Marie Duru-Bellat

Inequalities in Educational Opportunities in France: educational expansion, democratization or shifting barriers?

This paper is concerned with inequalities in educational opportunities in France and presents data collected from cohorts barn between 1919 and 1973, a period characterized by educational expansion. Data from large representative French surveys were compiled in such a way making accurate historical and international comparisons possible. The paper argues that there has been a shift in social inequalities, from entry into collège (lower secondary school) to entry into lycée (upper secondary school leading to the baccalauréat). This is so, even when, due to educational expansion, it is possible to chart a drop in inequalities, as far as the highest level of education is concerned. Finally, policy implications have been drawn from international comparisons of data on the reduction of social inequality.

Inequalities in Educational Opportunities in France: educational expansion, democratization

or shifting barriers-? 3In France as in most industrialised countries, equalising educational opportunities has been a long-standing political concern. It is also a traditional and persistent theme in French sociology of education. A distinctive feature of French researchers has been to generate macro-sociological theories which focus on social reproduction through schooling (such as [START_REF] Bourdieu | La reproduction[END_REF]. More recently, the interest of sociologists has shifted to the micro-level, as they have investigated the processes through which reproduction takes place in a predominantly empirical way.

Reference to Bourdieu' s social reproduction paradigm has become so well established in France, as elsewhere, that it does not seem necessary to elaborate it here. The predictive capacity of this model is well established. However, in relation to the evolution of social opportunities in schooling, Bourdieu's theoretical framework does not make it possible to arbitrate between two opposite predictions: either one could expect a reduction of social inequalities within schools, owing to the increase in parental levels of education, which would result in a lessening of cultural deprivation which characterizes many of the most under-privileged pupils; or one could also expect that inequalities will constantly evolve, as long as parents value some forms of certification more than others, in order to achieve advantage for their children.

Today, French sociologists working on the issue of equality, have either concentrated on methodological problems (with debates about pitfalls involved when analysing evolution: see [START_REF] Merllié | L'analyse de l'interaction entre variables : problème statistique ou sociologique ?[END_REF][START_REF] Vallet | L'évolution de l'inégalité des chances devant l'enseignement. Un point de vue de modélisation statistique[END_REF] for example), or have chosen to focus upon the links between social background, educational attainment and social outcomes, thus endorsing the classical viewpoint of social mobility [START_REF] Goux | Meritocraty and social heredity in France: Some aspects and trends[END_REF]. In doing so, they are nearer to Boudon's model, which considers individuals as social actors, who make meaningful choices within an institutional and economic context; the evolution of which may affect the characteristics of the social inequalities themselves. This perspective concentrates on the analysis of choices during schooling rather than on inequalities concerning academic achievement (for instance, [START_REF] Duru-Bellat | Social Inequalities in French Secondary Schools: from figures to theories[END_REF][START_REF] Duru-Bellat | Social Inequalities in the French Education System: The Joint Effect of Individual and Contextual Factors[END_REF]. In addition, it shifts attention to educational policies or reforms which are likely to modify the institutional context in which choices are made. However, until now, empirical studies have remained rather scarce in France. Still more rare are historical and international comparisons, even though they may prove an heuristic way to shed light on the persistence of social inequalities in schooling. Even where some comparative work has been undertaken, they rely mainly on old surveys. Using the 1970 French Education, Training and Professions Surveys (FQP Surveys, see appendix), [START_REF] Garnier | The Evolution of Equality of Educational Opportunity in France[END_REF], [START_REF] Müller | Social Selection Educational Systems in Europe[END_REF] and [START_REF] Müller | Class Inequalities in Educational Outcomes: Sweden in Comparative Perspective[END_REF] concluded that the association between social background and educational attainment has not undergone significant change in the post war period. However, their work covers generations born between 1910 and 1947, i.e. the youngest members had reached the age of eleven or twelve, making them eligible to enter a lower secondary school before the "Berthoin reform", which introduced universal access to that level of schooling. Using data including generations born in 1959-68, Goux and Maurin's findings (1997) confirm this conclusion. Recent studies undertaken in France by [START_REF] Brauns | Bildung in Frankreich. Eine Studie zum Wandel herkunfts-und geschlechtsspezifischen Bildungsverhaltens[END_REF] and [START_REF] Müller | Erwartete und unerwartete Folgen der Bildungsexpansion[END_REF] are taking contemporary data into account; their findings indicate some decline in inequalities but this needs to be confirmed by a specific study based upon repeated surveys, as with the work presented in this paper.

The question of shifts in educational opportunities is of special interest because the twentieth century has been a period of strong educational expansion at all levels in France, especially over the last thirty years. To give an example, the percentage reaching the "baccalauréat" (the French upper secondary school diploma) rose from 20% in 1966 to 40% in 1986, and to 68% in 1996. While one might have predicted that this expansion would result in some democratization, educational attainment remains strongly linked to pupils' social background: among 20-21 years old (in 1996), 51% of the children of manual workers reached the level of the "baccalauréat", compared with 87% of the children of top executives, teachers, or lower service class members (MENRT-DPD, 1998). To shed light on the issue "has some democratization been achieved?", it is inadequate to focus on the global association between social background and current positions at different periods of time, as is currently done by specialists of social mobility. What should be done is a comparison between the schooling careers of successive cohorts, within their socio-historical contexts. It is within the latter that the structures of opportunity have been changing. Among the factors which shape the different historical contexts of cohorts, educational structures certainly play some part, along with the policies implemented for change. If one focuses upon education (rather than upon social mobility), only a longitudinal approach makes it possible to assess precisely the evolutions between cohorts and by doing so, will shed some light on the effects of the educational policies launched during the period under study.

In this paper, we present some results that aim to assist historical and international comparisons. The first part contains information on the data and the empirical indicators used, and then the paper turns to describe briefly some important changes which have occurred in France concerning the educational attainment of cohorts born between the beginning of the century and early seventies. We will then examine whether during this period, education has appeared to be less and less linked to social background, thus trying to assess whether some democratization has taken place. Finally, we draw some comparisons with a sample of other European countries. It enables us draw up some policy implications.

This paper relies on data on men and women born between 1919 and 1973 (and even 1978 in the most recent, but distinct survey), observed in different data sets, that were constructed between 1970 and 1993. These data sets have been combined (see appendix). I. Trends in schooling careers in a changing educational system.

The cohorts attended school and entered the labour market at quite different periods, stamped by diversified historical contexts. It is important to remember that the French system has long been organised into two distinct tracks: five years of primary school (which could be followed either by "senior primary school" or by vocational courses) and secondary school (the lycées, including specific primary courses), leading to the baccalauréat after seven years. This dual set of institutions was not socially neutral, and from the forties onwards, several reforms were implemented to achieve some democratization. The first important one, in 1959 (the "Berthoin reform"), extended compulsory schooling by two years (from 14 to 16), and created a two-year "observation" (or "pre-sorting out") cycle at the secondary level, which every pupil was supposed to complete. However, this common path was implemented in different schools (in the traditional "lycées", but also in new lower secondary comprehensive structures, the collège), with very different opportunities to complete the four years of schooling which gave access to upper secondary school. A second important act was the "Haby reform" in 1978, which merged the existing tracks: every French pupil had to follow a common core curriculum in the "collège" of his or her neighbourhood.

We developed the following indicators in order to characterize schooling careers during this changing period, and thus make it possible to assess the evolution of social inequalities:

• The proportion of the cohort who reached the first form of the lower secondary (the "6ème"), rather than remaining in primary classes or leaving school;

• The proportion of the cohort who followed a complete four year lower secondary schooling (collège),

• rather than dropping out before the last level ("3ème");

• The proportion of the cohort who entered the first level of higher secondary, the lycée, which became at the end of the sixties, a distinct level, after collège ("2nde");

• The proportion of the cohort who obtained a baccalauréat, and a general baccalauréat (rather than a technical one), thereby gaining access to higher education.

Observations made at these educational levels will make it possible to track the social consequences of expansion in the system, i.e. do inequalities shift to higher levels as access to lower secondary expands? Consequently, each of these indicators has been computed either on a whole cohort or on those pupils who entered a" 6ème"; this allows us to assess the selection which has occurred within the secondary school itself. Owing to the changing historical context of this study, it was not possible to conduct the analysis solely on the basis of absolute ratios; these are of course interesting, especially as they show the social homogeneity of student bodies at different educational levels (this is what individuals see from the inside) ; however, absolute differences between the classes may decline as expansion proceeds, even if relative class inequality remains the same. Here, we do not focus on the opportunities of different groups but more upon their relative advantage or disadvantage, considering education as a positional good: we thus calculated relative indicators such as odds ratios and loglinear (or logit) regression models. These latter measures make it possible to focus on the association between schooling career and social background, while the former intermingles the evolution of this association with the global result of the development of schooling itself.

