Bruno Decreuse

André Zylberberg

Search intensity, directed search and the wage distribution

Keywords: Search e¤ort, Segmented markets, Equilibrium wage dispersion J.E.L. classi…cation: D83, J31, J41, J64

We propose a search equilibrium model in which homogenous …rms post wages along with a vacancy to attract job-seekers, while homogenous unemployed workers invest in costly job-seeking. The key innovation relies on the organization of the search market and the search behavior of the job-seekers. The search market is continuously segmented by wage level, individuals can spread their search investment over the di¤erent sub-markets, and search intensity has marginal decreasing returns on each sub-market. We show that there exists a non-degenerate equilibrium wage distribution. The density of this wage distribution is increasing at low wages, and decreasing at high wages. Under additional restrictions, it is hump-shaped, and it can be right-tailed. Our results are illustrated by an example originating a Beta wage distribution.

Introduction

This paper is a theoretical contribution to the literature on frictional wage dispersion. The size of residual wage dispersion in Mincerian wage regressions has motivated a major interest in understanding of wage distribution for homogenous labor. Controlling for individual characteristics typically leaves unexplained the two thirds of the variance of the (log of the) wage distribution [START_REF] Katz | Changes in the wage structure and earnings inequality[END_REF]. Modern theories of wage dispersion provide search-theoretic microfoundations for residual wage disparity. However, early search models have failed to generate empirically plausible wage distributions. In particular, they do not predict increasing density at low wages and decreasing density at high wages. They also fail to predict the single-peakedness This is the revised (and shorter) version of a paper that circulated under the title "Job search with ubiquity and the wage distribution". We thank Pierre Cahuc, Olivier Charlot, Pieter Gautier, Fabien Postel-Vinay, Etienne Wasmer, participants in University of Aix-Marseilles and University of Toulouse I seminars, and two referees and an editor of this review for their constructive comments.

y GREQAM, 2 rue de la charité, 13236 Marseille cedex 02, France. E-mail: decreuse@univmed.fr z CES-EUREQua, Maison des Sciences Economiques, 106-112 boulevard de l'Hôpital, 75647 Paris cedex 13, France. E-mail: azyl@univ-paris1.fr property. This particularity is usually pinned down by the explicit modelling of heterogeneity, but this is not the point. The challenge for this paper is to deliver the key properties of empirical wage distributions without the ad-hoc addition of heterogeneity.

Our paper introduces search intensity into the canonical directed search model. Directed search models have been examined by [START_REF] Hosios | On the e¢ ciency of matching and related models of search and unemployment[END_REF], [START_REF] Montgomery | Equilibrium wage dispersion and interindustry wage di¤erentials[END_REF], [START_REF] Peters | Ex ante price o¤ers in matching games: Non-steady states[END_REF], [START_REF] Moen | Competitive search equilibrium[END_REF], and Burdett, [START_REF] Burdett | Pricing and matching with frictions[END_REF]. They have been enriched by [START_REF] Shi | A directed search model of inequality with heterogeneous skills and skill-biased technology[END_REF] and [START_REF] Shimer | The assignment of workers to jobs in an economy with coordination frictions[END_REF] to account for …rm and worker heterogeneity. In such models, the search market is segmented by wage, workers can choose which jobs they apply for, and the probability of getting a job is a decreasing function of the length of the job queue. Wage competition takes place at the time of wage/market choice. In equilibrium, all wage o¤ers must yield the same utility: if not, the jobs would not be prospected. This implies that there is a unique wage o¤er balancing workers'marginal cost of seeking the highest wage o¤er (a lower employment probability) and their marginal bene…t (a better wage once employed).

In our framework, the search market is segmented by wage, workers can choose their search investment on each market place, and the technology that transforms search investment into matching probability has marginal decreasing returns. These assumptions sustain the existence of a continuous wage distribution. Workers can compensate a lower reward by a smaller e¤ort, thereby raising the marginal productivity of e¤ort. This provides them incentives to spread their e¤orts over several sub-markets at a time. Firms respond to such incentives by posting jobs on a continuum of markets. The assumption of decreasing marginal returns is essential. When the e¢ ciency of e¤ort function has constant marginal returns to search, workers concentrate their search e¤ort on the market that yields the highest reward. The equilibrium wage distribution collapses to the only wage featured by standard directed search models.

The equilibrium wage distribution re ‡ects the pattern of search investment and job o¤ers by wage. As the return to search …rst increases, and then decreases with wage, search investment evolves non-monotonically with the wage. We show that there is a unique maximum, corresponding to the only wage that would obtain if the marginal return to search investment were constant. The number of vacancies on each market tends to decrease with the wage, as paying higher wages must be compensated by lower search costs, and thus longer job queues. It also tends to increase with the number of e¤ective job-seekers, because recruitment rates depend on the ratio of vacancies to e¤ective number of job-seekers. The combination of these two e¤ects implies that the density of the wage o¤er distribution is increasing at low wages, and increasing at high wages. Those properties extend to the actual wage distribution -the one that counts. Provided the elasticity of the e¢ ciency of search e¤ort technology does not change too much with the amount of e¤ort, we show that both the wage o¤er distribution and the actual wage distribution admit a single peak.

The equilibrium wage distribution may also be consistent with another property of empirical wage distributions: they are right-tailed. We de…ne a long right tail by the requirement that the slope of the density of the distribution tends to zero as the wage becomes closer to the upper bound of the support of the distribution. We show that the density of the wage o¤er distribution is always right-tailed, while the density of the actual wage distribution may or may not be right-tailed. Those properties are illustrated by an example, in which the matching technology is Cobb-Douglas, and the e¢ ciency of search e¤ort is isoelastic. In that case, the actual wage distribution follows a Beta distribution and it is always hump-shaped.

The results heavily depend on our assumptions about market segmentation and search inten-sity. Market segmentation is a frequent assumption in the search literature. 1 Wage segmentation is a more abstract notion than segmentation by geographic or skill characteristics. However, considering other types of segmentation would fail in the goal to explain residual wage dispersion as econometricians could use observable characteristics to explain residual wage disparities. Similarly, search intensity is a popular way to model the search activity. Although we know very little about the e¢ ciency of e¤ort technology, the idea thereby search investments have decreasing returns is widely accepted. However, our model is more demanding in that we require that there is such a technology in each market segment. The e¢ ciency of e¤ort technology can receive a simple interpretation. In our model, workers are allowed to court the di¤erent jobs at various levels of aggressiveness. The degree of aggressiveness has decreasing returns on matching chances and varies with the wage o¤er as a result.

The reason why we consider a directed search model with endogenous search intensity can be inferred from the existing literature on frictional wage distributions.

On the one hand, undirected search models without a search intensity margin have a hard time explaining the salient features of empirical wage distributions. [START_REF] Diamond | A model of price adjustment[END_REF] shows that all …rms pay the monopsony wage in the basic model with homogenous agents. Burdett-Mortensen (1998) -hereafter BM -add on-the-job search. Their model admits a non-degenerate distribution. However, the density is strictly increasing: all workers apply to high-wage jobs with a positive probability, which provides incentives for …rms to create many high-wage jobs.

To obtain a single-peaked distribution, several papers add …rm heterogeneity, whether in the original BM model (see e.g. Van den Berg and Ridder, 1998, [START_REF] Mortensen | Equilibrium unemployment with wage-posting: Burdett-Mortensen meet Pissarides[END_REF], in models where employers can react to other …rms approaching their workers by making a countero¤er (Postel-Vinay and Robin, 2002a and 2002b), or in bargaining models with ex-post heterogeneity [START_REF] Moscarini | Job matching and the wage distribution[END_REF].

On the other hand, models with multiple applications easily feature equilibrium wage dispersion, despite agents' homogeneity and without on-the-job search. In papers with multiple applications, a worker can receive multiple o¤ers at the same time. Wage competition takes place at the time of choosing between di¤erent job o¤ers. In the random search model of [START_REF] Acemoglu | Wage and technology dispersion[END_REF], workers choose the number of o¤ers they receive. Acemoglu and Shimer show that there is equilibrium wage dispersion. The density of the wage o¤er distribution is strictly decreasing with a mass point at its upper bound. [START_REF] Galenianos | Directed search with multiple job applications[END_REF] consider the directed search model of Albrecht, [START_REF] Albrecht | Equilibrium directed search with multiple applications[END_REF] in which …rms post wages and workers send multiple applications. Galenianos and Kircher assume that …rms commit to pay the posted wage irrespective of the number of job o¤ers received by the applicant. They obtain a strictly decreasing density of the wage distribution. We believe that we modelize a di¤erent economic phenomenon. In our framework, workers court jobs with various levels of aggressiveness and they can only receive a single o¤er at a time despite they search on a continuum of markets. Wage competition takes place at the time of attracting the job-seekers, and not once they have received multiple o¤ers. The two approaches o¤er complementary views of the search activity: courting jobs is probably as important as deciding among several job o¤ers. Interestingly, this is because search aggressiveness varies with the wage that we can obtain a single-peaked wage distribution. [START_REF] Delacroix | Directed search on the job and the wage ladder[END_REF] consider a directed search version of BM. Their model features a strictly decreasing wage o¤er distribution, while the actual wage distribution can be strictly decreasing, strictly increasing, or even hump-shaped. However, individuals with the same income target the same job. The wage distribution conditional on former income is degenerate as a result. In our model, unemployed workers prospect all market segments simultaneously and may end up with any wage of the wage o¤er distribution.

