Marie Delnord
email: marie.delnord@inserm.fr

Béatrice Blondel

Nicolas Drewniak

Kari Klungsøyr

Francisco Bolumar

Ashna Mohangoo

Mika Gissler

Katarzyna Szamotulska

Nicholas Lack

Jan Nijhuis

Petr Velebil

Luule Sakkeus

James Chalmers

Jennifer Zeitlin

Euro-Peristat Preterm Group

Varying gestational age patterns in cesarean delivery: an international comparison

Keywords: Cesarean delivery (CD), Cross-national comparisons, Gestational age, Plurality, Mode of delivery, Euro-Peristat

Background: While international variations in overall cesarean delivery rates are well documented, less information is available for clinical sub-groups. Cesarean data presented by subgroups can be used to evaluate uptake of cesarean reduction policies or to monitor delivery practices for high and low risk pregnancies based on new scientific evidence. We studied differences and patterns in cesarean delivery rates by multiplicity and gestational age in Europe and the United States.

Methods: This study used routine aggregate data from 17 European countries and the United States on the number of singleton and multiple live births with cesarean versus vaginal delivery by week of gestation in 2008. Overall and gestation-specific cesarean delivery rates were analyzed. We computed rate differences to compare mode of delivery (cesarean vs vaginal birth) between selected gestational age groups and studied associations between rates in these subgroups namely: very preterm (26-31 weeks GA), moderate preterm (32-36 weeks GA), near term (37-38 weeks GA), term (39-41 weeks GA) and post-term (42+ weeks GA) births, using Spearman'sranktests.

Results: High variations in cesarean rates for singletons and multiples were observed everywhere. Rates for singletons varied from 15% in The Netherlands and Slovenia, to over 30% in the US and Germany. In singletons, rates were highest for very preterm births and declined to a nadir at 40 weeks of gestation, ranging from 8.0% in Sweden and Norway, to 22.5% in the US. These patterns differed across countries; the average rate difference between very preterm and term births was 43 percentage points, but ranged from 14% to 61%. High variations in rate differences were also observed for near term versus term births. For multiples, rates declined by gestational age in some countries, whereas in others rates were similar across all weeks of gestation. Countries' overall cesarean rates were highly correlated with gestation-specific subgroup rates, except for very preterm births.

Conclusions: Gestational age patterns in cesarean delivery were heterogeneous across countries; these differences highlight areas where consensus on best practices is lacking and could be used in developing strategies to reduce cesareans.

Background

Over the past decades, the percentage of cesarean deliveries has risen in almost all high-income countries, and the most recent European Perinatal Health Report with 2010 data confirms continuing increases [START_REF]women and babies in Europe in 2010[END_REF][START_REF] Declercq | Is a rising cesarean delivery rate inevitable? Trends in industrialized countries, 1987 to 2007[END_REF]. Researchers and clinicians have expressed concerns over these increasing rates and the impact on mothers' and newborns' health [START_REF] Declercq | Is a rising cesarean delivery rate inevitable? Trends in industrialized countries, 1987 to 2007[END_REF][START_REF] Villar | Maternal and neonatal individual risks and benefits associated with caesarean delivery: multicentre prospective study[END_REF][START_REF] Spong | Preventing the first cesarean delivery: summary of a joint Eunice Kennedy Shriver National Institute of Child Health and Human Development, Society for Maternal-Fetal Medicine, and American College of Obstetricians and Gynecologists Workshop[END_REF]. While cesarean delivery can be a lifesaving intervention for both mother and child, [START_REF] Betran | Rates of caesarean section: analysis of global, regional and national estimates[END_REF] it has been associated with significant neonatal and maternal adverse health outcomes in comparison with vaginal delivery [START_REF] Villar | Maternal and neonatal individual risks and benefits associated with caesarean delivery: multicentre prospective study[END_REF][START_REF] Spong | Preventing the first cesarean delivery: summary of a joint Eunice Kennedy Shriver National Institute of Child Health and Human Development, Society for Maternal-Fetal Medicine, and American College of Obstetricians and Gynecologists Workshop[END_REF][START_REF] Deneux-Tharaux | Postpartum maternal mortality and cesarean delivery[END_REF][START_REF] O'neill | Caesarean delivery and subsequent stillbirth or miscarriage: systematic review and meta-analysis[END_REF][START_REF] Jackson | Morbidity following primary cesarean delivery in the Danish National Birth Cohort[END_REF][START_REF] Macdorman | Neonatal mortality for primary cesarean and vaginal births to low-risk women: application of an "intention-to-treat" model[END_REF]. Studies have also linked cesarean delivery with complications in subsequent pregnancies for women (placentation disorders, uterine rupture and stillbirth) [START_REF] O'neill | Caesarean delivery and subsequent stillbirth or miscarriage: systematic review and meta-analysis[END_REF][START_REF] Jackson | Morbidity following primary cesarean delivery in the Danish National Birth Cohort[END_REF] as well as adverse long term child health outcomes such as asthma and type-1 diabetes [START_REF] Thavagnanam | A meta-analysis of the association between Caesarean section and childhood asthma[END_REF][START_REF] Cardwell | Caesarean section is associated with an increased risk of childhood-onset type 1 diabetes mellitus: a meta-analysis of observational studies[END_REF]. Given these risks and concerns about the optimal use of health resources, professional societies in many high income countries recommend strategies to reduce unnecessary cesareans [START_REF] Spong | Preventing the first cesarean delivery: summary of a joint Eunice Kennedy Shriver National Institute of Child Health and Human Development, Society for Maternal-Fetal Medicine, and American College of Obstetricians and Gynecologists Workshop[END_REF][START_REF]NICE: National Institute for Health and Clinical Excellence (NICE). Caesarean section[END_REF].

International comparisons of cesarean rates highlight differences in delivery practices and provide useful benchmarks for the evaluation of national practices. Such comparisons have been made using data in international databases run by Eurostat, OECD and WHO but these organizations collect overall cesarean rate data whereas comparisons stratified by risk groups may better inform cesarean reduction policies [START_REF] Betran | Rates of caesarean section: analysis of global, regional and national estimates[END_REF][START_REF] Althabe | Cesarean section rates and maternal and neonatal mortality in low-, medium-, and high-income countries: an ecological study[END_REF][START_REF] Katikireddi | A comparison of trends in caesarean section rates in former communist (transition) countries and other European countries[END_REF]. The Robson Ten-Group classification is a robust system which allows comparisons of cesarean rates based on characteristics of pregnancies, [START_REF] Brennan | Comparative analysis of international cesarean delivery rates using 10-group classification identifies significant variation in spontaneous labor[END_REF] but the data needed to identify cesareans based on this system are not available in any of the international databases and may be difficult to obtain in some countries on the national level.

Presenting rates by gestational age distinguishes between practices for higher risk (very preterm singletons and multiples) and lower risk infants (singletons at 39 and 40 weeks of gestation) and could be used by clinicians and health planners to evaluate strategies as well as uptake of cesarean reduction policies targeting specific sub-groupsfor instance, near term pregnancies. We thus compared singleton and multiple cesarean rates across the gestational age continuum using data from the European countries participating in the Euro-Peristat project and the United States to investigate cross-national differences in delivery practices for very preterm, moderate preterm, near term, term and post term pregnancies in 2008.

Methods

The Euro-Peristat project collects population-based aggregate data from European countries, on a set of core and recommended perinatal health indicators and also conducts ad hoc studies based on supplementary detailed data [START_REF] Zeitlin | Preterm birth time trends in Europe: a study of 19 countries[END_REF]. Scientific committee members (clinicians, statisticians, or researchers) from each participating country are responsible for data collection. The European data used in this analysis are from a study on preterm birth [START_REF] Zeitlin | Preterm birth time trends in Europe: a study of 19 countries[END_REF]. We requested data on the number of live births at or after 22 weeks for each completed week of gestation by multiplicity and mode of delivery for several years of which the most recent was 2008. We requested gestational age data based on the best obstetric estimate.