The first graph below (gr. 1) shows how these different ratios have shifted in the cohorts studied. Up to those born in 1939, all the figures are very stable; these cohorts went through the dual system, with very few pupils entering the secondary level (access to 6eme is about 28-30%). The figures begin to move among cohorts born between 1939 and 1948 (these individuals were 11-12 years old before the reforms were fully implemented). Access to 6ème increases (from 30 to 50% in ten years), and the same is true (but to a lesser extent) for access to upper secondary school. These changes become more substantial for cohorts born between 1949 and 1953, even though the "Berthain reform" had still not been completely implemented (only one out of two pupils entered into a 6ème in 1962). In the meanwhile, access to upper secondary school is still behind on this point, due to the different tracks existing at the lower secondary level. Changes became quite dramatic for the cohorts born after 1954. In a few years, access to 6ème spread widely, and so did access to the fourth year of lower secondary school. Expansion was much more limited as far as access into upper secondary school was concerned. Among the cohorts born after 1959, access to lower secondary became nearly universal, but again, expansion remained much weaker at the "lycée" level, except for the most recent cohorts, born in the early seventies.

Access to lower secondary thus became universal for cohorts born in the beginning of the sixties, but what must be stressed, and what appears clearly from Graph 2, is that, while access into 6ème spread, promotions within lower secondary school became more insecure. For pupils having entered into 6ème, it became more and more difficult to complete schooling up to the end of lower secondary school. This difficulty results from the development of new vocational tracks accessible to pupils after the two first years of collège (as a substitute for the training opportunities which followed primary school in the old system) or after the last year of collège. The so called "orientation" -"pre-sorting out"-process developed (with special official texts in 1971): instead of having pupils sorted out before lower secondary schools, the sorting process now took place within collège itself, as if the problem of the period was to absorb this larger pupil body. It became "politically incorrect" to put up barriers at entry to lower secondary school, however, the need to have or at least envisage some selection remained and resulted in the more acceptable guise of "orientation". Generally, have the changes been important? In the cohorts barn in the twenties, about 20% entered into a 6ème; among those happy few, 75% completed lower secondary schooling, and about half entered into an upper secondary school; 25% obtained the baccalauréat. In the most recent generations (barn in the beginning of the seventies), nearly 100% entered lower secondary school, 75% reaching the last form of collège (3ème); but again, only 50% entered a lycée and only about 33% will take the baccalauréat. Indeed, rates are of similar magnitude, even if the whole generation is now involved. However, this evolution is going on, and it seems to be accelerating: in the cohort born in 1978, access to the last year form of collège is nearly universal (93%), and the access rate to lycée is now about 59%. More recent information, if available, could well provide a more impressive picture of this recent and on-going evolution. Selection is following a similar pattern, but it tends to occur later on in the educational career.

II. Towards an equalisation of educational opportunities?

In the context of educational expansion, one can ask whether the relationship between social background and educational attainment has been declining. We shall focus here on access to both lower secondary s (6ème), and upper secondary school (2nde). Social background will be assessed taking into account both the father's and the mother's occupation (a class dimension), and their level of education (a "cultural" dimension). Access to 6ème gives a clear example of democratization through the ceiling (or saturation) effect (as Graph 3 shows). In the oldest generations, a high proportion of children of top executives entered into a 6ème (84%), while this was rare for children of manual workers (20%). Thus the former could not increase their participation at that level any further, while the latter had a certain amount of leeway: so the gap was necessarily reduced. Within the youngest cohorts, ratios are in both cases greater than 93%, so the decrease in inequalities seems to be a direct result of the expansion of the educational system. Models explaining access to 6ème (Table II) confirm this trend. The overall explanatory power of the models (as reflected by the G2 figures) goes down from the cohorts born between 1949 and 1953. This means that the capacity to predict access to 6ème for a child on the basis of his or her social background or gender falls sharply. Beyond the main trend, the models make it possible to assess the respective impact of the different characteristics of the pupil. However, this must be done with some care due to the fact that the coefficients drawn from different models are never perfectly comparable (especially because of the growth in the average access rates during the period in question).

For cohorts born during 1954-58, the net impact of the father's occupation becomes weaker. If some inequalities remain significant among younger cohorts, they now distinguish between children of all whitecollar workers and those from children of manual workers and farmers. The net impact of parents' education remains significant, but becomes less important than that of their occupations. Mothers' occupation does not have a clear influence, perhaps owing to the heterogeneity of the reference group (non working mothers could be either well educated spouses of top executives or poorly educated spouses of manual factory workers). Finally, girls, who were still slightly advantaged in the oldest cohorts, increased their advantage in their access of 6ème. Today, in the youngest cohorts, pupils not entering 6ème have access to special classes ("éducation spécialisée"); these classes are provided for children with specific problems and needs; these often prove to be boys from poorer backgrounds. It is one thing is to enter lower secondary school, it is another to survive until the end. Research focusing on the tracking process during the eighties showed [START_REF] Duru-Bellat | Social Inequalities in French Secondary Schools: from figures to theories[END_REF][START_REF] Duru-Bellat | Social Inequalities in the French Education System: The Joint Effect of Individual and Contextual Factors[END_REF] that streaming at the end of the second year of collège used to produce very strong social differentiation, due to inequalities in academic achievement, and also because of inequalities in self-selection: indeed, even when they are of similar academic achievement, pupils from lower groups ask for less prestigious tracks than pupils from higher social groups. A similar trend is observed at the end of collège, i.e. for a given level of academic achievement, parental (and pupils) demands for entering the lycée vary according to social groups; in particular, pupils from the most privileged groups prefer to repeat a year rather than moving into a vocational track.

One may thus expect access to the upper secondary school to be marked by social inequalities, of two-fold origin: inequalities in access to lower secondary school (6ème), and tracking inequalities within and at the end of this educational level. However, the models show (Table III) that the explanatory power of different social characteristics and gender, on the probability of reaching the 2nde is decreasing. This evolution is slightly less strong than at the 6ème level but it is comparable. With some care, one can also identify the impact of different variables through successive cohorts. The net impact of father's occupation remains critical during the period studied, and becomes very much stronger than the influence of parental level of education. If one computes models only on the sub-group who entered 6ème (Table N), an increase in social selection is evident, which is reflected in an increase in the G2 value. The impact of father's occupation becomes more important and the same is true of the impact of mother's level of instruction. However, this relative increase seems to have stopped for the youngest cohort. On the whole, social background remains influential for access to upper secondary school. This suggests that certain amounts of cultural and social capital are necessary to be able to go through the different transitions within and after collège. Another point to stress is that the democratization observed at the entry to lower secondary school has been counter-balanced by a stronger form of social selection within collège itself4 . Social inequalities seem to have shifted from access to lower secondary school to access to upper secondary school. This evolution has not been a steady one. Inequalities decreased in the cohorts born in 1949-1953, but the following cohorts witnessed the development of new vocational tracks at the end of the sixties; these absorbed a growing share of children of manual workers, resulting in growing inequalities, up to the youngest cohorts. Generally, if we compare access to 6ème in cohorts born between 1939-1948 (generation in which about 37% entered that level, before it was opened up), and access to 2nde in cohorts born between 1954-58 (generations in which a comparable proportion, 34%, entered that level), it shows that the whole pattern of inequalities is quite similar (see table I). This is except for children of farmers (compared with those of manual workers) and girls (compared with boys); both their situations have clearly improved.

Generally, there has been a shift in inequalities, as illustrated by graph 4.

heterogeneity in the population at risk; this is because prior selection is less stringent and accordingly unmeasured factors such as ability or motivation are less mediated by class effects. However one should stress that for cohorts barn in the first years of the seventies, the access rate to lycée becomes very high for the most privileged pupils (88% for children of top executives). This implies again a democratization through ceiling effects (as only the less privileged pupils can improve their access rate significantly). Here it was useful to examine the progress of recent cohorts using different data (obtained from the French Ministry of Education) for pupils barn in 1978 who entered collège in 1989. These longitudinal data can be compared to similar data for pupils who entered 6ème in 1972 and 1980. Between the two first cohorts, the comparison shows that no significant democratization took place, as far as access to upper secondary school is concerned. This is also true concerning the most recent cohort (1989). Ratios (comparing children of professionals to ones of manual workers) moved from 16,9 for the 1972 cohort, to 16,2 for the 1980 cohort, and collapsed down to 12,4 for the 1989 cohort. In the near future, this trend should bring about a certain amount of democratization at the baccalauréat level.

