F Le Normand

C S Co Jocaru

C Fleaca

J Q Li

P Vincent

G Pirio

L Gangloff

Y Nedellec

P Legagneux

A comparative study of the field emission properties of aligned carbon nanostructures films, from carbon nanotubes to diamond

Keywords: Carbon, diamond, graphite, Nanotubes, Field emission, ionization, evaporation, and desorption

The electron field emission properties of different graphitic and diamond-like nanostructures films are compared. They are prepared in the same CVD chamber on SiO 2 /Si(100) and Si(100) flat surfaces, respectively. These nanostructures are thoroughly characterized by scanning electron emission (SEM), transmission electron microscopy (TEM), Raman spectroscopy, X-ray photoelectron spectroscopy (XPS) and Auger electron spectroscopy (AES). Films of dense aligned carbon nanotubes by far display the low-est threshold fields around few V/µm and the largest emission currents. Carbon nanofibers, with platelet arrangement of the graphitic planes parallel to the substrate, exhibit higher emission thresholds around 10 V/µm. Diamond nanostructures, either modified through ammonia incorporation within the gas phase or not, exhibit the largest emission threshold around 25 V/µm. The high enhancement factors, deduced from the Fowler-Nordheim plots, can explain the low emission thresholds whereas limitations to the electron transport ever occur through different processes (i) surface modifications of the surface, as the transformation of the SiO 2 barrier layer into SiN x in the presence of ammonia evidenced by XPS; (ii) different orientation of the graphitic basal planes relative to the direction of electron transport (carbon nanofiber) and (iii) presence of a graphitic nest at the interface of the carbon nanostructure and the substrate, observed when catalyst is deposited through mild evaporation.

Introduction

The vacuum electronic devices based on electron emission use mostly two main emission processes depending on the application: the thermoionic emission or "hot emission", and the field emission by tunnelling effect or "cold emission". Nowadays requirements are to develop powerful, high brightness and miniaturised electron sources for flat panels display, or microwave amplifiers, as well as parallel electron beam for microscopy and nanolithography. In this case, thermoionic sources find some limitations such as thermal compatibility and large distributions, both in electron energy and emission direction. By contrast the cold sources can be highly miniaturized and display nearly punctual sources with thinner energy distribution [1]. Moreover the emission is expected to be much stable with a longer lifetime. Among possible candidates, the carbon nanotubes (CNTs) as well as other carbon sources such as doped CVD diamond, doped Diamond Like Carbon (DLC), amorphous carbon are presently intensively studied [1][2][3][4][5][6][7][8][9]. However the carbon nanotubes (CNTs) appear as the most promising emitters. They fulfill many advantages over other carbon sources. First, a high geometrical aspect ratio (diameter/length) with an emission radius that can be reduced to a size as low as 1-3 nm for a single-wall CNT (SWCNT) and in the 10-100 nm range for multi-wall CNT (MWCNT). This high aspect ratio allows switching on an electron emission with low extracting voltage. Second, the carbon is anyway an excellent electron emitter with a work function Φ around 5 eV. Third, CNTs exhibit both a strong hardness combined with a high flexibility, which is of great interest for the stability of the emitter. In addition a CNT without defects is expected to exhibit no dandling bonds, which infers an improved chemically stability and thus longer lifetimes compared to conventional field emission tips like Spindt tips. Recent results on CNTs show that excellent current stability for hours can be obtained [3,4]. Fourth, graphitic shells that make up the walls of the CNTs are excellent electrical conductors. Finally, the deposition of films and arrays of aligned CNTs is at the present time controlled on pretreated samples either by thermal or plasma-assisted CVD methods with an optimized and controlled density [START_REF] Ebbesen | Carbon Nanotubes: Preparation and Properties[END_REF][START_REF] Ajayan | [END_REF]. Thus high electron densities were reported in the literature, up to 1 A/cm 2 [12]. On the other hand electron emission from diamond nanostructures at low threshold voltage has been found. The process however is controversial. A low electron affinity for diamond [13], conduction at the diamond film-substrate interface [14], emission from states at the Fermi level by either graphite agglomerates or surfaces states or defects in the diamond film [15] have been invoked. Hence the field emission properties of diamond films are strongly dependent on the doping, either by boron or nitrogen. However these results are not always reliable due to different experimental measurement of the electron emission (either in a spherical or planar configuration) and they often suffer from some lack of characterization of the carbon nanostructures. Therefore a rationale is someway required between the field emission properties, on one part, and chemical, structural and morphological properties of the carbon nanostructures on another part.

We present here a comparative study of the field emission properties of vertically aligned films of different graphitic nanostructures and nitrogen-doped diamond films prepared in the same reactor by Direct Current Hot Filament Catalytic CVD (DC HF CCVD). This CVD process allows the growth of CNTs and many other aligned fiber nanostructures as well as diamond films [16,17]. This set up combines an activation of the gas mixture both by a DC gas discharge to produce ions, and by hot filaments to produce radicals, and specific high vacuum surface preparations of the substrate (thermal treatments, deposition of catalyst, . . . ). Thorough characterizations of the nanostructures formed include in this study surface analyses by in-situ electron spectroscopies (X-ray electron spectroscopy and Auger electron spectroscopy), Raman spectroscopy, MEB and TEM. This comparative study on widely different carbon nanostructures shows that (i) the threshold electron emission is largely dependent on the nature of the carbon nanostructures and (ii) the CNTs display better field emission characteristics both for turn-on field and current density but (iii) saturation of the electron emission is ever observed.

Experimental

Sample preparation

Diamond nanostructures

The substrate for diamond depositions were prepared from Si(100) substrate (thickness 265 µm, size 8 × 8 mm 2 ; Sb n-doped with ρ = 17 mΩ.cm). They were treated subsequently in a ultrasonic bath with acetone (5 min), ethanol (5 min), an aqueous diamond powder (1 µm) suspension at 343 K for 3 hours to increase the nucleation density, rinsed with deionized water (2 min) and finally ethanol once again (2 min).

Graphite-type nanostructures

Carbon nanostructures were prepared in three subsequent steps.

A SiO 2 layer of depth e (2 < e < 8 nm) was first deposited by a DECR process on a Si(100) sample (Sb n-doped with ρ = 3 mΩ.cm). The role of this SiO 2 layer was twofold: (i) to prevent the metal and silicon interdiffusion leading to silicide formation and catalyst deactivation and (ii) to minimize the wetting interactions between the transition metal and the substrate and thus to get an easier fragmentation into divided particles.

Transition metal (TM) film was deposited on these SiO 2 /Si(100) substrates by sputtering of either a Fe or Co target (grade 99.95% for both targets) at room temperature using either a magnetron DC (Co) or a RF (Fe) source (Ar + pressure: 4 × 10 -3 mbar, and gas flow: 15 cm 3 /min). The power energy was 110 W (DC source) and the ion current was 110 mA (RF source), respectively. The residual vacuum limit was around 10 -7 mbar. Within these conditions the calibrated deposition rates were in the range 0.05-0.10 nm/min, depending on the transition metal. One of the samples (Co5) was in situ prepared by evaporation in ultra high vacuum (UHV) at around 10 -10 mbar and 1083 K [18].

Carbon nanostructures growth

The growth of carbon nanostructures was carried out in the same experimental set up that has been previously described, whatever the diamond or graphite nature of carbon is [16,17].

Diamond deposition

The conditions for diamond deposition are summarized in Table 1. Briefly the deposition occurs in two steps by HF CVD (nucleation for 30 min at 973 K and 2% CH 4 , and growth for 10 hours at 1073 K and 1% CH 4 ) with variable ammonia concentrations in the gas phase ranging from 0 to 10 500 ppm.

Graphitic carbon deposition

Carbon was deposited in a UHV chamber (limiting vacuum 10 -9 torr) by a Direct Current-enhanced and Hot Filaments-enhanced Catalytic CVD (DC HF CCVD) process that had been previously described [17]. In this process the reactive gas mixture (C 2 H 2 :H 2 :NH 3 ) was activated both by hot filaments to produce neutral radical species, and by a DC gas discharge created by polarisation between two electrodes settled a few centimeters away from the sample to produce activated and ionic gas species. The sample was negatively biased relative to the cathode in the range 25-40 V. This allowed extracting a constant ionic beam that impinged the surface of the sample. The literature provides some other references of similar set up for carbon film growth [19][20][21][22][23][24][25][26][27][28][29][30][31][32]. It was shown elsewhere that the nature and the morphology of the nanostructures either grown or redeposited after sputtering and etching was strongly affected by the ammonia concentration in the gas mixture, by the nature, the thickness and the mode of deposition of the TM (either Fe or Co) [33]. Table 2 summarizes these parameters used for the preparation of these nanostructures.