The rest of the paper is organized as follows. Section 2 introduces and solves our model. Section 3 focuses on the properties of the equilibrium wage distribution. Section 4 concludes.

The model 2.1 Preferences and technology

We study the steady state of a continuous time economy. There are a continuum of identical and in…nitely-lived workers, and a continuum of …rms. Each …rm is associated with only one job. The measure of workers is normalized to one, while the measure of …rms is endogenously determined through entry. Both are risk neutral and discount time at instantaneous rate r. Jobs can be either …lled or vacant, while workers can be either employed or unemployed. There is no on-the-job search. A worker/…rm pair produces ‡ow output y until (exogenous) separation at rate q. Unemployed workers enjoy unemployment income z, 0 z < y, while …rms endowed with a vacancy bear the ‡ow cost k.

We make four key assumptions. First, the market is continuously segmented by wage. Segmentation follows the lines of the wage-posting search model developed, among others, by [START_REF] Moen | Competitive search equilibrium[END_REF] and [START_REF] Acemoglu | E¢ cient unemployment insurance[END_REF]. Firms post vacancies with non-negotiable wages. Workers, knowing all the posted wages, choose the amount of e¤ort they will spend to search for a job. If they decide to search for a job o¤ering a wage w, they compete with other workers seeking the same wage. Symmetrically, if a …rm posts a vacancy associated with the wage w, it will compete to attract workers with the other …rms o¤ering the same wage. For each wage w, workers face a speci…c queue length and vacant jobs have a speci…c probability of being …lled. For [START_REF] Acemoglu | E¢ cient unemployment insurance[END_REF], such a representation of the search process recognizes that the labor market is segmented by wages and that search frictions exist within each particular sub-market. Therefore, …rms advertise wages, and all …rms advertising a given wage and all workers applying for these jobs form a sub-market. By de…nition, the set of sub-markets is [0; 1).

Second, workers set their search intensity and allocate search e¤orts over the continuum of sub-markets. We denote by s : R + ! R + the function that maps the set of sub-markets into the set of possible values for the search intensity. The overall search e¤ort is S = R 1 0 s (w) dw, where s(w) is the search intensity on the sub-market o¤ering w: This assumption looks like the directed search assumption when workers play mixed strategies. In [START_REF] Burdett | Pricing and matching with frictions[END_REF], workers can apply to several jobs at a time with positive probabilities. A worker's opportunity cost of applying to a new wage is linear in the probability with which he applies there since he can only apply to the other wages with complementary probability. This cost turns out to be too high in the sense that the deviant …rms do not receive enough applications to make the same pro…ts as other …rms, even though workers would apply to a deviant wage with positive probability. However, our third assumption makes the model di¤erent from its predecessors.

Third, search intensity is costly and has marginal decreasing returns on each sub-market.

The opportunity cost of applying to a particular wage is not linear as a result, and this allows …rms posting any wage along a distribution to make equal pro…ts. We denote by c : R + ! R + the cost of e¤ort function, with typical element c (S). It depends on overall search e¤ort S.

We also de…ne x : R + ! R + the e¢ ciency of e¤ort function, with typical element x (s). The technology x transforms search e¤ort into e¤ective units of search on each sub-market.

Assumption A1

The cost of e¤ ort function c : [0; +1) ! [0; +1) is strictly increasing, convex, twice di¤ erentiable, and satis…es c (0) = 0, and c 0 (+1) = +1:

The e¢ ciency of e¤ ort function x : [0; +1) ! [0; +1) is strictly increasing, strictly concave, twice di¤ erentiable, and satis…es x (0) = 0, x 0 (0) = +1, and x 0 (+1) = 0: Job-seeking, like the search for ideas, has two components: tiredness and e¢ ciency. Tiredness depends on overall investment. This is why the marginal search cost depends on the aggregate search investment S. E¢ ciency depends on the amount of resources spent on the particular market that is prospected. The marginal productivity of search investment in terms of increased matching probability only depends on market-speci…c investment s. 2 Such marginal productivity is decreasing, which re ‡ects the fact that courting jobs more aggressively involves tasks of increasing di¢ culties.

Decreasing returns to search on each sub-market drive the existence of a non-degenerate equilibrium wage distribution. Intuitively, individuals are incited to spread their search investment over the di¤erent sub-markets (and …rms are incited to participate in the corresponding sub-markets). With constant returns to search, that is with x 0 (s) = x 0 , agents would direct their search investment toward the sub-market that yields the highest reward. There would be a single wage o¤er as a result, the competitive search equilibrium emphasized by the directed search literature with no on-the-job search (see [START_REF] Moen | Competitive search equilibrium[END_REF].

Fourth, agents meet according to a matching technology on each sub-market. If there are u unemployed persons in the economy, the overall search e¤ort on the sub-market o¤ering the wage w amounts to x(s (w))u where x(s (w)) denotes the market-speci…c mean e¢ ciency of search e¤orts. With v(w) vacancies o¤ering the wage w, the ‡ow number of matches on submarket w during the small time period dt is equal to M [x(s (w))u; v(w)] dwdt. The properties of the matching technology are standard. It is twice continuously di¤erentiable, strictly increasing in each of its arguments, strictly concave with M V V < 0 and linearly homogenous: It satis…es the boundary conditions M (U; 0) = M (0; V ) = 0, and lim

U !+1 M (U; V ) = lim V !+1 M (U; V ) = +1.
We impose more than strict concavity, as M V V < 0. As we discuss later, this condition is a su¢ cient condition for the uniqueness of an equilibrium in the standard directed search model.

Let (w) v (w) =x (s (w)) u be market-speci…c tightness and m ( ) M (1= ; 1). The ‡ow probability that a vacant job meets a job-seeker and the ‡ow probability that a job-seeker meets a vacant job per e¢ cient unit of search on the sub-market o¤ering the wage w are given by:

m [ (w)] dw M [x(s (w))u; v(w)] v(w) dw (w) m [ (w)] dw M [x(s (w))u; v(w)]
x(s (w))u dw;

In the sequel we will denote by ( ) m 0 ( )=m( ) the elasticity of the function m( ) with respect to : The elasticity ( ) belongs to [0; 1]:

Equilibrium

We denote by V u and V e (w) the expected utility of an unemployed worker and of a worker paid wage w. In the online Appendix, we show that workers can only receive one o¤er at a time. This allows us to write the standard value functions de…ning agents'expected gains. We have

rV u = max s(:) z c (S) + Z 1 0 x [s (w)] (w) m [ (w)] [V e (w) V u ] dw (1
)

rV e (w) = w + q [V u V e (w)] (2)
It follows from (2) that workers do not prospect sub-markets whose wage is lower than the reservation wage R = rV u .

The asset values of a vacancy advertised at wage w, denoted v (w), and of a …lled job paying w, denoted e (w), satisfy the arbitrage equations:

r v (w) = k + m [ (w)] [ e (w) v (w)] (3)
r e (w) = y w + q [ v (w) e (w)] (4)
An allocation is a tuple ( ; s; S; R; v ; ), where R + is the set of opened sub-markets, i.e. the sub-markets that are prospected by the unemployed and the …rms that have posted vacancies.

De…nition 1 An equilibrium is an allocation ( ; s; S; R; v ) such that (i) Pro…t maximization v (w) 0 for all w 2 R + ;

(5) with equality i¤ w 2 .

(ii) Optimal search intensity

s(w) = argmax s z c(S) + Z 1 R x(s) (w) m ( (w)) w R r + q dw (6) R = z c(S) + Z 1 R x [s (w)] (w) m ( (w)) w R r + q dw (7) with S = R 1 R s (w) dw.
Firms maximize pro…ts, and workers optimally spread their search investments over the di¤erent sub-markets. De…nition 1 shapes the set of opened sub-markets in equilibrium. Workers and …rms must have incentives to participate in those sub-markets. De…nition 1 also sets outof-equilibrium beliefs that rules out situations in which …rms may not create jobs paying a potentially pro…table wage, conjecturing that none (or, at least, too few workers) would apply.

Proposition 1 Equilibrium characterization

Under Assumption A1, (i) In equilibrium, we have

= [R; y] (8)
k m [ (w)] = y w r + q , 8w 2 [R; y] (9)
c 0 (S) = x 0 [s(w)] (w) m [ (w)] w R r + q , 8w 2 [R; y] (10)
R = z c(S) + Z y R x [s (w)] (w) m ( (w)) w R r + q dw (11) S = Z y R s (w) dw (12)
(ii) There exists a unique equilibrium.

All wages belonging to the interval [R; y] are prospected, so that [R; y] is also the support of the equilibrium wage distribution. We explain this result in three steps.