These changes result from higher rates of completion within collège, the first branching point (at the end of the second year of collège) having recently being abolished. This universal completion leads to democratization at the collège level, through a ceiling effect. Therefore, pupils from low SES have gained to a great extent: access to 3 ème (the last year of collège) has risen from 58 to 89% for manual workers' children between the two last surveys, even if access to the lycée bas not had the same levels of increased access (ratios moved from 30 to 42%).

As far as access to the lycée is concerned, there is also evidence of a ceiling effect: this is because for the 1989 cohort, access rate to 2nde is about 90% for the children of professionals (compared with 42% for the children of manual workers). However, it is worth noting that, when focusing on students going through lycée without repeating a year, the decrease in social inequalities is less significant: the figures for these pupils are respectively 60% and 22%. Again, it is the opening up of the system which boosts democratization, rather than a real change in the functioning of transitions: the functioning of transitions remains the same -in particular, the same pattern of social inequalities is at work-, but the process takes place later. So one may talk of a "quantitative" democratization, i.e. more schooling for all, rather than about a "qualitative" one, i.e. a school functioning in a fairer way.

It is worth stressing that this shift results in French pupils spending more and more time in school. As far as the highest level of education achieved is concerned, a decrease in social inequalities may also go along with stability in the social inequalities in transitions. As Mare showed (1981), linear regression models explaining the highest level of education intermingle, on the one band, the association between social background and transitions to different levels of the system, and, on the other band, the overall rates of access to those different levels. If it is heuristic to disentangle those distinct phenomena, one can propose that, if education is a good in itself, the evolution of average opportunities (as the absolute rates show)

should not be disregarded. lt is not trivial to assess to what extent the expansion of education brings about a decline in the association between social background and the level of education achieved.

Indeed, models show a drop in the percentage of variance explained by social background and gender, thus a drop in inequalities; there is also a sharp fall in the net impact of parents' level of education (see Table V).

Social background clearly explains less and less the final educational level of education; in particular, the effect of father's education has fallen off. Note: The construction of continuous variables used here are described in the appendix.

On the whole, the effects of father's (and even mother's) professions prove more resistant to change than father's and mother's education. This pattern does not fit well with reproduction theories, at least as far as a dominant and persistent influence of " cultural inheritance " is concerned. It is not very congruent either with modernisation theories, according to which, in societies where job allocation relies more strongly on credentialism, educational selection is supposed to become more meritocratic. Generally, the decline in parental influence on educational outcome can be explained by the opening up of the system, and not by real changes in the inequalities operating at successive transitions. However, what of democratization? Even if the successive transitions remain marked by social inequalities, it is not without social or political importance that pupils from underprivileged groups are more and more educated and that education is more equally distributed. Expansion rnay thus gradually eliminate inequalities in schooling, modifying the framework within which social bias operated in the allocation of pupils to education levels [START_REF] Hellevik | Class inequality and egalitarian reform[END_REF].

Il. France in comparative perspective.

In this study, we assess models which are as comparable as possible with those used in recent comparative research (such as by Shavit and Blossfeld, 1993); those models focus on cohort differences in the effects of social background on length of schooling (using linear regression), and on different transitions (using logit modelling). However, there are of course numerous reasons for being very cautious when comparing models constructed in different countries; the sample sizes may vary, some studies cover longer periods than others, some rely on pooled data sets, the way social origin is assessed may differ from one country to another, and so on... Accordingly, the comparisons should not be "too exacting" [START_REF] Jonsson | A half century of increasing educational openness? Social class, gender and educational attainment in Sweden, Germany and Britain[END_REF].

As far as the determinants of length of schooling are concerned, France is in an "average" position. The percentage of variance explained by the father's profession, the father's education and the mother's education are slightly weaker in France than in ltaly (in the most recent cohort, respectively about 20% and 27%).

These effects remain stronger than in countries such as the Netherlands (about 12%) and the United Kingdom, where the corresponding figure was still about 18% for cohorts born at the beginning of the century. The fact that education is less dependent on social characteristics in the UK, than in France could be explained by the fact that the British system remains more "meritocratic" than the French one [START_REF] Jonsson | A half century of increasing educational openness? Social class, gender and educational attainment in Sweden, Germany and Britain[END_REF]. lndeed, selecting pupils on the basis of academic tests, (more common in Britain) rather than letting them choose their tracks (as in France) would lessen inequalities in self-selection, one major source of social inequalities.

If we consider the evolution of the French indicators, the decrease in inequality during the second half of the century is comparable with ltaly. A similar decrease bas been observed in Germany, but the evolution seems to have been smoother in this latter country. A common feature, observed in France as well as in other countries (Britain, ltaly), is that the net impact of father' s level of education has been declining, while the net effect of his profession has remained unchanged. The problem is that in all those countries in which an important and long-term process of educational expansion bas occurred, regressions of number of years of school completed may be spurious, according to [START_REF] Mare | Change and Stability in Educational Stratification[END_REF]. He warns that they express both the association between social background and each transition, and the evolution of the probabilities attached to each of these transitions (i.e. the distribution of schooling). For this reason, the bulk of research focuses on the transitions which shape the "survival pattern" [START_REF] Müller | Social Selection Educational Systems in Europe[END_REF]) of an educational system.

We can begin with the entry into lower secondary school, even if it is no longer a true transition in those countries where the process of "comprehensivization" is complete. This is the case for Sweden, in which democratization bas been achieved through universal access, for the youngest cohorts (born in the beginning of the sixties); however, this trend was still visible for the oldest cohorts (born in the twenties). Compared with Sweden, France has been very late in comprehensivization (the fall began with the cohorts born in 1949-54, and became significant for the 1954-58 cohorts). However, after a delay of 10 years, the French evolution was the same (but of course at a more rapid pace). Germany has a very distinct pattern, since this country has kept a non comprehensive system (with three paths: Hauptschule, Realschule and Gymnasium).

The last one is more frequently attended by pupils from high SES, aiming for an Abitur, while pupils attending the other two paths will more often enter the vocational system. This biased structure (as far as social background is concerned) seems stable (as Kahler, 1992, showed). Inequalities associated with access to Gymnasium (assessed by odds ratios) are, in the youngest cohorts, comparable to the inequalities associated, in France, with the odds of attaining the third year of "collège" (rather than being channelled into a vocational track5 , the corresponding odds ratios being about 10 in both cases.

We must thus stress the fact that in spite of the implementation of a unified structure at the lower secondary level, France, because of persisting socially biased choices within collège, remained for a long time in a more unequal situation than educational systems differentiated from the outset, such as in Germany or Britain (with their different types of secondary schools), as long as the comprehensivization process was not complete. In the two latter countries, it has been observed that competition between pupils may be pushed forward by policies aiming at reducing inequalities. In Britain, in 1944, the Education Act, through removing financial barriers to secondary education, led to a period of harsh competition, which increased social inequalities [START_REF] Halsey | Origins and Destinations[END_REF]. In the same way, in France, the opening up of the new collège was counter-balanced by harsher selection during the formally unified curriculum, which resulted in more inequalities during the eighties.