Characterization of the nanostructures

In situ chemical surface analyses were carried out by XPS and AES, in a chamber directly connected to the experimental UHV chamber. Details on surfaces analyses by electron spectroscopies were reported elsewhere [34]. Raman spectra were recorded with a Renishaw spectrometer using a He-Ne laser at λ = 632.8 nm equipped with a Notch filter and working in a backscattering geometry.

The samples were furthermore observed by SEM for the general morphology of the deposition and by TEM and HRTEM for structural characterization of the nanostructures. SEM was performed on a XL30S-FEG PHILIPPS working at 3 kV and TEM was performed on a TOPCON 002B microscope operating at 200 kV. Carbon nanostructures were scratched by a diamond tip from the sample and directly pulled on a copper grid covered with amorphous carbon membrane and drilled with holes for accurate observations.

-The emission was calculated from the Fowler-Nordheim (FN) theory, strictly valuable for a flat and metallic emission center. The tip effect, in the case of either CNTs or any other strongly anisotropic material with a high aspect ratio, was accounted by a geometric enhancement factor β that measures the ratio of the local field at the tip to the mean apparent field E.

In the case of a pure whisker this shape factor can be approximated to the aspect ratio of the emitter, that means β ≈ h/d where h and d are the height and the diameter of the whisker and h × d, respectively [START_REF] Brodie | [END_REF]. -Temperature effects were neglected.

Therefore within the FN framework the current density j can be expressed as

j = I a = Aβ 2 E ( exp BΦ 3/2 \ (1)
(triode configuration).

Field emission measurements

The field emission measurements were performed at THALES-R&T (Domaine de Corbeville-Orsay, France) on an UHV set up (limiting vacuum 10 -9 torr) which is displayed in Figure 1 The samples were conditioned by progressively rising the extraction potential before performing a full cycle. This one was carried out by subsequently rising and dropping the extraction potential. Thus many cycles were performed on the same sample up to a saturation of the electron emission. Stable conditions were obtained after many of these cycles and we report only the results of the steady field emission properties of the sample. The emission centers were visualized on the screen and it was checked that the emission properties occurred from more than one single emission center. The threshold emission was arbitrarily defined as the field from which an emission can be detected, which presently meant an emission density above 0.1 nA/cm 2 . The turn-on field was arbitrarily fixed as the field with an emission of 0.1 µA/cm 2 . Let us remind that the reported values, when compared with data of the literature, have only a relative signification as they strongly depend on the experimental set-up.

The field emission intensity as a function of the applied field is mostly described in the literature within the Fowler-Nordheim (FN) theory [START_REF] Bonard | Understanding carbon nanotubes. From basics to application[END_REF]. It involves the following assumptions:

-A triangular barrier, neglecting the space charge effect.

S Φ βE

with constants A = 1.54 × 10 -6 and B = 6.49 × 10 9 for E in V.m -1 and Φ in eV. Φ was fixed to 5.0 eV, the value generally accepted for graphite. Within these assumptions and according to equation ( 1), a Ln(I/E 2 ) = f (1/E) plot allowed determining β from the slope of the FN plot.

Results

Field emission from carbon nanotubes grown with iron catalyst displaying different orientations of the graphitic shells

Figure 2A, B displays SEM images of samples Fe1 and Fe2, respectively. The difference stems from a high concentration of ammoniac (16%) in the gas mixture in the last case. Within the extraction plasma, an extraction power P is defined as the product of the extraction voltage V e times the intensity of the electrical current on the sample I e : this extraction power is indeed dependent on the nature of the gas and the ammonia content is among the most important parameter to control in order to grow either aligned or tangled CNTs. The former sample displays completely tangled nanostructures with an heterogeneous size distribution of iron particles. Biggest particles around 100 nm and some smaller ones are encapsulated with graphitic carbon shells (Fig. 3B) whereas the smallest ones (10-15 nm) are hold at one end of a CNT which displays an appreciable length (×1 µm) and few walls (around 4-7) (Fig. 3A). Most of them show a "bamboo" shape. By contrast sample Fe2 exhibits a very good alignment of CNTs in the direction normal to the substrate (Fig. 2B). They exhibit an uniform diameter distribution around 15-20 nm with few walls (around 4-5) (Fig. 3C). This dramatic enhancement in the alignment of the CNTs in the presence of ammoniac can be explained by a promoted sputtering yield by the heavier ammoniaderived ions of the gas phase. These ammonia-derived species maintain clean and free the catalytic iron surface for hydrocarbon adsorption and further CNTs growth [33].

In addition they also probably play a role in a more homogeneous fragmentation of the iron film into particles. The behavior has been observed whatever the sample is once both ammonia is added to the gas mixture and a biasactivated process is used for CNTs growth. In addition an iron reduction step is required, carried out at 973 K preliminary to the CNTs growth, in order to obtain metallic iron particles (Fe 0 ) which yields a Fe 2p3/2 single contribution at 707 eV instead of 711 eV (Fe III from oxide with an O1s contribution at 530 eV) before reduction (Figs. 4-IIA, IIB). This point has been checked systematically as the CNTs growth is strongly perturbed if iron particles are not preliminary reduced.

The field emission behavior of these Fe@CNTs nanostructures is shown in Figure 6 for some representative stabilized cycles. The electron emission exhibits in both cases an emission threshold lower than or equal to 2 V/µm with very large and close geometric enhancement fac-ment with the values reported in the literature for such CNTs configurations [1]. The ratio of tube lengths (in the order of few micrometers for the longest) to mean diameters (around 15-20 nm), exceeds two orders of magnitude. This means that the assumption of the nanotubes to whiskers is only a rough assumption. The slight difference observed in the threshold emission in favor of tangled nanotubes in sample Fe1 could be explained by the weaker perturbation of the Fe@CNT emitter by other ones in the vicinity on sample Fe1, while on sample Fe2 most of the CNTs with rather equivalent lengths are oriented in the same direction and the screening of the field is larger. However saturation of the emission is rapidly reached at higher field and a break to the FN plot is always observed, especially on sample Fe1. The maximum in emission density is quoted to 60 µA/cm 2 at 4.2 V/µm for the sample 2 tors β, estimated from the linear FN plots to ~1700 for Fe1 with disoriented CNTs, but 360 µA/cm at 4.8 V/µm both samples (Fig. 7). These values are in good agree-for the sample Fe2 with aligned CNTs. The origin of this saturation is not quite clear, but at high field, heating of the sample occurs resulting in some desorption of adsorbed species and even breaking or burnt off of the nanotube. As it is expected that the longest CNTs are the first emitter center, it is likely that either desorption of adsorbed species or irreversible consumption of a nanotube leads to the substitution by a new emission center in the vicinity of the emission center. However it is unlikely to find again a new emitter site in the vicinity on sample Fe1, whereas on sample Fe2 the high density of aligned tubes infers that a new center in the vicinity of the burnt off emitter will easily substitute to it. Moreover on sample Fe1 the carbon nanotubes initially tangled on the substrate surface, has to align along the direction of the electric field in order to be a potential emitter. This has been indeed shown for an electric field far lower than 1 V/µm [START_REF] Wei | [END_REF]40], but it results from this initial morphology that the density of potential electron emitters is much lower on sample Fe1. Saturation to the emission may also occur through limitations in the electron transport. This point is further discussed. In conclusion, the aligned or tangled character of the Fe@CNTs has a weak effect on the field emission threshold but a strong one on the saturation density.

