R Billard

J Marcel

C Weil

A Fostier

lnduced spawning (ovulation and spermiation) in carp and related species

à l'hypophysation chez les espèces telles que la carpe chinoise, la tanche. L'administration de LHRHa et de pimozide , facteur bloquant l'action de la dopamine (qui serait un inhibiteur de la décharge de GtH) améliore ces deux points. Ces essais sont encore au stade expérimental. D'autres techniques qui sont encore au stade de la recherche sont également revues.

SUMMARY.

In fish farms, hormonal induced ovulation and spermiation are commonly performed in carps and related species by hypophysation which consists of direct stimulation of the gonads by administration of crude carp pituitary homogenates.

The use of synthetic hypothalamic hormones as luteinizing hormone releasing hormone analogues (LHRHa) seems promising. LHRHa administration is used routinely in China for different species of carps. However, LHRHa treatments are partially efficient in inducing ovulation in female common carp and furthermore, in chinese carps and tench, the latency of the ovulatory response is delayed when compared to hypophysation. The use of LHRHa treatment combined with pimozide, a dopamine blocking agent,improves these two points. These trials are still at an experimental stage.

This paper also reviews other techniques for inducing spawning and which are still at a research stage.

RESUME,

Ovulation et spermiation induites chez la carpe et les espèces voisines.

La technique la plus couramment utilisée en pratique pour l'induction hormonale de l'ovulation et de la spermiation chez la carpe et les espèces voisines est l'hypophysation et consiste en la stimulation directe des gonades par des extraits bruts hypophysaires.

L'utilisation d'hormones synthétiques hypothalamiques induisant la décharge de l'hormone gonadotrope hypophysaire endogène tels des analogues du LHRH (luteinizing hormone releasing hormone LHRHa) semble prometteuse et est utilisée en routine en Chine pour différentes espèces de carpes. Cependant, d'une part ces hormones sont faiblement efficaces chez la carpe commune femelle et d'autre part elles entrainent une ovulation retardée par rapport INTRODUCTION Induced spawning is used in carp species which can mature but cannot spawn in captivity (major indian and chinese carp) as wel l as for those which can spawn when environmental required conditions are prov ided (common carp). However, in this latter case, occur rence and success of spontaneous spawning are aleatory and the use of induced spawning techniq ues allow fish farmers to optimize broodsto ck management by increasing t h e quantity of produced fry.

Hormonally induced spawning is commonly used for carp and related species and the knowledge of endocrinology of reproduction is required to i mprove the use of these techniques. Besides, the success of th ese te chniques can be improved by manipula ti ng environmental conditi ons (temperature, photoperiod, water quality, social factors).

In the present paper we will review the different hormonal techniques used in carp and related species, including those whi ch are routine ly used as well as thos e which are still at an experimental stage. We will emphasize the hormonal changes that occur and the influence of environmental fac tors on their success.

I -ENDOCRINOLOGY OF REPRODUCTION

I n fish environmental stimuli are known to synchronize the rhythms of reproduction v i a the brain by neural processes. The transmission is then communicated by the endocri n e system which is a link between these factors and the gonads.

It is well established that the pituitary controls the activity of the gonads (review of PICKFORD a n d [START_REF] Pickford | The physiology of the pituitary gland of fishes[END_REF] . In goldfish, hypophysectomy experiments have demonstrated the role of the pituita r y gland in vitellogenesis and meiosis [START_REF] Yamazaki | Endocrinological studies of the r epro d uction of th e fe male go ldfish, Carassius auratus L., with special r eference to the f unct ion of the p i t uitar y gland[END_REF], in spermatogenesis (BILLARD e t al. 1971) and in spawning behavior in female [START_REF] Stacey | Effects of indomethacin and prostaglandins on the spawning behaviour of female goldfish[END_REF] . In carp two pituitary hormones have been characterized and purified : a gonadotropic hormon e (GtH) purified by BURZAWA-GERAR D 1971, active on spermatogenesis [START_REF] Billard R | Régénération de la spermatogenèse du Cyprin hypophysec tomisé Carassius auratus L. par un facteur gonadotrope hautement purifié de Carpe[END_REF]) and on oocyte ma t uration [START_REF] Jalabert | Etude comparée de l'action des hormones hypophysaires et stéroides sur la maturation in vitro des ovocytes de la truite et du carassin (poissons téléosteens[END_REF] in go ldfish via steroid synthesis. A GtH radio immunoassay has been developped a llowing the study of gonadotropin secret ion d u ring the sexual cyc le in different cypr inid species.

a second gonadotropin has been iso la ted [START_REF] Idler | Studies on Two types of gonadotropins from both salmon and carp pituitaries[END_REF] and is thought to be necessary for completion of vitellogenesis although this fact has been questionned [START_REF] Burzawa- | Existe-t'il plusieurs gonadotropi nes (GtH) chez les poissons ? Données biochimiques et vite llogenèse exogène[END_REF] .

Pituitary activity is r egulated by the hypothalamus wh ich is a part of the ce ntra l nervous system. Functional evidence from lesioning studies has implicated the nuc leus l atera l tube ris (NLT) [START_REF] Peter R | Hypotha lamic control of thyro id g land activ ity and gonadal activity in the goldfish., Carassius auratus[END_REF] in a stimulatory regulation of GtH secretion via a gonadotropin releasing hormone (GnRH) . GnRH activity has been demonstrated in crude hypothalamic extracts i n c a rp both in vitro (BRETON et al. 1972a) and in v ivo [START_REF] Breton | Effets d'extraits hypothalamiques de carpe e t de LH/FSH-RH de synthèse sur la secrétion d'hormone gonadotrope c -GtH in vivo chez l a carpe Cyprinus carpio[END_REF] . GtH release was also induced by mammalian GnRH (LHRH) in vivo in carp (BRETON andWEIL 1973, ANONYMOUS 1978) and in goldfish [START_REF] Crim | Stimulati on of gonadotropin secretion by intraventricular injection of hypothalamic extracts in the goldfish Carassius auratus[END_REF] ) as well as by mamma lian superactive analogue [START_REF] Peter R | Serum gonadotropin leve ls in mature male goldfi sh in response to luteinizing hormone-releasi ng hormone (LH-RH) and des-Gl y10 -D-Ala -LHRH ethylamide[END_REF] . Recently fish GnRH has been isolated from chum salmon hypothalami identified and synthesized [START_REF] Sherwood | Characterization of a teleost gonadotropi n-releasing hormone[END_REF] . This salmon GnRH h as similar potency to nat ive LHRH in vi v o in go ldfi s h in terms of magnitude and duration of GtH response [START_REF] Peter R | Structure-activity relationships of mammalian, chicken, and salmon gonadotro p i n releas ing hormones in vivo in goldfish[END_REF].

Recently experiments of brain lesioning (PETER and PAULENCU 1980) have demonstrated the presence of a gonadotropin release-inhibiting factor (GRIF) in the anterior-ventral preoptic region. CHANG et PETER 1983a found that dopamine has GRIF activity in goldfish modulating spontaneous release of GtH as well as LHRHa-stimulated release of GtH. Finally, sexual steroids, which are under the control of GtH, act locally on the gonad to trigger gametogenesis. In the present paper we will focus on the regulation of the final aspects of oogenesis and spermatogenesis. Oocyte maturation (resumption of meiosis and structural and biochemical transformations which develop simultaneously) is induced in vitro by carp pituitary extracts or carp gonadotropic hormone in goldfish [START_REF] Jalabert | Etude comparée de l'action des hormones hypophysaires et stéroides sur la maturation in vitro des ovocytes de la truite et du carassin (poissons téléosteens[END_REF][START_REF] Jalabert | In vitro oocyte maturation and ovulation in rainbow trout (Salmo gairdneri), northern pike (Esox • lucius), and goldfish (Carassius auratus)[END_REF], EPLER et al. 1979) and in carp [START_REF] Epler | Effect of gonadotropin and steroid hormones on ovarian maturity in carp[END_REF], EPLER et al. 1979, 1982a). These in vitro study techniques have shown that gonadotropic action of the pituitary is probably mediated by direct steroid action on oocyte maturation in goldfish and in carp. Progestins and particularly 17OEhydroxy, 20 S-dihydroprogesterone (17a 20 B OH P) are active in inducing maturation in goldfish [START_REF] Jalabert | In vitro oocyte maturation and ovulation in rainbow trout (Salmo gairdneri), northern pike (Esox • lucius), and goldfish (Carassius auratus)[END_REF]in carp EPLER et al. (1980). Corticosteroids are also active in goldfish [START_REF] Jalabert | Etude comparée de l'action des hormones hypophysaires et stéroides sur la maturation in vitro des ovocytes de la truite et du carassin (poissons téléosteens[END_REF][START_REF] Jalabert | In vitro oocyte maturation and ovulation in rainbow trout (Salmo gairdneri), northern pike (Esox • lucius), and goldfish (Carassius auratus)[END_REF], EPLER et al. 1979) and in carp [START_REF] Epler | Joint action of various steroid hormones on maturation of carp oocytes in vitro[END_REF]. Since cortisol increases the potency of carp pituitary extracts in in vitro goldfish maturation [START_REF] Jalabert | In vitro oocyte maturation and ovulation in rainbow trout (Salmo gairdneri), northern pike (Esox • lucius), and goldfish (Carassius auratus)[END_REF], and desoxycorticosterone that of 17a 20 SOH P [START_REF] Epler | Joint action of various steroid hormones on maturation of carp oocytes in vitro[END_REF]) and that of carp hypophysial homogenate (EPLER et al. 1982b) in carp, it is evident that corticosteroids may have some synergistic action in the process of oocyte maturation. Other glands such as the thyroid may have a synergistic action in oocyte maturation as suggested by the results of [START_REF] Epler | Effects of triiodothyronine and of some gonadotropic and steroid hormones on the maturation of carp Cyprinus carpio L.) oocytes in vitro[END_REF] in carp. Triiodothyronine (T3) influences positively the maturational effect of gonadotropic and steroid hormones. Ovulation consists in follicular rupture and expulsion of the mature oocyte from the follicule. In goldfish ovulation was induced in vivo by prostaglandins [START_REF] Stacey | Effects of indomethacin and prostaglandins on ovulation of goldfish[END_REF] and in vitro in carp [START_REF] Epler | Effect of gonadotropin and steroid hormones on ovarian maturity in carp[END_REF][START_REF] Epler | Effect of temperature 17 a hydroxy-20 B dihydroprogesterone and Prostaglandin F2 a . on Carp oocyte maturation and ovulation in vitro[END_REF] as well as by adrenaline [START_REF] Epler | Effect of gonadotropin and steroid hormones on ovarian maturity in carp[END_REF]. The action of prostaglandins is temperature independant [START_REF] Epler | Effect of temperature 17 a hydroxy-20 B dihydroprogesterone and Prostaglandin F2 a . on Carp oocyte maturation and ovulation in vitro[END_REF] while in vivo ovulation is temperature dependant (HORVATH 1978).