First, several wages are o¤ered in equilibrium. Equilibrium wage dispersion relies on the fact that the e¢ ciency of e¤ort technology features marginal decreasing returns. This can be understood from equation [START_REF] Gautier | Unemployment and search externalities in a model of heterogeneous jobs and workers[END_REF] that describes the allocation of e¤orts over the sub-markets. The left-hand side of the equation corresponds to the marginal cost of e¤ort on sub-market w. The right-hand side corresponds to the marginal bene…t derived from searching on such a sub-market. The marginal bene…t is composed of two terms: the marginal productivity of e¤ort x 0 (:) times the expected reward from an additional unit of e¤ort in the sub-market. Workers can compensate a lower reward by a smaller e¤ort, thereby raising the marginal productivity of e¤ort. Workers are incited to spread their e¤orts over several sub-markets at a time. If the marginal productivity of search e¤ort were constant, workers would only invest on the submarket that yields the highest reward. With x(s) = x 0 s, one can verify that the equations ( 9) to [START_REF] Hosios | On the e¢ ciency of matching and related models of search and unemployment[END_REF] of Proposition 1 give a unique wage w and, consequently, a unique amount of search e¤ort spent on the unique sub-market posting the wage w.

Second, the support of the wage distribution is continuous. During the proof of Proposition 1, we show that if a wage a 2 (R; y) is not o¤ered in equilibrium, a deviant …rm can make positive pro…ts by creating such a job, which is impossible. Owing to the free-entry condition, such a pro…t opportunity must be exploited. Consequently, there are no holes in the equilibrium set of o¤ered wages. 3Third, the lower-bound of the wage distribution is R, while the upper-bound of the wage distribution is y. This results from the assumption that x 0 (0) = 1, and from the zero-pro…t condition featured by De…nition 1. When the wage w tends to y, the …rm's hiring rate has to go to in…nity for the free-entry condition [START_REF] Galenianos | Directed search with multiple job applications[END_REF] to hold. This implies that worker's job-…nding rate has to go to 0. Similarly, when the wage w tends to R, the wage di¤erential w R that governs search investment tends 0. Why does the worker apply to such wages? Equation ( 10) gives the answer. The marginal cost of such an application is c 0 (S), while the marginal bene…t is proportional to x 0 (s (:)). This marginal bene…t can increase without limit as search intensity tends to 0. 4The job-…nding rate is = R y R x [s (w)] (w) m [ (w)] dw, that is the sum of the di¤erent rates of contact over the di¤erent markets that the job-seekers prospect. Unemployment is computed from the equality between ‡ows in and out of unemployment: q(1 u) = u. The unemployment rate is u = q= (q + ).

Equilibrium wage distributions

The purpose of this section is to analyze the shape of the wage distribution that is implied by our model. We proceed in three steps. First, we describe the equilibrium pattern of search investment by wage. Second, we focus on the equilibrium wage distribution. Third, we focus on the Cobb-Douglas example.

Market-speci…c search investment

Workers prospect a continuum of sub-markets in equilibrium. In this sub-section, we analyze the pattern of equilibrium search investment by wage.

Proposition 2 Search intensity

Under Assumption A1, (i) The e¤ ort function s : [R; y] ! [0; +1) satis…es s (R) = s (y) = 0;

(ii) The e¤ ort function is \-shaped, i.e. there exists a unique w 2 (R; y) such that s 0 (w) R 0 i¤ w S w .

The key …nding of Proposition 2 is the non-monotonicity of the relationship between wage and search investment. The pattern of search investment re ‡ects the pattern of marginal reward to search. The wage has two con ‡icting e¤ects. On the one hand, a higher wage raises the return to search at given market tightness, thereby motivating search investment. On the other hand, market tightness decreases with wage. A higher wage deteriorates search prospects, thereby reducing search investment. The overall e¤ect is captured by the term (w) ( (w)) y + [1 ( (w))] R w. This term is positive at low wages, while it is negative at higher wages. The search investment then reaches a maximum on the market where the reward is the highest. This corresponds to the wage w , which is implicitly de…ned by (w ) = 0. The wage w is the only wage o¤er that would result if the marginal return to search investment were constant. It is also the only wage that would result if workers could not choose their search intensity as in the standard directed search equilibrium (see e.g. [START_REF] Moen | Competitive search equilibrium[END_REF][START_REF] Burdett | Pricing and matching with frictions[END_REF]). 5

Wage distribution

We distinguish the wage o¤er distribution from the actual wage distribution.

The wage o¤er distribution is the distribution of vacancies by wage. The number of vacancies advertised at wage w is v(w) = (w) x [s (w)] u, that is the product of market-speci…c tightness by the number of e¤ective job-seekers. The total number of vacancies is V = R y R v (w) dw. The cdf and the pdf of the wage o¤er distribution are then de…ned by

F (w) = R w R v ( ) d V ; F 0 (w) = v(w) V = (w) x [s (w)] u V (13)
Changes in the density F 0 re ‡ect changes in market tightness and search investment.

As workers court jobs paying di¤erent wages with di¤erent levels of aggressiveness, search intensity varies with the wage, and the actual wage distribution departs from the wage o¤er distribution. Let G(w) be the cdf of the actual wage distribution among employees. This cdf can be deduced from a standard ‡ow equilibrium reasoning. For each wage w 2 [R; y], the out ‡ow from the pool of those employed who earn less than w equals the in ‡ow from the pool of unemployed:

q(1 u)G(w) = u Z w R x [s( ] ( )m [ ( )] d (14)
Since q(1 u) = u, and remembering that v (w) = V F 0 (w) = x [s (w)] (w)u, it ensures that:

G(w) = V u Z w R F 0 ( )m [ ( )] d (15)
Thus, one has:

G 0 (w) = V u F 0 (w)m [ (w)] (16)
Changes in the density of the actual wage distribution are driven by changes in the density of the wage o¤er distribution and changes in market tightness.

Proposition 3 Wage distribution Under Assumption A1, (i) The wage o¤ er distribution

F 0 : [R; y] ! [0; 1] is non-monotonic and satis…es F 0 (R) = F 0 (y) = 0. If lim !0 ( ) > 0 and lim w!y (w) < 1, F 00 (y) = 0 (ii) The actual wage distribution G : [R; y] ! [0; 1] is non-monotonic and satis…es G 0 (R) = G 0 (y) = 0. Moreover, G (w) < F (w) for all w 2 (R; y)
The density of the wage o¤er distribution is non-monotonic. The density is zero at the bounds of the distribution. It is positive between those bounds. These properties mostly re ‡ect the pattern of search investment by wage. Workers have very few incentives to prospect jobs that o¤er slightly more than the reservation wage. In turn, …rms do not o¤er many such jobs. Similarly, workers do not seek high-wage jobs very intensively as they know that …rms do not o¤er many of them. This strengthens …rms'incentives not to create many high-wage jobs.

As workers sample job o¤ers from the equilibrium wage o¤er distribution, the actual wage distribution is non-monotonic, increasing at low wages and decreasing at high wages. The actual wage distribution also looks better than the wage o¤er distribution. Job-seekers observe the wage o¤er distribution and alter the wage they will be paid by modulating their search investment on each sub-market. This optimization process implies that the wage o¤er distribution …rst-order stochastically dominates the actual wage distribution.

Finally, the actual wage distribution can be right-tailed. However, unlike the wage o¤er distribution, the actual wage distribution is not always right-tailed.

Non-monotonicity does not guarantee that the wage distribution is single-peaked. The density may feature multiple peaks as a result.

Assumption A2 Let (s) = sx 0 (s)=x(s) sx 00 (s)=x 0 (s) . For all s 0, 0 (s) x 0 (s) =x (s).

Assumption A2 limits the set of possible e¤ort functions by restricting the upward volatility vis-à-vis changes in search intensity. We provide further details below.

Proposition 4 Single-peak property

Under Assumptions A1 and A2, (i) The wage o¤ er distribution is \-shaped, i.e. there exists a unique w 2 (R; y) such that F 00 (w) R 0 i¤ w S w .

(ii) The actual wage distribution is \-shaped, i.e. there exists a unique w 2 (R; y) such that G 00 (w) R 0 i¤ w S w .

The wage distribution is single-peaked provided additional restrictions on the e¢ ciency of e¤ort function hold. The e¤ect of the wage on the density of the wage o¤er distribution is given by F 00 (w) F

+ x 0 [s (w)] x [s (w)] s 0 (w) 0 (w) 0 (w) (w)
Changes in the density re ‡ect the pattern of market tightness and search e¤ort by wage. The former e¤ect is negative, as …rms post fewer jobs when the wage increases. The latter e¤ect is non-monotonic. Its relative weight depends on the elasticity of the e¤ort function. This elasticity may vary so much that there may be several peaks. Assumption A2 restricts the set of e¤ort functions by imposing bounds on such variations. Such bounds are not very demanding. For instance, A2 is compatible with 0 (s) 0, which encompasses the case of isoelastic e¢ ciency search functions with 0 (s) = 0. In addition, A2 is a su¢ cient condition, which means that there are cases where A2 does not hold and, still, the wage o¤er distribution is hump-shaped.

These results are in sharp contrast with the literature discussed in the introduction, which predicts either increasing or decreasing density of the wage distribution6 .