Field emission from carbon nanotubes grown with cobalt catalyst at different temperatures

The elaboration process and the characteristics of electron emission from samples Co1 and Co2 are displayed in Table 2. The two films were grown at different temperatures, 973 K and 1173 K, respectively. In both cases multiwalled carbon nanotubes with a cobalt particle on top of them (Co@CNT) are grown and are aligned in the direction normal to the substrate (Figs. 2C andD). Small CNTs (around 300 nm) are obtained on the sample Co1 grown at lower temperature but both samples exhibit heterogeneous length distributions. The XPS spectra (not shown) confirm the silicon nitride formation at the surface of the substrate. The graphitic nature of the deposit is also evidenced by Raman spectroscopy (Fig. 8). Many defects are present on both samples as exhibited by an intense D Raman band at 1332 cm -1 compared to the G band at 1585 cm -1 . This G band clearly exhibits a shoulder at 1620 cm -1 which is much better resolved for sample Co2. The G band is weak and broad on sample Co1, inferring that the carbon nanotubes are much more defective on this sample. The two samples start to emit at an emission threshold around 3 V/µm (sample Co1) and 2 V/µm (sample Co2) with high enhancement factors β around 1300 (Figs. 6,7). However both samples rapidly reach a saturation of the electron emission at higher fields. The maximum of elec-tron emission densities are 105 µA/cm 2 at 10.4 V/µm and 85 µA/cm 2 at 4.9 V/µm on samples Co1 and Co2, respectively. Moreover, on sample Co1 one needs a long set of pretreatments before finding stable emission conditions. The emitting sites could be the long filaments merging above the dense forest of small Co@CNTs. These scarce long tubes are progressively burnt off in the course of the first cycles and the smaller value of β then obtained stems for a high density of nanotubes of nearly the same length. The sample Co2 displays more interesting field emission properties with rather a high β � 1500. This difference compared to Co1 can be explained by the fact that the filamentous nanostructures, whose shape is about the same, are more regularly ordered above the CNTs forest, decreasing screening effects by vicinal nanotubes. The low threshold field around 2 V.µm -1 is conjugated with a very stable emission. However a strong saturation is observed at a low value of ~85 µA/cm 2 and a low field around 5 V/µm.

Field emission from carbon nanotubes with Co catalysts with different ammonia concentrations

The samples Co3 and Co4 are grown at 0 and 16% ammonia concentrations, respectively (Tab. 2). It has been shown elsewhere that the nature of the nanostructures is highly sensitive to the ammonia concentration [33]. Thus without ammonia within the gas phase, new carbon nanostructures can be grown, whereas at low ammonia concentration (0.5 < NH 3 < 3%) CNTs are readily obtained. At larger NH 3 concentrations, the etching of the surface overcomes the deposition of carbon, and silicon-based nanostructures are now grown. This is illustrated in Figures 2F, C, E andFigures 3F, E andG for samples Co4, Co1 and Co3 with ammonia concentrations 0%, 3% and 16%, respectively. From the SEM and TEM images all the nanostructures exhibit high density and aligned morphology oriented along the direction normal to the substrate with a metallic particle on top of them. The nature of these nanostructures is however strongly different. On sample Co3 with 16% ammonia within the gas phase, the absence of any C1s signal in the XPS wide scan, as well as the absence of any carbon signal (D and G bands) in the Raman spectrum indicates that the chemical nature of the nanostructure is not made up of carbon. By contrast, intense N1s and Si2p single lines can be ev- idenced with a Si/N ratio around 0.8 (corrected for the photoionization cross sections), not far from the expected 0.75 one for the stoichiometric silicon nitride Si 3 N 4 . Moreover the initial Si2p signal initially split into silicon and silicon oxide contributions coalesces after the CVD process into one single contribution around 101.5 eV (Fig. 4-IIIB). From structural investigations it can be seen that these nanocones are mainly amorphous (Fig. 3E). It is therefore tempting to assimilate these nanostructures to amorphous silicon nitrides, possibly slightly oxidized. The distribution of these nanocones is quite homogeneous with a base diameter around 25 nm and a curvature radius around 10 nm. On the other hand the XPS wide scan recorded on the Co4 sample, grown without ammonia within the gas mixture, exhibits now a noticeable C1s signal. In addition Intensity (Arb Units) Fig. 7. FN-plots Ln(j/E 2 ) = f(1/V) of the nanostructures, for a set of some stabilized cycles. D band 8000 the nanocone is not yet strictly valid [41]. This can explain the sensible drop of this factor to around 200 as deduced from the linear FN behavior (Fig. 7). More important, the emission thresholds occur at much higher electric fields than the carbon nanotubes: 8.2 V/µm for the platelet graphitic nanocones and 14.5 V/µm for the silicon-based nanocones with many cycles in both cases before reaching reproducibility. Moreover saturation densities are rapidly reached. The emission densities at saturation decrease by more than one order of magnitude (4.8 µA/cm 2 at 17 V/µm) on graphitic nanocones and by more than two orders of magnitude (0.5 µA/cm 2 at 43 V/µm) on silicon-based nanocones, respectively. This can be easily understood in the case of silicon nanostructure whose electron conduction is not known but can be estimated to be low. In the case of the platelet graphitic morphology, the electron conduction occurs from an electron transport from one graphitic base plan to another one, instead of an electron transport inside curved basal shells in the case of carbon nanotubes. In the former case the very low electrical conductivity leads to a rapid saturation of the electron transport to the emitting surface for HOPG graphite [START_REF] Bernier | Le carbone dans tous ses états[END_REF] in the directions parallel and normal to the graphitic base plan, respectively.

Field emission from carbon nanotubes with Co catalysts with different deposition techniques

We have compared the field emission properties of Co samples in which Co particles are prepared through different deposition methods. In sample Co5, Co is in situ deposited by vacuum evaporation instead of magnetron the Raman spectrum exhibits two bands at 1328 cm -1 (D band) and 1585 cm -1 (G band). TEM examinations provide a clear picture of these nanostructures which are now entirely filled with carbon. The graphitic planes, distant by 0.34 nm each other, are oriented parallel to the substrate surface with a platelet morphology (Figs. 3F,G) and an isotropic cobalt particle is encapsulated on top of it (Figs. 2E,3F,3G). Finally the much broader Raman G and D bands also support a different structural arrangement of the carbon nanostructures (Fig. 8). In conclusion these nanostructures might be described more as carbon fibers (CFs) rather than carbon nanotubes. In the absence of ammonia the different morphology obtained for the carbon nanostructures, as well as the different morphology of the catalytic particles on top of it (isotropic instead of anisotropic), suggest that the nature of the plasma created is quite different and induced a different mechanism of growth [17]. These aligned nanostructures display an anisotropic morphology which is prone for electron field emission, albeit the conical shape observed in Co3 sample is far from the model shape of a whisker. Thus the assumption of β to the aspect ratio (length/lateral diameter) of sputtering on Co4 sample. Unlike carbon nanofibers on Co4 sample, CNTs are now grown with an uniform diameter around 15 nm (Fig. 2G and Fig. 3H). We have shown elsewhere that the nature of the carbon nanostructure may be strongly dependent on the mode of deposition of the catalytic layer deposition, especially through a different adhesion of the metallic particle to the substrate [17,18]. The C KVV Auger signal is displayed in Figure 5. It is not structureless and quite characteristic of carbon nanotubes. Field emission properties of the CNT films are similar to those above obtained with low emission threshold and high enhancement factor (Tab. 2).

Field emission from diamond films doped with nitrogen

The diamond films grown with 6000 ppm and without ammonia in the gas phase are shown in Figures 2I and2H, respectively. The morphology of the diamond films exhibits ma jor changes with addition of ammonia in the gas phase. Whereas the polycrystalline diamond film shows facetted grains with sizes smaller than 1 µm and a general [111] growth orientation, the ND6000 sample exhibits quite rounded-shape grains of "cauliflower" morphology, indicating either nitrogen incorporation or many localized stable and the emission threshold rapidly increases to values equivalent to and even higher than those observed on ND0 sample. defects formation with graphitic carbon formation. Furthermore in situ XPS investigation (not shown) clearly indicates that a nitrogen signal is hardly evidenced. More details on the properties of the nitrogen-doped diamond films as a function of the ammonia content within the gas phase are reported elsewhere [START_REF] Li | [END_REF]. The Raman spectrum of the diamond films grown in the presence of ammonia also displays quite ma jor modifications (Fig. 9). Whereas the diamond film ND0 exhibits the sharp D contribution at 1334 cm -1 corresponding to the A1g vibration mode and a broad and weak G band around 1560 cm -1 corresponding to the E2g vibration mode developing over a rising photoluminescence background, the spectrum on sample ND6000 exhibits many other bands while the D peak almost vanishes. Contributions at 1142 cm -1 , 1191 cm -1 , 1510 cm -1 and 1555 cm -1 may be assigned to the presence of new carbon species (surface carbon, trans-acethylenic carbon, graphitic-like clusters, . . . ). The discussion about these Raman spectra is presented in more details elsewhere [START_REF] Li | [END_REF]. The main conclusion from these studies of the properties of diamond films is that the morphology of the diamond films is deeply transformed through the introduction of ammonia into the gas phase. However this will not infer that stable nitrogen is deeply incorporated into the films but rather the nitrogen-containing gas species are more prone to create an intense concentration of lattice defects throughout the growth and to generate graphitic carbon species.