Spermiation in carp and related species _(thinning of the semen) is pituitary dependant [START_REF] Clemens | The seminal thinning response of carp (Cyprinus carpio) and ra inbow trout (Salmo gairdneri ) after i njections of pituitary extracts[END_REF]. In goldfish hypophysectomy and replacement therapy studies indicate that spermiation seems to depend on androgen via GtH. Different steroids are active : 11-ketotestosterone, testosterone, methyltestosterone, testosterone propionate, progesterone (reviewed in BILLARD et al. 1982).

II -HORMONAL TECHNIQUES USED FOR INDUCING SPAWNING.

The use of hormones to induce fish spawning has been reviewed many times and recent reviews dealing with this topic include those of [START_REF] Chaudhuri | Use of hormones in induced spawning of carps[END_REF]) HARVEY and HOAR (1979[START_REF] Lam | Applications of endocrinology to fish culture[END_REF].

A -Selection of breeders

Induced spawning is always performed on mature animals and fully ripe animals are selected for treatments.

Ripe males are selected based upon the possibility of collecting a small quantity of sperm by hand stripping. For females different criteria have been adopted and most depend upon external characteristics such as the swelling, softness and elasticity of the abdomen, colour and state of swelling of the cloacal region and body girth. However these are only rough indications of the stage of development of the ovary and attempts have been made and must be made to improve the estimation of egg development. This implies firstly the use of a technique for egg sampling and secondly the determination of criteria for egg maturity. Samples of eggs can be obtained by aspiration either by inserting a tubing into the ovary through the genital pore as described by [START_REF] Chen | Induced spawning of the three major Chinese carps in Malacca, Ma laysia -Malay[END_REF] for chinese carp or by inserting a needle through the abdominal wall cavity as described by BIENARZ and EPLER 1976 in common carp. Criteria of egg maturity used in carp are : the colour of the oocytes, their size or the position of the nucleus and the nucleolus after clearing the egg as described by [START_REF] Brzuska | Méthodes de déterminatin du stade de maturité des carpes femelles e n relation avec des injecti ons d'homogena ts d'hypophyses de carpes[END_REF] B -Administration of exogenous gonadotropins. 1) Hypophysation. Hypophysation is the traditional method of induced fish breeding and is widely practiced with the major indian carp, chinese carp and the common carp. In these species it consists in the injection of homoplastic or heteroplastic pituitary extracts. However pituitaries collected from marine catfish have been used for indian carps [START_REF] Varghese | Preliminary observations on the use of marine catfish pituitary glands for induced spawning of the Indian major carps[END_REF].

Details of techniques employed in the collection, preservation of pi tuitaries and preparation of extracts have been presented in [START_REF] Pickford | The physiology of the pituitary gland of fishes[END_REF]and often reviewed (SHEHADEH 1975[START_REF] Chaudhuri | Use of hormones in induced spawning of carps[END_REF], HARVEY and HOAR 1980[START_REF]Pré paration et utilisation de broyats hypophysaires pour l'induction de la reproduction des poissons. La piscicul ture en Etang[END_REF][START_REF] Rothbard | Induced reproduction in c ultivated cyprinids. The common carp and the group of chinese carps : 1. The technique of induction, spawning and hat ching[END_REF].

In summary the following dosages, expressed in mg of dry weight,are used: -common carp : In females 1 pituitary/kg is injected in two injections (equivalent to about 3mg/kg). The first injection or priming consists in 1/10 of the dose, the remaining 9/10 are given in a second injection. For màles, 2mg/kg are given in one injection at the time of the second injection of females.

-indian carp : the female is given a priming dose of 2-3mg/kg b.w. and after an interval of 4-6h a final dose of 5-8mg/kg b.w. At the time of the second injection males are given 2-3mg/kg b.w. chinese carp : 4 to 4.5mg/kg b.w. is given in two injections for females, 2mg/kg b.w in one injection for males. A more precise determination of the dose can be made by estimating the gonadal status of the females. This is done by measuring the girth of the female in the deepest part of the body. A dose relative to the body girth is given and the diagram is reported in [START_REF]Pré paration et utilisation de broyats hypophysaires pour l'induction de la reproduction des poissons. La piscicul ture en Etang[END_REF][START_REF] Rothbard | Induced reproduction in c ultivated cyprinids. The common carp and the group of chinese carps : 1. The technique of induction, spawning and hat ching[END_REF] .

Hormonal profiles and oocyte evolution

Results obtained on carp (ANONYMOUS 1978) and goldfish (STACEY et al. 1979a) have shown that the periovulatory GtH surge is very brief during spontaneous ovulation. Classical hypophysation induces GtH increase of longer dura tian i.e. 3-4 days [START_REF] Jalabert | A new tool for induced spawning : the use of 17ahydroxy-20B-dihydroprogesterone to spawn carp at low temperature[END_REF][START_REF] Fostier | Stimulation hypophysaire de la secrétion oestradiol-17 chez la carpe commune, Cyprinus carpio L[END_REF], BIENIARZ et al. 1980). Priming dose induces a significant GtH increase one hour after the injection and then GtH levels continue to increase for 3 hours and remain high during the next 12 hours [START_REF] Weil | Profiles of plasma gonadotropin and 17 -estradiol in the common carp, Cyprinus carpio L; as related to spawning induced by hypophysation or LHRH treatment[END_REF] . The massive pituitary dose (given as a second injection or as a single injection) induces higher GtH levels (250-400ng/ml) which can last 3-4 days according the case as described by BIENIARZ et al. 1980. In fact these authors recorded high levels for 3 days in spawning fish and for 4 days in unspawned ones. This fact suggests that, following hypophysation, circulating gonadotropin levels may depend on the physiological conditions of the avaries and in the case of spawning the avaries can take up gonadotropin for maturation process. Priming dose induces shifting of the germinal vesicle towards to the micropile in responding and unresponsive fish whereas the second injection induces maturation and ovulation in responding fish and resorbtion without maturation in the non responding ones [START_REF] Jalabert | A new tool for induced spawning : the use of 17ahydroxy-20B-dihydroprogesterone to spawn carp at low temperature[END_REF].

Although ovaries are ready to spawn they are able to respond to a GtH stimulation since an increase in plasma estradiol (17B E2) is noticed when compared to levels before injection [START_REF] Fostier | Stimulation hypophysaire de la secrétion oestradiol-17 chez la carpe commune, Cyprinus carpio L[END_REF][START_REF] Weil | Profiles of plasma gonadotropin and 17 -estradiol in the common carp, Cyprinus carpio L; as related to spawning induced by hypophysation or LHRH treatment[END_REF], KIME and DOLBEN 1985). Serial samplings have shown that 17B E2 are low during the first 3 hours following priming and then begin to increase 5 hours after the injection. Maximal values are attained 12 hours later [START_REF] Weil | Profiles of plasma gonadotropin and 17 -estradiol in the common carp, Cyprinus carpio L; as related to spawning induced by hypophysation or LHRH treatment[END_REF]. However mean 17 B E2 levels are not increased following the second injection (WEIL et al . 1980, KIME and[START_REF]Hormonal changes during induced ovulation of the carp, Cyprinus carpio[END_REF]. Thi~ increase in 17 B E2 might be due to the presence of vitellogenic oocytes since at the time of ovulation oocytes of all stages are present [START_REF] Brzuska | Méthodes de déterminatin du stade de maturité des carpes femelles e n relation avec des injecti ons d'homogena ts d'hypophyses de carpes[END_REF]. The increase in 17 B E2 levels following priming is preceded by an increase in testosterone which is consistent with a precursor product relationship. The second injection of pituitary extract did not lead to a massive increase in testosterone related to the increase in GtH but greater amounts of glucuronides were observed (KIME and DOLBEN 1985).

Concerning progestins KIME and DOLBEN (1985) observed detectable levels of 17a hydroxyprogesterone (17a0H P) before injection and an increase after hypophysation. Peak height varied from 1 .6 to 13,3ng/ml and dropped rapidly after ovulation . On the other hand they observed no detectable levels of 17a 20B OH P except in one fish. Levels were 1 ,08ng/ml at the time of ovulation and 0, 11 and 1 ,14ng/ml respectively 6 and 3 hours before ovulation. We must emphasize the fact that the treatment of hypophysation induced only partial ovulation in all the fish.

In a work conducted in our laboratory blood samples were taken every 3 hours for 30 hours on 7 females submitted to hypophysa tion (100 µg/kg of partially purified carp GtH). Two fish have completely ovulated, 2 partially and the remaining three have not ovulated. (fig. 1). In the non ovulated fish, 17a 2 0 B OH P levels were low var yi ng between undetectable value and 0,5ng/ml. In the partially ovulat ed fish, a low rise in 17a 20B OH P was observed between 18 and 21 hours, followed by a decrease by 24 hours when ovulation was moni tored. On the other hand higher l e v els were monitored in ovulating fish but the highest value was 7ng/ ml, val u e much lower than that one indicated by LEVAVI-ZERMO NSKY and YARON ( summar y this volume). These authors report a dramatic (111ng/ml) but transient increase followed by GV breakdownn, corroborating the idea that 17a 2 0 B OH P is involved in oocyte maturation. Furthermore duri ng spontaneous ovulation transient increase of different progestins (proges t erone, 17 a OH P, 17a 20 B OH P) have been recently described in goldfish [START_REF] Kagawa | Changes in plasma steroid hormone levels during gonadal maturation in female goldfish Carassius auratus[END_REF] . Since these increases are transitory, higher values of 17a 20 B OH P ma y h a v e bee n attained between the sampling periods in our experiment.

Hormonal profiles and spermia tion

In the literature, no data are available on circulating GtH a nd steroid levels following hypophysation. In a work conduct ed in our laboratory , (figure 2) plasma GtH, androgens and 11-ketotestosterone were mon i tored every 3 hours for a 24 hour period following a single i nj ection of partially purified carp gonadotropin (lOOµg /kg). Collecta ble mi l t volume as well as spermatocrit were measured before the injection a nd 12 and 24 hours aft e r the injection. The treatment induce d a larg e increa se in GtH levels (64ng/ml) 3 and 6 hours after the injection . The levels decreased g radual ly from 9 to 24 h ours t o reach 40ng/ml by 24 hours. These circulating levels are much higher than those described during spontaneous spawning (ANONYMOUS 1978) . 11 k etotes t osterone levels followed the same profile than GtH with an increase the firs t 6 hours followed by a decrease to reach by 24 hours levels which are higher than those oberved just before the injection and those previously described for androgens (about 3ng/ml) in a study conducted all the year round with males held in a concrete tank and which released a small quantity of milt [START_REF] Weil | La fonction gonadotrope de l'hypophyse au cours du cycle sexuel chez deux poissons téleosteens, la carpe commune (Cyprinus carpio) et la truite arc-en-ciel (Salmo gairdneri) ; son controle par l'hypothalamus, les gonades et les facteurs externes[END_REF]. After hypophysation collectable milt volume increased considerably during the 24 hour experimental period ( from 0, 190ml to 9, 5ml/ d' ) while spermatocrit was declining from 75 % to 53 % indicating a sperm dilution. This increase in GtH and 11-ketotestosterone confirms the role of GtH via steroid synthesis in spermiation (see above). These data and works from KIME and MANNING (summary this volume) and from [START_REF] Takashima | Stimulation of spermiation by LHRH analogue in carp[END_REF] indicate that in carp 11-Ketotestosterone is closely associated with male reproductive development.