If we turn now to the second important transition, between lower and upper secondary, the figures lead to rather similar conclusions. Sweden and Britain again appear to be the most egalitarian countries (the odds ratios being respectively 6.2 and 7.5, CF Jonsson, Mills and Müller, 1996), whereas for Germany and France, the corresponding figures are 9.6 and 9 6 • The "survival pattern" is thus quite different in the countries mentioned. Even if comprehensivization is universal in all of them, France remains more selective than Sweden because of the streaming which still takes place during comprehensive school. While such streaming has declined for the youngest cohorts, even if it was officially ended in 1978 (Haby Act) recent research shows that some form of streaming is often implemented in " collège " through the class grouping process according to option choices; the latter are related to pupils' academic level and social background [START_REF] Duru-Bellat | La constitution de classes de niveau dans les collèges, les effets pervers d'un pratique à visée égalisatrice[END_REF]. Through this process new forms of social inequality are implemented within an officially common core curriculum. We note that a similar pattern of social inequalities through option choices bas been observed in other countries such as Scotland [START_REF] Croxford | Equal Opportunities in the Secondary-School Curriculum in Scotland, 1977-1991[END_REF]. On the whole, the level of social inequalities at this level in France is comparable with Germany, but this occurs through different mechanisms: early differentiation takes place in Germany, while a more continuous one takes place in France. Up to very recently, there were two points of choices (at the end of the second and the fourth year of collège), and they were often socially selective. A similar shift in social inequalities that occurred in France from the first year of" collège" to the first year of" lycées", occurred also in Britain. [START_REF] Heath | Educational Reform and Changes in the Stratification Process in Great Britain in[END_REF] wrote: " if we regard O-level today as the functional equivalent of selective schooling before the war we can• conclude that the relative class chances are the same for the post 1960 cohort as they were for the 1900-1909 cohort ". In upper secondary school, even if, through a ceiling effect, "quantitative" inequalities may decrease, some "qualitative" inequalities may become more significant. In France, even if access to the "baccalauréat" is becoming more widespread in the most recent cohorts, the success in a scientific "baccalauréat", and the level of the grades remain marked by social background. A comparison can be made with Scotland [START_REF] Gamoran | Curriculum Standardization and Equality of Opportunity in Scottish Secondary Education: 1984-1990[END_REF], where even if social inequalities concerning awards in English or Mathematics have been declining, these same inequalities prove to be more stable as far as passes/no passes are concerned. In the two countries, the gateway to higher education appears more resistant to change, due to unequal access to different academic fields of study, or unequal levels of attainment.

However, in France, once pupils have gone through these successive points of selection, they have higher probabilities of entering higher education. Our results, as well as those of [START_REF] Brauns | Bildung in Frankreich. Eine Studie zum Wandel herkunfts-und geschlechtsspezifischen Bildungsverhaltens[END_REF], show that in France social inequalities are less important at this level, for recent cohorts, than in countries such as Germany, and would be comparable to Sweden (the odds ratios are less than 5, while they amount to 9.6 in Germany, cf. [START_REF] Jonsson | A half century of increasing educational openness? Social class, gender and educational attainment in Sweden, Germany and Britain[END_REF]. The French situation can be explained by the expansion of higher education in recent years. If we compare the access rate to higher education for cohorts born in the fifties in Sweden and France, we observe that the rates are higher in France, whatever the social background of the students. In other words, children of unskilled workers have in France a higher probability of entering higher education than children of service-class workers in Sweden, a country in which the development ofhigher education has been weaker. Comparison with Britain leads to the same conclusion: the relatively slow development of British higher education is linked to the stability of social inequalities (cf. for example [START_REF] Blackburn | Changing Inequalities in Access to British Universities[END_REF]. Furthermore, we must stress that there exist in France two very distinct paths: manual workers (and not, as in the other countries, to the children of unskilled workers only).

the very selective "grandes écoles", where children of top executives are strongly over-represented, and

Universities, which are open to all students who have passed the baccalauréat, and these have a larger representation of working class children (even more in short vocational tracks). In the most prestigious "grandes écoles", children of "cadres moyens et supérieurs" make up more that 90% of the student body, compared with about 60% in the Universities, and 30% in the short vocational tracks (Euriat and Thélot, 1995). This interna! diversity of higher education exists in Britain too, but with a different pattern, as it is within Universities that the pupils from the service class are more over-represented compared with Polytechnics or Colleges. Indeed, the opening of Universities, as witnessed in France (where free access of all the "bacheliers" is the rule) has modified their way of functioning, and the purchasing power of degrees.

They are certainly very different today from their British counterparts.

IV. Discussion: Sorne policy implications By and large, France appears less atypical than some studies have suggested [START_REF] Müller | Social Selection Educational Systems in Europe[END_REF][START_REF] Müller | Class Inequalities in Educational Outcomes: Sweden in Comparative Perspective[END_REF]. These studies have relied on old data, including generations born between 1910 and 1947, who had entered collège before the "Berthoin reform" was implemented. During this period, the French system was characterized by a very selective survival pattern, which led to strong social inequalities.

However, if one takes into account more recent cohorts, then France is in a more average situation, even if it remains more socially selective than countries such as Sweden, the Netherlands, or the United Kingdom.

The French distinctive features are:

1) the delay with which France introduced an opening up (and thus a democratization) of lower secondary school, for instance about ten years after Sweden, about four or five years after Italy;

2) the expansion in recent years of higher education, (access rates to higher education rose from 5,4% in the oldest cohorts to 20,3% for those born between 1959 and 1968).

Some democratization occurred mainly because of the opening up of the system. Education expansion has been at the core of French policy since the end of the fifties, not only because a larger cake is easier to share, but also because of supposed economical needs. Until the beginning of the eighties, this policy has been characterized by the basic belief that equality meant providing identical educational structures for all.

The successive reforms implemented from 1959 onwards strived to abolish structural streaming, so that every pupil had access to the classical culture of the old "lycée". Thus, the most obvious way to achieve at least an apparent democratization was to lessen selection. Data suggests that this "quantitative" policy does result on the whole in a lower association between schooling career and parental status. However, the selection process is pushed to a higher level, and the social pattern of transition rates remain unchanged, thus inequalities are only shifted along. Nevertheless, the fact that access to education is more equally spread is not insignificant. Even if the way social class affects the probability of obtaining a given level of education remains largely the same (only shifting from one level to another), the way education is distributed among social groups has become less unequal. In that way, policies aiming at opening up the system (or at removing barriers) may prove effective. One should thus not confuse different concepts of inequality in education: the uneven distribution of education (the outcome of competition between groups)

with bias in the process of allocating pupils to different educational levels (the process of the competition; cf. [START_REF] Hellevik | Class inequality and egalitarian reform[END_REF].

However, there is an alternative way to consider democratization, i.e. focusing upon the social and economic use of one's diploma. In this respect, the French situation becomes more problematic, as far as democratization is concerned. A policy of educational expansion may have important economic and social

• costs. Analyzing the relations between the degrees obtained and the jobs the degree holders actually gain leads to some depreciation of the worth of certification. Recent research [START_REF] Forsé | French Trends in Social and Educational Opportunities[END_REF] shows that the retums to any diploma are " slipping because of the aggregate backlash effect of this trend into more schooling "; the only exception is for the élite school diplomas that maintain their "job-buying power".

This development stems from the discrepancy between the respective speed of change in the educational structure and the speed of change in the social class structure (which confirms Boudon's theory). This trend may be stronger in France, than in other European countries such as Germany or Rolland, for institutional reasons [START_REF] Müller | The institutional Embeddedness of the Stratification Process. A comparative Study of Qualifications and Occupations in Thirteen Countries[END_REF]. In France, general training is the dominant way to rank people and to allocate them to jobs: this "level'' logic prevails upon a logic of vocational training. However, the large expansion of the system is certainly the most important factor, especially as far as higher education is concerned. Of course, educational expansion may have produced some more positive effects, such as better occupational integration of women or the development of qualified jobs.

Nevertheless, democratization through the opening up of the system, beyond the intrinsic benefits of education itself, clearly bas some limits when it is a question of the social and economic integration of students. In this respect, policies aiming at modifying the functioning of the educational system would not be accompanied by these kind of unexpected and undesirable effects. However, in France, reforms have been primarily organizational. Some changes were implemented concerning school choices, but they were oriented towards more power being given to families; indeed, we know that free choices can foster inequalities. Pedagogy bas been much less affected by policy changes. This is because, in France, this would have involved an important ideological shift, similar to that which emerged in the eighties. As long as a persistent cohort of a generation kept performing poorly, it became more and more obvious that offering them "more of the same" (i.e. repeating a year) would not be successful. In an ideological context where the notion of equity was spreading, the French Ministry of Education argued that to achieve social justice, it might be necessary to provide differentiated pedagogical provision, according to different types of needs.

The Ministry encouraged teachers to implement some forms of individual help to pupils, instead of making them repeat a year. Schools were also given more leeway to implement by themselves the pedagogy they considered to be the most suitable for their pupils. This new concern for adapting teaching arose when distinct tracks were being abolished at the collège level. However, this policy, although it opened up collège without letting the academic level of pupils fall, seemed to be ineffective in respect of the bard core of underachieving pupils. And it could be that this new freedom given at the "local level'' may well result in more differentiation and social polarization between schools.