The field emission properties on diamond samples with different ammonia content in the gas phase in the range 0-10 500 ppm are summarized in Table 1. The threshold emission occurs at large electric field (25 V/µm) with a low emission density, a rather good stability of the emission after many cycles. Reversely in the presence of ammonia the emission can start initially at a lower threshold down to 5 V/µm for ND6000 sample, but this emission is never

Discussion

Thresholds for field emission are quite variable according to the sample but these variations are clearly dependent on the nature of the carbon material. Here carbon materials with different structural configurations (doped or undoped diamond, graphite), different crystalline states (amorphous or crystalline) and different morphologies (flat films, oriented films normal to the surface with graphite basal planes either parallel or normal to the surface) were prepared in the same experimental set up.

Thus for the CNT films, threshold fields are within the 1-4 V/µm range, rather independent on the nature of the metal catalyst or the growth temperature. These values are in agreement with other data of the literature (in the order of some V/µm for nanotubes), considering however that the definition of the emission threshold and the experimental set up might be different [1]. We do not find at this stage any significant effect of the transition metal as the difference observed between Fe and Co is within the experimental errors. The slightly larger emission threshold observed on sample Co1 might be explained from the Raman spectra (Fig. 8). From these spectra the intensity ratio of the G band to the D band (ID /IG ) is calculated to 0.37, 0.60 and 0.58 on samples Co1, Co2 and Co5, respectively. Thus the shift of the threshold towards larger fields can be explained by a higher density of defects in the nanotubes on sample Co1.

By contrast the threshold fields on the nanofibers and the nanocones, in the graphitic as well as in the siliconbased forms, are strongly shifted towards higher fields. This is accompanied with much lower enhancement factors β. As the densities and the mean size of the nanostructures are rather similar within one order of magnitude, it can be concluded that these difference can be due to a difference in the nature of these nanostructures. Another factor can be found in the insulator character of these nanostructures. This is obvious for the silicon nitride nanocones, but also for the carbon nanofibers. When the basal graphitic planes are oriented perpendicularly to the direction of the electron transport, it is expected that the electron conduction will be impeded. This conductivity also explains the saturation of the emission observed on all the samples with a deviation of the Fowler-Nordheim plot (Fig. 7). It must be recalled that, when ammonia is present within the gas phase, the SiO2 barrier is fully transformed into silicon nitride SiN x , probably as an amorphous phase (Fig. 4-IIIB). Thus the electron conduction occurs through channeling through an amorphous SiO2 or SiNx layer. Despite the absence of conductivity data through these layers, it is no doubt that the nature, as well as the thickness or the rugosity, of these layers will influence the saturation of the electron emission at high fields. Generally it is observed that the emission density at saturation closely relates to the field enhancement factor (Tab. 2). However adhesion of the nanostructure may explain some discrepancy to this behavior. The very low saturation density on sample Co5, in spite of the low threshold emission, can be related to a poor adhesion of the nanostructure to the substrate via the formation of a nest of graphic planes parallel to the surface during the nucleation of the nanotube [17]. This mode of adhesion results from the mild deposition conditions of catalytic particles through thermal vacuum evaporation (MBE) used in the case of the Co5 sample, whereas by the sputtering process used for other Co and Fe samples stronger adhesion impedes the formation of such graphitic nest and improves the conductivity at the interface.

By contrast the field emission from diamond films nanostructures is expected to occur from less sharp nanostructures and this is partly checked by the much lower enhancement factors observed. However significant changes occur in the diamond films when NH3 is added within the gas phase for grain size and grain morphology, from facetted surfaces to cauliflower morphology due to secondary nucleation [44][45][46]. These changes cannot be explained only by some incorporation of nitrogen within the diamond films as from XPS investigations it has been shown that the nitrogen content remains vanishingly small. More probably morphological modifications and new graphitic carbon species are formed, as depicted by SEM (Fig. 2-I) and Raman spectra (Fig. 9).

Finally some non stabilized electron field emission occurs according to the cycles. The reasons of these variations must be found in a continuous process of consumption of the sites that leads to a different distribution of the sites from one cycle to another one. In addition it is possible that the high flexibility of numerous nanotubes leads to modifications of the distribution of the emitter sites. This is especially the case for thin nanotubes which are known to be highly flexible and some of them can orientate along the field lines. In the case of doped-diamond films a shift towards higher emission thresholds is systematically observed. One explanation could be a nitrogen removal from the samples as it is shown that the emission properties of the doped-ND samples after many cycles are similar to the emission properties of the stable undoped ND0 sample.

Conclusion

The electron field emission properties of different graphitic and diamond-like nanostructures films are compared. Films of dense and aligned carbon nanotubes by far display the lowest threshold fields around few V/µm and the largest emission current density. Carbon nanofibers, with platelet arrangement of the graphitic planes parallel to the flat substrate, exhibit higher emission thresholds around 10 V/µm. Diamond nanostructures, either modified through ammonia incorporation within the gas phase or not, exhibit the largest emission threshold around 25 V/µm. The high enhancement factors, deduced from the Fowler-Nordheim plots, can explain the low emission thresholds whereas limitations to the electron transport ever occur through different processes (i) surface modifications of the surface, as the transformation of SiO 2 barrier layer into SiN x in the presence of ammonia as evidenced by XPS, (ii) different orientation of the graphitic basal planes relative to the direction of electron transport in carbon nanofibers and (iii) presence of a graphitic nest at the interface of the carbon nanostructure and the substrate, observed when catalyst is deposited through mild evaporation.

Anyway the density of CNTs or other nanostructures here obtained is by far too large to optimize the screening of the tubes. Therefore it is important to work with catalysts spread on a surface as an array and to work with conductive surface that are both a barrier to the diffusion and able to get and to stabilize small metallic particles at high temperatures. Much effort is performed on these issues.

Fig. 1 .

 1 Fig. 1. Experimental set up for field emission measurements

  . A planar triode configuration was used, in which the sample was fixed in front of an extraction grid biased to the potential V and distant by d = 110 µm by mean of a quartz spacer. The mean apparent electric field around the sample was then E = V/d. The emitted electrons were collected on an ITO (indium tin oxide)/phosphorus flat anode screen integrating filtration grids to discriminate the secondary electrons. The emitted current I a was measured with a picoamperometer. The current density j = I a / S was calculated from an emitting surface S estimated to 0.7 mm 2 , accounting for the opaqueness of the extraction grid[35].

Fig. 2 .Fig. 3 .

 23 Fig. 2. SEM images of (A) sample Fe1; (B) sample Fe2; (C) sample Co1; (D) sample Co2; (E) sample Co3; (F) sample Co4; (G) sample Co5; (H) sample ND0 and (I) sample ND6000. The samples are imaged at variable incidence angles of the electron beam.

Fig. 4 .

 4 Fig. 4. XPS spectra on samples (A) Fe1; (B) Fe2. Panel displays (I) wide scans; (II) Fe2p; (III) (next) Si2p and (IV) (next) O1s (sample Fe1) and N 1s (sample Fe2) core levels, respectively.

Fig. 4 .

 4 Fig. 4. Continued.

Fig. 4 .

 4 Fig. 4. Continued.

Fig. 5 .

 5 Fig. 5. AES on the C KVV level: Fe1 (full); Co5 (dotted).

Fig. 6 .

 6 Fig. 6. j = f (E) plots of the nanostructures prepared for a set of some stabilized cycles.

Fig. 8 .

 8 Fig. 8. Raman spectra.

F

  . Le Normand et al.: A comparative study of the field emission properties of aligned carbon nanostructures films... 125

Fig. 9 .

 9 Fig. 9. Raman spectra on diamond films (A) ND0 and (B) ND600.

Table 1 .

 1 Conditions of nanodiamond films preparation and field emission properties. Nucleation : T = 973 K; P = 23.5 mbars; [CH 4 ] = 2%; t = 0.5 h. Growth: T = 1073 K ; P = 30 mbars; [CH 4 ] = 1%; t = 10 h. Otherwise throughout the deposition process, filaments power = 185 W; filaments distance = 5 mm and gas flow = 200 sccm.