The US data used in this analysis were downloaded from the Center for Disease Control website (http://www.cdc. gov/nchs/data_access/Vitalstatsonline.htm). We used birth certificate data for live births in 2008 and extracted the variables on gestational age, multiplicity and mode of delivery. The clinical estimate of gestational age has been shown to include fewer birth weight extremes [START_REF] Callaghan | Differences in birth weight for gestational age distributions according to the measures used to assign gestational age[END_REF] and has been used successfully in other cross-country studies [START_REF] Mohangoo | Gestational age patterns of fetal and neonatal mortality in Europe: results from the Euro-Peristat project[END_REF].

Seventeen European countries participated in the study and provided data on births in 2008, or the closest available time point. Data from France and Spain are from 2010; in Malta and Sweden data were provided for 2009. Three countries provided data on selected regions only: Germany provided data from three Länder, Belgium for Flanders, and the United Kingdom for Scotland only. Births with unknown mode of delivery were excluded from the analyses; births for which gestational age was missing were included in the computation of overall cesarean rates only. In total, we included 4,092,381 live singleton births and 143,777 live multiple births from the United States and 1,501,575 live singleton births and 55,550 live multiple births from Europe. For less than 0.5% of cases, gestational age data were missing. Our analysis was done on live births to improve the comparability of our estimates; stillbirths were excluded from the study due to differences in registration criteria across countries [START_REF] Mohangoo | Gestational age patterns of fetal and neonatal mortality in Europe: results from the Euro-Peristat project[END_REF], and the distinct obstetric management of stillbirths versus live births [START_REF] Kollee | Obstetric interventions for babies born before 28 weeks of gestation in Europe: results of the MOSAIC study[END_REF].

For the analyses, we computed countries' overall cesarean rates per 100 live births as well as rates for singleton and multiple births; we further calculated rates per completed week of gestation starting at 26 weeks [START_REF] Kollee | Obstetric interventions for babies born before 28 weeks of gestation in Europe: results of the MOSAIC study[END_REF][START_REF] Blondel | Preterm birth and multiple pregnancy in European countries participating in the PERISTAT project[END_REF] in order to reduce variability resulting from small sample sizes and differences in management at the earlier gestational ages. We did not display data with less than five cesareans in a given cell. For singletons, we presented rates for each individual week of gestation and for very preterm births: 26+0 -31+6 weeks GA, moderate preterm births: 32+0 -36+6 weeks GA, near-term: 37+0 -38+6 weeks, term: 39+0 -41+6 weeks and post-term births: 42+0 weeks GA and over. For multiples, we used different gestational age cut-offs to account for the increased risk of premature delivery and for the small number of births at the later gestational ages. Due to small sample sizes for twins in our participating countries, we only presented rates at grouped weeks of gestational age: below 34 weeks GA, 34+0 -36+6 weeks, and 37+0 weeks and over [START_REF] Blondel | Preterm birth and multiple pregnancy in European countries participating in the PERISTAT project[END_REF].

Next, we computed rate differences in each country to compare cesarean practices for term pregnancies to those for near term and very preterm pregnancies. We accounted for variations in the number of births by presenting rate differences with their 95% confidence intervals. We calculated the mean cesarean rate difference between very preterm and term singleton births and between near term and term singleton births. We also computed the mean cesarean rate difference between multiple births before 34 weeks and after 36 weeks of gestation. Last, we examined the associations between cesarean rates in singletons and multiples and the overall cesarean rate using Spearman's rank tests; we carried out this analysis in each of our gestational age subgroups. Data were analyzed using STATA 10.0 software (StataCorp LP, College Station, TX, USA).

Results

Overall cesarean rates ranged from 15.7% to 32.5% of all live births (Table 1). For singleton births, Germany and the United States had the highest rates, 31.0% and 30.8% respectively, while Slovenia and The Netherlands had the lowest rates (15.5% and 14.8% respectively). For multiples, Austria and Malta had the highest rates, 83.4% and 95.4% respectively while The Netherlands and Norway had the lowest rates, 39.6% and 46.7% respectively.

In Figure 1 we display cesarean rates by week of gestation for singleton births; the dotted line shows the overall rate. Rates were highest for very preterm deliveries ranging from 37.0% in Lithuania to 83.3% in Germany (Table 2). Rates decreased with increasing gestational age to a nadir at 40 weeks of gestation, ranging from lows of 8.0% in Sweden and 8.4% in Norway to a high of 22.5% in the US, before rising at 41 and 42 weeks. In a majority of countries, there was also an increase in cesarean deliveries at 38 weeks; this rise was pronounced in Ireland, Sweden, and Estonia. Rates declined everywhere with increasing gestational age but depending on the country, rate differences between preterm, near term and term births differed.

The rate differences presented in Figure 2 measure the range in delivery practices by gestational age across countries for singletons which were observed in Figure 1. We compared rates for two groups at higher risk of cesarean, namely very preterm and near term births, with rates in term births. The average rate difference between very preterm and term births was 43 percentage points, with a range from 14.4% in Lithuania to 61% in Germany as illustrated in Figure 2A. Figure 2B shows that a great heterogeneity also existed between near term and term singleton births; while the average rate difference was 12 percentage points, it ranged from 0.4% in Lithuania to 25.2% in Austria. Some of the variability in early gestational ages also reflects random variation among countries with smaller number of births. Data on multiple births in Table 2 show rates of cesarean section by grouped weeks of gestation. In Finland, Germany, Norway, and Sweden, rates declined by gestational age whereas in others: Slovakia, UK, and The Netherlands, rates were more similar across all weeks of gestation. The average rate difference between babies born before 34 weeks and those born after 37 weeks was 7 percentage points and ranged from 0.2% in Slovakia to 23.0% in Germany.

Table 3 displays the correlation between cesarean rates in singletons and multiples and the overall rate; we examined these associations in each of our gestational age categories. Across countries, the overall cesarean rate was very highly correlated with all other subgroup rates (r = 0.8, p = 0.0), with the exception of very preterm singleton deliveries (r = 0.16, p = 0.52). Singleton and multiple subgroup rates were moderately to highly correlated with each other. Notably in singleton births, rates in moderate preterm births were most correlated with rates in near term births, and rates in post-term pregnancies were most correlated with rates in term pregnancies. Cesarean rates for very preterm singletons were not correlated with any other subgroup rates.

Discussion

Our results reveal broad heterogeneity in use of cesarean delivery by week of gestation for both singleton and multiple births in Europe and the United States. Overall rates were correlated with most other sub-group rates, with the exception of very preterm births. Our analysis illustrates that there is consistency in levels of use of cesarean beyond 32 weeks GA, but there were distinct patterns in rate differences by gestational age groups across countries.

Variations in cesarean rates may result from differences in the distribution of population characteristics such as mothers' age, parity, body mass index or country of origin, [START_REF] Rio | Caesarean section rates in immigrant and native women in Spain: the importance of geographical origin and type of hospital for delivery[END_REF][START_REF] Poobalan | Obesity as an independent risk factor for elective and emergency caesarean delivery in nulliparous women-systematic review and meta-analysis of cohort studies[END_REF][START_REF] Parrish | Effect of changes in maternal age, parity, and birth weight distribution on primary cesarean delivery rates[END_REF] which we were unable to take into consideration in this study. Even so, studies that have analyzed underlying differences in women's risk profiles found that significant variations in cesarean use between countries or regions of a same country remained after risk adjustment [START_REF] Luthy | Physician contribution to a cesarean delivery risk model[END_REF][START_REF] Baicker | Geographic variation in the appropriate use of cesarean delivery[END_REF][START_REF] Zeitlin | Variability in caesarean section rates for very preterm births at 28-31 weeks of gestation in 10 European regions: results of the MOSAIC project[END_REF]. Our results support the conclusions of these studies as variations in cesareans for singleton births at 40 weeks of gestationwhen mortality is lowestwere at least as wide in both absolute and relative terms as that for higher-risk groups such as twins or preterm births.