The technique of hypophysation has certain disadvantages as : its relatively high cost due to the unpredictable supply of pituitary and the necessity to maintain water temperature above 18°C. its partial success. Only about 70 % of the animals ovulate and unspawned fish undergo oocyte atresia leading to an increase in mortality even under ideal conditions (good oxygenation of the water and food sùpply).

2) Use of mammalian hormones. HCG (Human Chorionic gonadotropin) and Synahorin (mammalian pituitary extract) can be reliably supplied and attempts have been made to use them instead of hypophysation. HCG efficiency depends on the species. It's effective in inducing ovulation in goldfish [START_REF] Yamamoto | Hormonal control of spermiation and ovulation in goldfish[END_REF]. Chinese studies report that it is weakly effective in grass carp and mud carp but more efficient in silver carp and bighead. However it has harmful effects on unspawned fish leading to the death of the animals. In indian carp, the relatively low efficiency of HCG and Synahorin can be improved by using them in combination with a lower dose of pituitary extract (CHAUDHURI 1976).

C -Discharge of endogenous gonadotropin hormone.

1) Administration of gonadotropin releasing hormone (GnRH)~

As mentionned above LHRH or LHRH analogues (LHRHa) evoke the release of endogenous pituitary gonadotropin supplies. A seasonal pituitary sensitivity to GnRH have been demonstrated but at the time of inducing spawning the sensitivity is maximal in carp [START_REF] Weil | Physiologie, Etude de la réponse hypophysaire à l'administration de GnRH exogène au cours du cycle reproducteur annuel chez la Carpe Cyprinus carpio[END_REF]) and in goldfish [START_REF] Sokolowska | Seasonal effects of pimozide and des Gly10 (D-Ala6) LHRH Ethylamide on Gonadotrophin secretion in goldfish[END_REF] •

Action in female

In goldfish, spontaneous ovulation does not occur at 12°C when fish are held in aquaria without vegetation. Relatively high doses of LHRH given in one injection administered either by intraperitoneal (lµg/g) or by intra cranial injection (200ng/g) were partially effective in inducing ovulation (respectively 50 % and 33 % of ovulation, LAM et al. 1975[START_REF] Lam | Effect of synthetic luteinizing hormone -releasi ng hormone (LHRH) on ovulation and pituitary cytology of the goldfish Carassius auratus[END_REF]. A weak percentage of ovulation was also recorded with the total dose of 200ng/g of a potent LHRHa when given in two injections 12 hrs apart at 12 -14°C (CHANG et PETER 1983) or 3 hrs apart at 18 -20°C (SOKO LOWSKA et al. 1984). Sustained GnRH release a l lowed by the implantation of pellets containing 25µg or 125µg of LHRHa were also partially effective (20 -50 % of ovulation [START_REF] Sokolowska | Induction of ovulation in goldfish, Carassius auratus, by pimozide and analogues of LHRH[END_REF]). Both treatments with LHRHa induced high plasma GtH levels which were in the range of those reached during spontaneous ovulation but were of longer duration.

In carp, nine daily LHRH intrahypophseal injections at lµg/kg stimulated oocyte maturation (germinal vesicle migration and breakdown) at 18 -20°C but the same dosage given by intracardiac or intraventricular injection produced any effect [START_REF] Sokolowska | Synthetic releasing hormones effect of intracerebral and intramuscular carp (Cyprinus carpio L.) maturation[END_REF]. Germinal vesicle migration but no ovulation was obtained after 5 daily intracardiac LHRH injections at 1µg/kg [START_REF] Sokolowska | The influence of luteinizing h ormone -releasing hormone (LH-RH) on serum gonadotropin levels and oocyte maturation in female carp (Cyprinus carpio L.)[END_REF] or two intracardiac injections at 3 µg/kg given 3 hrs apart [START_REF] Weil | Profiles of plasma gonadotropin and 17 -estradiol in the common carp, Cyprinus carpio L; as related to spawning induced by hypophysation or LHRH treatment[END_REF].

Single dose of LHRHa, 1µg/kg given by intracardiac injection in females held in aquaria or spawning ponds [START_REF] Billard R | Effects of LHRH and LHRHa on plasma GtH levels and maturation/ovulation in the common carp, Cyprinus carpio, kept under v arious environmental conditions[END_REF] or 50 g given intraperitoneally in females held in concrete tank (BILLARD et al. 1983a) were also ineffective in inducing ovulation. Repeated doses, 5 µg/Kg plus 50 µg/Kg given 6 hrs apart, given by intraperitoneal injection in females held in aquaria with or without vegetation were ineffective in inducing ovulation while they were partially effective in females held in cages maintained in a pond. The authors think that the response was faciliated by the occurrence of circadian changes in dissolved 02, C02 and pH, changes similar to those observed in spawning ponds. This point will be discussed in a next paragraph . In this latter experiment plasma GtH levels were similar in ovulated and non ovulated females but were higher than those of females held in aqua-.ria. However measured levels were lower than those circulating during spontaneous ovulation. This lack of efficiency of LHRH in carp was also reported by [START_REF] Kouril | Application LHRH and its anal og for reaching ovulation in female tench, Grass Carp, Carp and Sheat fish[END_REF] for doses ranging from 10 to 200 µg/kg. The same authors report successful results for tench at a dose of 100-200 µg/kg of LHRH and 10-40 µg of LHRHa and for grass carp at a dose of 200 µg of LHRH and 20µg/Kg of LHRHa. In China LHRH analogue is routinely used to induce spawning of different species of chinese carp. Trials performed in 24 fish farms and conducted on 500 animals are reported in ANONYMOUS 1977a -1977b. In summary the recommended doses for the different species held at temperature lying within the normal range (20 -28°C) are the following. For blaGk carp a single dose of 10 µg/kg is efficient ; effectivness is increased when injection of LHRHa is followed by that of 1 -2mg of pituitary. In grass carp 5 -10 µg/kg 6f LHRHa in one injection is prescribed while in bighead a priming dose of 1 -2 µg/kg followed by 8-9 µg/kg 12 hrs later for matured fish or 24 hours later for less matured fish, is suggested. For silver carp 5 -10µg/kg of LHRHa in one injection is recommended for first time spawners or 10-25 µg in two injections for repeated spawners (10µg/kg + 0,1 -1mg cAMP/kg in two injections spaced 12h apart enhances effectivness).

From the literature it is noteworthy that there is a prolonged latency in the ovulatory response to LHRH or LHRHa, in goldfish, tench, chinese carp when compared with hypophysation or HCG treatment. This treatment presents the advantage of decreasing the rate of mortality in chinese carp (ANONYMOUS 1977b, ROTTMANN and[START_REF] Rottmann R | The use of syntheti c LHRH analogue to spawn Chinese carps[END_REF].

Recent promising works from [START_REF] Epler | Effect of temperature 17 a hydroxy-20 B dihydroprogesterone and Prostaglandin F2 a . on Carp oocyte maturation and ovulation in vitro[END_REF] report that salmon GnRH analogue is very active in inducing ovulation in common carp as compared to LHRHa.

Action in male

Chinese studies (ANONYMOUS 1977a) report that LHRHa in very effective in inducing spermiation in carp. Injection of 10-30µg induces milt release in the black carp after 12 -24h. The effect is remarkable in spotted silver carp which shed notile spermatoza whithin 6 1/2 hrs following the injection.

In common carp LHRH and LHRHa analogue induce also an increase in collectable milt volume. One injection of LHRH at the dosage of 50 µ;g/kg is more effective than at the dosage of 10µg/kg while with LHRHa both dosages have the same effect (BILLARD et al. 1983b). Following an injection of 50µg/kg collected milt volume is increased the first two days and then begin to decrease by day 3 for reaching basal levels on day 5 (BILLARD et al. 1983a[START_REF] Takashima | Stimulation of spermiation by LHRH analogue in carp[END_REF] report that smaller dosage of the analogue administered by 5 daily injections of 1µg/kg does not stimulate the increase of collectable milt volume while collected milt volume is increased with 5 daily injections o f 10µg/kg. The same authors demonstrated that the efficiency of LHRH administration depends on the solvent used (affecting the kind of release of LHRH) since spermiation was not stimulated by 5 daily injections of 10µg /kg of LHRHa dissolved in propylene glycol. In th is l atter case, plasma GtH and 11-ketotestosterone were not modified compared to •control. On the other side, following the injections of 10µg/ kg of LHRHa dissolved in saline an increase in 11-ketotestosterone levels was monitored as long as spermiation was stimulated i.e. from 24 h after the first injecti on until 24 hrs following the fifth.

3) Release of an inhibition influencing spontaneous and GnRH induced GtH release. PETER and CRIM 1978 and PETER and PAULENCU 19 80 have suggested the presence of a gonadotropin release inhibitory factor (GRIF) in goldfish on the basis of brain lesioning experiments. On the bas is of i n vivo studies on goldfish CHANG et PETER (1982, 1983a) demonstrated that dopamine has GtH release inhibitory activity by acting directly on gonadotrophs to inhibi t spontaneous secretion of GtH and to black the action of GnRH. Removal of the dopamine inhibition of GtH release by pimozide treatment, a dopamine antagonist, potentiates the ability of LHRHa to stimulate GtH release and ovulation in female goldfish at 12-1 4°C (CHANG et PETER 1983b ) as wel l as 18 -20°C [START_REF] Sokolowska | Induction of ovulation in goldfish, Carassius auratus, by pimozide and analogues of LHRH[END_REF]. Indeed the low percentage of ovulation (between 25% 50 %) obtained at 12 -14°C following LHRHa injec t ion and at 18 -20°C following LHRHa treatment (injection or pellet implantation) is improved by action of pimozide and can reach 80-90 %. The authors studied the timing of administration of pimozide and LHRHa allowing to optimize the rate of ovulation. Best results were obtained when pimozide is g iven prier to the injection of LHRHa or with the first of two injections of LHRHa or following the pellet implantation. These observations indi cates that prier release from dopamine inhibition , greatly potentiates the ovulatory response to LHRHa as well as GtH release since higher GtH leve ls were obtained in this case. A transient increase in 17a 20B OH P and 17a OH P was observed in such treated fish before ovulation [START_REF] Peter R | Secretion of progestogens during induced ovulation i n go ldfish[END_REF].