A Cobb-Douglas example

We consider explicit forms for the matching function and the e¢ ciency of e¤ort function. The following Proposition shows that the actual wage distribution is strongly linked with the Beta distribution.

Proposition 5 The Cobb-Douglas example

Assume that m( ) = M 0 ; M 0 > 0; 2 (0; 1) and x(s) = s 1+ ; > 0. Let ! = (w R) = (y R) be the normalized wage, and let H be the cdf of the actual normalized wage distribution. Then, H is the cdf of a 1 ( + 1) + 1; + 1 distribution with

H 0 (!) = (1 !) 1 ( +1) ! B 1 ( + 1) + 1; + 1 , 8! 2 [0; 1]
where B is the Beta function such that

B (t 1 + 1; t 2 + 1) = Z 1 0 (1 ) t 1 t 2 d
The Cobb-Douglas example displays several features. First, we can …nd a normalization of the wage such that the actual distribution of such a normalized wage follows a Beta distribution. Second, the parameters of the Beta distributions only involve the elasticity of the matching function and the elasticity of e¤ort function. Third, the wage distribution is single-peaked. Fourth, we can highlight the parameter circumstances under which the actual wage distribution has a ‡at right tail. Indeed, G"(y ). Therefore, the wage distribution is right-tailed when the parameters of the matching function and the search function satisfy > (2 1) = (1 ). This restriction always holds when the elasticity = 1=2. Similarly, we can show that G 00 (R) = 0 if > 1 and G 00 (R) = 1 if < 1. Therefore, the Cobb-Douglas case is consistent with an actual wage distribution characterized by a single peak, a ‡at right tail, and no left tail. This is so when (2 1) = (1 ) < < 1. Computing the actual wage distribution requires to set all the parameters of the model. Table 1 gives the parameter values. We have the French economy in mind, where the unemployment rate u is close to 10% over the past two decades, the job loss rate q is 10% and the mean unemployment duration 1= is one year. Output is normalized to one, and unemployment income (plus value of non-labor time) accounts for 40% of output per worker. The cost of e¤ort is c (S) = c 0 s , 1. We choose = 2, so that the cost of e¤ort is strictly convex. The other parameters have been set to simple values. r q c0 M 0 k y z :05 :10 1:0 2:0 3:0 :25 1 :4 :5 1:0

) = 0 if > (2 1) = (1 ) and G"(y) = 1 if < (2 1) = (1

Table 1: Parameter values of the baseline simulation

In the baseline simulation where = :5 and = 1:0, the reservation wage is R = :9 and the unemployment rate is u = 9:5%. Table 2 presents the e¤ects of changes in and on the equilibrium reservation wage and the equilibrium unemployment rate. In each case, we consider 5 di¤erent values of the parameter. When we increase the elasticity of the matching technology, the reservation wage goes down from .94 to .86, while the unemployment rate goes up from 5.5% to 13.5%. When we raise the elasticity = (1 + ) of the e¢ ciency of e¤ort technology, the reservation wage goes up from .89 to .94, while the unemployment rate goes down from 11.5% to 4.0%. The parameter values are given by Table 1.

Figures 1 and 2 depict the equilibrium wage distributions. Qualitatively, those distributions feature the properties described by Proposition 5. Indeed, parameters and govern the shape of the wage distribution. Figures 1 and2 allow us to see the impacts of and on the equilibrium reservation wage. As increases, the wage distribution becomes less concentrated. Moreover, the long right tail progressively disappears. As increases, the wage distribution becomes more concentrated. When tends to in…nity, the e¢ ciency of e¤ort function has constant marginal returns. As a result, workers concentrate their search investment in the sub-market where the return is the highest. The actual wage distribution collapses to the Hosios wage w , the only wage o¤er featured by the standard directed search model.

Conclusion

This paper examines the choice of search intensity in the standard directed search model with homogenous agents. We make two assumptions: the search place is segmented by wage, and search investment has decreasing returns in each market segment. While setting search e¤ort in each market, workers can compensate a lower marginal return to search by a lower search investment that raises the marginal productivity of e¤ort. The model features a non-degenerate equilibrium wage distribution as a result. This equilibrium wage distribution is non-monotonic. With additional restrictions, it is also hump-shaped and right-tailed.

Unlike geographic or skill segmentation, wage segmentation is an abstract notion and the market segments can hardly be identi…able. Similarly, the shape of the search technology is exogenous and does not derive from elementary principles. However, the predicted wage distribution has appealing empirical properties. The model could be enriched so as to account for on-the-job search and worker/…rm heterogeneity. In turn, the enriched model could be structurally estimated. Such a structural estimation could help revisit the respective roles of heterogeneity and frictional dispersion in the wage distribution.

The estimates could be used so as to evaluate the impacts of policy experiments like changes in payroll tax rates or in unemployment bene…ts. Compared to alternative models of the search market, our model makes a key prediction: owing to decreasing returns to search in each market segment, labor market policies have heterogenous e¤ects across market segments. Such policies should not only a¤ect employment and the mean wage, they should also a¤ect the whole wage distribution. Parameter goes from .3 to .7, while the other parameters are given by Table 1. 1.

APPENDIX A Proof of Proposition 1

Part (i). Equations ( 9) to ( 12) directly result from De…nition 1. It remains to show that = [R; y]. First, sub-markets paying more than y involve v (w) < 0, and they must be closed. Second, …rms cannot o¤er wages lower than R. Those jobs would not be prospected by the unemployed, implying that (w) = 1. Consequently, v (w) < 0. Third, suppose that there is some wage a 2 (R; y) that is not o¤ered in equilibrium, and consider a particular employer who o¤ers wage b 6 = a in equilibrium. This employer obtains r

v (b) = k(r+q)+m[ (b)](y b) r+q+m[ (b)]
= 0 by equilibrium de…nition. The employer may think about o¤ering wage a. However, if this wage is not o¤ered in equilibrium, this must be so because the job-seekers do not invest in this market, i.e. s (a) = 0. This is compatible with equation ( 9) provided that (a) = 0. In such a case, the matching probability is one in any time interval for the …rm. Ex-ante pro…t is then worth r v (a) = y a > 0. The deviant …rm can make positive pro…ts, which is impossible. Wages R and y belong to the equilibrium support by continuity.

Part (ii). Using the properties of the matching function, equation ( 9) de…nes a unique speci…c market tightness function (w). Then, equation ( 10) can be solved in s as a function of w; S and R. Let e (w; R; S) be this unique solution. From Assumption A1 and the implicit function theorem, the partial derivatives of e (w; R; S) are such that:

e S =
c"(S) c 0 (S)

x 0 (e)

x"(e) 0; e R = 1 w R x 0 (e)

x"(e) < 0

Now, substitute e (w; R; S) for s(w) in equation [START_REF] Halko | Search direction and wage dispersion[END_REF], and consider the function S 1 (R) such that R = z c(S 1 ) +

Z y R
x [e (w; R; S 1 )] (w) m [ (w)] w R r + q dw (19)

The derivative of the function S 1 (R) reads: x [e (w; z; S 1 (z))]

S 0 1 (R) =
x 0 [e (w; z; S 1 (z))] dw

Moreover, when e (w; R; S) is substituted for s(w) in equation [START_REF] Hosios | On the e¢ ciency of matching and related models of search and unemployment[END_REF], we obtain another equation de…ning a unique S as a function of R. We call this function S 2 (R). Di¤erentiating this latter equation with respect to R gives: The properties of the matching function and equation [START_REF] Gautier | Unemployment and search externalities in a model of heterogeneous jobs and workers[END_REF] imply that e(R; R; S) = 0. Consequently

S 0 2 (R) = R y R e R dw 1 R y R e S dw < 0 (22)
So far, we have shown that S 0 1 (R) < S 0 2 (R) < 0. In addition, S 2 (y) = 0. Lastly, S 2 (z) is given by: which proves that S 1 (z) > S 2 (z). The properties of functions S 1 (R) and S 2 (R) entail that they cross once at a point such that R 2 (z; y). Thus, the equilibrium values of R and S are unique. It follows that the equilibrium functions (w) and s (w) given respectively by ( 9) and ( 10) are also unique.

B Proof of Proposition 2

Part (i). Relation [START_REF] Galenianos | Directed search with multiple job applications[END_REF] and the properties of the matching technology imply that (y) = 0. Therefore, relation [START_REF] Gautier | Unemployment and search externalities in a model of heterogeneous jobs and workers[END_REF] and Assumption A1 imply that s(y) = 0. Furthermore, [START_REF] Galenianos | Directed search with multiple job applications[END_REF] shows that (R) is …nite, and [START_REF] Gautier | Unemployment and search externalities in a model of heterogeneous jobs and workers[END_REF] and Assumption A1 then entail that s(R) = 0. Part (ii). Let (w) [ (w)] y + [1 [ (w)]] R w. Since s is continuous and s (w) > 0 for all w 2 (R; y), the function s is non-monotonic. Di¤erentiating the logarithm of both sides of equation [START_REF] Gautier | Unemployment and search externalities in a model of heterogeneous jobs and workers[END_REF] with respect to w, we get:

x 00 [s(w)] x 0 [s(w)] s 0 (w) = 0 (w) (w) [1 ( (w))] + 1 w R (23)
The equilibrium pattern of tightness by wage is given by equation [START_REF] Galenianos | Directed search with multiple job applications[END_REF]. Di¤erentiating the latter equation with respect to w gives:

( (w)) 0 (w) (w) = 1 y w (24)
Then, eliminating 0 (w)= (w) between ( 23) and ( 24) one obtains:

s 0 (w) x 0 [s(w)]
x 00 [s(w)] (w) ( (w)) (y w)(w R) [START_REF] Van Den Berg | An empirical equilibrium search model of the labor market[END_REF]

.