Variations in delivery practices have been hypothesized to reflect population-level differences in cultural values, legal liability, specific perinatal health care characteristics and women's preferences [START_REF] Katikireddi | A comparison of trends in caesarean section rates in former communist (transition) countries and other European countries[END_REF][START_REF] Baicker | Geographic variation in the appropriate use of cesarean delivery[END_REF][START_REF] Habiba | Caesarean section on request: a comparison of obstetricians' attitudes in eight European countries[END_REF][START_REF] Mcpherson | International variations in a selected number of surgical procedures[END_REF][START_REF]ACOG committee opinion no. 559: Cesarean delivery on maternal request[END_REF]. Countries' structural health factors including level of subsidy for cesarean, delivery settings, and resources may also influence cesarean rates [START_REF] Zeitlin | Variability in caesarean section rates for very preterm births at 28-31 weeks of gestation in 10 European regions: results of the MOSAIC project[END_REF][START_REF] Stafford | The impact of nonclinical factors on repeat cesarean section[END_REF][START_REF] Coonrod | Nulliparous term singleton vertex cesarean delivery rates: institutional and individual level predictors[END_REF][START_REF] Blondel | Organisation of obstetric services for very preterm births in Europe: results from the MOSAIC project[END_REF][START_REF] Coulm | Obstetric interventions for low-risk pregnant women in France: do maternity unit characteristics make a difference[END_REF][START_REF] Roberts | Pathways to a rising caesarean section rate: a population-based cohort study[END_REF][START_REF] Reime | Do maternity care provider groups have different attitudes towards birth?[END_REF]. The importance of common country-level factors is supported by the strong correlations we found between overall and sub-group rates.

For instance, Nordic countries such as Finland, Sweden and Norway tended to have lower rates than other countries such as Germany, Austria, and the United States. Among singletons, cesarean delivery rates mapped onto a general pattern which mirrors risks of adverse birth outcomes by gestational age. This pattern translates into a significant decline in cesarean rates with increasing gestational age until 40 weeks followed by a rise at 41 and 42 weeks. Additionally, in many countries, we observed an increase at 38 versus 37 weeks, most probably explained by a rise in elective cesarean delivery. In the event of a complicated pregnancy, near term extraction may be judged to confer less risk for the fetus than the benefits of a longer duration of gestation [START_REF] Alfirevic | Caesarean section versus vaginal delivery for preterm birth in singletons[END_REF]. Interestingly, this rise occurred everywhere at 38 weeks and not at 37 weeksalthough a birth at 37 weeks is not considered pretermand not at 39 weeks, although many professional societies have issued recommendations advising against elective delivery before 39 weeks [START_REF] Salim | Health implications resulting from the timing of elective cesarean delivery[END_REF].

Our results show that reporting cesarean rates by gestational age and multiplicity provides useful additional information for countries seeking to understand and compare cesarean practices. For example, whereas cesarean rates were generally low in Sweden and Finland, very preterm rates were high. On the other hand, the United States displayed the highest overall cesarean rate but very preterm rates were lower than in many other countries. In general, there was no correlation between rates at term and at very preterm gestations, suggesting that interpretation of research on the benefits of cesarean for very preterm delivery may be independent from general attitudes towards cesarean. In fact, the benefits of cesarean delivery for very preterm babies in the absence of other obstetric indications are debated [START_REF] Kollee | Obstetric interventions for babies born before 28 weeks of gestation in Europe: results of the MOSAIC study[END_REF][START_REF] Hogberg | Infant mortality of very preterm infants by mode of delivery, institutional policies and maternal diagnosis[END_REF]. Moreover, different ethical decisions related to active treatment for extremely preterm births might enhance heterogeneity between countries in obstetricians' decisions to choose cesarean delivery [START_REF] Chervenak | The professional responsibility model of obstetric ethics and caesarean delivery[END_REF][START_REF] De Leeuw | Treatment choices for extremely preterm infants: an international perspective[END_REF].

Similarly, there were differences in delivery practices for multiples. Thus, for instance, two countries with low overall cesarean rates, The Netherlands and Sweden, had very different rates for multiples. In addition, whereas Austria, Estonia and The Netherlands displayed similar cesarean rates for multiples throughout all weeks of gestation, Finland, Sweden and Norway had declining rates with increasing gestational age. The use of systematic cesarean sections for preterm and term twin births is also an area where scientific evidence has been largely debated [START_REF] Hogle | Cesarean delivery for twins: a systematic review and meta-analysis[END_REF]. In a 2013 randomized control trial, researchers showed that planned cesarean delivery did not increase or decrease the occurrence of adverse birth outcomes compared to vaginal delivery for twin pregnancies [START_REF] Barrett | A randomized trial of planned cesarean or vaginal delivery for twin pregnancy[END_REF]. Stratified data by gestational age and multiplicity can be used to refine policies aiming to reduce unnecessary cesareans, or to evaluate uptake of national or state policies targeting certain risk groups. In the United States, for example, financial quality incentives have been implemented to discourage elective delivery before 39 weeks in hospitals and Medicaid coverage for near term elective deliveries has been phased out in states like New York, South Carolina and New Mexico [START_REF]Reducing early elective deliveries in Medicaid and CHIP[END_REF]. Moreover, presenting cesarean rates by subgroups can be useful to assess changes in practices resulting from the publication of new studies and trials on best mode of delivery.

Our study compared cesarean delivery rates by risk groups using data from routine population-based registers in 18 countries and very few data items were missing for variables included in our analyses. The use of subgroup data from routine population registers represents a strength over other international studies based on overall cesarean rates only or institution-level data. We asked for data on gestational age in completed weeks based on the best clinical estimate, but did not have further information on how this estimate was derived. Although mode of onset and previous mode of delivery may also contribute to overall cesarean patterns, these data were unavailable. Our study covered only one year and thus our estimates are less precise in smaller European countries, especially at the extremes of the gestational age distribution.

Conclusions

The presentation of cesarean rates by gestational age in cross-national comparisons is feasible and these data were readily available in a large number of countries. Overall cesarean rates provide a valuable summary measure of practice given the strong correlations between subgroups, with the exception of very preterm births. However, cesarean data presented by gestational age and multiplicity could make it possible to design more targeted cesarean reduction policies, and to assess changes in practice for subgroups at higher risk of cesarean delivery namely preterm, near term deliveries and multiple births.

Figure 1

 1 Figure 1 Cesarean rates for singleton births overall and by gestational age at delivery in 2008. Legend: ----------------overall cesarean rate cesarean rate by GA in completed weeks.