In female common carp, experiments of brain lesioning (PETER, SOKOLOWSKA, BILLARD and CRIM unpublished) suggest that a GRIF is also present. Preliminary promising results were obtained with combined treatment with pimozid~ (10mg/kg) and LHRHa (50 µg/kg) 6 hours later [START_REF] Billard | [END_REF]. This combination induced ovulation in some fe males (about 50 %) while only oocyte maturation was induced by LHRHa alone. Furthermore, promising recent results on t he success of ovulation have been obtained in female chinese carp with pimozide + LHRHa and in female c ommon carp with reserp ine (a catcholamine depletor) + LHRHa [START_REF] Lin | The effects of LHRH analogue and drugs which block the effects of dopamine on gonadotropin secretion and ovulation i n fish cultured in China in Aquaculture of Cyprinids R[END_REF]). On the other hand, in male common carp, a combination of LH RHa + pimozide had only a slight effect on the stimulation of spermiation wh en compared to LHRHa treatment alone (BILLARD et al. 1983a). 4) Action of factors abating the negativ e feed back effects of endogenous sex steroids on hypothalamo-hypophys ial axis. Antioestrogens such as clomiphene citrate ( clomid) or tamoxifen (ICI 46474) induce gonadotrop in release in carp (BRETON et al. 19 75) and go ldfish (BILLARD and PETER 1977). BRETON et a l. 1975, comparing different dosages (0,1mg -1mg -10mg/kg) found that the dosage of 1mg/kg was the best to induce high GtH levels similar to those present during ovulation in goldfish and carp. The use of clomid (10mg/kg) was successful in inducing ovulation in go ldfish (PANDEY andHOAR 1972, PANDEY et al 1973). In carp KUMAR and CHANDRASEKHAR 1980 accelerated spawning (4 months ) by tri-weekly injections of clomid (25 and 50 µg/animal) for a period of 3 months during the prepatory per iod. On mature animals three daily injections of 1mg/kg fa il ed to induce ovulation (BIENIARZ et al. 1979) while two successive i n jections of higher dosage (10mg/kg) were successful in overcoming the blockade induced by indomethacine (an inhibitor of prostaglandin synthesis) treatment [START_REF] Kapur | The effect of clomiphene citrate on ovulation and spawning in indomethaci n treated carp, Cyprinus carpio[END_REF].

In cyprinid this technique has been used only at a research level.

D -Action of factors acting on oocyte maturation.

As mentionned above corticosteroids as well as progestins and particularly 17a 20 B OH P have a role in inducing in vitro oocyte maturation in goldfish and carp. Corticosteroids and progestins were tested for induction of ovulation activity in vivo.

In goldfish trials were successful with deoxycorticosterone (DOC), cortisone and corticosterone (100µg/g) (KHOO 1974) while only partial ovulation was obtained with deoxycortisol (LAM and PANDEY cited in LAM 1982). Metopirone, an inhibitor of 11 -hydroxylase induced ovulation in goldfish [START_REF] Pandey | Effects of dexamethasone and metopirone on ovulation in the goldfish, Carassius auratus[END_REF]. With regard to progestogens, progesterone (100µg/g) was effective (KHOO 1974) while 17a20B OH P (12µg/g daily for 5 days) was ineffective in goldfish. (PANDEY et al cited in LAM et al. 1978). In carp, neither 17 a 20BOH P (2mg/kg) nor DOC (1mg/Kg) alone or in combination, induced ovulation at 13-15°C [START_REF] Jalabert | A new tool for induced spawning : the use of 17ahydroxy-20B-dihydroprogesterone to spawn carp at low temperature[END_REF]. 17a 20 B OH P gave positive results when administered 1 day after priming with a low dose of carp pituitary extract (0.6mg/kg). At such temperatures classical hypophysation yielded worse results since only partial ovulation followed by resorption was observed. Priming with a low dose of pituitary extract induced migration of the GV toward the periphery of the oocyte, indicating that responsivness to 17a20 B OH P seems dependent on GV migration. However [START_REF] Epler | Effect of temperature 17 a hydroxy-20 B dihydroprogesterone and Prostaglandin F2 a . on Carp oocyte maturation and ovulation in vitro[END_REF] obtained in vitro oocyte maturation only in some females altough ovarian fragments were in the same stage i.e. with oocytes before migration of the GV. This indicates that oocytes in the same stage as determined by position of the nucleus have variable responsivness to the same hormonal treatment.

Altough thyroid hormones act synergistically with gonadotropin hormone to induce vitellogenesis [START_REF] Hurlburt | Role of the thyroid gland in ovarian maturation of the goldfish, Carassius auratus L. Can[END_REF] and T3 influences the maturational response to gonadotropic and steroid treatments [START_REF] Epler | Effects of triiodothyronine and of some gonadotropic and steroid hormones on the maturation of carp Cyprinus carpio L.) oocytes in vitro[END_REF] have not yet been tested in vivo.

These techniques are not used in fish farm.

E -

Influence of factors promoting ovulation by action on follicular contraction and rupture.

As mentioned above prostaglandins and catecholamines can trigger ovulation in vitro.

In vivo only the role of prostaglandins (PG) was investigated.

Indomethacine, an inhibitor of prostaglandins biosynthesis, blocks the ovulatory response of goldfish to a combination of HCG and 20°C warmed water while a single IP injection of PGE1, PGE2 or PGE2 a (5µg/g) restores the response [START_REF] Stacey | Effects of indomethacin and prostaglandins on ovulation of goldfish[END_REF]. In carp [START_REF] Kapur | The effect of clomiphene citrate on ovulation and spawning in indomethaci n treated carp, Cyprinus carpio[END_REF] demonstrated that a single injection of indomethacine (10µg/g) blocked ovulation for 12 days in females held in conditions in which control fish ovulated spontaneously. The role of prostaglandins mediated by and stimulation of GtH secretion is unlikely since [START_REF] Peter R | Effects of third ventri c le injection of prostaglandins on Gonadotropin secretion in goldfish, Carassius auratus[END_REF] found that injection of PGF2 a or PGE2 but not PGE1 into the third ventricle of the brain decreased serum GtH in the female. A direct action on follicular rupture via smooth muscle contraction seems more likely [START_REF] Stacey | Effects of indomethacin and prostaglandins on the spawning behaviour of female goldfish[END_REF]PANDEY 1975, KAGAWA and[START_REF] Kagawa | In vitro effects of protaglandins on ovulation in goldfish Carassius auratus[END_REF] for goldfish [START_REF] Epler | Effect of gonadotropin and steroid hormones on ovarian maturity in carp[END_REF][START_REF] Epler | Effect of temperature 17 a hydroxy-20 B dihydroprogesterone and Prostaglandin F2 a . on Carp oocyte maturation and ovulation in vitro[END_REF] for carp. Furthermore prostaglandins have a role in spawning behavior in goldfish [START_REF] Stacey | Effects of indomethacin and prostaglandins on the spawning behaviour of female goldfish[END_REF]. Thus it may.be possible to use prostaglandins to induce ovulation as well as spawning behavior in fish with mature oocytes.

III -INFLUENCE OF ENVIRONMENTAL AND SOCIAL FACTORS ON SPONTANEOUS AND HORMONAL INDUCED SPAWNING.

It is well estab lished that in cyp rinids water temperature plays an important role in gametogenesis development (review of PETER and CRIM 1979) . In common carp, sexual maturi ty is reached earlier in warm c limates than in temperate conditions and GUPTA 1975 obtained a precocious maturity in carp held in warm water aquaria (23°C). Male and female tench maintained all y ear in a heated pond (6°C above amb i ant temperature) showed an accelerated spermatogenesis and vitellogenesis compared to the control (BRETON et al. 1980a et b). However daylenght may interfe re wi th tem perature si nc e in an i nd ian carp a constant short daylength retarded the mat u rity of gonads in female [START_REF] Verghese | Preliminary experiments on the modification of the reproductive cycle of an indian carp cirrhina reba (HAM.) by control of light and temperature[END_REF] and in male of common carp [START_REF] Weil | La fonction gonadotrope de l'hypophyse au cours du cycle sexuel chez deux poissons téleosteens, la carpe commune (Cyprinus carpio) et la truite arc-en-ciel (Salmo gairdneri) ; son controle par l'hypothalamus, les gonades et les facteurs externes[END_REF]-showed in BILLARD et al. 1982).

In practice the influenc e of environmenta l factors on inductio n of a rematuration of the gonad after hormonal induced spawning is used in common carp, chinese carp and indian car p (resp ective ly HORVA TH 1980a ;[START_REF] Lin | Polycultural system of freshwater fish in China[END_REF][START_REF] Freeze | Fall Spawning of silver carp[END_REF][START_REF] Freeze | Fall Spawning of silver carp[END_REF]BHOWNICK et al 1977 ). The combination• of high water tempe rature and good f eed ing induces within 2 months the matura t ion of the gonad and a second spawning may be i nduced with hormonal treatment when temperature required for ovulation is provided.

With respect to the end of gametogenesis i.e. spermiation and ovulation, different factors may improve the spontaneous process. In goldfish spermiation is promoted a progressive r ise in temperature which is in correlation with an increase in plasma GtH (GILLET et al~ 197 7) ; ovulation occurs at 20-21°C (YAMAMOTO et a l. 1966., STACEY et al. 1979b) in presence of absence of aquatic vegetation while it occurs at 12°C only in the presence of vegetation (STACEY et al 1979b). Furthermore ovulation in goldfish is highly synchronized with photoperiod (STACEY et al 19 79b) . Social factors may act as well. In female goldfish [START_REF] Yamazaki | Endocrinological studies of the r epro d uction of th e fe male go ldfish, Carassius auratus L., with special r eference to the f unct ion of the p i t uitar y gland[END_REF][START_REF] Yamamoto | A me t h od to induce artificia l spawning of go ldfish all through the year[END_REF] indicate that the presence of sexually active males induce ovula tion . For males [START_REF] Experien-Kyle A | Elevations i n gonadotropin concentrations and milt volumes as a result of spawning behavior in the goldfish[END_REF] report that the presence of a r eceptive female or stimulus pairs of spawning goldfish induce an increase in expressible milt volumes and plasma GtH. NIKITINA and GODOVICH 1983 have demonstrated that the dynamics of sex hormones in the blood of ma le common carp depend on the presence or absence of spawning in the females placed wi th th em. In pratice in China it is possible to induce spawning of female mud carp by the presence of active female spawners (injected with pituitary ex tract ) when they are held in a common spawning pond [START_REF] Lin | Polycultural system of freshwater fish in China[END_REF] .

Environmental factors modulate the response to hormonal treatment . Temperature is an important factor for -the injection i nterval the latenc y of t he ovulatory and milting response as demonstrated in female and male common and chinese carp ( HORVA TH 1978 b) and in female go l dfish (STACEY et a l. 19 79 b) fo llowing pituitary or HCG treatment.

the time of overrippening of ovulated oocytes. It depends on ca rp species. (HORVATH 1978b) -the success of spawning i.e. the matura tion of breeders. Oxygen concentration also conditi ons the success of hypophysation [START_REF] Horvath | The effect of oxygen content of water on the ovulation of carps[END_REF]. These authors demonstrated that bath, water temperature and the amount of dissolved oxygen, strongly influence the maturation of spawners. If one of these parameters is lower than require d the adverse . effect cannot be counter balanced by the higher level of the otber factor.

Time of inj ection (since there is evidence of a possible c i rcadian rythm of oocyte sensitivity, BIENIARZ et al., summary this vo lume) as we ll as the presence of vegetation, or a precise photoperiod, or spawning fish may be also important to ensure a maximal efficiency of hormonal treatment .

IV -CONCLUSION

The hormonal treatment most commonl y used for inducing spawning in common carp, chinese and major indian carp is hypophysation. This technique presents some disadvantages such as the unpredictability of pituitary supply, the problem of standardization of the pituitary extracts and its partial effectivness (around 70 %). The use of GnRH promoting endogenous GtH release seems_ prom1s1ng. LHRH analogs have different advantages : they are commercially supplied and they don't evoke antibody formation. On the other hand their efficiency on ovulation varies with the species and the latency to ovulatory response is delayed compared to hypophysation. Removal of the dopamine inhibition improves these two points as demonstrated by LIN 1986 and studies, still at an experimental stage, are made in this field. Furthermore , the use of salmon GnRH analogues alone seems promising and further studies must be done in this direction.