Figure 1 :

 1 Figure 1: Equilibrium wage distribution and changes in the elasticity of the matching technology.Parameter goes from .3 to .7, while the other parameters are given by Table1.

Figure 2 :

 2 Figure 2: Equilibrium wage distribution and changes in the elasticity of the e¢ ciency of e¤ort technology. Parameter goes from .5 to 20, while the other parameters are given by Table1.

For

  R y R e R x 0 [e (w; R; S 1 )] (w) m [ (w)] w R r+q dw 1 = (r + q) c 0 (S 1 ) R y R e S x 0 [e (w; R; S 1 )] (w) m [ (w)] w R r+q dw = R y R e R dw 1 R y R e S dw 1 + = (r + q) c 0 (S 1 ) 1 R y R e S dwHence, equation (19) implicitly de…nes S 1 (R) with R = y, one has c(S 1 (y)) = z y < 0; which means that the function S 1 (R) crosses the horizontal axis before R = y: For R = z; the value of S 1 (z) is given by:c [S 1 (z)] = Z y z x [e (w; R; S 1 )] (w) m [ (w)] w R r + q dw = c 0 [S 1 (z)]Z y z

S 2

 2 (z) = Z y z e [w; z; S 2 (z)] dwAssumption A1 implies that x (e) =x 0 (e) > e and c(s)=c 0 (s) s, therefore[START_REF] Postel-Vinay | Wage dispersion with worker and employer heterogeneity[END_REF] givesS 1 (z) c [S 1 (z)] c 0 [S 1 (z)] > Z y z e[w; z; S 1 (z)] dw;

Table 2 :

 2 Impacts of and on equilibrium reservation wage and unemployment rate.

		3	.4	.5	.6	.7
	R	.94 .92 .90 .87	.86
	u (in%)	5.5 7.5 9.5 11.5 13.5
		.5	1	5	10	20
	R	.89 .90 .92 .93	.94
	u (in %) 11.5 9.5 5.5 4.5	4.0

In mismatch models, the search place is segmented by geographic location and workers are imperfectly mobile across locations (seeShimer,

2007). In models with two-sided heterogeneity, the search place is segmented by job type and workers choose the subset of sub-markets they participate in (see e.g.[START_REF] Moscarini | Excess worker reallocation[END_REF][START_REF] Gautier | Unemployment and search externalities in a model of heterogeneous jobs and workers[END_REF][START_REF] Decreuse | Choosy search and the mismatch of talents[END_REF].

Of course, search intensity may also have decreasing returns across the sub-markets. This could be captured by assuming that x x (s; S). In this paper we only focus on the novel aspects induced by the dependence vis-à-vis s.

The fact that the mass of …rms responds to a free-entry condition is not very important for the result of a continuous distribution. What is important is the equal-pro…t condition that must hold on each sub-market. This condition is compatible with the assumption of a …xed number of jobs/…rms.

If x 0 (0) were bounded, the marginal bene…t from search e¤ort could not increase without limits. This would restrict the interval of opened sub-markets. The bounds of the distribution would be the two roots wmin and wmax of c 0 (S) =x 0 (0) = (w) m [ (w)] (w R) = (r + q) with R < wmin < wmax < y. The assumption that x 0 (0) = 1 preserves simplicity without losing much insight from the case where x 0 (0) is bounded.

Our model predicts a unique directed search equilibrium, while[START_REF] Moen | Competitive search equilibrium[END_REF] shows that there may be multiple equilibria. The reason is that we impose MV V < 0, while Moen only assumes strict concavity. The condition MV V < 0 is a su¢ cient uniqueness condition.

[START_REF] Halko | Search direction and wage dispersion[END_REF] assume that two search places coexist. In the wage demand market, unemployed contact vacancies; in the wage o¤er market, vacancies contact unemployed. The density of the wage distribution is increasing at low wages, decreasing at high wages, but U-shaped in the middle.

Taking the derivative of function and using [START_REF] Shimer | Mismatch[END_REF] gives:

The properties of the matching technology imply that [ m ( )]

we have

It follows that:

Relations ( 26) and (28) imply that

Therefore, 0 (w) < 0 whenever (w) 0. Consequently, the equation (w) = 0 has a unique root w .

C Proof of Proposition 3

where is de…ned during the proof of Proposition 2 and (s) = (s) = sx 0 (s)=x(s) sx 00 (s)=x 0 (s) . According to proposition 2, one has s (R) = s (y) = 0. Remembering that (y) = 0 and that (R) is …nite, relation [START_REF] Katz | Changes in the wage structure and earnings inequality[END_REF] arrives at F 0 (R) = F 0 (y) = 0. Non-monotonicity results from the facts that F 0 is continuous and F 0 (w) > 0 for all w 2 (R; y). Using ( 17), ( 24) and (25), we have:

Equation (30) shows that:

Using ( 9) and ( 13), one arrives at:

The result follows from the facts that (w) m ( (w)) and x [s (w)] tend to zero when w ! y. Part (ii). As G 0 (w) = V F 0 (w)m [ (w)] = u, part (i) of Proposition 3 implies that G 0 (R) = G 0 (y) = 0. Non-monotonicity follows from the fact that G 0 (w) > 0 for all w 2 (R; y) and the continuity of G 0 . As m [ (w)] = k(r + q)= (y w) -see (9) -, relation [START_REF] Mortensen | Equilibrium unemployment with wage-posting: Burdett-Mortensen meet Pissarides[END_REF] becomes:

Let w 0 denote the unique wage such that V u k(r+q) y w 0 = 1. Suppose …rst that w 0 R. One has G 0 (w) < F 0 (w) for w < w 0 and G 0 (w) > F 0 (w) for w > w 0 : Therefore, when w < w 0 one has:

While, when w > w 0 , one has:

Hence, G(w) < F (w) when w 0 R. Now, assume that w 0 < R. Then, G 0 (w) > F 0 (w) for all w R, and (31) holds for all w R: Consequently, G(w) < F (w) when w 0 < R.


Bastien Lyonnet
email: bastien.lyonnet@gipsa-lab.grenoble-inp.fr

Cyrille Siclet
email: cyrille.siclet@gipsa-lab.grenoble-inp.fr

Jean-Marc Brossier
email: jean-marc.brossier@gipsa-lab.grenoble-inp.fr

J.-M Brossier

Maximum Likelihood Estimator For Doppler Parameter And Cramer Rao Bound In ZP-OFDM UWA Channel

A Doppler estimation system using a maximum likelihood criterion is presented in the context of underwater acoustic communications between moving transmitter/receiver. We simulate the method for the estimation of the Doppler effect induced by an underwater acoustic channel (UWA) using Zero Padded-Orthogonal Frequency Division Multiplexing (ZP-OFDM). Among the wide range of physical processes that impact OFDM communications through the underwater environment, Doppler effect is an important cause of loss of orthogonality and must be compensated before discrete Fourier transform based demodulation can be performed. The proposed approach is based on the perfect knowledge of the emitted signal during one block and rely on the assumptions of a small variation of the Doppler effect induced by the canal between two ZP-OFDM blocks and, above all, during one block.

I. INTRODUCTION

During the last three decades, underwater acoustic communications have received a growing interest. In the same time, OFDM modulation has been extensively developed in the context of wireless communication systems for its robustness against frequency selective channels. The idea comes naturally to use OFDM in underwater channel. But the ocean is a temporally and spatially varying propagation environment. The Doppler effect, induced by the channel, both affects the carrier tracking and the symbol synchronization. Relative motion between transmitter and receiver is usually unavoidable. This complicates the use of wide band, high speed acoustic communication techniques. The performance of a conventional OFDM receiver is severely limited by the intercarrier interference (ICI) due to the Doppler effect. Thus, several techniques have recently been developed in order to estimate and compensate it [START_REF] Li | Non-uniform doppler compensation for zero-padded OFDM over fastvarying underwater acoustic channels[END_REF], [START_REF] Sharif | A computationally efficient doppler compensation system for underwater acoustic communications[END_REF].

In this paper, we focus on the receiver design. We developed a Doppler estimation method using a Maximum Likelihood (ML) criterion. This method relies on the perfect knowledge of the transmitted signal during one OFDM block. Considering communications between two underwater vehicles, we may realize the assumption that the Doppler will still be the same on the following blocks. Simulations are made using a channel with same Doppler effect on each path. Moreover, we also assume that the Doppler effect is constant from the beginning to the end of one OFDM block. Finally, measures of the estimator variance are realized and compared to the Cramer Rao Bound (CRB).