	Sample	% NH 3 (ppm) Morphology	Initial	Long term	Emission density	β	Stability
				threshold (V/µm)	threshold (V/µm)	(µA/cm 2 ) (for the field E (V/µm))
	ND 0 ND 45	0 45	Film Film	25 >25	25 -	20(40) -	350	Stable -
	ND 500	500	Film	>25	-	-		-
	ND 2000	2000	Film	14	-	-		Unstable
	ND 4000	4000	Film	15	25	20 (22)		Unstable
	ND 6000	6000	Film	5	>10	10 (7)	500 Unstable
	ND 10500	10500	Particles	7	-	15 (9)		Unstable

Table 2 .

 2 Some important parameters of the nanostructures grown and field emission properties. Other growth conditions are: overall flux: 100 sccm; P = 15 mbar; t = 15 min; Filaments power = 150 W corresponding to a temperature of around 2223 K; distance between the cathode of the plasma and the substrate = 5 mm; plasma activation potential within 300 and 400 V.

	Sample	Preparation

J. Faerber, G. Ehret and G. Schmerber are acknowledged for SEM, TEM characterizations and film preparations, respectively.


Ming Liu

Matthieu Crussière

Jean-Franc ¸ois Hélard

Enhanced Two-Dimensional Data-aided Channel Estimation for TDS-OFDM

In time domain synchronous (TDS)-OFDM, the channel estimation is conventionally carried out based on the pseudo noise (PN) sequence. The PN sequence based channel estimation however suffers interference from adjacent OFDM data symbols. This paper proposes a new low-complexity dataaided channel estimation method with two-dimensional (2-D) estimate refinement and interpolation. Data-aided channel estimation is carried out using the rebuilt OFDM data symbols as virtual training symbols. In contrast to the classical turbo channel estimation, interleaving and decoding functions are not used when rebuilding OFDM data symbols thereby reducing the complexity. 2-D estimate refinement and interpolation are proposed to improve the data-aided channel estimation. Simulation results show that the performance of TDS-OFDM based DTMB system using the proposed method is very close to that with perfect channel estimation in terms of bit error rate (BER).

I. INTRODUCTION

Orthogonal frequency division multiplexing (OFDM) attracts much attention for high data rate communications due to its merits of multipath immunity and bandwidth efficiency. Traditionally, cyclic prefix (CP) is inserted as guard interval (GI) between two adjacent OFDM symbols to mitigate intersymbol interference (ISI) and enable simple equalization at receiver side. Recently, an alternative type of GI which is composed of a known pseudo noise (PN) sequence is proposed for the purpose of improved spectrum efficiency and fast tracking. This new OFDM signal waveform is referred to as time domain synchronous (TDS)-OFDM1 [START_REF] Tang | Iterative channel estimation for block transmission with known symbol padding -a new look at TDS-OFDM[END_REF]. TDS-OFDM has been adopted by the Chinese digital terrestrial multimedia broadcasting (DTMB) system [START_REF]Framing Structure, Channel Coding and Modulation for Digital Television Terrestrial Broadcasting System[END_REF].

In TDS-OFDM, the PN sequence in GI is reused as training sequence for channel estimation. Since the PN sequence based channel estimation is affected by the adjacent OFDM data symbols [START_REF] Steendam | Different guard interval techniques for OFDM: performance comparison[END_REF], several approaches were proposed to remove the influence of data symbol from PN sequence [START_REF] Tang | Iterative channel estimation for block transmission with known symbol padding -a new look at TDS-OFDM[END_REF] [3] [START_REF] Tang | Novel decision-aided channel estimation for TDS-OFDM systems[END_REF]. However, the performance is degraded either by ISI from previous data symbol in [START_REF] Tang | Iterative channel estimation for block transmission with known symbol padding -a new look at TDS-OFDM[END_REF] or by the error of interference estimation in [START_REF] Steendam | Different guard interval techniques for OFDM: performance comparison[END_REF] or by error propagation resulting from the hard decision of data symbols in [START_REF] Tang | Novel decision-aided channel estimation for TDS-OFDM systems[END_REF]. A better solution is the iterative turbo channel estimation [START_REF] Zhao | Iterative turbo channel estimation for OFDM system over rapid dispersive fading channel[END_REF] which rebuilds the data symbols using the decoded information bits and then uses the rebuilt data symbols as known training symbols for channel estimation. The computational complexity and time delay of this method are however very high especially for systems with sophisticated decoder and deep interleaver. For example, the DTMB system includes a low density parity check (LDPC) encoder followed by an interleaver of depth 170 (or 510) OFDM symbols, which makes the turbo channel estimation impracticable for real use. [START_REF] Liu | A combined time and frequency algorithm for improved channel estimation in TDS-OFDM[END_REF] proposed a dataaided channel estimation method that excludes the channel decoding and interleaving from the feedback loop to reduce the computational complexity. However, the use of this method is limited to uniform-power modulations e.g. BPSK and QPSK.

In this paper, we propose a new two-dimensional (2-D) channel estimation method which not only enables the use with non-uniform-power (higher order) modulations but, more importantly, improves the channel estimation performance with reduced complexity. More concretely, refined channel estimates are first obtained for some 'virtual pilot' positions located in the time-frequency plan by averaging the instantaneous data-aided channel estimates within the coherence region. 2-D Wiener filtering based interpolation is then performed to get an overall data-aided channel estimation. Several virtual pilot patterns are also studied in this paper. Simulation results show that the proposed method can provide nearoptimal performance.

The rest of the paper is organized as follows. In section II, the mobile channel and TDS-OFDM signal models are described. In section III, the proposed 2-D data-aided channel estimation method is presented. Simulation results are shown in section IV. Conclusions are drawn in section V.

In this paper, (•) * denotes the conjugate of complex number, E{•} is the expected value, (•) T and (•) H are the matrix transpose and Hermitian transpose, respectively.

II. SYSTEM MODEL

A. Channel Model

The time-variant channel impulse response (CIR) with L distinct paths is described as:

h(τ, t) = L-1 l=0 γ l (t)δ(τ -τ l ) (1)
where τ l and γ l (t) are the delay and the gain of the l th path, respectively. γ l (t) is assumed to be zero mean complex Gaussian process which is mutually independent for different paths. In this paper, the channel is assumed to be quasi-static, namely constant within one OFDM symbol but changing from one symbol to another. The channel frequency response (CFR) for the k th subcarrier of the i th OFDM symbol is:

H[i, k] H(iT, k∆f ) = L-1 l=0 γ l (iT )e -j2πk∆f τ l , (2)
where H(t, f ) is the Fourier transform of the CIR, T is the time duration of one OFDM symbol and ∆f = 1

T is the subcarrier spacing. Assuming the channel has the same correlation function for all paths, the time-frequency correlation function of the CFR can be represented by the product of time domain and frequency domain correlations [START_REF] Li | Pilot-symbol-aided channel estimation for OFDM in wireless systems[END_REF]:

φ H [p, q] E {H[i + p, k + q]H * [i, k]} = σ 2 h r t [p]r f [q], (3)
where σ 2 h is the total average power of CIR. Without loss of generality, we assume σ 2 h = 1. r t [p] and r f [q] are the time and frequency domain correlation functions of the CFR, respectively. Concretely, taking the Jakes' mobile channel model, r t [p] is:

r t [p] = J 0 (2πpf d T ), (4)
where J 0 (•) is the zeroth-order Bessel function of the first kind, and f d is the Doppler frequency related to velocity v and carrier frequency f c by f d = vf c /c, where c is the speed of light. The frequency domain correlation r f [q] is:

r f [q] = L-1 l=0 σ 2 l e -j2πq∆f τ l . ( 5
)
where σ 2 l is the variance of the i th path amplitude.

B. Baseband TDS-OFDM Signal Model

Fig. 1 presents the baseband model of the TDS-OFDM. The i th OFDM symbol is formed by N -point inverse fast Fourier transform (IFFT):

x(i, n) = 1 √ N N -1 k=0 X[i, k]e j 2π N nk , 0 ≤ n ≤ N -1. (6)
A ν-length PN sequence {c(i, n)} -1 n=-ν is then inserted before {x(i, n)} N -1 n=0 as GI. The i th transmitted time domain signal is thus:

t(i, n) = c(i, n) -ν ≤ n < 0 x(i, n) 0 ≤ n ≤ N -1. (7)
After passing the multipath fading channel, the received signal is a linear convolution of the transmitted signal and the CIR. After PN sequence removal and overlap and add (OLA) process, the linear convolution becomes a circular one [START_REF] Liu | A combined time and frequency algorithm for improved channel estimation in TDS-OFDM[END_REF]. The received frequency domain TDS-OFDM signal Y has finally a similar representation as the CP-OFDM:

Y [i, k] = H[i, k]X[i, k] + W ′ [i, k], (8)
where W ′ is the equivalent noise component which is slightly colored after OLA process with an equivalent variance of σ 2 W ′ = N +ν N σ 2 w given σ 2 w the variance of the additive white Gaussian noise [START_REF] Steendam | Different guard interval techniques for OFDM: performance comparison[END_REF]. The TDS-OFDM signal can thus be easily equalized by a one-tap equalizer.