Table 1

 1 Cesarean delivery rates by plurality in 2008

			Overall		Singletons		Multiples
	Country/Region	Live births (N)	Rate %	95% CI	Live births (N)	Rate %	95% CI	Live births (N)	Rate %	95% CI
	Austria	77 728	28.0	27.7-28.3	75 072	26.0	25.7-26.4	2656	83.4	81.9-84.8
	Belgium: Flanders	69 187	20.2	20.0-20.5	66 672	18.8	18.5-19.1	2515	56.7	54.7-58.6
	Czech Republic	119 455	22.0	21.7-22.2	114 722	19.7	19.5-20.0	4733	75.4	74.1-76.6
	Estonia	16 032	21.2	20.5-21.8	15507	19.3	18.7-19.9	525	77.5	73.7-81.0
	Finland	59 594	17.2	16.9-17.5	57 872	16.2	15.9-16.5	1722	51.9	49.5-54.2
	France 1	14 594	21.1	20.4-21.7	14 161	20.1	19.4-20.7	433	54.3	49.4-59.0
	Germany 2	211 889	32.5	32.3-32.7	204 746	31.0	30.8-31.2	7143	74.6	73.6-75.6
	Ireland	75 249	26.7	26.3-27.0	72 592	25.3	25.0-25.6	2657	64.0	62.1-65.8
	Lithuania	31 287	23.9	23.4-24.4	30 510	22.9	22.4-23.3	777	63.3	59.8-66.7
	Malta 3	4152	30.2	28.8-31.6	4020	28.1	26.7-29.5	132	95.4	90.4-98.3
	The Netherlands	174 828	15.7	15.5-15.6	168 619	14.8	14.6-15.0	6209	39.6	38.4-40.8
	Norway	61 206	17.2	17.0-17.5	59 075	16.2	15.9-16.5	2131	46.7	44.6-48.9
	Slovakia	53 971	25.7	25.3-26.0	52 520	24.2	23.9-24.6	1451	77.5	75.2-79.6
	Slovenia	21 806	16.9	16.4-17.4	21 041	15.5	15.0-16.0	765	54.9	51.3-58.5
	Spain 4	398 922	27.4	27.3-27.5	382 141	25.5	25.3-25.6	16 781	71.3	70.6-72.0
	Sweden 3	108 923	18.0	17.8-18.3	105 855	17.0	16.7-17.2	3068	54.6	52.8-56.3
	UK: Scotland	58 302	26.1	25.8-26.5	56 450	24.9	24.5-25.2	1852	64.7	62.5-66.9
	USA	4236158	32.4	32.3-32.4	409 2381	30.8	30.8-30.9	143777	76.4	76.2-76.6

1 Data from France came from a nationally representative survey of all births in 2010. 2 Data from Germany are from 3 Landers. 3 Data from Malta and Sweden are from 2009.

Table 2

 2 Cesarean delivery rates by plurality and gestational age subgroups in 2008

	Singletons

3

Data from Malta and Sweden are from 2009.

  Cesarean rate differences by gestational age groups for singletons in 2008. A. Rate differences between very preterm (26-31 weeks GA) and term (39-41 weeks GA). B. Rate differences between near term (37-38 weeks GA) and term (39-41 weeks of GA).

	A	Country		RD (95% CI) RD (95% CI)
		Lithuania		0.14 (0.08, 0.21) 0.14 (0.08, 0.21)
		Spain		0.29 (0.27, 0.31) 0.29 (0.27, 0.31)
		USA		0.32 (0.31, 0.32) 0.32 (0.31, 0.32)
		Estonia		0.33 (0.23, 0.43) 0.33 (0.23, 0.43)
		Slovenia		0.33 (0.24, 0.43) 0.33 (0.24, 0.43)
		UK: Scotland		0.36 (0.31, 0.40) 0.36 (0.31, 0.40)
		Slovakia		0.42 (0.37, 0.46) 0.42 (0.37, 0.46)
		Malta		0.42 (0.25, 0.60) 0.42 (0.25, 0.60)
		The Netherlands		0.43 (0.40, 0.46) 0.43 (0.40, 0.46)
		Czech Republic		0.45 (0.41, 0.48) 0.45 (0.41, 0.48)
		Ireland		0.45 (0.40, 0.49) 0.45 (0.40, 0.49)
		Sweden		0.51 (0.47, 0.55) 0.51 (0.47, 0.55)
		Belgium: Flanders		0.51 (0.46, 0.56) 0.51 (0.46, 0.56)
		France		0.53 (0.43, 0.64) 0.53 (0.43, 0.64)
		Finland		0.55 (0.49, 0.61) 0.55 (0.49, 0.61)
		Austria		0.58 (0.54, 0.62) 0.58 (0.54, 0.62)
		Norway		0.58 (0.54, 0.63) 0.58 (0.54, 0.63)
		Germany: 3 Länder		0.61 (0.59, 0.63) 0.61 (0.59, 0.63)
		0 -.1 0	.3	.7
	B	Country		RD (95% CI) RD (95% CI)
		Lithuania		-0.00 (-0.02, 0.01) -0.00 (-0.02, 0.01)
		USA		0.06 (0.06, 0.06) 0.06 (0.06, 0.06)
		Slovenia		0.06 (0.04, 0.07) 0.06 (0.04, 0.07)
		Finland		0.07 (0.06, 0.08) 0.07 (0.06, 0.08)
		Czech Republic		0.09 (0.08, 0.10) 0.09 (0.08, 0.10)
		Slovakia		0.09 (0.08, 0.10) 0.09 (0.08, 0.10)
		Spain		0.09 (0.09, 0.10) 0.09 (0.09, 0.10)
		France		0.09 (0.08, 0.11) 0.09 (0.08, 0.11)
		The Netherlands		0.11 (0.11, 0.11) 0.11 (0.11, 0.11)
		UK: Scotland		0.12 (0.11, 0.13) 0.12 (0.11, 0.13)
		Estonia		0.13 (0.11, 0.14) 0.13 (0.11, 0.14)
		Ireland		0.14 (0.13, 0.15) 0.14 (0.13, 0.15)
		Norway		0.15 (0.14, 0.16) 0.15 (0.14, 0.16)
		Belgium: Flanders		0.15 (0.15, 0.16) 0.15 (0.15, 0.16)
		Sweden		0.20 (0.20, 0.21) 0.20 (0.20, 0.21)
		Germany: 3 Länder		0.24 (0.24, 0.25) 0.24 (0.24, 0.25)
		Malta		0.25 (0.21, 0.28) 0.25 (0.21, 0.28)
		Austria		0.25 (0.24, 0.26) 0.25 (0.24, 0.26)
		0 -.1 0	.3	.7
	Figure 2

Table 3

 3 Spearman correlation coefficients for cesarean rates by plurality and gestational age subgroups in 2008Notes: Asterisks indicate p < 0.05. Gestational age (GA) in completed weeks (wks).

	Subgroups	Overall ≥22 wks	26-31 wks	32-36 wks	Singletons 37-38 wks	39-41 wks	≥42 wks	<34 wks	Multiples 34-36 wks	≥37 wks
	Singletons
	26-31 GA	0.16	1
	32-36 GA	0.86*	0.45	1
	37-38 GA	0.83*	0.39	0.89*	1
	39-41 GA	0.86*	-0.05	0.60*	0.51*	1
	≥42 GA	0.81*	-0.00	0.70*	0.63*	0.76*	1
	Multiples
	<34 GA	0.80*	0.24	0.67*	0.64*	0.59*	0.58*	1
	34-36 GA	0.80*	-0.03	0.73*	0.62*	0.59*	0.60*	0.76*	1
	≥37 GA	0.77*	-0.10	0.70*	0.57*	0.58*	0.57*	0.71*	0.98*	1

Data from Spain are from 2010.

Acknowledgements

Data for this study were provided by the Euro-Peristat Preterm Study Group: Gerald Haidinger (Austria), Evelyne Martens (Flanders), Peter Velebil (Czech Republic), Luule Sakkeus (Estonia), Mika Gissler (Finland), Béatrice Blondel (France), Nicholas Lack, Björn Misselwitz, Paul Wenzlaff (Germany), Sheelagh Bonham (Ireland), Jone Jaselioniene (Lithuania), Miriam Gatt (Malta), Ashna Mohangoo, Jan Nijhuis (The Netherlands), Kari Klungsøyr (Norway), Katarzyna Szamotulska (Poland), Henrique Barros (Portugal), Ziva Novak (Slovenia), Francisco Bolumar (Spain), Karin Gottvall (Sweden), James Chalmers (UK, Scotland). This analysis was partially funded by a grant to the Euro-Peristat project from the European Commission (2010 13 01). The funding agency was not involved in the study. This study used aggregated national indicators, ethical approval was not required.

Abbreviations GA: Gestational age; Wks: Weeks.

Competing interests

The authors declare that they have no competing interests.