II. GENERAL CONTEXT

The framework of our study is the transmission of digital information in shallow water environment. We first recall here the fundamentals of the multicarrier communication system we use, the OFDM with Zero Padding (ZP), and we describe then the channel model we have used.

A. OFDM with Zero Padding

The principle of ZP-OFDM is to use several carriers in parallel,

f m (f m = f 0 + mF 0 ), -M/2 ≤ m ≤ M/2 - 1,
with M the number of carriers, with a symbol duration slightly higher than necessary. Indeed, since the frequency spacing F 0 is identical between each carrier frequency, then the minimum symbol duration is equal to T 0 = 1/F 0 . And yet, in ZP-OFDM, a symbol duration is in fact equal to T = T 0 + T g > T 0 . That is why T g is called a guard time. Then, for -T g -T 0 /2 ≤ t ≤ T 0 /2 (one symbol duration), the ZP-OFDM baseband signal writes:

s(t) = M 2 -1 m=-M 2 c[m]e j2πmF0t g(t) (1)
and the transmitted signal is

s HF (t) = {s(t)e j2πf0t } (2)
with c[m] the transmitted data (QPSK 1 symbols in here) and with g(t) = 1/ √ T 0 for t ∈ [-T 0 /2, T 0 /2[, and g(t) = 0 otherwise. Thus, s(t) = 0 for -T g -T 0 /2 ≤ t < -T 0 /2, which results in a Zero Padding operation after sampling of the signal. It is worthwhile recalling that this causes a loss in spectral efficiency (no information is transmitted during this time interval), but it considerably simplifies the equalization procedure. If the guard interval is long enough, there will be no interference between OFDM symbol at the reception [3, chapter 11].

B. Underwater acoustic channel model

Our modelisation relies on several assumptions. First of all, we suppose that the acoustic channel can be described thanks to a time-varying multipath model. We moreover assume that the time-varying nature of the channel during one OFDM block is mainly caused by the relative motion between the receiver and the transmitter [START_REF] Li | Non-uniform doppler compensation for zero-padded OFDM over fastvarying underwater acoustic channels[END_REF]. This assumption 1 QPSK : Quaternary Phase Shift Keying is reasonable considering that the coherence time of an UWA channel is about one second [START_REF] Preisig | Acoustic propagation considerations for underwater acoustic communications network development[END_REF] and a typical duration for T is 200ms, depending on the number of subcarriers we choose. Then, assuming a constant relative speed v between the emitter and the receiver, and denoting c the sound velocity and θ l the arrival angle of path l, the received modulated signal r HF (t) writes:

r HF (t) = L l=1 A l s HF t - v cos(θ l ) c t -τ l + n HF (t), (3)
L being the number of propagation paths, A l the amplitude of the l-th path and τ l the corresponding path delay, and b HF (t) being an additive white gaussian noise in the band of interest. What is more, if the receiver and the transmitter are sufficiently far away from each other, each propagation path seems to come from the same arrival angle, which means that

θ l ≈ θ 1 for 1 ≤ l ≤ L, so that, denoting d = 1 -v cos(θ l ) c : r HF (t) = L l=1 A l s HF (dt -τ l ) + n HF (t) (4)
Thus, using equations ( 2) and ( 4) we get:

r HF (t) = L l=1 A l {s(dt -τ l )e j2πf0(dt-τ l ) } + n HF (t) (5)
= L l=1
A l {s(dt -τ l )e j2πf0((d-1)t-τ l ) e j2πf0t } + n HF [START_REF] Stojanovic | On the relationship between capacity and distance in an underwater acoustic communication channel[END_REF] so that the lowpass version r(t) of the received signal becomes:

r(t) = L l=1
A l s(dt -τ l )e j2πf0((d-1)t-τ l ) + n(t) [START_REF] Liu | A high-efficiency carrier estimator for OFDM communications[END_REF] with n(t) the equivalent lowpass gaussian noise. Given that the intercarrier space is equal to F 0 and that we use M carriers, the signal occupancy bandwidth is about M F 0 + B, with B the frequency occupancy bandwidth of g(t) which is about F 0 . Thus, for M sufficiently high and discarding the first and last carriers, we can reduce the signal occupancy bandwidth to W < M F 0 so that the sampling period T s = 1 M F0 respects the sampling theorem. So finally, after sampling at the instant t k = t 0 + kT s , we have the signal :

r[k] = L l=1 α l s(dkT s + t 0 -τ l )e j2πf0(d-1)(kTs+t0) + n[k] (8) with r[k] = r(t 0 + kT s ), n[k] = n(t 0 + kT s )
, and α l = A l e -2πjf0τ l , and -

Tg+ T 0 2 +t0 dTs ≤ k < T 0 2 -t0
dTs . It is worthwhile noting that contrary to the radio-mobile case, the Doppler effect does not only causes a frequency shift but also a time compression/dilatation that can not be neglected. Indeed, in underwater acoustic communication, c = 1500m/s and the maximal speed is about v max = 6m/s, which means that maximal Doppler value is about d max = 1 ± 0.004, whereas for radio-mobile communication we rather have v max = 60m/s and c = 3.10 8 m/s so that d max = 1 ± 2.10 -7 . This means that, during a 1024 symbols OFDM block, underwater acoustic Doppler causes a 4 samples shift, whereas radiomobile one causes a delay equal to 2.10 -4 times one sample duration.

III. ML ESTIMATOR AND CRB

In here, we restrict to the simplified case where r[k] = s(dkT s )e j2πf0(d-1)kTs + n[k]. In other words, we consider only one path and we assume that the received signal has already been equalized and synchronized so that the only perturbation remaining is due to the gaussian noise and to the Doppler effect. Without loss of generality, we can assume that t 0 = -(T g + T0

2 ), and moreover, given that each block begins with a guard time constituted of a signal equal to zero, we can assume that -T g -T0 2 ≤ t ≤ T0 2 in spite of the dilatation/compression effect, by adjusting the guard time duration. So, assuming also that T g = M g T s , we have

0 ≤ k ≤ M + M g -1 = K -1. Let us denote R = [r[0], . . . , r[K -1]] T , N = [n[0], . . . , n[K -1]] T and S(d) = [s[0](d), . . . , s[M -1](d)]
T , with T the transpose operator and s[k](d) = s(dkT s + t 0 )e j2πf0(d-1)(kTs+t0) . Thus, we have:

R = S(d) + N (9)
We assume that S( 1) is known at the receiver side, so that we want to estimate d with the knowledge of R and S(1).

The likelihood of d is therefore:

Λ = f (R/S(1); d) = 1 √ 2πσ 2 K e -1 2σ 2 (R-S(d)) H (R-S(d))
(10) given that the components of N are gaussian, independent, zero-mean and with variance σ 2 , with H the transpose conjugate operator and K = M + M g . Then, the best estimation of d is:

d = argmax d {f (R/S(1); d)} (11)
Then, under the assumption that the maximum likelihood estimator is unbiased (E d -d = 0), its Cramer Rao bound is given by [5, pp 65-72].

CRB = 1 -E ∂ 2 ∂d 2 ln Λ (12)
Using (10), we get:

ln Λ = - K 2 ln(2πσ 2 ) - 1 2σ 2 (R -S(d)) H (R -S(d)) and ∂ ∂d ln Λ = - 1 2σ 2 ∂ ∂d R H R -S(d) H R -R H S(d) +S(d) H S(d) ≈ - 1 2σ 2 ∂ ∂d -S(d) H R -R H S(d) + S(d) H S(d)
for K sufficienlty high. Indeed, R H R is equal to the received energy of the signal during one OFDM block, which is approximately constant for K sufficiently high, and thus independent from d. Then,

∂ 2 ∂d 2 ln Λ = 1 2σ 2 ∂ 2 ∂d 2 S(d) H R + R H ∂ 2 ∂d 2 S(d) - ∂ 2 ∂d 2 S(d) H S(d) - ∂ ∂d S(d) H ∂ ∂d S(d) - ∂ ∂d S(d) H ∂ ∂d S(d) -S(d) H ∂ 2 ∂d 2 S(d) = 1 2σ 2 ∂ 2 ∂d 2 S(d) H (R -S(d))+ (R -S(d)) H ∂ 2 ∂d 2 S(d) -2 ∂ ∂d S(d) H ∂ ∂d S(d)
so that, with E{R -S(d)} = E{B} = 0 and using (12):

CRB = σ 2 ∂ ∂d S(d) H ∂ ∂d S(d) = σ 2 K-1 k=0 ∂s[k](d) ∂d 2
Let us recall that s[k](d) = s(dkT s + t 0 )e j2πf0(d-1)(kTs+t0) . Therefore, according to (1):

∂s[k](d) ∂d = ∂s(dkT s + t 0 ) ∂d e j2πf0(d-1)(kTs+t0) + j2πf 0 (kT s + t 0 )s(dkT s + t 0 )e j2πf0(d-1)(kTs+t0)
= j2π(kT s + t 0 )e j2πf0(d-1)(kTs+t0) f 0 s(dkT s + t 0 )+ depends upon the initial sampling instant t 0 , the CRB is in fact independent of t 0 since it is computed thanks to a sum over k and the set of values kT s + t 0 , which is in fact independent of t 0 . We also underline that g(t) is constant equal to 1 √ T0 or to 0 so that its derivative is equal to 0. This bound depends on the complex signal S derived according to the continuous real parameter d. It is worthwhile noting that the higher is the number of samples, the lower is the CRB.