It is evident from new hormonal treatment hypophysation technique appropriate time of its REFERENCES the results reported above that the development of a for inducing spawning as well as the improvment of implie the knowledge of criteria determining the administration and this particularly in female. Cyprinids constitute the largest group of cultured fish in the world and include species such as carp whose domestication dates from ancient times. The ability to adapt to a wide range of temperatures and high fluctuations in water oxygen content as well as to diverse modes of feeding in various biotopes has made this species the most highly produced in the world : 2 million tons of cyprinids out of a total of 3.5 million tons of fish, including all species. This book reviews the actuel state of knowledge and of management techniques in the fields of nutrition, reproduction, genetics, management of larvae, growth and rearing, pathology, economics and processing of cyprinids.

Les Cyprinidés constituent le groupe de poissons le plus largement élevé dans le monde et comptent des espèces comme la carpe dont la domestication est très ancienne. Ses caractéristiques d'élevage : tolérance à une large gamme de températures, de Fortes fluctuations d'oxygène dissous, modes d'alimentation diversifiés à partir des différentes niches du réseau trophique, Font de cette espèce la première production de l'aquaculture mondiale (2 millions de tonnes de cyprinidés sur 3,5 millions de tonnes de poissons, toutes espèces confondues). Ce volume présente l'état actuel des connaissances et des techniques d'élevage en matière de nutrition,reproduction,génétique,élevage larvaire,croissance et élevage,pathologie,

  Figure 1Plasma 17a 20$ OH P in female carp following hypophysation.

  Figure 2

  ANONYMOUS, 1977a . A new highly effective ovulating agent for fish reproduction. Practical application of LHRH analogue for the induction of spawning of farm fishes. Sei. sin., 20 : 469-474. ANONYMOUS, 1977b. A further investigation on the stimulatory effect of a s ynthetic analogue of hypothalamic luteinizing hormone releasing hormone ( LRHa) on spawning in "Dornes tic fishes". Acta Biochim. Biophys. Cyprinus carpio L.). J. Fish. Biol., 8 : BIENIARZ K., EPLER P., POPEK W., 1977. Histological changes in the avaries of mature female carp in summer time. Inv. Pesq., 41 : 95-102. BIENIARZ K., EPLER P., THUY L.N. and KOGUT E., 1979. Changes in the avaries of adult carp. Aquaculture, 17 : 45-68. BIENIARZ K., EPLER P . , THUY L.N. and BRETON B., 1980. Changes in blood gonadotropin level in mature female carp following hypophysial homogenate injections. Aquaculture, 20 : 65-69. BIENIARZ K., EPLER P., POPEK W., BILLARD R., SOKOLOWSKA M. , 1985. In 7th conference of European Society for Comparative Physiology and Biochemistry. Barcelona Spain.

	Sinica, 9	15-23.
	ANONYMOUS, 1978. Radioimmunoassay on serum gonadotropin of carp Cyprinus
	carpio L. Acta Biochim. Biophys. Sinica, 10 : 399-407.
	BIENIARZ K., EPLER P.,	1976. Preliminary results of the in vivo studies on
	ovarian resorption 449-451. in carp (BILLARD R., PETER R.E., 1977. Gonadotropin release after implantation of
	antiestrogens in pituitary and hypothalamus of goldfish Carassius
	auratus. Gen Comp. Endocrinol., 32 : 213-220.

Aquaculture of Cyprinids

L'aquaculture des Cyprinidés

Actes du Colloque International sur I' Aquaculture de la Carpe et des espèces voisines tenu à Evry (France) du 2 au 5 septembre 1985 ORGANISATION GENERALE J. MARCEL et R. BILLARD Institut National de la Recherche Agronomique avec la collaboration de : J. BERTIN, M. BILLARD, M.G. HOLLEBECQ, R. LE GOUVELLO, M. et N. MARCEL, G. de MONTALEMBERT, P. RENARD, A. SAAD, les services techniques du C.N.R.Z. , de FIAPEVRY, des mairies d'Evry et Jouy-en-Josas Traduction simultanée : INCAA ( Mmes CHA VE, PERD ER EAU et SUKTHANKAR) Excursion: SRAE Centre (J .F. MARTIN), Comité Régional de Promotion Centre -Val de Loire -Berry( S. FOURSIN ), MAJ Les Clouzioux, Pisciculture de M. HENNEQUART, SCI du Château de Cheverny, SARL de !'Orangerie COMITE SCIENTIFIQUE G. BALVAY, G. BENOIT, R. BILLARD, R. BRETON , B. CHEVASSUS, P. de KINKELIN, B. LANOISELEE, P. LUQUET, J. MARCEL, J.F. MARTIN, P. PROTEAU, M. VUILLOT, C. WEIL EDITION SCIENTIFIQUE Dessins M. WEBER Dactylographie M.E. MARMILLOD, C. TOMASSI Traduction anglaise A. DAIFUKU Les organismes suivants ont apporté leur aide matérielle pour l'organisation de ce colloque : Conseil Régional Centre IFREMER INRA Ministère de !'Agriculture Ministère de la Coopération Ministère de la Recherche et la Technologie Office Régional de la Mer Syndicat Régional des exploitants d'étangs du Centre Union Nationale des Syndicats de l'Etang

Aquaculture of Cyprinids


Roland Billard

Christian Bry

Christian Gillet

Stress, environment and reproduction in teleost fish


Roland Billard

Control of reproduction and fish farming


Bernard Jalabert

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


Bernard Breton

Roland Billard

Photoperiod and gonadotropin control of the reproductive cycle in the female rainbow trout

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


Nadine Martin
email: nadine.martin@gipsa-lab.inpg.fr

Michelle Vieira
email: michelle.vieira@gipsa-lab.inpg.fr

Frequency and amplitude tracking for short nonstationary and nonlinear signals Keynote address

Keywords: Signal processing, time-frequency, nonstationary, instantaneous frequency and amplitude, polynomial phase signal, simulating annealing, tracking

Whatever the application domains, when dealing with strongly nonstationary and nonlinear signals, Fourier-based methods and even classical time-frequency methods are no longer able to estimate and track the instantaneous amplitudes and frequencies of each signal component. One possibility to get away from the Heisenberg incertitude constraint is to set a model, which has to be as general as possible in order to be valid in many domains. In that context, we have proposed and studied a new model, which writes as a time-varying polynomial-amplitude and polynomial-phase signal. Parameter estimation is casting as the maximisation of the likelihood function, which is overcome by meta-heuristic approaches, such as simulating annealing. In that paper, based on simulations, we detail the performance of the Short Local Polynomial model-based method (Short-LP) we proposed for a number of particular cases when analysing one or two components on very short-time duration. We first assess the frequency resolution limit when analysing close frequency-modulated components. The performance is explained according to classical Fourier limits and more particularly the dynamic signal bandwidth, the signal being nonstationary. The analysis of amplitude-modulated components is then discussed according to the presence or not of the carrier frequency. Finally, we stress upon the performance of the amplitude estimation in the particular case where the instantaneous frequencies are known a priori.

INTRODUCTION

Surveillance is based on indicator monitoring. When an indicator exceeds a threshold, an alarm is raised. An absolute threshold on scalar indicators is the most widespread approach but is not adapted to the detection of some damages. For instance in rotating machines, unbalances, misalignments, anisotropic rotors, flexible coupling or magnetic bearings infer excitation forces, which amplitude and frequency are modulated. In that case, typological indicators are essential for a reliable and early detection. A highly evolved indicator is capable to directly represent the vibration image of a default and to track its evolution.

A time model of a measure provides an accurate description of this observation, which parameters can be such typological indicators. In previous publications [3-4-5-6-7-8-9], we have proposed a time model for signals having strong nonstationarities. Furthermore this model based on a Short Local Polynomial model (Short-LP) is relevant for processing short signals. In all the methods belonging to the Cohen class, the resolution is limited by the Heisenberg incertitude. To get away from this constraint and to be able to estimate nonlinear modulated signals with a better resolution need to add some hypothesis mostly if the signal analysis is of short duration.

We will not handle the statistical properties of the Short-LP estimator proposed, this topic has been thoroughly covered in previous publications [START_REF] Jabloun | Estimation of the Amplitude and the Frequency of Nonstationary Short-time Signals[END_REF][START_REF] Jabloun | A New Flexible Approach to Estimate Highly Nonstationary Signals of Long Time Duration[END_REF]. We will now focus on some important issues for the analysis of multicomponent signals.

Based on simulations, we detail the performance of the algorithm for a number of particular cases when analysing components on very short-time duration. We first assess the frequency resolution limit when analysing close frequency-modulated components. The performance is explained according to classical Fourier limits and more particularly the dynamic signal bandwidth, the signal being nonstationary. The analysis of amplitude-modulated components is then discussed according to the presence or not of the carrier frequency. Finally, we stress upon the performance of the amplitude estimation in the particular case where the instantaneous frequencies are known a priori This paper is outlined as follows. Section II will recall the model used for modelling short nonstationary signals. Section III briefly describes the optimization algorithms chosen to estimate the model parameters whereas section IV deals with the multicomponent case. The last sections present new results elaborated for a number of simulated signals in order to highlight important properties of the algorithm proposed: the frequency resolution limit in section V, the behaviour of the algorithm face to amplitude modulation without carrier in section VI and the improve of performance with prior upon frequency modulation in section VII. Section VIII looks at the conclusions that can be drawn from this work.

II. THE MODEL WITH A SPECIFIC BASE

Let us consider a model [ ]

x n of a discrete signal defined as the sum of C components [ ]

where amplitudes a c [n] are strictly positive, phases [ ] c n φ are differentiable and show no discontinuity. The constraint on the amplitudes and the phases guaranty the unicity of the model [START_REF] Cohen | On an ambiguity in the definition of the amplitude and phase of a signal[END_REF][START_REF] Loughlin | On the amplitude-and frequency-modulation decomposition of signals[END_REF].

A way of modelling the signal with a better accuracy is to increase the number of parameters to describe it without being over the point number of the signal. In [START_REF] Jabloun | Estimation of the Amplitude and the Frequency of Nonstationary Short-time Signals[END_REF], we propose a polynomial modelling of each amplitude a c [n]

[ ] [ ] [ ] [ ] [ ] [ ] [ ] , 0 , 0 0 2 2 , , 2 , c c Ma c c m m m Mf c c m m m n c c c c k N k N a n a p n f n f p n n f k f k φ ϕ π = = =- =- = =   = + -     ∑ ∑ ∑ ∑ (2)
where Ma c and Mf c are the approximation orders of the amplitude modulation and frequency modulation respectively. c ϕ stands for the initial phase of the signal in (1).