Authors' contributions MD, JZ, and BB conceived the study, MD and ND carried out statistical analysis, KK, FB, ADM, MG, KS, NL, JN, PV, LS, and JC contributed to the interpretation of the results and revised successive versions of the article. Members of the Euro-Peristat preterm birth group were responsible for the provision, accuracy, and interpretation of data in their country: they commented on initial and final versions of the article. All authors approved the final article. 4 Department of Public Health Sciences, University of Alcalá, Madrid, Spain. 5 Department Child Health, TNO Netherlands Organization for Applied Scientific Research, Leiden, The Netherlands. 6 Department of Information, THL National Institute for Health and Welfare, Helsinki, Finland. 7 NHV Nordic School of Public Health, Gothenburg, Sweden. 8 Department of Epidemiology, National Research Institute of Mother and Child, Warsaw, Poland. 9 BAQ, Bavarian Institute for


G Goavec-Mérou

N Chrétien

J.-M Friedt
email: jmfriedt@femto-st.fr

P Sandoz

G Martin

M Lenczner

S Ballandras

Fast contactless vibrating structure characterization using real time FPGA-based digital signal processing: demonstrations with a passive wireless acoustic delay line probe and vision

Keywords: numbers: 84.40.-x, 43.20.Ye, 43.38.Rh strain gauge, wireless, battery-less, surface acoustic wave, RADAR, signal processing, image processing

Vibrating mechanical structure characterization is demonstrated using contact-less techniques best suited for mobile and rotating equipments. Fast measurement rates are achieved using Field Programmable Gate Array (FPGA) devices as real-time digital signal processors. Two kinds of algorithms are implemented on FPGA and experimentally validated in the case of the vibrating tuning fork. A first application concerns in-plane displacement detection by vision with sampling rates above 10 kHz, thus reaching frequency ranges above the audio range. A second demonstration concerns pulsed-RADAR cooperative target phase detection and is applied to radiofrequency acoustic transducers used as passive wireless strain gauges. In this case, the 250 ksamples/s refresh rate achieved is only limited by the acoustic sensor design but not by the detection bandwidth. These realizations illustrate the efficiency, interest and potentialities of FPGA-based real-time digital signal processing for the contact-less interrogation of passive embedded probes with high refresh rates.

I. INTRODUCTION

Vibrating structure characterization is classically performed to assess mechanical equipment characteristics or as part of preventive maintenance. On rotating or moving parts, a wired solution is not robust for long term monitoring, with sliding contact or rotating optical couplers 1 being demonstrated in the literature for laboratory experiments. One alternative to structure bending and vibration analysis -whether applied to macroscopic 2,3 or microscopic structures 4,5 -is image processing 6 in which the structure deformation and motion is detected either through motion of its natural surface properties (roughness yielding varying reflected light intensity) or artificially patterning the surface with a structure best suited to motion detection 7 . Embedding strain gauges on the mobile part with wireless communication yet on-board energy source 8 reduces the life expectancy of the measurement setup to the battery life of the powered sensor and hence requires periodic maintenance 9 : the broad field of wireless sensor networks (WSN) is concerned with such an approach [10][START_REF] Kim | 6th International Symposium on Information Processing in Sensor Networks(IPSN)[END_REF][START_REF] Ni | [END_REF][START_REF] Hashemian | [END_REF] . Only very few studies are concerned with embedding passive sensors in the vibrating structure and using variations of the reflected signal when illuminated by RADAR signals [START_REF] Gentile | Radar Technology[END_REF] -either through a RADAR Cross-Section (RCS) variation or resonance frequency variation [START_REF] Jeon | [END_REF][16][17] -to measure strain in mechanical structures.

We here consider two contact-less vibrating structure characterization means: digital image processing and using piezoelectric acoustic delay lines acting as passive strain gauges probed through a wireless link. In the context of control, real time analysis of the recorded data is mandatory and postprocessing of large records is not an option. Furthermore, in the context of vibrational analysis, measurement speed is a significant indicator of sensor performance in order to extend to the highest vibration modes the analysis of the mechanical structure under investigation. As such, a real time fast measurement technique, as opposed to as stroboscopic approach yielding high equivalent sampling rate but assuming a periodic signal, allows for the detection of transient events with short durations 18 . While the envisioned algorithms are basic and have been described previously, the implementation on hardware compatible with real time feedback control is challenging, and is the topic of this paper.

Real time signal processing has long been devoted to analog processing, with the utmost bandwidth being achieved by piezoelectric and optical physical signal processing including convolution, correlation and Fourier transform [START_REF] Morgan | Surface acoustic wave filters[END_REF] . The advent of digital signal processing yielded long term stability (no dependence of the analog component value to environmental conditions) and reconfigurability [START_REF] Mindell | Digital Apollo: Human and Machine in Spaceflight[END_REF] has become the mainstream implementation strategy, with the loss of real time signal processing when software and operating systems are involved. Field programmable gate arrays (FPGA) provide the means of achieving the best of both worlds, with digital signal processing running at radiofrequency rates. One current application of such an approach is concerned with Software Defined Radio aiming at sampling the baseband radiofrequency signal and performing all signal processing as software defined algorithms. In these demonstrations, we implement one example of digital image processing for motion detection with frame rates reaching the ksample rate (i.e. many thousand images recorded and processed every second) and a pulsed mode RADAR operating in the 2.45 GHz aimed at probing a surface acoustic transducer acting as a passive, wireless strain sensor. The latter measurement technique is the result of a lengthy evolution 21 in which a piezoelectric substrate is used to convert an incoming electromagnetic pulse to a mechanical (acoustic) wave: the environmental (temperature, strain) effect on the acoustic wave velocity modifies the time of flight of the acoustic pulse which, once converted back to an electromagnetic wave, is recorded by the RADAR receiver 22 . A measurement of the time of flight allows for the recovery of the acoustic velocity and hence the physical property most significantly affecting this velocity as selected by proper sensor design (minimizing unwanted physical property effect while emphasizing the dependence with the considered quantity).

While most RADAR approaches have been developed with power consumption or computational limitations considerations, with the Frequency Modulated Continuous Wave approach (FMCW) 23 being the most basic implementation, high bandwidth measurements aimed at matching at best the performance of the sensor have not been presented in the literature.

Our pulsed RADAR setup aims at using at best digital signal processing implemented on FPGA for performing a wireless measurement technique of probing acoustic delay lines, only limited in measurement bandwidth by the acoustic propagation time (physical measurement principle), and not by the means of measuring the signal. Pulse mode interrogation of wireless acoustic sensors has been demonstrated previously in academic settings [START_REF] Schmidt | Ultrasonics symposium[END_REF][START_REF] Scherr | IEEE Ultrason. Symp[END_REF] but is still not used in commercial embodiments of such interrogation units applied to acoustic delay lines [START_REF] Hütter | Wireless Surface Acoustic Wave Sensor Symposium[END_REF] . While passive target motion has been detected wirelessly with high bandwidth by using Doppler RADAR [START_REF] Pieraccini | [END_REF]28 , actual remote strain measurement has not been achieved by such means with the use of a dedicated transducer attached to the probed structure.

Torque measurement using acoustic transducers acting as passive strain gauges attached to the structure under investigation has been investigated in-depth by Transense [START_REF] Beckley | IEEE International Frequency Control Symposium and PDA Exhibition[END_REF][START_REF] Kalinin | Proc. IEEE IFCS[END_REF] and in [START_REF] Stoney | [END_REF] .

II. VISUAL DETECTION OF IN-PLANE DISPLACEMENTS

A. Measurement principle

Optical measurement of vibrating structure is the most intuitive approach to contactless displacement detection. Various out of plane displacement measurement methods range from speckle interferometry detection to holographic displacement detection, while in-plane displacement is tackled by imaging 32 and photogrametry. The latter approach will be of interest to us here, with the specific case of a quantitative displacement analysis by using a periodic pattern attached to the vibrating structure 33,34 : the known periodicity yields a quantitative in-plane displacement measurement.