However, even if it was possible to put an end to social inequalities in academic achievement, the French system would not be perfectly fair (or "meritocratic"): inequalities in social destinations of youngsters with identical educational attainment, but of different social backgrounds, are very stable and remain important (as wide as inequalities in the school system itself). Recent research shows similar trends in Ireland [START_REF] Breens | The persistence of class origin inequalities among school leavers in the Republic of Ireland, 1984-1993[END_REF]. However, social inequalities in academic achievement are not the only mechanism of social reproduction, in which educational attainment will directly determine social destination (following Bourdieu and Passeron' s thesis). This is because, in the system, families and pupils are not completely passive agents; they behave like actors, and try to use the system according to what they perceive as their personal interest (according to Boudon's thesis). More and more, French sociologists, like their European colleagues, refer to action theory: the emphasis shifts from" push factors" to "pull factors" [START_REF] Gambetta | Where they Pushed or did they Jump? Individual Decision Mechanism in Education[END_REF]. In other words, the focus is on actor's choices embedded in constraints, such as educational and jobs structures [START_REF] Müller | The institutional Embeddedness of the Stratification Process. A comparative Study of Qualifications and Occupations in Thirteen Countries[END_REF] although it must be stated that there is no intention to rely entirely upon "rational action theory". We merely need to appreciate that the social meaning and functioning of educational structures are not only shaped from the "top" but are also produced from the "bottom".

This certainly puts clear limits on educational policies, as illustrated through international comparisons.

Everywhere in the "developed" world, educational expansion has accelerated. However, this has been matched with patterns of "persisting inequalities" (Shavit and Blossfeld, 1993). Many countries have implemented unified structures, in order to delay the moment when choices are made in an attempt to avoid early social differentiation. Nevertheless, particular social actors (especially middle and upper-class families) have shown great capacity to use educational reforms to preserve their advantage (for example [START_REF] Kerckhoff | Staying ahead: The Middle Class and School Reform in England and Wales[END_REF]. For the State, the most successful changes, as far as democratization is concerned, are those directed toward the institutional set-up of the school system, especially the choice structure. It has proved worthwhile to attempt to move choice to a later stage, to make choices less irreversible, to eliminate dead-ends and to develop alternative routes to reach the same final level. An alternative policy would be to develop a "life-long learning" process, to compensate later on for the inequalities in school, which prove so resistant to action, and which induce, in countries such as France, strong inequalities during the whole of life.

However, another important lesson to be drawn from international comparisons is that we may have, up to now, over-estimated the role played by the educational system as a means of transmitting class advantage and disadvantage. The countries which have been successful in reducing social inequalities in education (such as Sweden or the Netherlands), whatever their educational structures, are those which have achieved some reduction in social inequalities in everyday life. This is particularly so because less inequalities in life result in less inequalities stemming from socially biased self-selection (taking into account inequalities in resources and attitudes towards the future). One thing seems certain, we tend to exaggerate the role of school upon the reduction of social inequalities; "educational reform should not serve as a political substitute for direct social reform" (Moore, 1996). This of course is a long standing perspective (Cf. Baudon, 1973). It is however of prime importance as far as policy considerations are concerned. Overall then, to progress on a sound basis, there is nothing like the heuristic power of international comparisons.

Figure 1 .

 1 Figure 1. Access rates to different levels of education according to birth year. France.

Figure 2 .

 2 Figure 2. Access rates to last year of college according to birth year .

Figure 3 .

 3 Figure 3. Access rates to first year of college for worker s' and professionals' children.

Figure 4 .

 4 Figure 4. Inequalities in access to college (cohort born 1939± 48) and access to lyce es (cohort born 1954-58).

Table I

 I Access rates to different levels of education for children of workers and professionnals

		Bef. 129	29-38	39-48	49-53	54-58	59-63	64-73
	Entry into 6ème (lower secondary education)
	whole cohort	26,7	24,1	36,9	46,7	75,4	91,6	96,1
	professionals' children	83,9	83	86,5	89,2	96,5	97,3	99
	workers' children	19,8	16,2	23,7	33,9	67,8	87,5	93,2
	difference of %	64,2	66,8	68,8	55,3	28,7	9,8	5,8
	odds ratio for entry into 6eme	21,1	25,3	20,6	16,1	13,1	5,1	7,2
	Entry into 2nd (upper secondary education)
	whole cohort	12,1	13,1	24,3	28,2	34,5	39,4	49,2
	professionals' children	64,8	69,4	81,6	80,9	81,5	85,6	87,4
	workers' children	5,37	5,9	11,4	14,9	19,0	21,2	28,0
	difference of %	59,43	63,5	70,2	66	62,5	64,4	59,4
	odds ratio for entry into 2 nd	32,4	36,2	34,5	24,2	18,8	22,1	17,8
	baccalauréat
	whole cohort	5,1	5,9

Table 2 . Net effects of parents' characteristics and gender on access to 6ème

 2

	Cohort

father's educational level ref. complete primary education or less

Table 3 . Net effects of parents' characteristics and gender on access to 2 nd .

 3

	cohort	av. 39	39-48	49-53	54-58	59-63	64-73
	intercept	-2.55*** -1.86***	-1.73***	-1.69***	-2.09***	-1.6537
	father's socio-economic category ref. manuel worker
	farmer	70.54*** 70.09(ns)	0.15(ns)	0.86***	0.15(ns)	0.55**
	petit bourgeoisie	1.02***	0.89***	0.53***	0.97***	0.85***	0.79***
	Higher professionals	1.91***	1.88***	1.65***	1.91***	2.03***	1.78***
	Lower professionals	1.22***	1.34***	1.04***	1.11***	1.11***	0.94***
	routine non manual	0.92***	0.80***	0.36***	0.89***	0.75***	0.64***
	mothers' socio-economic category ref. out of work
	farmer	71.16*	71.62**	716.05(ns)	70.27*	0.55***	70.07(ns)
	Petit bourgeoisie	70.06(ns) 0.18*	0.51***	0.26*	0.49***	0.32*
	Higher professionals	0.27(ns)	0.52(ns)	0.88(ns)	1.24**	0.52(ns)	1.08**
	Lower professionals	0.42**	0.45***	0.70***	0.54***	0.64***	0.78***
	routine non manual	70.13	0.01	0.01	0.16*	0.27***	0.16*
	fathers' educational level ref. complete primary education or less
	Intermediate diploma	0.64***	0.44***	0.49***	0.53***	0.32***	0.37***
	baccalauréat or more	1.81***	1.37***	1.07***	1.08***	0.93***	0.93***
	mothers' educational level ref. complete primary education or less
	Intermediate diploma	1.09***	0.85***	0.83***	0.82***	0.77***	0.56***
	baccalauréat or more	1.73***	1.54***	1.31***	1.28***	1.28***	0.81***
	gender	ref.	men
	women	0.02	0.20***	0.33***	0.48***	0.83***	0.64***
	G2 (df.19)	1863	2686	1101	1497	1261	1078
	N	7460	9319	5293	5424	4674	4042
	Mean access rates	12.9	24.27	28.26	34.50	39.39	49.23

Sources: FQP77, FQP85,FQP93 (own calculations).

Table 4 . Net effects of parents' characteristics and gender on access to 2nd . (population who entered 6 ème ).

 4

	cohort	av. 39	39-48	49-53	54-58	59-63	64-73
	intercept	-0.25*	0.13	-0.25*	-1.15***	-1.9***	-1.654***
	fathers' socio-economic category ref. manual worker
	farmer	0.01ns	0.41*	0.11ns	0.73***	0.12ns	0.546***
	petit bourgeoisie	0.64***	0.41***	0.06ns	0.89***	0.78***	0.794***
	higher professionals	1.28***	1.31***	0.96***	1.73***	2.06***	1.776***
	lower professionals	0.60***	0.73***	0.51***	0.98***	1.05***	0.944***
	routine non manual	0.15**	0.44***	0.12ns	0.82***	0.69***	0.637***
	mothers' socio-economic category ref. out of work
	farmer	0.91ns	-1.57*		-0.05ns	0.52*	-0.073
	petit bourgeoisie	-0.48**	-0.03ns	0.42*	0.16ns	0.49**	0.317*
	higher professionals	0.89ns	0.18ns	0.53ns	1.11*	0.69ns	1.080***
	lower professionals	0.19ns	0.46*	0.28ns	0.48*	0.66***	0.785***
	routine non manual	-0.17ns	-0.29*	-0.04ns	0.14ns	0.24**	0.161*
	fathers' educational level ref. complete primary education or less
	intermediate diploma	0.26*	0.15ns	0.39***	0.36***	0.29***	0.368***
	baccalauréat or more	1.34***	0.93***	1.01***	1.01***	0.92***	0.930***
	mothers' educational level ref. complete primary education or less
	intermediate diploma	0.61***	0.15***	0.41***	0.70***	0.79***	0.557***
	baccalauréat or more	1.1***	1.29***	1.17***	1.18***	1.23***	0.811***
	gender ref. men
	women	-0.31**	-0.06ns	0.18*	0.41***	0.79***	0.643***
	G2 (df. 19)	422	574	315	992	1146	1079
	N	2120	4076	2527	4278	4295	4029
	Mean access rates	55	65.8	60.5	45.7	43	51.2

Sources: FQP77, FQP85,FQP93 (own calculations).