C. PN based Channel Estimation

As specified in [START_REF]Framing Structure, Channel Coding and Modulation for Digital Television Terrestrial Broadcasting System[END_REF], the ν-length PN sequence in the GI is composed of a N P N -length PN sequence and its CP. If this CP is longer than the length of CIR, the N P N -length PN sequence is ISI free. Using the ISI free PN sequence, a least square (LS) channel estimation is made for the i th OFDM symbol:

H1 [i, k] = S[i, k] P [i, k] = H[i, k] + W [i, k] P [i, k] , 0 ≤ k < N P N , (9)
where P and S are obtained through N P N -length fast Fourier transform (FFT) applied on the transmitted and received ISI free PN sequences, respectively. The N -length CFR estimation Ĥ1 is obtained by FFT-based interpolation. The mean square error of Ĥ1 is:

ε Ĥ1 = 1 N N -1 k=0 E |H[i, k] -Ĥ1 [i, k]| 2 = Lσ 2 w N P N NP N -1 k=0 1 |P [i, k]| 2 . ( 10
)
III. PROPOSED 2-D DATA-AIDED CHANNEL ESTIMATION METHOD The proposed 2-D data-aided channel estimation method consists of three steps. First, a PN sequence based channel estimation as described above is carried out. The estimated CIR is used to remove the PN from the received TDS-OFDM signal and then make the equalization. Secondly, the 2-D dataaided channel estimation is made to provide a new channel estimate. Finally, the two channel estimates are combined using the minimum mean squared error (MMSE) criteria.

A. Proposed Data-aided Channel Estimation

In contrast to a classical iterative turbo channel estimation algorithm [START_REF] Zhao | Iterative turbo channel estimation for OFDM system over rapid dispersive fading channel[END_REF], the proposed method excludes the deinterleaving, channel decoding and interleaving processings from the feedback loop as shown in Fig. 1. In other words, the estimated data symbols used for data-aided channel estimation are rebuilt directly using the output information of the demapper.

The soft-output demapper demodulates the complex data symbols into Log-likelihood ratio (LLR) of bits [START_REF] Tosato | Simplified soft-output demapper for binary interleaved COFDM with application to HIPERLAN/2[END_REF]. The LLR λ l [i, k] corresponding to the l th bit of the (i, k)th equalized data symbol Z[i, k] is defined as:

λ l [i, k] log P (b[i, k, l] = 1|Z[i, k]) P (b[i, k, l] = 0|Z[i, k]) , (11)
where the

P (b[i, k, l] = 1|Z[i, k]) is the conditional proba- bility of the l th bit equal to 1 given Z[i, k].
The sign of the LLR decides the corresponding bit equal to 1 or 0, and its absolute value gives the reliability of the decision.

Based on the LLR, the probabilities of a bit equal to 1 and

0 are P (b[i, k, l] = 1) = e λ l (k) 1+e λ l (k) and P (b[i, k, l] = 0) = 1 -P (b[i, k, l] = 1)
, respectively. They are then used as a priori probabilities to estimate the data symbol. First, the probability that the transmitted symbol X[i, k] is equal to a specific constellation point α j is computed as the product of the probabilities of all the bits belonging to this constellation:

P (X[i, k] = α j ) = log 2 µ l=1 P b[i, k, l] = κ l (α j ) , ( 12
)
where Ψ is the set of the constellation points of a given modulation scheme, µ is the modulation order and κ l (α j ) ∈ {0, 1} is the value of the l th bit of the constellation point α j . The soft data symbol is an expected value taking the a priori probabilities (12) into account:

X[i, k] = αj ∈Ψ α j • P (X[i, k] = α j ). (13)
Note that, being different from the classical decision feedback channel estimation methods [START_REF] Tang | Novel decision-aided channel estimation for TDS-OFDM systems[END_REF], [START_REF] Li | Pilot-symbol-aided channel estimation for OFDM in wireless systems[END_REF], no decision is made here in order to prevent error propagation. Directly using the soft data symbols, an instantaneous channel estimate is obtained over all active subcarriers by: where

H2 [i, k] = 1 E X [i, k] X[i, k] * Y [i, k] = X[i, k] * X[i, k] E X [i, k] H[i, k]+ 1 E X [i, k] X[i, k] * W ′ [i, k], (14)
E X [i, k] = | X[i, k]| 2
is the power of the (i, k)th soft data symbol which is used as power normalization factor. In contrast to the uniform-power modulation cases e.g. BPSK and QPSK where the normalization factor is an accurately known value [START_REF] Liu | A combined time and frequency algorithm for improved channel estimation in TDS-OFDM[END_REF], the normalization factor should be estimated in the non-uniform-power modulation cases e.g. 16QAM and 64QAM. If the data symbols are perfectly rebuilt (e.g. signalto-noise ratio is high), namely X[i, k] = X[i, k], (14) turns to an LS estimator:

H2 [i, k] = H[i, k] + W ′ [i, k] E X [i, k] . (15)
Since the rebuilt soft data symbols X are affected by the noise as well as the channel fading, the instantaneous CFR estimates (14) are inaccurate for some subcarriers. (15) indicates that the estimation error is reciprocally proportional to the power of the data symbol used to make the estimation. It means that the CFR estimates obtained using the data symbols with higher power are more reliable. Therefore, in order to achieve better estimation results, we propose to set a power threshold D to select data symbols with higher power for channel estimation as shown in Fig. 2. Note that the threshold is used to select the reliable data symbols but not to make hard decision to the data symbols. The side effect of this selection is that the CFR estimates are not available for all subcarriers and the available CFR estimates locate on random positions.

It is known that given sufficient time-frequency samples, the CFR can be rebuilt using some interpolation techniques, e.g. linear, FFT-based, MMSE interpolations. Among all methods, the MMSE interpolation based on Wiener filtering provides the best mean squared error (MSE) performance [START_REF] Dong | Linear interpolation in pilot symbol assisted channel estimation for OFDM[END_REF]. However, the randomly located CFR estimates require to compute the coefficients of the Wiener filter by means of matrix inversion for each OFDM symbol. The computational complexity is unaffordable when the FFT size is large. Fortunately, as both the time delay spread and Doppler spectrum of the channel are limited, the CFR is highly correlated, i.e. almost identical within the coherence bandwidth and coherence time [START_REF] Rappaport | Wireless Communications: Principles and Practice[END_REF]. Moreover, the coherence bandwidth spreads over several adjacent subcarriers in the OFDM system with large FFT size, and the coherence time is normally longer than several OFDM symbol durations in low and medium velocity cases. This enables us to carry out a 2-D estimate refinement and interpolation to provide a better channel estimation.

B. 2-D CFR Estimate Refinement

The 2-D estimate refinement and interpolation is carried out over a number of OFDM symbols within an interpolation block as shown in Fig. 3. The block size B, i.e. the number of OFDM symbols in the block, is chosen according to the latency, storage and complexity considerations. Suppose that the set of indices of available CFR estimates in an interpolation block is Θ. For low-complexity consideration, some indices (i p , k p ) ∈ Ξ are pre-selected as 'virtual pilots', where Ξ is the set of time-frequency indices of virtual pilots. Then, a refined CFR estimate for a virtual pilot is computed by averaging all the available CFR estimates within the coherence region bounded by the coherence bandwidth and time, as shown in Fig. 3. The size of the coherence region is chosen less than or equal to the coherence bandwidth and time. It can be either empirically pre-selected according to the worst case, or adaptively chosen by using the information obtained in the initial PN based channel estimation. The refined CFR estimate for the pilot located at (i p , k p ) is:

H2 [i p , k p ] = 1 M p,q∈Θ ip ,kp H2 [p, q] = H[i p , k p ] M p,q∈Θ ip ,kp X[p, q] * X[p, q] E X [p, q] + W ′ M p,q∈Θ ip ,kp X[p, q] * E X [p, q] ≃ H[i p , k p ] + W ′ M p,q∈Θ ip ,kp X[p, q] * E X [p, q] , (16)
where Θ ip,kp ∈ Θ is the subset of available CFR estimates within the coherence region with (i p , k p ) its central position and M is the cardinality of Θ ip,kp . The variance of the estimation error in ( 16) is then:

σ 2 H2 [i, k] = E{|H[i, k] -H2 [i, k]| 2 } = E{| W ′ M p,q X[p, q] E X [i, k] | 2 } = σ 2 W ′ M 2 p,q | X[p, q] E X [p, q] | 2 = σ 2 W ′ M 2 p,q 1 E X [p, q] (17)
Particularly, if only the data symbols with power higher than 1, i.e. E X [p, q] ≥ 1, are used for estimation, the estimation error is bounded by σ 2 H2 ≤ σ 2 W ′ . By repeating this process over all virtual pilots in the interpolation block, a number of refined CFR estimates are obtained. With these reliable estimates, the overall CFR estimate can thus be obtained via interpolation.