IV. SIMULATION

The emission of a ZP-OFDM symbol through the channel presented before is simulated. The UWA channel is doubly dispersive. The coherence time is close to the second, while the typical delay observed in shallow water environment is around 40 ms for a distance between source and receiver close to the kilometer. A study of the relation between capacity, distance and frequency as been done in [START_REF] Stojanovic | On the relationship between capacity and distance in an underwater acoustic communication channel[END_REF]. Regarding the typical bandwidth in underwater acoustic communication, we choose a carrier frequency f 0 = 20 kHz, T = 200 ms and the guard interval has a duration T g = 53.5 ms. We choose to simulate a communication over 1500 subcarriers. the guard interval is composed of 548 samples, so we have K = 2048. The subcarrier spacing is 6.8 Hz. QPSK modulation is used. We chose these values regarding those used in [START_REF] Li | Non-uniform doppler compensation for zero-padded OFDM over fastvarying underwater acoustic channels[END_REF]. Typical transmission will concern underwater vehicles moving around 6 to 12 knots, this corresponds approximately to a speed between 3 m/s and 6 m/s and a Doppler parameter of d = 1 ± 0.002 and d = 1 ± 0.004, depending on the movement direction. The Doppler shift will be between 40 and 80 Hz. A compensation is necessary to mitigate the Doppler effect. In order to illustrate the effect of the Doppler on the OFDM signal, we have realized several simulations illustrated in the figure [START_REF] Li | Non-uniform doppler compensation for zero-padded OFDM over fastvarying underwater acoustic channels[END_REF]. In these simulations, the number of subcarriers and the Doppler parameter is growing and affect the Binary Error Rate. Theses simulations allow us to determine from which value of the Doppler the information carrier by the OFDM signal is destroyed. For example here, with the configuration of M = 1500 and f 0 = 20 kHz, the BER is up to 10 -2 for a Doppler parameter of 1 + 6.10 -5 . Considering this information, the desired estimator should have a standard deviation inferior to 2.10 -5 in order to guarantee the decoding of the information after demodulation with a limited error rate. At the receiver, we treat the sampled passband signal. First of all, a quick estimation of the Doppler parameter is realized using the method developed in [START_REF] Sharif | A computationally efficient doppler compensation system for underwater acoustic communications[END_REF]. This method is based on a measure of the dilatation/compression effect due to the Doppler effect. Then, an estimation of the Doppler parameter using the maximum likelihood estimator is done. The knowledge of the emitted signal is necessary for this part. The research of the maximum is realized using the Nelder-Mead Simplex method [START_REF] Proakis | Digital Communications[END_REF]. The compensation of the Doppler effect is based on the same idea as used in [START_REF] Li | Non-uniform doppler compensation for zero-padded OFDM over fastvarying underwater acoustic channels[END_REF]. A rescaling of the signal is performed to compensate the influence of the Doppler on the signal r. Then, the residual offset is also compensated using null subcarriers [START_REF] Liu | A high-efficiency carrier estimator for OFDM communications[END_REF]. With these two operations, the Doppler effect is completely compensated.

In order to test the estimator, we simulated two kinds of channels. In channel 1, we only implemented Doppler and additive white gaussian noise. We receive the signal r To illustrate our words, we show in figure (3) the variance of the estimator in the two channels and the theoretical Cramer Rao Bound. In order to fully understand these results, it is necessary to follow the demonstration made on the figure [START_REF] Li | Non-uniform doppler compensation for zero-padded OFDM over fastvarying underwater acoustic channels[END_REF]. The threshold value that the variance should not exceed is 4.10 -10 , corresponding to a standard deviation of 2.10 -5 . Looking at the figure (3), the CRB is below the threshold for a SNR inferior to 6.10 -3 . In theory, it is not possible to obtain an estimation of the Doppler parameter precise enough to recover the information of our OFDM signal with an SNR up to -22dB. This limited SNR value will change with other signals and with the Doppler parameter. The variance of our ML estimator falls below the threshold for an SNR up to 1.6 × 10 -2 . The results, with and without multipaths, are close. Between a SNR of 10 -2 and 10 -1 , the variance of the estimator follows the Cramer Rao Bound. Up to 10 -1 , the estimator is limited by the algorithm of the simplex search method. We can compare our results with those of [START_REF] Li | Non-uniform doppler compensation for zero-padded OFDM over fastvarying underwater acoustic channels[END_REF] and [START_REF] Sharif | A computationally efficient doppler compensation system for underwater acoustic communications[END_REF], using block Doppler estimation. In these articles, the precision of the estimator directly depends on the sampling period. It is shown that the resolution of their estimator is 1/BT with B the bandwidth of the signal. In our case, with a bandwidth of 10.2kHz, the minimum variance of the estimator proposed in [START_REF] Sharif | A computationally efficient doppler compensation system for underwater acoustic communications[END_REF], will be 4.48 × 10 -7 . As we have said it before, it is not precise enough for our computation. But, if we compare the ML estimator with the one used in [START_REF] Li | Non-uniform doppler compensation for zero-padded OFDM over fastvarying underwater acoustic channels[END_REF] for the same number of samples, the ML estimator seems to be more robust and more precise. V. CONCLUSION In this paper, we have investigated the application of an estimator of the Doppler shift and the Doppler distortion using a Maximum Likelihood criterion applied to ZP-OFDM in UWA channel. The receiver is based on block-by-block processing. Estimation needs the perfect knowledge of the transmitted signal during one OFDM block. We have to assume that the Doppler parameter during the following block will not change and the previous Doppler estimated will still be correct. The method proposed is tested in simulations for a channel with and without multiple paths. The estimation is correct in both cases. Simulations show the variance of the estimator compared to Cramer Rao Bound. Further improvements will focus on an estimator using partial knowledge of the signal allowing an estimation of the Doppler parameter during each block and communication in the same time.

M 2 - 1 m=-M 2 mF 2 = 4π 2 (M 2 - 1 m=-M 2 mF 2

 212222122 0 c[m]e j2πmF0d(kTs+t0) g(dkT s + t 0 ) and ∂s[k](d) ∂d kT s + t 0 ) 2 f 0 s(dkT s + t 0 )+ 0 c[m]e j2πmF0d(kTs+t0) g(dkT s + t 0 ) It is worthwhile noting that even if ∂s[k](d) ∂d 2

Fig. 1 .

 1 Fig. 1. Binary Error Rate for different Doppler parameter without any Doppler compensation.

  [k] = s(dkT s + t 0 )e j2πf0(d-1)(kTs+t0) + b[k]. We have tested it with several values for the Doppler parameter d and we show the Likelihood function for a value of d = 0.997 in the figure (2). The maximum is clearly identified even with strong noise (SNR equal to 0 dB). Channel 2 is a three paths channel with coefficient A 1 = 0.6, A 2 = 0.1 and A 3 = 0.3 and delays τ 1 = 0, τ 2 = 10ms and τ 3 = 40ms. But, during our estimation of d, we don't know these values. In fact, we still consider the model of s[k](d) unchanged, s[k](d) = s(dkT s + t 0 )e j2πf0(d-1)(kTs+t0) . The likelihood function is shown in the figure (2). The maximum is still accurate. But the theoretical computation of the CRB is not valid anymore.

Fig. 2 .

 2 Fig. 2. Maximum Likelihood as a function of the parameter d , d = 0.997, SNR = 0 dB.

Fig. 3 .

 3 Fig. 3. Cramer Rao Bound and Variance of the estimator as a function of the Signal to noise ratio d = 0.997.

This work was partly supported by the French ANR (Agence Nationale de la Recherche) LURGA project.


Bastien Lyonnet
email: bastien.lyonnet@gipsa-lab.grenoble-inp.fr

Cyrille Siclet
email: cyrille.siclet@gipsa-lab.grenoble-inp.fr

Jean-Marc Brossier
email: jean-marc.brossier@gipsa-lab.grenoble-inp.fr

DOPPLER ESTIMATORS USING PILOT TONES FOR OFDM COMMUNICATION SIGNAL IN SHALLOW WATER ENVIRONMENT

Keywords: Underwater communication, shallow water, OFDM, Doppler estimation, Pilot tones

Underwater acoustic communications constitute a very difficult issue since the transmission channel is severely doubly dispersive. Indeed, radio waves suffer from strong attenuation so that the only solution is to use acoustical modems. Thus, achievable bandpass is very reduced (typically less than 20kHz) and narrowband signal approximation cannot be assumed since sound velocity is around 1500 m/s and frequency carrier is around 10kHz. A natural choice to fight against time dispersivity is the use of OFDM (Orthogonal Frequency Division Multiplex). Unfortunately, this choice increases the sensitivity to the Doppler effect. That is why the subject of this paper, the estimation of this Doppler effect, is so crucial. For this purpose, we use ZP-OFDM (OFDM with Zero Padding) modulation in order to consider independent time blocks with duration chosen such that the Doppler effect can be considered as approximately constant. Our study is restricted to the case of a single path. Firstly, Doppler is estimated thanks to a correlation between the received signal and a pilot signal. Then, another estimator is also studied, using the minimization of a cost function based on the exponential base of the OFDM signal.