Table 5 . Regression model of net effects of father socio± economic category and both parents' levels of education on highest level of education completed.

 5

		Av. 1939	1939-48	1949-53	1954-58	1959-63	1964-73
	Intercept	0.05	0.58***	0.59***	1.14***	1.41***	2.06***
	father's socio economic category	0.53***	0.63***	0.77***	0.67***	0.80***	0.62***
	father's level of education	1.12****	0.87***	0.76***	0.64***	0.39***	0.29***
	mother's level of education	0.34***	0.41***	0.48***	0.44***	0.42***	0.47***
	R 2	26.4%	25.8%	25.2%	22.5%	19.9%	18.2%
	N	5559	7409	5362	5201	2176	4029

We are grateful to Marie-Odile Lebeaux and Louis-André Vallet (CNRS-Lasmas-IdL) for their careful reading and their critical comments on first drafts of this paper.

15,7 19,8 23,9 27,9 33,5 professionals' children 35,1 40,7 67,4 67,9 68,9 74 61,7 workers' children 1,2 1,9 6,2 8,8 11,2 12,8 19,6 difference of % 33,9 38,8 61,2 59,1 57,7 61,2 42,1 odds ratio for obtaining baccalauréat 44,5 35,4 31,3 21,9 17,6 19,4 6,4 Access to tertiary education * Whole cohort 5,4 12,5 16,8 20,3 professional' s children 41,8 53,4 61,1 50,5 worker' s children 1,6 4,2 6,2 9,6 difference of % 40,2 49,2 54,9 40,9 odds ratio for access to tertiary ed. 53,3 26,1 23,8 9,6 Pupil s having entered 6 eme only. Entry into 2 nd (upper secondary education) whole cohort 45,3 54,5 65,8 60,5 45,7 43,0 51,2 professionals' children 77,2 83,6 90,8 88,6 84,4 87,7 88,2 workers' children 27 36,5 47,8 48,1 28,0 24,3 30,1 differences of % 50,2 47,1 43 40,1 56,4 63,4 58,1 odds ratio for entry into 2 nd 9,2 8,9 10,8 8,1 13,9 22,2 17,4 baccalauréat whole cohort 27,7 33,5 42,6 42,5 31,7 30,5 34,9 professionals' children 41,8 49 70,2 69,9 71,4 74 62,4 workers' children 6,1 11,4 26 31,4 16,5 14,6 21,1 difference of % 35,7 37,6 44,2 38,5 54,9 59,4 41,3 odds ratio for obtaining baccalauréat 11,1 7,5 6,7 5,1 12,6 16,6 6,2 Note: Rates concerning cohorts 64± 73 are underestimated, due to the fact that the schooling is not ended for the whole cohort yet. Concerning tertiary education, 10 year groups of cohorts have been constructed for sampling reasons: before 1939, 39-48, 49-58, 59-68. Sources: cohorts born before 1939 (FQP77) cohort born 1939-48 (FQP77) cohort born 1949-1953 (FQP77) cohort born 1954-1958 (FQP85) cohort born 1959-1963 (FQP85) cohort born 1964-1973 (FQP94)

This increase in social selection may also be related, as[START_REF] Shavit | Educational Stratification in Italy: Reforms, Expansion and Equality of Opportunity[END_REF] suggest, to the growing

However, these two figures are not perfectly comparable. In Germany, pupils entering Gymnasium usually obtain Abitur ; in France the process of selection is more continuous, and it is relevant to consider the pupils who made the first transition, and consequently having reached the last form of collège

These figures are not exactly constructed in the same way, since in France the children of top executives are opposed to children of

Appendix

Our study is based upon the different FQP surveys (Education, Training and Occupations) Because the sample of the 1993's survey has been reduced to half (n=18000), we preferred to widen the last cohort to 10 year (born between 1964 and 1973).

The classification of occupations used here is the French PCS (socio-economic categorisation). As this classification changed over the period, we recoded the occupations in the PCS to that of 1982.

We also used the so-called panel of pupils of secondary education, carried out by the French Ministry of Education. These successive surveys follow a representative sample (about 22000 and 37000 according to the survey) of pupils from their entry into college (6ème) to when they leave secondary education, whatever the track followed (general or vocational). The files contain a lot of information about students' social background and the last file includes data concerning academic performance. In this paper we use the three existing panels which were started respectively in 1972, 1980 and 1989.


Leonardo S Cardoso
email: leonardo.cardoso@supelec.fr

Mari Kobayashi
email: mari.kobayashi@supelec.fr

Ryan Oyvind
email: oyvindry@ifi.uio.no

Merouane Debbah
email: merouane.debbah@supelec.fr

Mérouane Debbah

Vandermonde Frequency Division Multiplexing for Cognitive Radio

à la 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.

MOTIVATION

We consider a 2 × 2 cognitive radio model where both a primary (licensed) transmitter and a secondary (unlicensed) transmitter wish to communicate with their corresponding receivers simultaneously as illustrated in Fig. 1. When both transmitters do not share each other's message, the information theoretic model falls into the interference channel [START_REF] Debbah | Interference channels[END_REF][START_REF] Sato | Two-user communication channels[END_REF] whose capacity remains open in a general case. A significant number of recent works have aimed at characterizing the achievable rates of the cognitive radio channel, i.e. the interference channel with some knowledge of the primary's message at the secondary transmitter [START_REF] Devroye | Achievable rates in cognitive radio channels[END_REF][START_REF] Jovicic | Cognitive radio: An informationtheoretic perspective[END_REF][START_REF] Maric | Capacity of Interference Channels With Partial Transmitter Cooperation[END_REF][START_REF] Maric | On the capacity of interference channels with partially-cognitive transmitter[END_REF]. These include the pioneering work of [START_REF] Devroye | Achievable rates in cognitive radio channels[END_REF], the works of [START_REF] Jovicic | Cognitive radio: An informationtheoretic perspective[END_REF], [START_REF] Maric | Capacity of Interference Channels With Partial Transmitter Cooperation[END_REF] for the case of weak, strong Gaussian interference, respectively, and finally a recent contribution of [START_REF] Maric | On the capacity of interference channels with partially-cognitive transmitter[END_REF] with partial knowledge at the secondary transmitter. In all these works, the optimal transmission scheme is based on dirty-paper coding that pre-cancels the known interference to the secondary receiver and helps the primary user's transmission. Unfortunately, this optimal strategy is very complex to implement in practice and moreover based on rather unrealistic assumptions : a) the secondary transmitter has full or partial knowledge of the primary message, b) both transmitters know all the channels perfectly. Despite its cognitive capability, This work was partially supported by Alcatel-Lucent. the assumption a) seems very difficult (if not impossible) to hold. This is because in practice the secondary transmitter has to decode the message of the primary transmitter in a causal manner by training over a noisy, faded or capacitylimited link. The assumption b) requires both transmitters to perfectly track all channels (possibly by an explicit feedback from two receivers) and thus might be possible only if the underlying fading channel is quasi-static.

The above observation motivates us to design a practical transmission scheme under more realistic assumptions. First, we consider no cooperation between two transmitters. The primary user is ignorant of the secondary user's presence and furthermore the secondary transmitter has no knowledge on the primary transmitter's message. Second, we assume that transmitter 1 knows perfectly h (11) , while transmitter 2 knows its local channels h (21) and h (22) . This assumption is rather reasonable when the channel reciprocity can be exploited for time division duplexing systems. Also, each receiver i is assumed to estimate perfectly its direct channel h (ii) . Finally, assuming frequency selective fading channels, we consider OFDM transmission. The last assumption has direct relevance to the current OFDM-based standards such as WiMax, 802.11a/g, LTE and DVB [START_REF]Ansi/ieee standard 802.11[END_REF]. Under this setting, there is clearly a tradeoff between the achievable rates of the two users. For the cognitive radio application, however, one of the most important goals is to design a transmit scheme of the secondary user that generates zero interference to the primary receiver.