C. 2-D Wiener Filtering based Interpolation

The 2-D Wiener filtering outperforms other interpolation techniques, e.g. linear interpolation, FFT-based interpolation etc. However its computational complexity is very high. The use of two concatenated 1-D Wiener filters, i.e. one in frequency domain and then the other in time domain, significantly reduces the computational complexity with negligible performance degradation compared to the optimum 2-D Wiener filter [START_REF] Hoeher | Pilot-symbol-aided channel estimation in time and frequency[END_REF]. The 2×1-D Wiener filtering is represented as:

Ĥ2 [i, k] = ip ω t [i, k, i p ] time domain kp ω f [k, i p , k p ] frequency domain H2 [i p , k p ], (18)
where ω t 's and ω f 's are the coefficients of the 1-D Wiener filter in time and frequency domains, respectively, while

K f = ⌊ N L f ⌋ and K t = ⌊ B
Lt ⌋ are the number of virtual pilots within one OFDM symbol and the number of OFDM symbols including virtual pilots, respectively. L f and L t are the pilot spacings in frequency and time domains. The coefficients of frequency domain Wiener filter can be computed as:

ω f = Φ -1 f θ f , (19)
where

Φ f = E{ H2 HH 2 } = (R (1)
f + σ 2 H2 I) is the K f × K f autocorrelation matrix of CFR estimates on pilot positions obtained in (16), I is identity matrix. θ f = E{ H2 H} = R [START_REF]Framing Structure, Channel Coding and Modulation for Digital Television Terrestrial Broadcasting System[END_REF] f is the K f × N cross-correlation matrix of CFR estimates H2 and the real CFR H. R [START_REF] Tang | Iterative channel estimation for block transmission with known symbol padding -a new look at TDS-OFDM[END_REF] f and R [START_REF]Framing Structure, Channel Coding and Modulation for Digital Television Terrestrial Broadcasting System[END_REF] f are the autocorrelation matrix of the real CFR in frequency domain, and its components can be computed by [START_REF] Tang | Novel decision-aided channel estimation for TDS-OFDM systems[END_REF]. The MSE of the CFR estimate after frequency domain filtering is:

ε f = 1 N Tr R (3) f -R (2)T f Φ -1 f R (2) * f , ( 20
)
where Tr(•) is the trace operation, R

f (3)
is the N × N autocorrelation matrix of CFR. Similarly, the coefficients of the time domain Wiener filter can be computed as:

ω t = Φ -1 t θ t , (21)
where Φ t and θ t have the same definition as Φ f and θ f except that the components of autocorrelation matrix of CFR are computed according to [START_REF] Muck | A pseudo random postfix OFDM modulator and inherent channel estimation techniques[END_REF]. The MSE of the CFR estimate after frequency domain filtering is:

ε Ĥ2 = 1 B Tr R (3) t -R (2)T t Φ -1 t R (2) * t . ( 22
)
Since the positions of the virtual pilots are pre-selected, the matrix inversion Φ -1 f and Φ -1 t in the filter coefficients computation can be pre-computed and used for all OFDM symbols to reduce the computational complexity.

D. MMSE Combination

When the PN based and data-aided channel estimates are obtained, a linear combination is proposed to get an improved CFR estimate:

Ĥ = β Ĥ1 + (1 -β) Ĥ2 . ( 23
)
For the DTMB system, since all subcarriers are active, this combination is carried out over all subcarriers. Otherwise, the combination is only made for the active subcarriers and the CFR estimates for null subcarriers are kept as the results obtained from the PN based one. The optimum weight values β opt can be obtained using the MMSE criteria:

β opt = arg min β E{|H -Ĥ| 2 } = arg min β β 2 ε Ĥ1 + (1 -β) 2 ε Ĥ2 . ( 24
)
The above equation uses the fact that Ĥ1 and Ĥ2 are obtained from different sources and thus uncorrelated. Since the MSE function is convex, the β opt is obtained by setting the derivative with respect to β equal to zero and the solution is:

β opt = ε Ĥ2 ε Ĥ1 + ε Ĥ2 . ( 25
)
After a given maximum number of iterations, the combined CFR (23) estimate is output as the final CFR estimate. Otherwise it is used as the initial channel estimate in the next iteration for PN subtraction and equalization.

E. Complexity Analysis

The additional computational complexity of the proposed channel estimation method is evaluated in terms of required real multiplications and real additions for each OFDM symbol. The data rebuilding process including [START_REF] Dong | Linear interpolation in pilot symbol assisted channel estimation for OFDM[END_REF], ( 12) and ( 13

K f Lt + 2K t ) multiplications and N • (2 log 2 µ + 6 + 2 K f
Lt + 2K t ) additions are needed, which is between O(N ) and O(N 2 ). Interestingly, the proposed blocktype 2-D channel estimation method has a complexity less than the symbol-by-symbol 1-D processing manner [START_REF] Liu | A combined time and frequency algorithm for improved channel estimation in TDS-OFDM[END_REF] which needs N • ((µ + 2) log 2 µ + 14 + 2K f ) + 2K f multiplications and N • (2 log 2 µ + 8 + 2K f ) additions. Even lower complexity can be achieved by reducing the size of Wiener filter as in [START_REF] Dong | Linear interpolation in pilot symbol assisted channel estimation for OFDM[END_REF].

IV. SIMULATION RESULTS

Simulation parameters are chosen from the DTMB system [START_REF]Framing Structure, Channel Coding and Modulation for Digital Television Terrestrial Broadcasting System[END_REF]. The length of the GI is set to ν = 420. Both the convolutional interleaver and LDPC code are included in the Fig. 4. MSE of the proposed data-aided channel estimation for 16QAM after 2 iterations in TU-6 channel with velocity of 30 km/h. simulation in order to perform a system level evaluation. The COST207 Typical Urban (TU-6) channel model is employed in the simulation. The coherence bandwidth of TU-6 channel is 18.8 kHz equivalent to 9 subcarrier spacings. The mobile speed is set to 30 km/h and 120 km/h which correspond to maximum Doppler frequency 13.89 Hz and 55.56 Hz, respectively, given the carrier frequency 500 MHz. The coherence time is 0.0305 s and 0.0076 s equivalent to 61 and 15 OFDM symbols, respectively. Without loss of generality, the size of the coherence region is set equal to the pilot spacing. The data symbol selection threshold D is chosen to be 0.9 for 16QAM and 0.75 for 64QAM in the simulation, respectively. Fig. 4 presents the MSE performance of proposed method adopting rectangular shape, diamond shape and random shape pilot patterns. All the three cases use the same time domain Wiener filter. The difference lies in the pilot arrangement in frequency domain. In the rectangular shape case, the pilot position is fixed with a spacing of L f subcarriers for all OFDM symbols that carry virtual pilots, while in the diamond shape case, the pilot position circularly shifts L f 2 from one OFDM symbol to the next as shown in Fig. 3. Note that the latter is known as the optimum pilot pattern for channel estimation [START_REF] Choi | Optimum pilot pattern for channel estimation in OFDM systems[END_REF]. As for the random pilot shape case, all available instantaneous CFR estimates are fed to the frequency domain 1-D Wiener filter to compute the CFR over all subcarriers, and no averaging is performed before filtering. It is the case that uses the maximum number of instantaneous CFR estimates.

