 -4) 4 - 8λ m = 0, (89)
C 1 D ′ ζ f n • (m 2 -4) 2m - 2C 2 f 6 n • m 2 -4
(92) Finally, by using (92), we do some manipulations with (88), the system of equations reduces to (70), this equation has 11 roots that we can obtain by Computer-aided calculation. The condition of m (real positive and m > 2) returns a unique available value, that is m = 3.19. We obtained then ζ = 0.39 by (87), and also f n by (92).

Fig. 2 :

 2 Fig. 2: Transfer functions |L| of RW3-CATL (45) (the three blue continuous curves) and of the corresponding 3rd-order analog PLL (55) (blue dashed curves) versus f T , with fixed natural frequency f n T = 10 -3 , while varying the parameters (m, ζ) respectively in (a) and (b). Red continuous curves represent the corresponding high-pass transfer function |1-L| and the green dot-dashed lines are the linear approximation defined in (64), the parameters marked by a star (⋆) are those satisfying the constraint (65)

448m 4 +

 4 1024m 3 + 1024m 2 -3072 = 0, (70) yielding m ≈ 3.19. ζ is then computed with (65), yielding ζ ≈ 0.39, which are slightly different from the values used in [1] 5 . Then f n , which depends on f d and the SNR =

Fig. 4 :

 4 Fig. 4: Theoretical and simulated MSEs versus f n /f d (RW3-CATL with m = 3.19, ζ = 0.39, SNR = 0, 20, 40 dB, f d T = 10 -3 )

3 Fig. 5 :Fig. 6 :Fig. 7 :

 3567 Fig. 5: MSE versus SNR of the RW3-CATL compared to reference estimators, (a) f d T = 10 -4 (b) f d T = 10 -3

Fig. 8 :

 8 Fig. 8: Effect of a mismatch on the knowledge of f d : MSE versus f d T , f d T ′ (used to tune the RW3-CATL) shifted 10%, 20%, 50% from the true value and SNR = 20 dB

Fig. 9 :

 9 Fig. 9: BER versus SNR for QPSK modulation, f d T = 10 -3

3 ζ 5

 5 + 4m 3 ζ 3 + 48m 2 ζ 5 + 68m 2 ζ 3 + 64mζ 3 + 20mζ + 12ζ, I = 8m 3 ζ 4 + 64m 2 ζ 4 + 8m 2 ζ 2 + 32mζ 4 + 56mζ 2 + 16ζ 2 + 8, J = 16m 2 ζ 3 + 32mζ 3 + 16ζ. APPENDIX B ASYMPTOTE APPROXIMATION OF |1 -L(e j2πf T )| 2

  B L = A(ω n T ) 6 + B(ω n T ) 5 + C(ω n T ) 4 + D(ω n T ) 3 + E(ω n T ) 2 + F (ω n T ) A(ω n T ) 6 + B(ω n T ) 5 + C(ω n T ) 4 + G(ω n T ) 3 + H(ω n T ) 2 + I(ω n T ) + J (77)analytical expression of the integral, it is necessary to simplify the expression of |1 -L(e j2πf T )| 2 .

m 4 ζ 5 + 4m 3 ζ 5 + 8m 2 ζ 5 + 8mζ 3 -5 ζ 6 + 16m 4 ζ 6 + 32m 3 ζ 6 + 16m 2 ζ 6 +

 55536666 mζ + 2ζ 2m 4 ζ 4 + 8m 3 ζ 4 + 8m 2 ζ 4 + 4m 2 ζ 2 + 8mζ 2 • • • 4m 4 ζ 6 + 16m 3 ζ 6 + 16m 2 ζ 6 + 8m 2 ζ 4 + 16mζ 4 + 4ζ

4 +ζ 3 C 1

 431 2λ(m 2 -4) = 0, (90) with C 1 = 2πσ 2 w T and C 2 = 5 16 • f 6 d • σ 2 α . The terms B ′ , are respectively obtained from B (61), D m (85) and D ζ (86) where ζ is replaced by (87). Then by combining (89) and (90), λ and f n can be expressed as a function of m as follows: (m 2 -4) D ′ m m 2 (m 2 -4) + 2D

  • • • + 20m 2 ζ 4 -3m 2 ζ 2 + 16mζ 4 + 4ζ 2 -1 . (86)

Note that in practice, our channel estimator can easily be coupled with an efficient detector in order to perform joint channel estimation and decision tasks, for example via the Expectation-Maximization algorithm framework (see[START_REF] Al-Naffouri | An EM-Based Forward-Backward Kalman Filter for the Estimation of Time-Variant Channels in OFDM[END_REF]). In another already mentioned context, it can simply also be used to track the channel gain at pilot frequencies in an OFDM system as in[START_REF] Chen | Kalman-filter channel estimator for OFDM systems in time and frequency-selective fading environment[END_REF].

The proposed estimator (denoted Or3-CATL) in[START_REF] Shu | Third-order complex amplitudes tracking loop for slow fading channel estimation[END_REF] is similar to the one of this paper, but they stem from different structures, the Or3-CATL is given directly from the third-order DPLL while the RW3-CATL is deduced from a RW3-KF; Comparing with the structure of Or3-CATL (Fig.1of[START_REF] Shu | Third-order complex amplitudes tracking loop for slow fading channel estimation[END_REF]), we could also find a difference between the loop filters.

The transfer function L(z) w.r.t. ωn, ζ and m in (45) is same as the one of Or3-CATL in[START_REF] Shu | Third-order complex amplitudes tracking loop for slow fading channel estimation[END_REF], but the relationships between (µ 1 , µ 2 , µ 3 ) and (ωn, ζ, m) are different since the loop filter derived in this paper is different from the one used in[START_REF] Shu | Third-order complex amplitudes tracking loop for slow fading channel estimation[END_REF] (see footnote 2).

This formula can be applied in different channel models by changing the complex gain spectrum, e.g., for a 3-D scattering model[START_REF] Clarke | 3-D mobile radio channel statistics[END_REF], Γα(f ) becomes a constant which yields a much simpler result

Note that in[START_REF] Shu | Third-order complex amplitudes tracking loop for slow fading channel estimation[END_REF], we have used m = 3, ζ = 0.37 as a sub-optimal set, which is obtained from the numerical optimization, and then we proceeded a one-dimension optimization on the natural frequency fn. Obviously, the Lagrange multiplier approach used here is more accurate because it is a 3dimension optimization.

In this simulation, the results of RW3-KF is not illustrated basically because the steady-state RW3-KF is equivalent to the RW3-CATL, as long as the state noise variance σ 2 u is well tuned, according to Section III-A.