We propose a linear Vandermonde precoder that generates zero interference at the primary receiver by exploiting the redundancy of a cyclic prefix and name this scheme Van-dermonde Frequency Division Multiplexing (VFDM). The precoder exploits the frequency selectivity of the channel to form a frequency beamformer (analogy to the classical spatial beamformer). More precisely, our precoder is given by a Vandermonde matrix [START_REF] Golub | Matrix Computations[END_REF] with L roots corresponding to the channel h (21) from the seconday user to the primary receiver. The orthogonality between the precoder and the channel enables the secondary user to send L symbols, corresponding to the size of a cyclic prefix, while maintaining zero interference. This is contrasted with the approach of [START_REF] Jovicic | Cognitive radio: An informationtheoretic perspective[END_REF] where the zero interference is limited to the case of weak interference. To the best of our knowledge, a Vandermonde precoder to cancel the interference has never been proposed. The use of the Vandermonde filter together with a Lagrange spreading code was proposed to cancel the multiuser interference on the uplink of a CDMA system [START_REF] Scaglione | Lagrange/Vandermonde MUI eliminating user codes forquasisynchronous CDMA in unknown multipath[END_REF]. However, this scheme is conceptually different in that its interference cancellation exploits the orthogonality between the spreading code and the filter. Moreover, it does not depend on the channel realization.

Since the size L of the cyclic prefix is typically fixed to be much smaller than the number N of OFDM symbols (sent by the primary user) [START_REF]Ansi/ieee standard 802.11[END_REF], VFDM is highly suboptimal in terms of the achievable rate. Nevertheless, we show that the secondary user can improve its rate by appropriately designing its input covariance at the price of additional side information on the interference plus noise covariance seen by the secondary receiver. Numerical examples inspired by IEEE 802.11a setting show that VFDM with our proposed covariance design enables the secondary user to achieve a non-negligible rate of 8.44 Mbps while guaranteeing the primary user its target rate of 36 Mbps with the operating SNR of 10 dB. Finally, although this paper focuses on the zero interference case desired for the cognitive radio application, VFDM can be suitably modified to provide a tradeoff between the amount of interference that the secondary transmitter cancels and the rate that it achieves. We discuss in Section 4 some practical methods to generalize VFDM.

SYSTEM MODEL

We consider a 2×2 cognitive model in Fig. 1 over frequency selective fading channels. By letting h (ij) denote the channel with L + 1 paths between transmitter i and receiver j, we assume that entries of h (ij) are i.i.d. complex circularly symmetric AWGN ∼ N C (0, σ ij /(L + 1)) and moreover the channels are i.i.d. over any i, j. In order to avoid blockinterference, we apply OFDM with N subcarriers with a cyclic prefix of size L. The receive signal for receiver 1 and receiver 2 is given by

y 1 = F T (h (11) )x 1 + T (h (21) )x 2 + n 1 y 2 = F T (h (22) )x 2 + T (h (12) )x 1 + n 2 (1)
where

T (h (ij) ) is a N × (N + L) Toeplitz with vector h (ij) T (h (kj) ) = 2 6 6 6 6 6 4 h (kj) L • • • h (kj) 0 0 • • • 0 0 . . . . . . . . . . . . . . . . . . . . . . . . 0 0 • • • 0 h (kj) L • • • h (kj) 0 3 7 7 7 7 7 5
F is an FFT matrix with [F] kl = exp(-2πj kl N ) for k, l = 0, . . . , N -1, and x k denotes the transmit vector of user k of size N + L subject to the individual power constraint given by tr

(E[x k x H k ]) ≤ (N + L)P k (2)
and n k ∼ N C (0, I N ) is AWGN. For the primary user, we consider DFT-modulated symbols

x 1 = AF H s 1 ( 3
)
where A is a precoding matrix to append the last L entries of F H s 1 and s 1 is a symbol vector of size N . For the secondary user, we form the transmit vector by x 2 = Vs 2 where V is a linear precoder and s 2 is the symbol vector (whose dimension is be specified later). Our objective is to design the precoder V that generates zero interference, i.e. satisfies the following orthogonal condition

T (h (21) )Vs 2 = 0, ∀s 2 . ( 4
)

VFDM

In this section, we propose a linear Vandermonde precoder that satisfies (4) by exploiting the redundancy L of the cyclic prefix or equivalently the degrees of freedom left by the system. Namely, we let V to be a (N + L) × L Vandermonde matrix given by

V = 2 6 6 6 6 6 4
1 • • • 1 a1 • • • aL a 2 1 • • • a 2 L . . . . . . . . . a N+L-1 1 • • • a N+L-1 L 3 7 7 7 7 7 5 (5)
where {a l , . . . , a L } are the roots of the polynomial S(z)

= L i=0 h (21) i
z L-i with L+1 coefficients of the channel h (21) . Since the orthogonality between the precoder and the channel enables two users to transmit simultaneously over the same frequency band, we name this scheme Vandermonde Frequency Division Multiplexing (VFDM) . Clearly, the secondary user needs to know perfectly the channel h (21) in order to adapt the precoder. This can be done easily assuming that the reciprocity can be exploited under time-division duplexing systems. The resulting transmit vector of the secondary user is given by

x 2 = αVs 2 (6)
where s 2 is a symbol vector of size L with covariance S 2 and α is determined to satisfy the power constraint (2)

α = s (N + L)P2 tr(VS2V H ) .
The following remarks are in order : 1) Since the channels h (21) and h (22) are statistically independent, the probability that h (21) and h (22) have the same roots is zero. Therefore the secondary user's symbols s 2 shall be transmitted reliably; 2) Due to the orthogonality between the channel and the precoder, the zero interference condition (4) always holds irrespectively of the secondary user' input power P 2 and its link σ 2,1 . This is in contrast with [START_REF] Jovicic | Cognitive radio: An informationtheoretic perspective[END_REF] where the zero interference is satisfied only for the weak interference case, i.e. σ 2,1 P 2 ≤ P 1 and σ 1,1 = σ 2,2 = 1; 3) To the best of our knowledge, the use of a Vandermonde matrix at the transmitter for interference cancellation has never been proposed. In [START_REF] Scaglione | Lagrange/Vandermonde MUI eliminating user codes forquasisynchronous CDMA in unknown multipath[END_REF], the authors proposed a Vandermonde filter but for a different application. By substituting ( 3) and ( 6) into y 1 , we obtain N parallel channels for the primary user given by

y 1 = H (11) diag s 1 + ν 1 (7)
where

H (11) diag = diag(H (11) 1
, . . . , H

N ) is a diagonal frequency domain channel matrix with i.i.d. entries H

y 2 = H 2 s 2 + H (12) diag s 1 + ν 2 (8)
where we let H 2 = αFT (h (22) )V denote the overall N ×L channel, H

diag = diag(H

, . . . , H

N ) denotes a diagonal frequency domain channel matrix with i.i.d. entries H (12) n ∼ N C (0, σ 12 ), and ν 2 ∼ N C (0, I N ) is AWGN. From ( 7) and ( 8), we remark that VFDM converts the frequency-selective interference channel (1) into one-side vector interference channel (or Z interference channel) where the primary receiver sees interference-free N parallel channels and the secondary receiver sees the interference from the primary transmitter. Notice that even for a scalar Gaussian case the capacity of the one-side Gaussian interference channel is not fully known [START_REF] Sason | On achievable rate regions for the Gaussian interference channel[END_REF][START_REF] Etkin | Gaussian Interference Channel Capacity to Within One Bit[END_REF]. In this work, we restrict our receiver to a single user decoding strategy which is clearly suboptimal for the strong interference case σ 12 > σ 11 .