In-plane detection is usually performed as off-line processing due to the computationaly intensive processing requirements of correlation methods 7 . Furthermore, reaching sub-pixel resolution additionally requires some curve fitting or interpolation. We here first describe the basics of applying a cross-correlation method for detecting the motion of a patterned target (Fig. 1) fixed to the moving prong of a vibrating tuning fork 33 . We then demonstrate the implementation of this algorithm in a FPGA for real time computation of the target position.

For a given magnification factor, the analysis function is defined by the length L of the sample (image length in pixels) and the period p of the pattern (in pixels). We furthermore window the analysis function by a Gaussian envelope in order to generate the analysis function:

analysis = exp - x 4p √ 2 2 × exp i × x 2π p + ϕ 0
with a pixel range x = [1..L] -x with L the line length being analyzed and x the midpoint of this line. In our case, L = 1024, p = 32 and x = 512. ϕ 0 is a constant selected to avoid 2π rotation when computing the phase ϕ of the cross-correlation which is performed in the spatial domain as ϕ = arg Σ k= [1..L] measurement k × analysis k . This phase is representative of the spatial position of the pattern, with a 2π phase jumps associated with a one-period displacement of the reference grid. Hence, this analysis is quantitative with p only given by the optical setup magnification factor and the actual grid periodicity. Typical resolutions achieved by phase analysis and interpolating the unwrapped phase are in the range of 10 -2 pixel or 10 -3 grid period 33 , i.e. providing 10 nm positioning resolution when using visible illumination and low magnification lenses (×10 or ×20). This sub-pixel resolution is achieved by the integral (sum) over mulitple periods of the convolution of the image with the analysis function which has been tuned to match the period for a known pattern observed at a known magnification: the longer the observed line length (parameter L), the better the achievable resolution, for a given camera resolution and a given illumination condition (i.e. given signal to noise ratio of the recorded pixels).

B. Real time processing: FPGA implementation and experimental results

A Spartan 6 FPGA shares common data, address and control busses with an i.MX27

CPU as provided on a custom design based on the Armadeus Systems (Mulhouse, France) APF27 board. The FPGA is in charge of real time processing of the data as the pixels flow in through the serializer-deserializer interface, while the CPU is only used for post-processing the resulting data, namely computing the ratio of the imaginary to the real part of the complex output of the algorithm, computing the arctan of the result to reach an angle, and transfering the data to the user through an ethernet link.

[FIG. 1

about here.]

The imaging source is a PhotonFocus MVD1024E160 camera communicating through a CameraLink interface. Thus, the data source is the LVDS interface in charge of feeding the processing chain with 8-bit wide pixel values transferred as pairs at every clock cycle.

Considering that the CameraLink digital interface clock speed is 80 MHz and 2 bytes are provided at every clock cycle, the use of a 12-pixel wide (for averaging and thus noise reduction) and 1024-pixel long region of interest out of the available 1024×1024 pixel size yields a data transmission duration of 76.8 µs per frame. This camera hardware transfer time, in addition to the exposure duration, limits the frame rate. In addition, each line ends with a 100 ns delay, while before each frame the exposure time is set to the minimum value thanks to a bright LED light source, namely 10 µs. The practical frame rate meets the theoretical threshold of 5285 frame/s (Fig. 2), only limited by the camera data transfer rate since all processing operations are performed along the pipeline described in Fig. 1. This implementation is able to receive two consecutive pixels at each clock cycle and produce one phase output value one cycle after the end of image transmission. However, this analysis By rotating the camera by an angle of 90 o , the latencies due to the end of lines are reduced to the number of pixels parallel to each periodic pattern line. In our case, by averaging over 12 pixel wide lines, the 102.4 µs delay is reduced to 1.2 µs. Both approaches yield similar implementations and since phase computation derived from pixel averages and phase averaging computed on each column are linear operations, the two approaches yield results with the same resolution. However, with the latter approach, the frame rate increases up to 11360 fps. Removing the averaging along the 12-pixel wide line still increases the update rate up to 60610 fps, at the cost of reduced signal to noise ratio. In the latter case, the exposure duration of 10 µs accounts for more than half of the frame acquisition duration and becomes the limiting factor defining the acquisition rate. The camera orientation allowing for maximum bandwidth is used to generate Fig. 3.

The implementation of the signal processing block aims at using at best the parallel processing capability of a FPGA (Fig. Since the hardware multiplier and accumulator are optimized to operate on 18 bit values, the analysis function is scaled by 32765 to fit in a 16 bit signed integer value.

In order to avoid any limitation associated with digital data transfer to a general purpose CPU, the sink of the data is a digital to analog (D/A) converter output providing voltages proportional to the I and Q components of the cross correlation. Fig. 2 exhibit oscilloscope outputs of these computations as a music tuning fork is vibrated at 440 Hz through a magnetic excitation by a voice coil 32 . The digital oscilloscope traces are here recorded and the phase extraction is performed as post-processing, avoiding the need to implement online the arctan() lookup table function. In case of the use of such information for feedback control, the 2π phase jump induced by the displacement of the target by more than one period must additionally be handled to avoid the discontinuity as observed on Figs. 2 and4, right.

[FIG. 2

about here.]

Having achieved periodic oscillation condition observation of the tuning fork prong motion at a frequency known to be close to one of the eigenmodes of the device, we consider the impulse response of the tuning fork as observed by means of digital image processing (Fig. 3).

In this case, two modes are well identified, the fundamental mode around 440 Hz and one overtone close to 10 kHz.

[FIG. 3

about here.]

We have previously described the implementation of a fast measurement algorithm identifying the limitation on the measurement bandwidth when using an acoustic resonator transducer acting as a strain gauge 35 : a practical implementation reaching this refresh rate has been demonstrated by another group in [START_REF] Kalinin | European Freq. & Time Forum[END_REF] . Comparison of the resonator based measurement and the optical measurement, both of which are performed at 7 ksamples/s, is provided in Fig. 4. The quantitative analysis of the resonator based strain gauge measurement to generate an estimate of the prong displacement has been provided previously 35 , while the optical periodic pattern image processing analysis directly provides a quantitative estimate of the prong end displacement. Fig. 4 exhibits on the one hand good agreement between the two measurement methods performed simultaneously, and emphasizes the limitation of the optical measurement when large displacements are considered, yielding 2π phase rotations in the analysis (Fig. 4, right). For large displacements, the quartz strain gauge seems to overestimate the displacement, as has also been observed independently in quasi-static conditions, possibly due to the assumption on the tuning fork prong stress transfer to the piezoelectric substrate not being valid at high stress values. Indeed, in this test, the goldcoated backside of the acoustic resonator substrate was soldered using SnPb to the tuning fork, possibly inducing creep at high strain rates.

[FIG. 4 However, typical delay line tags exhibit a maximum echo delay of 4 µs, or a 1.6 cm two-way acoustic trip assuming a 4000 m/s acoustic velocity on lithium niobate, and should thus be compatible with a 250 kHz refresh rate. Indeed, in the most basic embodiment of a RADAR interrogation unit, a short (as will be defined later) radiofrequency pulse is generated, echoes are recorded up to the maximum duration of the acoustic delay, and this process is repeated.

The repetition rate is thus only limited by the acoustic delay line length which defines the maximum delay. In this example, the 8-bit tag is assumed to reflect all incoming energy within the 4 µs after excitation by the incoming pulse, and we will demonstrate the practical implementation of a sampling strategy providing the maximum possible refresh rate.

[FIG. 5

B. FPGA implementation and results

The major implementation challenge of a fast reader is in recording the returned echoes, extracting the relevant information representative of the acoustic velocity, and repeating the process fast enough to provide the user with a fast refresh rate. Recent availability of fast analog to digital converters and arrays of reconfigurable logic gates as provided in FPGAs provides the basic tools for implementing the considered measurement technique (Fig. 6). The FPGA controls all interrogation sequences: an internal counter triggers the switch for generating a 40 ns pulse, the dataflow from the 1 GS/s analog to digital converter (ADC, Hittite HMCAD1511) is fed to a serializer-deserializer stage as available in the Xilinx Spartan6 FPGA, and either the full time domain record of 2048 samples is transferred to the iMX27 CPU for processing and display in order to select the echo delay, or the sample of interest is stored in RAM for further processing. This acquisition and processing strategy is reminiscent of the previous camera-based data acquisition approach. Notice that 2048 samples recorded at 500 Msamples/s provide a record lasting 5 µs or the total time delay of the acoustic transducer, while such a dataset fits within the RAMs internal to the FPGA.