The set

[ ] { } ( ) 0,max , c c m m Ma Mf
p n = is a polynomial base. This base could be computed from the discretization of a continuous-time base, such as the canonical polynomial base or the orthogonal polynomial bases, such as Legendre, Tchebychev or Hermite. However the orthogonal property is lost when discretizing these functions. This property is fundamental to guarantee the independence between the parameters estimated. Therefore, in Short-LP, we use a discrete base we derived in [START_REF] Jabloun | Estimation of the Amplitude and the Frequency of Nonstationary Short-time Signals[END_REF] and which corresponds to the application of the Gram Schmidt procedure in discrete-time directly. The expressions for the set

[ ] { } ( ) 0,max , c c m m Ma Mf p n = are recall in Appendix.
Finally the parameters to estimate can be gathered in a vector V

    T T 1 C V = V ...V (3) with
,0 , ,0 , c c T c c Ma c c c Mf a a f f ϕ   =   c V … … with c=1,C (4)
The dimension of the vector V is equal to C×M with M=Ma c +Mf c +3, the dimension of each vector V c .

In addition to the choice of the base, the choice of modelling the frequency instead of the phase and the centring of the polynomial at the signal middle to ensure a minimum variance, the idea was to also consider only small approximation orders in Short-LP [START_REF] Jabloun | Estimation of the Amplitude and the Frequency of Nonstationary Short-time Signals[END_REF]. Ma c and Mf c are less than or equal to 2. This rule does not induce a constraint since the signal analysed is assumed to be short enough to be adapted to this approximation. The signal duration should contain one or two periods at least.

In order to generate the model (1), two approaches have been considered to estimate the vector V.

One possible approach is to consider the maximisation of the likelihood function, which is equivalent to a minimization of the least square function

( )

LS l V when the error can be assumed to be a Gaussian noise. This minimization is ( )

arg min LS l = V V V ⌢ , (5)
with

( ) [ ] [ ] 2 2 2 N LS n N l s n x n =- = - ∑ V , (6)
where [ ] s n is the observation and [ ]

x n the model defined in [START_REF] Aburdene | On the computation of discrete Legendre polynomial coefficients[END_REF]. The error between the observation and the model, namely the difference [ ] [ ] ( )

s n x n -, represents the sum of the noise, in which the deterministic signal we want to estimate is embedded, and the model error. We verify a posteriori that this error is distributed as a Gaussian law.

Direct minimization of ( 6) is extremely difficult due to the high non-linearity of the function and the parameter number. Classical optimization techniques such as gradient descent, Gauss-Newton and EM algorithm do not ensure convergence to the global minimum when local minima are numerous. This problem can be overcome with meta-heuristic approaches, and in particular, with the simulated annealing algorithm. The simulated annealing technique is efficiency, when a desired global extremum is hidden in many local extrema. The simulated annealing has an analogy with thermodynamics where metal cools and anneals. In the same way, after an initialization of the parameter vector to estimate, an iterative loop, controlled by a scalar referred to as a temperature, generates a new candidate of the vector that minimizes the cost function, namely the least square function ( ) LS l V . This candidate is accepted or not according the acceptation Metropolis rule.

In lieu of a likelihood maximum approach, a second approach can also be investigated upon a Bayesian point of view. In that case, the parameters in V and the variance of the error [ ] [ ] ( )

s n x n are viewed as random variables for which a prior distribution has to be assigned. A straightforward calculation leads to an expression of the posterior distribution up to a normalisation constant. This function being highly nonlinear, we propose to use a Metropolis-Hasting MCMC algorithm to sample from this distribution. Once this distribution is estimated, the parameters are estimated by a Minimum Mean Square Error method [START_REF] Jabloun | Estimation of the Amplitude and the Frequency of Nonstationary Short-time Signals[END_REF].

By comparing both estimation algorithms [START_REF] Jabloun | Estimation of the Amplitude and the Frequency of Nonstationary Short-time Signals[END_REF], we concluded that the simulating annealing approach, in addition to a simple implementation, has provided a better compromise between the bias and mean square errors of the parameter estimates and the computation time. In this paper, we will then only use the simulating annealing based approach with the Short-LP.

The method was extended to a monocomponent signal whatever its length and its modulations by considering a local approximation on nonsequential parts of the signal. This issue is not considered in this paper, the reader should refer to [START_REF] Jabloun | A New Flexible Approach to Estimate Highly Nonstationary Signals of Long Time Duration[END_REF] for details about this extension.

III. OPTIMIZATION TECHNIQUE FOR A MONOCOMPONENT SIGNAL

Let us come back to the approximation of a short monocomponent signal [ ] c

x n defined in [START_REF] Aburdene | On the computation of discrete Legendre polynomial coefficients[END_REF]. In that case only one vector V c of dimension M, as defined in (4), has to be estimated. Table 1 gives the tuning of the Short-LP running with a simulating annealing based algorithm. a rough estimation of the additive noise variance -z α α-quantile for the Gaussian distribution (by default α=31% and z α =1)

Step 1. Initialization

-( ) ( ) ( ) ( ) 0 0 0 0 ,0 ,0 0 0 0 0 T c c c c a f ϕ   =   V … … the parameter vector with ( ) ( ) 0 ,0 arg max c c s f FT f ν = ( ) ( ) ( ) 0 0 ,0 0 c c s a FT f = ( ) ( ) ( ) 0 0 0 arg c c s FT f ϕ = where ( ) [ ] 2 2 2 N c N n jfn s c n FT f s n e π =+ - = - = ∑ -( ) [ ] ( ) [ ] 0 0 , 0 c Ma c c m m m a n a p n = = ∑ , ( ) [ ] ( ) [ ] , 0 0 0 c c m Mf c m m f n f p n = = ∑ , ( ) [ ] ( ) ( ) [ ] ( ) [ ] 0 0 0 0 0 2 2 2 n c c c c k N k N n f k f k φ ϕ π = - =-   = + -     ∑ ∑ -( ) [ ] ( ) [ ] ( ) [ ] ( ) 0 0 0 cos c c c
x n a n n φ = the model evaluated from ( )

0 c V - ( ) ( ) [ ] ( ) [ ] 2 2 0 0 2 N LS c c n N l s n x n =- = - ∑ V the LS function -( ) ( ) ( ) ( ) 0 2 0 M i diag σ = Σ
the M×M diagonal matrix of the initial parameter perturbation,

-( ) ( ) ( ) 0 0 LS c l τ = V the initial temperature
Step 2. Iterations from 1, t iter = -Generate a new parameter vector deviated from a Gaussian perturbation

( ) ( ) ( ) gen t t c c c = + V V δV with ( ) ( ) ( ) 0, t t c M = Σ δV N -Compute the model from ( ) gen c V ( ) [ ] ( ) [ ] ( ) [ ] ( ) cos gen gen gen c c c x n a n n φ = -Compute the LS function ( ) ( ) gen LS c l V -Draw a random number according to a uniform distribution [ ] 0,1 u U ∼ -Apply the acceptation Metropolis rule If ( ) ( ) ( ) ( ) ( ) u < exp t gen LS c LS c t l l τ   -         V V set ( ) ( ) 1 t gen c c + = V V , ( ) ( ) ( )
( )

1 t gen LS c LS c l l + = V V otherwise ( ) ( ) 1 t t c c + = V V , ( ) ( ) ( )
( )

1 t t LS c LS c l l + = V V -Draw a random number according to a Bernouilli distribution ( ) B u B p ∼ -If u=1 set ( ) ( ) ( ) ( ) ( ) 2 1 2 1 i i t t σ σ ε σ + = - ∀i and ( ) ( ) ( ) 1 1 t t τ τ ε τ + = - otherwise set ( ) ( ) ( ) ( ) 2 1 2 i i t t σ σ + = ∀i and ( ) ( ) 1 t t τ τ + = -While t<iter set t=t+1 and iterate step 2 Stop criterion -If [ ] ( ) ( ) ( ) 2 2 1 1 iter b b e n v N z v N α - + < + set [ ] c x n from ( ) iter c = c V V ⌢
and stop the algorithm otherwise repeat step 2

IV. THE CASE OF MULTICOMPONENT SIGNALS

In the case of a multicomponent signal, which is the purpose of this paper, the model is always defined in (1) but with C>1. The dimension of the parameter vector V, equal to C×M, is high and represents the dimension of the parameter space describing the signal. In order that the modelling makes sense, this dimension has to be lower than the degree of freedom of the signal. Subsequently, this dimension has to be lower than N+1, the length of the signal. We then deduce the largest number of components,

N+1 C , M ≤ (7)
that the model can take into account. This constraint is strong in the context of short signals with non-linear modulations in frequency and in amplitude. Namely, this means that for a classical modelling with polynomial functions of order 2, M is equal to 7 and the number of components cannot be greater than 4 for a signal length of 32 points. This strong limit is due to the high precision of the model, which, as we will see in the following, will induce high performance at the expense of the computation time.

In [START_REF] Jabloun | Maximum Likelihood Parameter Estimation Of Short-Time Multicomponent Signals With Nonlinear Am/Fm Modulation[END_REF] and [START_REF] Jabloun | Multicomponent Signal: Local Analysis And Estimation[END_REF], two approaches have been considered for estimating the parameters.

The first is referred to as the global algorithm and is a direct generalization of the previous algorithm. All of the parameters of all the components are estimated together. The algorithm derived is then optimal regarding the maximum likelihood, but it is important to notice that this property is true asymptotically only.

The second one is a method by deflation, where the highest-energy component is first estimated, then extracted from the observation in order to estimate the following one. Due to its design, this algorithm is suboptimal but with a less computational time.

The two following sections sum up these two approaches by pointing out the steps modified with respect to the algorithm developed in the monocomponent case.

Global algorithm

The global algorithm is close to the one presented in Table 1. In the initialization step, named step 1, the parameter vector V c of each component is estimated by a deflation method applied on a Fourier transform, the number C of components being known. The perturbation matrix is now of dimension C×M and is initialized in the same way.

In order to increase the convergence rate and to reduce the computational time, which can be very high according to the observations, the adaptation has consisted in reducing the random perturbation in the iteration step 2 of Table 1.

Table 2 shows the new iteration step, referred to as step 2' where the Metropolis rule is no more applied.

Step 2' -Iterations from 1, t iter = -Generate a new parameter vector deviated from a Gaussian perturbation ( ) ( ) ( )

gen t t = + V V δV with ( ) ( ) ( ) 0, t t C M × = Σ δV N -Compute the model from ( ) gen V ( ) [ ] [ ] ( ) 1 C gen gen c c x n x n = = ∑ -Compute the LS function ( ) ( ) gen LS l V -If ( ) ( ) ( ) ( ) 0 t gen LS LS l l - > V V set ( ) ( ) 1 t gen + = V V , ( ) ( ) ( )
( )

1 t gen LS LS l l + = V V
otherwise generate a new parameter vector again -Draw a random number according to a uniform distribution

[ ] 0,1 u U ∼ -If u < 2/3 set ( ) ( ) ( ) ( ) ( ) 2 1 2 1 i i t t σ σ ε σ + = - ∀i otherwise set ( ) ( ) ( ) ( ) ( ) 2

Sub-optimal Algorithm by deflation

The deflation approach consists of a component-by-component algorithm. At iteration c, the c th component only is estimated through a vector we denote

c → V ⌢ . Instead of the parameter vector V defined in (3), c → V ⌢ , defined as 1 ( ) c c c C c times → - -       T T T T T 1 V = V ...V V 0 ...0 ⌢ ⌢ , ( 8
)
with 0 T the null vector of dimension M is partially filled in according to the iteration number. For the same reasons as in the previous section, a modified simulated-annealing technique is applied for estimating this vector.