INPUT COVARIANCE OPTIMIZATION

This section considers the maximization of the achievable rates under the individual power constraints. First, we consider the primary user. Since the primary user sees N parallel channels [START_REF]Ansi/ieee standard 802.11[END_REF], its capacity is maximized by Gaussian input and a diagonal covariance, i.e. S 1 = diag(p 1,1 , . . . , p 1,N ). The rate of the primary user is given by

R 1 = max {p1,n} 1 N N n=1 log(1 + p 1n |H (11) n | 2 ) (9)
with the constraint N n=1 p 1,n ≤ N P 11 . The set of powers can be optimized via a classical waterfilling approach.

p 1,n = µ 1 - 1 |H (11) n | 2 + ( 10
)
where µ 1 is a Lagrangian multiplier that is determined to satisfy N n=1 p 1n ≤ N P 1 . The received signal of the secondary user (8) when treating the signal from the primary transmitter as noise is further simplified to

y 2 = H 2 s 2 + η
where η denotes the noise plus interference term whose covariance is given by

S η = H (12) diag S 1 H (12) diag H + I N
Under the Gaussian approximation of η, the rate of the secondary user is maximized by solving

maximize 1 N log ˛IN + (N + L)P2 tr(VS2V H ) GS2G H subject to tr(S2) ≤ LP2
where we define the effective channel as

G = S -1/2 η H 2 ∈
C N ×L . Notice that the above problem can be solved with perfect knowledge of the covariance S η at the secondary transmitter, which requires the secondary receiver to estimate S η during a listening phase and feed it back to its transmitter. The above optimization problem is non-convex since the objective function is neither concave or convex in S 2 . Nevertheless, we propose a two-step optimization approach that aims at finding the optimal S 2 efficiently. The first step consists of diagonalizing the effective channel in order to express the objective function as a function of powers. We apply singular value decomposition to the effective channel such that G = U g Λ g P H g where U g ∈ C N ×N , P g ∈ C L×L are unitary matrices and Λ g is diagonal with r ≤ L singular values {λ 1/2 g,l }. Clearly, the optimal S 2 should have the structure P g Ŝg P H g where Ŝg = diag(p 2,1 , . . . , p 2,r ) is a diagonal matrix, irrespectively of the scaling tr(VS 2 V H ). For a notation simplicity let us define the signal-to-interference ratio of channel i

SIRi = (N + L)P2ci βip2,i P r j=1 βj p2,j
where we let

β i ∆ = [P H g V H VP g ] i,i and c i = λg,i
βi . By using these notations, it can be shown that the the rate maximization problem reduces to

maximize f (p2) = 1 N L X i=1 log (1 + SIRi) subject to r X i=1 p2,i ≤ LP2 (11)
where we let p 2 = (p 2,1 , . . . , p 2,r ). The second step consists of solving the above power optimization problem. Unfortunately the objective function is not concave in {p 2,i }.

Let us first assume that the high SIR approximation is valid for any i, i.e. SIR i ≫ 1 (this is the case for large (N + L)P 2 c i and in particular when the secondary user's channel is interference-free). Under the high SIR assumption, the function f can be approximated to

J(p 2 ) = 1 N r i=1 log (SIR i ).
It is well known that this new function can be transformed into a concave function through a log change of variable [START_REF] Chiang | Geometric Programming for Communication Systems[END_REF]. Namely let define pi = ln p i (or p i = e pi ). The new objective function is defined by

J(p2) = 1 N r X i=1 (log(ai) + p2,i) - r N log r X j=1 λj e p2,j ! (12)
where we let a i = (N + L)P 2 c i β i . The function J is now concave in p2 since the first term is linear and the second term is convex in p2 . Therefore we solve the KKT conditions which are necessary and sufficient for the optimality.

It can be shown that the optimal power allocation reduces to a very simple waterfilling approach given by

p ⋆ 2,i = LP 2 β i r j=1 1 βj (13)
which equalizes

β 1 p ⋆ 2,1 = • • • = β r p ⋆ 2,r and yields SIR i = (N +L)P2ci r
. The resulting objective value would be

f r = 1 N r i=1 log 1 + (N + L)P 2 c i r
It is worth noticing that the high SIR approximation is not necessarily satisfied due to the interference from the primary user and that the optimal strategy should select a subset of channels. One possible heuristic consists of combining the waterfilling based on high SIR approximation with a greedy search. Let us first sort the channels such that

c π(1) ≥ c π(2) ≥ • • • ≥ c π(r) (14)
where π denotes the permutation. We define the objective value achieved for a subset {π(1), . . . , π(l)} using the waterfilling solution [START_REF] Ibragimov | On roots of random polynomials[END_REF] with cardinality l

f l = 1 N l i=1 log 1 + (N + L)P 2 c π(i) l
The greedy procedure consists of computing f l for l = 1, . . . , r and sets the effective number of channels r ⋆ to be the argument maximizing f l . As a result, the secondary user achieves the rate given by

R 2 = 1 N r ⋆ i=1 log 1 + (N + L)P 2 c π(i) r ⋆ (15)
From the rate expression [START_REF] Ryan | Random Vandermonde matricespart I,II: Fundamental results, applications[END_REF], it clearly appears that the rate of the secondary user (the pre-log factor) depends critically on the rank r of the overall channel H 2 , which is determined by the rank of V since F, T (h (22) ) are full-rank. It turns out that the rank of V is very sensitive to the amplitude of the roots {a l } especially for large N, L. Although the roots tend to be on a unit circle as N, L → ∞ while keeping L/N = c for some constant c > 0 [START_REF] Ibragimov | On roots of random polynomials[END_REF], a few roots outside the unit circle (with |a l | > 1) tend to dominate the rank. In other words for a fixed fraction c. Fig. 2 shows the averaged number of ranks of a 5L × L Vandermonde matrix (corresponding to c = 1/4) versus L. The figure shows that for a fixed c there is a critical size L ⋆ above which the rank decreases and this size decreases for a larger N . This suggests an appropriate choice of the parameters to provide a satisfactory rate to the secondary user with VFDM. When the size of the cyclic prefix is larger than L ⋆ , VFDM can be suitably modified so as to boost the secondary user's rate at the price of increased interference (or reduced rate) at the primary user. This can be done for example by normalizing the roots computed by the channel or by selecting L columns from (N +L)×(N +L) FFT matrix. The design of the Vandermonde precoder by taking into account the tradeoff between the interference reduction and the achievable rate is beyond the scope of this paper and will be studied in a separate paper [START_REF] Sampaio | Vandermonde frequency division multiplexing[END_REF] using the theory of Random Vandermonde Matrices [START_REF] Ryan | Random Vandermonde matricespart I,II: Fundamental results, applications[END_REF].

NUMERICAL EXAMPLES

This section provides some numerical examples to illustrate the performance of VFDM with the proposed power allocation. Inspired by 802.11a [START_REF]Ansi/ieee standard 802.11[END_REF], we let N = 64, L = 16. Fig. 3 shows the average rate of the secondary user as a function of SNR P 1 = P 2 in dB. We let σ 11 = σ 22 = 1 and vary σ 12 = 1.0, 0.1, 0.01, 0.0 for the link h (12) . Notice σ 12 = 0 corresponds to a special case of no interference. We compare the VFDM performance with equal power allocation S 2 = P I L and with the waterfilling power allocation enhanced by a greedy search. We observe a significant gain by our waterfilling approach and this gain becomes even significant as the interference decreases. This example clearly shows that the appropriate design of the secondary transmitter's input covariance is essential for VFDM. Although not plotted here, the optimization of the primary user's input covariance has a negligible impact on the rates of two users. Finally, it can be shown that the secondary user's rate becomes bounded as P → ∞ for any σ 12 > 0 independently of the input covariance. Next we consider the scenario where the system imposes a target rate R ⋆ 1 to the primary user and the primary transmitter minimizes its power such that R ⋆ 1 is achieved. The system sets the transmit power to its maximum P 1 if the rate is infeasible. Fig. 4 shows the achievable rates of both users as a function of the target rate R ⋆ 1 in bps/Hz with P 1 = P 2 = 10 dB. Again we observe a significant gain due to the appropriate design of the secondary input covariance.

We wish to conclude this section with a simple numerical example inspired by the IEEE 802.11a setting [START_REF]Ansi/ieee standard 802.11[END_REF], showing that VFDM with the appropriate input covariance design enables the secondary user to achieve a considerable rate while guaranteeing the primary user to achieve its target rate over interference-free channels. For example, for the target rate of R 1 = 2.7, 1.8 [bps/Hz] that yields the two highest rates of 54, 36 [Mbps] over a frequency band of 20MHz, the secondary user can achieves 6.06, 8.44 [Mbps] respectively with operating SNR of 10 dB.

Fig. 1 . 2 × 2

 122 Fig. 1. 2 × 2 cognitive model

  0, σ 11 ) and ν 1 ∼ N C (0, I) is AWGN. The received signal of the secondary user is given by

Fig. 2 .

 2 Fig. 2. Rank of (N + L) × L Vandermonde matrix vs. L

Fig. 3 .

 3 Fig. 3. Rate of user 2 vs. SNR

Fig. 4 .

 4 Fig. 4. R 1 , R 2 vs. a target rate R 1

The power constraint considered here is different from (2). However, the waterfilling power allocation of (10) satisfies (2) in a long-term under the i.i.d. frequency-domain channels.