In the example we are demonstrating here, 2048 successive samples of the interrogation sequence are stored in RAM for display to the user as a delay line acting as a strain gauge glued on a 440 Hz musical tuning fork is probed. At 200 kHz refresh rate, 2048 records of the I and Q components last 10.24 ms or about 4-periods of the vibrating tuning fork.

This result is demonstrated in Fig. 7 in which a SAW delay line is glued on one prong of a tuning fork 38 using Epotek H760E epoxy glue, hence illustrating the full speed measurement capability only limited by the delay line length. Faster refresh rates are accessible with shorter delay lines: since here an 8-bit tag is used the length is defined by the number of echoes, but sensing capability only does not require such long delay lines and shorter designs (with maximum echo delays of 500 ns for example) with only two reflectors as needed for a single physical quantity monitoring will provide higher refresh rates.

[FIG. 6

about here.]

Signal processing for physical quantity extraction is based on the classical phase detection of the echoes: since the two way trip of the acoustic wave along a path of length 2d is 2d/c, the phase shift ϕ with respect to the local oscillator (as obtained through the I/Q demodulator during the mixing process) is

ϕ = 2π • 2d • f /c
where f is the probed pulse central frequency. Getting rid of any drift of the local oscillator (f ) as well as of the distance between the reader and the sensor in case of mobile target applications requires a differential approach in which the phase difference between two echoes is processed. This computation is challenging since the phase extraction from the I and Q components requires obtaining the arctan function, even though a feedback control aimed at keeping a constant strain with a quadrature condition only requires to keep the Q component close to 0 and the I component maximized, with no need to actually compute the associated phase. In the previous example, I and Q data were considered sufficient to provide an estimate of the acoustic velocity c since over the short time scales of a few tens of milliseconds, the local oscillator drift is neglected.

[FIG. 7 about here.]

The need for high bandwidth measurement of vibrating structure mode identification is emphasized in Fig. 8 which exhibits the spectral analysis of the impulse response of the tuning fork fitted with the acoustic delay line strain gauge. In such a case, beyond the periodic analysis of the forced regime response of the tuning fork at its fundamental resonance frequency, high frequency modes are excited as well during the impulse response generation. As seen in the spectral analysis (Fig. 8, main graph) of the time-domain record (Fig. 8, insets), higher order modes 38 are well observed up to 40700 kHz, at frequencies consistent with a tuning-fork emitted sound record sampled using a personal computer sound-card acquiring at a rate of 96 kS/s. Identifying the prong motion characteristics of each one of these modes with respect to tabulated overtones is challenging due to the asymmetry between the two prongs introduced by the instrumentation process (adding the delay line transducer on the base of one prong and the target at the end of the other prong) which moves the resonance modes away from the expected frequencies modeled for a symetric tuning fork 38 . Nevertheless, a comparison of Figs. 8 and3 illustrates consistency within the common frequency range (fundamental mode around 440 Hz and overtone around 10 kHz)

and an inconsistency for a mode located around 5.5 kHz which is observed by the SAW strain gauge (and the sound card records, not shown) but not by the optical methods. We attribute this mode to a non-flexural mode of the tuning fork, which is detected by acoustic means since the SAW strain gauge is sensitivite to multiple components of the strain field, while the image processing method is only sensitive to the displacement component orthogonal to the patterned lines. Hence, any displacement along the lines cannot be detected by the image processing method.

C. Strain measurement resolution assessment

A significant hindrance of pulsed mode RADAR operating at temperature T is the wide integration bandwidth B which brings thermal noise k B • T • B on the low noise amplifier receiver stage, with k B = 1.38 • 10 -23 J.K -1 the Boltzman constant, over a much wider frequency range than FMCW or narrowband resonator based measurement techniques. One might wonder then whether the gain in refresh rate was achieved at the loss of resolution. In both FMCW and pulsed mode RADAR, the phase of the signal within the returned echoes is extracted by mixing with the local oscillator which has drifted over the propagation delay by a quantity σ ϕ defined as the phase noise of the local oscillator whose spectral density over a bandwidth BW is defined as S ϕ = σ 2 ϕ /BW rad 2 /Hz.

Two conditions are considered: since the phase noise of a broadband amplifier (in our case the low noise receiver amplifier -LNA) rises with the received power, at close range the mixer output signal phase noise is defined by the local oscillator phase noise, while at longer range the received power is low enough that the LNA output phase noise drives the phase detection capability. In the latter case in which the receiver LNA overwhelms the local oscillator phase noise, a power P received by an antenna connected to an amplifier at temperture T and characterized by a noise factor F induces a white phase noise floor of

S ϕ = k B • T • F/P .
Since no signal extraction is performed (e.g. using a cross correlation technique) to extract the returned signal from the noise, the received power is necessarily above the thermal noise

= 2π • 2d • f • dc/c 2 = 2π • τ • dc/c
where τ = 2d/c is the echo delay, so by normalizing with τ expressed in µs, dϕ/τ = 0.0656 rad.MPa -1 .µs -1 and the measurement resolution at the detection limit of the returned signal is 1/0.0656=15 MPa for an echo delayed by 1 µs. In the case of our experiment in which +18.5 dBm were emitted and 43 dB loss were exhibited by the acoustic delay line, the returned power to the first low noise amplifier is -24.5 dBm, hence a phase resolution of 0.21 mrad and the stress resolution is improved to 0.21 • 10 -3 /0.0656 = 3.10 -3 MPa=3 kPa or much less than the expected hundreds of kPa induced by the tuning fork vibration 35 .

In the case of a significant power returned by the sensor, the LNA phase noise is not the driving factor on the phase detection uncertainty which is then driven by the local oscillator.

Assuming a flat, white, phase noise in the 100 kHz (refresh rate and hence lower boundary of the integral limits) to above 10 MHz (upper limit of the phase noise characterization) decade equal to a value b 0 , then the phase fluctuation due to the local oscillator is

σ ϕ = √ b 0 × BW .
This analysis is numerically applied to the case of an Agilent E5071B network analyzer whose phase noise is measured (using an Agilent E5052B signal source analyzer) to be around -100 dBc/Hz in the 100 kHz-1 MHz region when providing a continuous output signal at 2.475 GHz (0 Hz span). Under such considerations (signal to noise ratio being unity), the phase measurement resolution is σ ϕ = 0.06 rad and the stress resolution is 1 MPa, the poor performance being associated with a rather low quality source provided by the network analyzer. A better source exhibiting a noise floor of b 0 = -140 dBc/Hz would provide a resolution of 10 kPa.