At each iteration c, the c th component of

c → V ⌢
is initialized as in Table 1 but from the Fourier Transform of the residue of the previous iteration.

Table 3 shows the principle of this algorithm. Inside the deflation loop, the iteration step, referred to as step 2", is close to step 2' of Table 2 but for one component only. The perturbation matrix is of dimension M only. Step 1 from Table 1 → Initialization ( )

0 c → V ⌢ from ( ) c r FT f
Step 2" -Iterations from 1, t iter = -Generate a new parameter vector deviated from a Gaussian perturbation

( ) ( ) ( ) gen t t c c c = + V V δV with ( ) ( ) ( ) 0, t t c M = Σ δV N -Compute the model from ( ) gen c V ( ) [ ] ( ) [ ] ( ) [ ] ( ) cos gen gen gen c c c x n a n n φ = -Compute the LS function ( ) ( ) gen LS c l V -If ( ) ( ) ( ) ( ) 0 t gen LS c LS c l l - > V V set ( ) ( ) 1 t gen c c + = V V , ( ) ( ) ( )
( )

1 t gen LS c LS c l l + = V V
otherwise generate a new parameter vector again -Draw a random number according to a uniform distribution

[ ] 0,1 u U ∼ -If u < 2/3 set ( ) ( ) ( ) ( ) ( ) 2 1 2 1 i i t t σ σ ε σ + = - ∀i otherwise set ( ) ( ) ( ) ( ) ( ) 2 1 2 1 1 i i t t σ σ ε σ + = - ∀i
-While t<iter t=t+1 and iterate step2"

Step 3 -Construction and deflation of the c th component

-Set ( ) iter c V in c → V ⌢ -Compute [ ]
= - - = =   = - - = =   V V for Testset 1. (9)
N is equal to 32 and the signal to noise ratio is 15 dB. The sampling period is 1s. At each date, the frequency distance between each component is a constant number ∆f, which lies in the range [0.0870 Hz, 0;0087 Hz]. This distance is set by means of the value of 2,0 f , see Table 4. For an easier comparison with Fourier-based method, the last line of p n to 1 1 N + with N=32 according to the discrete base used. The colour qualifies the results of the Short-LP: red, orange and green for bad, so-so and good respectively.

In the algorithms proposed, the modulation orders are assumed to be unknown. As already specified in section II, we set the modulation orders to the maximum value that is 2. Figure 1 and Figure 2 show the results for some characteristic values of the frequency distance. Down to a frequency distance equal to 0.06 Hz, the algorithm is able to separate both components with an accurate estimation of the modulation functions. As already mentioned in the statistical study [START_REF] Jabloun | Maximum Likelihood Parameter Estimation Of Short-Time Multicomponent Signals With Nonlinear Am/Fm Modulation[END_REF], estimating the magnitude is always more difficult especially at low frequencies. Despite this, the time model is close to the theoretical signal as shown by the curves at the top of Figure 1.

right). Top to bottom, both theoretical (-) and estimate (--): time signal, frequency modulations and amplitude modulations of both components

Then, as illustrated in Figure 2, the algorithm behaves worse for a frequency distance lying in the range [0.03 Hz, 0.06 Hz]. At 0.0348 f Hz ∆ =

, the frequency modulations are properly estimated, whereas the amplitude modulation presents a higher standard deviation. Then, at lower f ∆ , such as 0.0261 f Hz ∆ = , one component is estimated at a frequency equal to the frequency mean, the other one with a frequency modulation of any type rather in the low frequencies. Curiously, the modulation errors compensate each other in order that the time model always lies close to the theoretical signal as shown by the curves at the top of Figure 2. Under 0.03 Hz, this behaviour is accentuated. The algorithm estimates only one component at the mean frequency with amplitude equal to the sum of the two components. Given that the algorithm has one supplementary degree of freedom, the second component is estimated at low frequencies and amplitudes. Even at these frequency distances, the time model always lies close to the theoretical signal. Moreover, the estimation error is the lowest at these frequencies. The three frequency ranges observed experimentally can be also detected on the mean square error evaluated between the time model and the theoretical signal. The error increases whereas the frequency distance decreases until a critical value, 0.0609 Hz, from which the errors decreases again. As expected, the mean square errors between the frequency or the amplitude modulations increase while the frequency distances decrease. In fact, this behaviour can be explained by the fact that the modulation orders are constant in the model. The degrees of freedom of the model permit an accurate modelling of the signal. Moreover, these simulations allow us to conclude that the error between the model and the observation should not be used as the only stop criterion when the modulation orders are constant. A whiteness criterion should be added.

The critical value 0.0609 Hz equal to 2 bins can be assumed to be the frequency resolution of the algorithm at the given signal to noise ratio (15 dB) and for the given signal length (33 points). The location of the frequency modulation, in the signal frequency band, or the arch back of this modulation have no effect upon the result. We get the same result with a convex frequency modulation (Testset 2 with f 1,0 =1.2, f 1,1 = f 2,1 =+0.18, f 1,2 = f 2,2 =+0.3 and always N equal to 32 and the signal to noise ratio to 15 dB).

In order to understand this value according to the theoretical modulation law of the signal, it may be interesting to determine the degree of energy concentration of the signal in the time-frequency plane.

Rihaczek [START_REF] Rihaczek | Signal Energy distribution in time and frequency[END_REF] has investigated this question by means of a time-frequency cell, T r ×B d , within most of the signal energy is concentrated, and defined as the deviation of the phase from the linearity. The interval T r , referred to as the relaxation time, is the time over which the signal phase may be considered linear, or the instantaneous frequency constant. The frequency band B d , referred to as the dynamic signal bandwidth, is the frequency band swept during this time. Rihaczek shows that ( )

1 1 2 c d r df t B dt T π = = ( 10
)
with f c (t) the instantaneous frequency of the signal. The dynamic bandwidth depends on the rate of change of the instantaneous frequency. For nonlinear modulations the cell size varies with time.

In the case of signals belonging to Testset 1 defined in (9), Figure 4 shows the variation of B d , which lies in the range [0.0253 Hz, 0.1951 Hz]. Consequently, T r varies between 5.1s when the modulation rate change is high, and 39.5s at the middle of the window when the modulation rate change is nearly null. The Short-LP frequency model is of order 2, so has no limit in time. In frequency, the Short-LP gives a good indication given that B d is almost greater than the frequency resolution limit 0.0609 Hz we observed in simulations. To compare with a classical analysis, a spectrogram with a Hanning window should attain 0.09 Hz only and over 16s, which stands for 2.97 bins of the full signal. Figure 5 should be compared with Figure 1 (right).

These results have been obtained for a signal length of 33 points and at a signal to noise ratio equal to 15 dB. For a longer length and, works in progress seems to corroborate that the critical value of 2 bins could be the resolution limit whatever the length. But further works are required to confirm this assumption and to assess the robustness to the noise level.

VI. DISCRIMINATION AM VERSUS TWO FMS

In this section we aim at evaluate the behaviour of the algorithm when analysing one component only according to the amplitude modulation properties. Indeed the spectrum of such a signal has two adjacent sidebands. We asked about the behaviour of the algorithm in this case. How many degrees of freedom does the algorithm require? Does the algorithm need one or two components for modelling one AM component? As in the previous section, our conclusions will be drawn from simulations. Two properties have to be considered: First the frequency distance between the two sidebands of the amplitude modulation. Second the sign of the amplitude. Two testsets, Testset 3 and Testset 4 have been considered in order to answer this question, with always N equal to 32 and the signal to noise ratio to 15 dB. Features of theses Testsets and a sum up of the results of the Short-LP are given in Table 5.

When the signal is made of two components and as expected, the Short-LP is not able to model it with one degree of freedom (C=1) as shown with Testset 3.1 and 3.2 in Table 5 and in Figure 6.

When the signal is made of one amplitude-modulated component, the Short-LP will be able to manage whatever the degree of freedom. But, in this case, the sign of the amplitude is essential. We assumed that the model in (1) should have positive amplitude in order to maintain the uniqueness of the solution. This constraint can be a limitation in some cases. When the amplitude is positive, see Testset 4.1 and 4.3 in Table 5 and Figure 7, the Short-LP provides a good model, more accurate with C = 1 in opposition to the error values as already specified in the previous section. When C=2, given that the frequency distance between the two sidebands, referred as to ∆f m , are lower than the frequency resolution, the Short-LP models the carrier frequency transferring the modulation in the amplitude. The other component is estimated with any frequency modulation and zero-amplitude modulation.

Features of the

When the amplitude is negative as in Testset 4.2, 4.2 and 4.5 of Table 5 and as shown in Figure 8, the Short-LP will produce a better approximation with the highest degree of freedom. However and as expected, the amplitude modulation cannot be tracked in the negative parts and so the normalized mean square error is high whatever the degree of freedom. If the modulation index is high as it was the case in the last example, Testset 4.5 of Table 5, the model has to be applied on shorter parts of the signal and the extension of the algorithm proposed in [START_REF] Jabloun | A New Flexible Approach to Estimate Highly Nonstationary Signals of Long Time Duration[END_REF], [START_REF] Jabloun | A AM/FM Single Component Signal Reconstruction using a Nonsequential Time Segmentation and Polynomial Modeling[END_REF] should be applied.

In conclusion, the simulations tend to show that an amplitude-modulated component with a positive amplitude will be better estimated with a right degree of freedom, that is C=1. The normalized mean square error does not reflect this accurate modeling given that a higher degree-of-freedom model allows a diminution of the error thanks to a second component. Nevertheless the amplitude modulation and the frequency modulation of the component considered is closer to the theoretical one with C=1.

Let us try to explain this behaviour. Indeed the positivity constraint on the amplitude corresponds to a signal with a time duration T that has to be shorter than half the period of the amplitude modulation for retaining the positive part of the modulation only, ( )

2 1 2 m m f T f ∆ < = ∆ . ( 11
)
with ∆f m being twice the frequency of the amplitude modulation. However, if we admit that the frequency resolution of the Short-LP is about 2 bins, that is the conclusion of the simulation study of section V, a model with a frequency resolution higher than this limit should verify

2 2 f T T f ∆ > ⇔ > ∆ (12)
It is straightforward to conclude that constraints [START_REF] Rihaczek | Signal Energy distribution in time and frequency[END_REF] and (12) do not match. Then setting positive amplitude necessarily leads to a frequency distance between the two sidebands, ∆f m , which is lower than the frequency resolution of the method. A model with one degree of freedom is then the best one. We have not simulated the case with a carrier frequency, which should need a higher degree of freedom.

In the case of non-positive amplitude, the frequency distance between the two spectral sidebands can be either lower or higher than the frequency resolution, and in both cases, the model is not appropriate. A higher degree of freedom allows a better approximation. The model obtained can be quite good but the amplitude modulation will never be correct. The frequency modulations are close to the sidebands of the modulation.