GENERAL CONTEXT

Underwater acoustic communication channel being severely doubly dispersive, the use of ZP-OFDM in order to mitigate time dispersivity also imposes to develop a strategy in order to evaluate and compensate Doppler effect. Several methods based on pilot tones have already been published, but with some limitations. Thus the method developed in [START_REF] Kibangou | Doppler Estimation and Data detection for Underwater Acoustic ZF-OFDM Receiver[END_REF] applies to Doppler shift inferior to the intercarrier spacing. In [START_REF] Li | Pilot-tone based ZP-OFDM Demodulation for an Underwater Acoustic Channel[END_REF] the Doppler shift in frequency is considered to be the same over all subcarriers and this approximation involves errors during Doppler compensation and demodulation.

The principle of ZP-OFDM is to use several carriers in parallel, m f (

0 0 = mF f f m  ), 1 /2 /2     M m M
, with M the number of carriers, with a symbol duration slightly higher than necessary. Indeed, since the frequency spacing 0 F is identical between each carrier frequency, then the minimum symbol duration is equal to

0 0 1/ = F T
. And yet, in ZP-OFDM, a symbol duration is in fact equal to

0 0 > = T T T T g  where g
T is called a guard time. If this guard interval is long enough, there will be no interference between OFDM symbols at reception [START_REF] Muquet | Cyclic Prefix or Zero-Padding for Multi-Carrier Transmissions[END_REF]. Among the M carriers, we choose pilot tones for whom we know the transmitted symbols ] [ m c

. We note V the set of natural numbers between

2 / M  and 1 2 /  M . p
V represents the set of natural numbers for the pilot carriers and i V the set for information carriers:

V V V i p =  and  = i p V V  . Then, for /2 /2 0 0 T t T T g    
(one symbol duration), the ZP-OFDM baseband signal writes:

         ) ( 0 2 0 2 ) ( ] [ ) ( ] [ = ) ( t s t mF j p V m t mF j V m p t g e m c t g e m c t s                 ) ( 0 2 ) ( ] [ t s t mF j i V m i t g e m c     (1)

UNDERWATER ACOUSTIC CHANNEL MODEL

In our model, the emitter-receiver distance is supposed to be close to one kilometer and the deep is about 100 meters. Considering the geometry of the problem, we use the ray theory for sound propagation [START_REF] Munk | Ocean Acoustic Tomography[END_REF]. First of all, we suppose that the acoustic channel can be described thanks to a time-varying multipath model. We moreover assume that the timevarying nature of the channel during one OFDM block is mainly caused by the relative motion between the receiver and the transmitter [START_REF] Sharif | A computationally efficient Doppler compensation system for underwater acoustic communications[END_REF]. Then, assuming a constant relative speed v between the emitter and the receiver, the received low pass signal

) (t r writes: ) ( ) ( = ) ( 0 2 1) ( 0 2 1 = t n e e t d s A t r l f j t d f j l l l L l          (2)
with L the number of propagation paths,

DOPPLER ESTIMATION

From this part, we restrict to the simplified one path case :

] [ ) ( ) ( = ] [ k n dt s dt s k r k p k i  
. We consider that the received signal has been temporally synchronized with the beginning of the OFDM symbol. Without loss of generality, given that each block begins with a guard time constituted of a signal equal to zero, we can assume that

1 = 1 0      K M M k g , with s g g T M T =
. The number of pilot and information carriers are denoted p M and i M , respectively.

3.1.CORRELATION MAXIMISATION

In this section, pilot subcarriers are based on P-QAM symbols known by the receiver. We denote

  T K r r 1] [ , [0], =   r ,   T K n n 1] [ , [0], =   n ,   T i i d K s d s d ) 1]( [ , ), [0]( = ) (   i s and   T p p d K s d s d ) 1]( [ , ), [0]( = ) (   p s , with ) ( = ) ]( [ k p p dt s d k s , ) ( = ) ]( [ k i i dt s d k s
and T the transpose operator. At reception, we suppose the signal . Given that the components of n are complex Gaussian, independent, zero-mean and with variance 2  , we have :

                2 2 ) ( 2 ) ) ( ( exp ) ( 1 = ) (   s s r p i S s d P d d K M i (4)
Thus, we have to compute a sum containing i M P terms which may be prohibitive. It can be simplified considering that

   2 2 ) ( s s r p d 2 ) ) ( ( s s r p   d 2 2 ) ( s s r p    d . Indeed, for ) (d i S s  , K AM c i V m K k m i        1 0 2 2 s (with A c m  2
) so that we get, after a few calculus

                                        2 2 2 2 2 2 ) ( exp ) ( ) ( ) ( exp ) ( 2 2       d e d d e K K AM K K AM i i p p s r s r
. Therefore, we can compute a suboptimal approximation of d with the formula: of the OFDM signal are known by the receiver, equation ( 5) meets the expression of the Maximum Likelihood (ML) estimator of the Doppler parameter d . This estimator has been developed in a previous publication [START_REF] Munk | Ocean Acoustic Tomography[END_REF] with the calculation of the Cramer Rao bound in this case.

                      2

3.2.COST MINIMIZATION USING EXPONENTIAL BASE

In this part, the pilot subcarriers contain no energy: . Thus, d may be approximated by:

1 ) , ( argmin = ˆ      G r d f V m p (7) 2
The estimator consists in minimizing the energy on null subcarriers. The time computation of this estimator is strongly affected by the inversion of matrix Considering that the velocity of the mobile will always be really inferior to the speed of sound in water (1500 m/s), this condition will always be fulfilled.

lA

  the amplitude of the l -th path, l  and l d the the corresponding initial path delay and Doppler parameter, respectively, ) (t n an additive white Gaussian noise in the band of interest, 0 f the central carrier frequency. Note that if we denote l  the arrival angle of the l -th path and c the sound velocity, then respect the sampling theorem even if the Doppler effect enlarge the bandwith of the signal. We consider the beginning time of sampling parameter d we can encounter. So finally, after sampling at

  known. The signal r is affected by an additive white Gaussian noise in the band of interest. The first idea to get an estimate of d is to compute the likelihood and to maximize it: the likelihood, we look for the value of d which is the most likely to appear, knowing one realization of r . If we note ) messages have the same probability of appearance. Using P-QAM modulation we have

.

  In the special case where all the emitted symbol ] [ m c

,

  with c the vector composed of the emitted QAM symbols. For the special case where

.

  on the value of  . When the Doppler effect provokes a time compression of the signal, we may have aliasing. We have aliasing if: Then a way to avoid bad conditioned matrix is to oversample the signal by a factor 2 . Then

SIMULATION RESULTS

The emission of a ZP-OFDM symbol through the channel presented before is simulated. The UWA channel is doubly dispersive. As presented in [START_REF] Lyonnet | Maximum likelihood estimator for Doppler parameter and cramer rao bound in ZP-OFDM UWA channel[END_REF] the coherence time is close to the second for communication distance over one kilometer, while the typical delay observed in shallow water environment is around 40 ms. We choose a carrier frequency , depending on the movement direction. The Doppler shift will be between 40 and 80 Hz. As presented in [START_REF] Lyonnet | Maximum likelihood estimator for Doppler parameter and cramer rao bound in ZP-OFDM UWA channel[END_REF], considering the number of subcarriers and the value of 0 f , we need an estimate of d very accurate. The desired estimator should have a standard deviation inferior to 5 2.10  in order to guarantee the decoding of the information after demodulation with a limited error rate. A one path channel is simulated with a Doppler parameter 0.9974 = d . Both estimators are compared with 100 (Fig. 1) and 300 (Fig. 2) pilot or null subcarriers. Each figure also recalls the Cramer-Rao Bound of the first estimator when all subcarriers are kwown that is to say for 1500 = M M p  (dashed black line), as well as the variance of this estimator in this latter case (black, solid line). The optimization of ( 5) and ( 7) is realized using Nelder-Mead algorithm in Matlab. Our purpose is to have a variance inferior to

CONCLUSION

In this paper, we have investigated the application of two estimators of the Doppler effect using perfect knowledge of pilot tones of a ZP-OFDM signal in UWA channel. The receiver is based on block-by-block processing. In the first method, pilot subcarriers yield QPSK symbols. In the second method, null subcarriers are used. These methods allow us to estimate the channel parameters and communicate information in the same time. We have to assume that the Doppler parameter will not change during one OFDM block. The proposed method is tested in simulations for a channel with different intercarrier spaces for the pilot tones. Simulations show the variance of the estimator compared to theoretical Cramer Rao Bound. Further improvements will focus on perfect Doppler compensation and application to multipath channel.