It might be worth considering at which distance the transition from one regime to another occurs. The RADAR equation applied to a wireless acoustic sensor exhibiting accoustic loss IL -acting as an equivalent RADAR cross section when combined with the signal wavelength to account for the antenna aperture -states that the receiver power P R is related to the emitted power P E when a target is located at distance d from the reader by

P R = P E λ 4 IL • (4π) 4 • 1 d 4
with λ the incident pulse wavelength, so the transition distance occurs when the LNA phase noise k B T /P E reaches the local oscillator noise b 0 : b

0 = k B •T •d 4 •IL•(4π 4 ) P E λ 4 which is solved for a distance d 4 = P E •λ 4 b 0 k B •T •IL•(4π) 4 .
A numerical application for b 0 = -100 dBc/Hz and λ = 12.2 cm shows that the critical distance is d = 12 cm. A high quality oscillator exhibiting b 0 = -150 dBc/Hz reduces this range further to 7 cm, so the LNA will be the dominant issue in most practical applications. The maximum interrogation range is reached when the returned signal reaches the thermal noise, hence a distance d so that P R = k B • T • BW with again BW = 40 MHz the transducer bandwidth, or 50 cm when emitting P E = 10 mW (Fig. 9). This range extends to 88 cm when emitting 100 mW and 1.2 m if the insertion loss is reduced from 40 dB to 35 dB. All these considerations assuming isotropic antennae fitting the interrogation unit and the sensor: practical applications must account for antenna gain which, at such a frequency, typically reaches 8.5 dB for antennae with λ 2 area (e.g Huber-Suhner model SPA-2400, Switzerland) and additionnaly extends the interrogation range for dedicated setups (with the last numerical application, to 3.1 m). Considering the 1% duty cycle, compliance with outdoor emission regulations in the 2.45 GHz band (mean power of 10 dBm) allows for a peak power of P E = 1 W, yielding a 5.5 m expected measurement range, consistent with experimental demonstrations.

[FIG. 8 allows for the identification of the velocity of a mechanical wave in an acoustic delay line acting as a passive, wireless sensor configuration with a measurement speed only limited by the acoustic wave echo delay. We demonstrate a 250 kHz refresh rate, only limited by the 4 µs acoustic delay due to the 8 mm long acoustic path of the SAW tag used in this experiment, which could be increased further by using a dedicated SAW sensor design with smaller acoustic path length. Digital image processing for extracting pattern motion analysis using a cross-correlation technique was implemented digitally and achieved a refresh rate of 27 kHz. Both measurements performed on a same vibrating structure, a musical tuning fork, provided consistent vibrational mode analysis results. Real time performance yielding sub-10 µs latencies when generating the computation results can only be achieved by implementing the processing algorithms in Field Programmable Gate Array (FPGA), with latencies only limited by the physical measurement principle (image acquisition or acoustic delay line probing) and not the computation time.

  Fig. 1 (c) requires two different analysis coefficients since adjacent pixels are not located on the same line. This difference yields some inversion in the multiplier and accumulator pattern as seen by comparing Figs. 1 (b) and (c).

  1, right): since two adjacent pixels are transferred by the camera at each clock cycle, the averaging is first performed by the adder. The multiplication by the analysis function is performed on the result, which is then accumulated to compute the cross-correlation function along a line. The computation is performed in parallel on the real part (I-component) and the imaginary part (Q-component), with only the analysis function changing from cos(2πx/p) to sin(2πx/p).

  about here.]III. REAL TIME ACOUSTIC DELAY LINE PROBINGA. Pulsed-mode RADAR interrogation of an acoustic delay line Acoustic delay lines22 are one possible implementation of the broader range of acoustic sensors manufactured by patterning interdigitated electrodes on a single-crystal piezoelectric substrate. Various sensor designs -broadband delay lines, chirped delay lines, resonatorsexhibit various advantages and drawbacks, providing the freedom of selecting design rules depending on the targeted property. When considering vibrating structure monitoring, refresh rate appears as the most significant parameter, with interrogation range often a secondary issue. While a resonator configuration exhibits low loss and hence extended interrogation range in a wireless sensing application in which the resonance frequency is representative of the physical quantity under investigation and monitored through a radiofrequency link, the time constant for loading and unloading energy limits the update rate to a few kilohertz as was discussed earlier35 when operating in the 434 MHz range, while the faster expected refresh rate achievable when using resonators in the 2450 MHz range has not been demonstrated yet. On the other hand, an acoustic delay line converts an incoming electromagnetic pulse to an acoustic wave propagating on the single-crystal piezoelectric substrate at a velocity dependent on the physical property under investigation, and the two-way trip is recorded as a phase information of the reflection coefficient of the transducer probed through a wireless link[START_REF] Bulst | [END_REF] . The lack of energy confinement and the poor efficiency of electromechanical conversion define the acoustic delay line losses which are typically in the 35 to 40 dB range, hence reducing the interrogation range with respect to the resonator configuration (Fig5).

  about here.] The emitted pulse length is defined in the time domain by the separation of the echoes, which itself is typically set in order to comply with radiofrequency emission regulation in the frequency domain. Since the 2.45 GHz ISM band is 80 MHz wide, typical pulse widths of 25 ns will occupy a 40 MHz spectral width or half of the allocated band. In our experiments, 40 ns long pulses are generated as a tradeoff between radiofrequency switch toggling time constant (in our case, Minicircuits ZASWA-2-50DR+), maximizing transmitted energy, and separation of echoes. Ideally, the number of periods of the oscillating radiofrequency wave within the pulse should be equal to the number of interdigitated electrode pairs in the transducer. Although the FMCW approach has been classically selected for the low sampling rate of the receiver since audio-frequency beat signals between the emitted wave and the time-delayed echoes are generated, reaching the utmost bandwidth achievable by the transducer acoustic delay requires increasing the sweep rate, yielding increased beat frequency and hence increased sampling rate of the receiver analog to digital converter. Hence, whatever the considered architecture, high bandwidth signal processing necessarily requires high acquisition bandwidth, and we hence consider the pulsed mode RADAR approach as best suited to reach the maximum measurement speed since no additional processing (Fourier transform) is needed beyond recording the echoes received from the sensor.

  power integrated over the analyzed bandwidth: in the case of a 500 MS/s A/D converter recording the 80 MHz-wide signal, BW = 40 MHz and k B × T × BW = -98 dBm. In such a condition, the phase noise floor is S ϕ = -76 dB.rad 2 /Hz or a phase standard deviation σ ϕ = S ϕ • BW , of 1.0 rad. Since the relative stress sensitivity along the propagation axis of lithium niobale YXl/128 o cut we considered is 39 df /f = dc/c = -4.26 ppm/MPa, operating at 2450 MHz yields a phase to stress sensitivity dϕ

  about here.] [FIG. 9 about here.] IV. CONCLUSION In the context of contactless vibrating structure characterization, we demonstrate real time image or radiofrequency signal processing for measuring displacements up to several kHz sampling rates with real time processing compatible with feedback control. The implementation of the pulsed mode RADAR with 500 MHz sampling rate (2 ns sample period)

FIG. 1 .FIG. 2 .FIG. 3 .

 123 FIG. 1. (a) 800×480 subset of the 1024×1024 image of the periodic pattern used for tuning fork displacement measurement. During the calculation, a 1024-pixel long and 4 to 12-pixel wide horizontal swath of the image is selected with an offset maximizing image quality.

FIG. 4 .FIG. 5 .

 45 FIG.4. Comparison of the displacement measurement using the optical in-plane detection method (bottom curves), and the acoustic resonator acting as a strain gauge measurement (top curves) at a 7 kS/s sampling rate. Left: driving voltage of the voice coil is 2 V pp . Right: driving voltage of the voice coil is 10 V pp .

FIG. 6 .FIG. 7 .FIG. 8 .d = 4 P E •b 0 λ 4 k 4 P 2 distanceFIG. 9 .

 67844429 FIG.6. Schematic of the experimental setup for probing an acoustic delay line response at 250 ksamples/s.

ACKNOWLEDGEMENTS

Funding for this research was partly provided by the European Territorial Cooperation Programme INTERREG IV A France-Switzerland 2007-2013 and the OscillatorIMP Labex grant. The MIMENTO technological facility of FEMTO-ST, part of the RENATECH network, is acknowledged for providing technical support, especially V. Petrini for skillful handling of the microsystem transducers. The SAW delay lines were manufactured by É. Courjon at Frec'n'Sys SASU (Besançon, France). E. Foltête (Applied Mechanics department, FEMTO-ST, Besançon, France) performed initial sound-card records of the signal emitted by a vibrating tuning fork in order to identify some of the vibration modes. E. Rubiola and the members of the Time & Frequency department of FEMTO-ST are acknowledged for motivating and fruitful discussions.