VII. PRIOR UPON FREQUENCY MODULATION

In the foregoing examples the amplitude modulation turns out to be the most difficult part of the Short-LP as in all model-based methods. However, the frequency modulation can be known in some particular applications. For instance, in mechanical systems such as turbines or gearboxes, time variations of the instantaneous frequency can be known a priori or estimated from independent synchronous measurements provided by tachometers. Assuming no error in this frequency estimation, this section aims at investigating the properties of the Short-LP for the estimation of the amplitude only, a crucial point in most applications.

Testset 5 is made of two components with two quadratic-frequency modulation components, each with a quadratic-amplitude modulation. The frequency modulations are known and are set in the initialization step of the algorithm. The amplitude modulations cross each other over the time duration of the signal. The Short-LP is then investigated according to the signal to noise ratio.

Figure 9 shows the result of the amplitude estimation averaged over 50 runs of an additive white noise for a signal to noise ratio equal to 15 dB and 5 dB. The exact frequency modulation is given just for information. At 15 dB the estimation of the amplitude estimation is accurate whereas a little bias appears at 5 dB. Table 6 shows the values of the mean square errors without and with a priori information on the frequency modulation. First and as already visible in Figure 3, the error comes essentially from the amplitude estimation. Second, when the frequency modulation is known, the error can come only from the amplitude estimation and is lower than in the previous case: 0.1685 instead of 0.8254 at 15 dB. The difference is really noticeable. Figure 10 shows the normalized mean square errors for a signal to noise ratio lying in the range 0 to 20 dB. The behavior of the Short-LP is nearly linear according to the variation of the signal to noise ratio.

VIII. CONCLUSIONS

This paper focuses on some important properties of the Short-LP method previously published by the authors. The Short-LP method estimates all the parameters of a time model, which provides an accurate description of a measured signal. The signals under consideration are multicomponents, nonlinear, nonstationary, and have short duration.

A first set of signal-test allows us to give an approximation of the frequency resolution, which seems to be equal to 2 bins, without forgetting that the signal is nonstationary and nonlinear over a short time duration. The Short-LP is then over performing the Fourier-based method. Further works are in progress for validating this results according the length the signal and the signal to noise ratio. We also attempt to explain the value obtained according to the dynamic signal bandwidth, the signal being nonstationary.

A second set of signal-test explains the number of degrees of freedom necessary for modelling one amplitude-modulation signal. Indeed, the constraint of positive amplitude set by the model definition guides the behaviour of the algorithm. This constraint induces that the amplitude modulation frequency is lower than the frequency resolution of the Short-LP and then the optimal model is a one-component model. If the amplitude is not positive the model is not optimal and the signal is better approximated by a higher degree of freedom. Increasing the degree of freedom will improve the time approximation but non necessarily the frequency and amplitude modulation.

A third set of signal-test shows that an a priori knowledge of the frequency modulations, as it is possible in mechanical systems, allows the Short-LP method to reach higher performance for the amplitude estimation.

  and of each frequency f c[n]. The phases obtained by a discrete integration of f c [n] up to 2π . For each c varying from 1 to C, the approximation is given by

  variances (by default setting to max (s[n]) for the amplitude parameters, 2π for the initial phase and half of the sampling frequency for the frequency parameters, theses values being formulated in the chosen base)

  the orders of the amplitude polynomial for each c=1,C (by default 2) -Mf c the order of the frequency polynomial for each c=1,C (by default 2) -iter the number of iterations (by default 500) σ ε the decrement of the parameter perturbation variance (by default 0.01) Initialization of the residue [ ] [ ] 1 r n s n = Deflation loop -Iterations from c=1,C

Figure 1 -

 1 Figure 1 -Good results of Short-LP with Testset 1 for 0.087 f Hz ∆ = (left) and 0.0609 f Hz ∆ = (right). Top to bottom, both theoretical (-) and estimate (--): time signal, frequency modulations and amplitude modulations of both components

Figure 2 -

 2 Figure 2 -Results of Short-LP with Testset 1 for 0.0348 f Hz ∆ = (left) and 0.0261 f Hz ∆ = (right). Top to bottom, both theoretical (-) and estimate (--): time signal, frequency and amplitude modulations of both components.

Figure 3 -

 3 Figure 3 -Mean square error of the time signal (left), frequency modulation (middle) and amplitude modulation (right), normalized by the signal power, of Testset 1 versus the frequency distance between the two components for 100 noise runs.

Figure 3

 3 Figure3presents the mean square errors versus the frequency distance and illustrates this behaviour. The three frequency ranges observed experimentally can be also detected on the mean square error evaluated between the time model and the theoretical signal. The error increases whereas the frequency distance decreases until a critical value, 0.0609 Hz, from which the errors decreases again. As expected, the mean square errors between the frequency or the amplitude modulations increase while the frequency distances decrease. In fact, this behaviour can be explained by the fact that the modulation orders are constant in the model. The degrees of freedom of the model permit an accurate modelling of the signal. Moreover, these simulations allow us to conclude that the error

Figure 4 -Figure 5 -

 45 Figure 4 -Dynamic signal bandwidth of the components of Testset 1 over the time support of 33 s (33 points)

Figure 6 -

 6 Figure 6 -Results of Short-LP with Testset 3.1 (left) and Testset 3.2 (right), 2 constant and linear FM parallel components (∆f = 0.1741 Hz ) with constant AM. Top to bottom, both theoretical (-) and estimate (--): time signal, frequency and amplitude modulations of both components. C=2.

Figure 7 -

 7 Figure 7 -Results of Short-LP with Testset 4.1, 1 constant FM component with positive AM, with C=2 (left) and C=1 (right). Top to bottom, both theoretical (-) and estimate (--): time signal, frequency and amplitude modulations of both components.

Figure 8 -

 8 Figure 8 -Results of Short-LP with Testset 4.2, 1 constant FM component with non-positive AM, with C=2 (left) and C=1 (right).Top to bottom, both theoretical (-) and estimate (--): time signal, frequency and amplitude modulations of both components.

Figure 9 -

 9 Figure 9 -Results of short-LP with Testset 5, two quadratic frequency and amplitude modulation with a priori known frequency modulation (top). Both theoretical(-) and mean estimate (--) amplitude modulation over 50 runs of an additive white noise with asignal to noise ratio equal to 15 dB (middle) and 5dB (bottom).

Figure 10 -

 10 Figure 10 -Mean square error of the time signal (left) and of the amplitude modulation (right), normalized by the signal power, of Testset 5 versus the signal to noise ratio for 50 noise runs.

Table 1 -

 1

The Short-LP with a simulating-annealing-based algorithm for a monocomponent signal

Table 2 -

 2

	σ	i	t	1 + =	1 1	-	σ ε σ	2 i	t	∀i

Step 2' of the global algorithm of the Short-LP with an adapted simulating-annealing-based algorithm for a multicomponent signal -In this case and compared with step 2 of the monocomponent case, the decrement σ ε of the parameter perturbation variance has a default value equal to 0.04 and there is no more temperature parameter.

Table 3 -

 3 The deflation algorithm of the Short-LP with an adapted simulating-annealing-based algorithm for a multi-component signalIn this section, the limit of the frequency resolution of the Short-LP is evaluated by means of simulations. A set of signals, referred to as Testset 1, is simulated according model[START_REF] Aburdene | On the computation of discrete Legendre polynomial coefficients[END_REF] with two close quadratic-frequency modulations. In order to break up the influence of the amplitude modulation, the amplitudes of both components are set to the same constant value. The parameter vectors V c as defined in (4) are set up for each component to

	V.	FREQUENCY RESOLUTION LIMIT
	1	[ 16 0.4 1.90 0.2 0.5	] T	1 with Ma	0,	1 Mf	2
	2	16 1.0	f	2,0	0.2 0.5	T	2 with Ma	0,	Mf	2	2

  Table 4 gives ∆f in frequency bins.

						f 2,0 in Hz	1.85	1.80	1.75	1.70	1.65	1.60	1.55	1.50	1.40	1.45
	f ∆ =	(	1,0 f	-	f	2,0	) [ ] 0 p n	in Hz 0.0087 0.0174 0.0261 0.0348 0.0435 0.0522 0.0609 0.0696 0.0783 0.0870
	f ∆ ×	(	N	1) + in frequency bins 0.2871 0.5742 0.8613 1.1484 1.4355 1.7226 2.0097 2.2968 2.5839 2.871
	Table 4 -Theoretical frequency resolutions of the Testset 1, 1,0 f being equal to 1.90 and	0	[ ]

Table 5 -

 5 Features and mean square error, normalized by the signal power, of Testset 3 et Testset 4 when considering two different degrees of freedom of the Short-LP. The colour qualifies the results: red, orange and green for bad, so-so and good respectively.

		Testsets	Peak number	Normalized mean square error
		33 points, signal to noise ratio 15 dB FM is for Frequency Modulation AM for Amplitude Modulation	in the theoretical spectrum	of the Short-LP C=2 C=1
		1. 2 constant FM parallel components (∆f = 0.1741	2 peaks	0.1435	0.7042
	Testset 3	Hz > 0.0609 Hz) with constant AM
		2. 2 linear FM parallel components (∆f = 0.1741 Hz>	Two spectral	0.1186	0.6909
		0.0609 Hz) with constant AM	patterns with
			several peaks
		1. 1 constant FM component with positive AM	2 sidebands	0.1389	0.1547
		(∆f m between the two sidebands =2×0.00727 Hz =	non-resolved
		0,01454 Hz < 0.0609 Hz)	by a global
			Fourier
			spectrum
		2. 1 constant FM component with non-positive AM	2 sidebands	0.3327	0.6204
		(∆f m between the two sidebands = 2×0.027 Hz =	resolved by a
	Testset 4	0.054 Hz < 0.0609 Hz)	global Fourier spectrum
		3. 1 linear FM component with positive AM	1 spectral	0.1061	0.2490
		(∆f m between the two sidebands = 2×0.0058 Hz =	pattern with
		0.0116 Hz < 0.0609 Hz)	several peaks
		4. 1 linear FM component with non-positive AM	2 sidebands	0.3279	0.7408
		(∆f m between the two sidebands = 2×0.043 Hz =	resolved by a
		0.0868 Hz > 0.0609 Hz)	global Fourier
			spectrum
		5. 1 constant FM component with non-positive AM	2 sidebands	0.3881	0.6930
		and the carrier	and the carrier
		(∆f m between the two sidebands = 2×0.0428 Hz =	peak resolved
		0.0856 Hz > 0.0609 Hz and the difference between	by a global
		the carrier frequency 0,15 Hz and the AM wave	Fourier
		frequency 0.0428 Hz = 0.1072 Hz > 0.0609 Hz)	spectrum

Normalized mean square error of the Short-LP 15 dB 5 dB

		Estimated FM	A priori known FM	Estimated FM	A priori known FM
	Time signal	0.2338	0.1648	0.3870	0.5087
	Amplitude modulation	0.8254	0.1685	0.8074	0.6089
	Frequency modulation	0.0540	0	0.0533	0

Table 6 -

 6 Mean square error of the time signal and the amplitude modulation, normalized by the signal power, of Testset 5 at two signal to noise ratio.

Appendix: Expression of the discrete base

In [START_REF] Jabloun | Estimation of the Amplitude and the Frequency of Nonstationary Short-time Signals[END_REF] we derived the base