It can be seen that the performance of random pilot pattern method is worse than that of fixed pattern while the computational complexity of the random pilot pattern method is much higher than that of fixed pattern method. It can be explained by the fact that in the fixed pattern case, the instantaneous CFR estimates are averaged before interpolation, and the accuracy of CFR estimates on virtual pilot positions significantly affects the MSE of overall CFR after Wiener filtering. The rectangular shape pilot pattern has almost the same performance as diamond shape pattern. Hence the rectan- gular shape pattern is used in the following simulations for the sake of implementation simplicity. When signal-to-noise ratio (SNR) is low, since the rebuilt data symbols are not reliable, performing average in larger averaging region provides better performance, e.g. the cases of L t = 8, L f = 9 are better than L t = 4, L f = 9. On the contrary, the rebuilt data symbols are quite accurate when SNR is high. Thus, small average region can sufficiently reduce the estimation error, while smaller pilot spacing provides better interpolation results. For instance, the performance floor, which mainly comes from the interpolation error and the channel variation within the coherence region, is lower in L t = 4, L f = 9 case than that in L t = 8, L f = 9 case. Interestingly, as no averaging over several consecutive OFDM symbols is performed in the random pilot shape case, the channel variation does not introduce estimation error. Therefore, there is no performance floor at high SNR with the random pilot shape.

Fig. 5 and6 give the MSE performance of the proposed method in TU-6 channel with a velocity of 30 km/h with Fig. 8. BER of the DTMB system using the proposed data-aided channel estimation for 16QAM and 64QAM after 2 iterations in TU-6 channel with velocity of 120 km/h. Rectangular shape pilot pattern with Lt = 2, L f = 9.

Interpolation block length B = 21. LDPC code rate 0.6.

16QAM and 64QAM, respectively. For both cases, the proposed method provides satisfactory performance after only two iterations. The performance with iterations more than three converges. Therefore, the iteration time is set to two in the BER performance evaluation in this paper. Not surprisingly, the proposed method performs better with 16QAM given the same SNR, because the rebuilt data symbols are more reliable in this case. The proposed method provides 7.8 dB gain in terms of required SNR to achieve MSE of 1 × 10 -3 over PN based channel estimation method with 16QAM. From Fig. 6, the gain is accordingly 6.9 dB with 64QAM given MSE of 1 × 10 -3 . As higher order modulation normally requires more accurate channel estimation, the gain is also evaluated with a lower MSE level, say 2 × 10 -4 . The gain over PN based channel estimation increases to 8.7 dB. Fig. 7 and 8 present the bit error rate (BER) of the DTMB system using the proposed method with velocities of 30km/h and 120km/h, respectively. Interestingly, performance using the proposed method is very close to that using perfect channel estimation. Only 0.1 to 0.2 dB gaps can be found. The proposed method provides more than 0.5 dB gain over the PN based estimation method in terms of required SNR to achieve a BER of 5×10 -5 . Considering that the LDPC code with code rate 0.6 is used in the simulation, this improvement is quite significant. Even more gain can be expected if a weaker channel coding is adopted. The latest channel estimation method dedicated to TDS-OFDM proposed in [START_REF] Tang | Novel decision-aided channel estimation for TDS-OFDM systems[END_REF] is also simulated for comparison. It needs a computational complexity equivalent to the proposed method. As [START_REF] Tang | Novel decision-aided channel estimation for TDS-OFDM systems[END_REF] does not give the partial decision boundary for 64QAM, for the fairness of the comparison, only the performance for 16QAM is presented. The results are obtained with three iterations in the channel estimation. It can be found that the proposed method outperforms the method in [START_REF] Tang | Novel decision-aided channel estimation for TDS-OFDM systems[END_REF] 0.7 dB and 0.5 dB with velocities of 30km/h and 120km/h, respectively.

V. CONCLUSION

This paper proposes an iterative data-aided channel estimation algorithm with reduced complexity. In contrast to turbo channel estimation, the proposed algorithm does not include the decoding and interleaving functions when rebuilding the data symbols. A new joint 2-D channel estimate refinement and interpolation is proposed to provide efficient estimation performance. Moreover, the proposed algorithm is adaptable to the non-uniform-power modulation cases as exemplified by the given 16QAM and 64QAM results. Interestingly, the BER performance obtained when applying our method to the DTMB system is not only superior to that using the classical PN based method and the latest method in literature [START_REF] Tang | Novel decision-aided channel estimation for TDS-OFDM systems[END_REF],

but also close to that with perfect channel estimation, which demonstrates the efficiency of the algorithm.

Fig. 1 .

 1 Fig. 1. Block diagram of TDS-OFDM system. The shaded blocks are the processings used for the proposed method. The dashed blocks are the additional processings needed for the turbo channel estimation.

D

  (b) 64QAM (first quadrant).

Fig. 2 .

 2 Fig.2. Select the constellation points with high power (marked black). The values in the bracket are the normalized power of the constellation points.

Fig. 3 . 2 -

 32 Fig. 3. 2-D average in time and frequency, with diamond shape pilot pattern.

  ) needs (N • µ • log 2 µ + N • 2 log 2 µ) multiplications and N • 2 log 2 µ additions. The instantaneous data-aided estimation (14) needs 6N multiplications and 2N additions. The averaging in coherence region (16) requires 2N additions and a few times of division which are neglected. The frequency domain Wiener filtering needs 2N • K f Lt additions and multiplications, while the time domain Wiener filtering requires 2N • K t additions and multiplications. The MMSE combination needs 4N multiplications and 2N additions. In all, N • ((µ + 2) log 2 µ + 10 + 2

  Fig. 5. MSE performance of the proposed data-aided channel estimation method with 16QAM in TU-6 channel with velocity of 30km/h. Rectangular shape pilot pattern with Lt = 4, L f = 9. Interpolation block length B = 21.

Fig. 6 .

 6 Fig.[START_REF] Zhao | Iterative turbo channel estimation for OFDM system over rapid dispersive fading channel[END_REF]. MSE performance of the proposed data-aided channel estimation method with 64QAM in TU-6 channel with velocity of 30km/h. Rectangular shape pilot pattern with Lt = 4, L f = 9. Interpolation block length B = 21.

  BER of the DTMB system using the proposed data-aided channel estimation for 16QAM and 64QAM after 2 iterations in TU-6 channel with velocity of 30 km/h. Rectangular shape pilot pattern with Lt = 4, L f = 9. Interpolation block length B = 21. LDPC code rate 0.6.

	10 -1
	10 -2
	BER
	10 -3				16QAM			64QAM
					perfect CIR
					data-aided channel estimation
					PN based channel estimation
	10 -4				method in [5]
	12	13	14	15	16	17	18	19	20	21	22
						SNR (dB)
	10 -4 10 -2 10 -1 Fig. 7. 12 BER 10 -3	13	14	15	16 perfect CIR 17 data-aided channel estimation 18 19 PN based channel estimation method in [5] 16QAM 64QAM	20	21	22
						SNR (dB)

It is also termed as known symbol padding (KSP)-OFDM[START_REF] Steendam | Different guard interval techniques for OFDM: performance comparison[END_REF] and pseudo random postfix (PRP)-OFDM[START_REF] Muck | A pseudo random postfix OFDM modulator and inherent channel estimation techniques[END_REF].


Jean-Baptiste Bellet
email: bellet@cmap.polytechnique.fr

Gérard Berginc
email: gerard.berginc@fr.thalesgroup.com

Imagerie laser

Nous élaborons une méthode originale d'imagerie directe d'un objet. Il s'agit d'une méthode de type dérivée topologique, destinée à inverser des mesures contenant l'amplitude des ondes rétrodiffusées à la suite d'illuminations laser.

Introduction

L'identification d'objets par un système d'imagerie laser est un sujet aux domaines d'application variés, comme la détection d'engins explosifs improvisés (IED) ou de tumeurs dans le derme. Dans ce contexte motivant, nous élaborons une méthode d'imagerie directe (non itérative) par un tel système. Ainsi, nous développons un algorithme d'inversion de mesures d'ondes rétrodiffusées par l'objet à identifier, à la suite d'illuminations laser. Cette technique est monofréquentielle : une seule fréquence est exploitée ; elle est monostatique : les sources d'illumination et les récepteurs sont confondus. Comme en pratique la vue est partielle (une partie de l'objet n'est pas illuminée), nous supposerons que les mesures sont prises sur une droite. Enfin, comme pour les applications, nous supposerons que les mesures contiennent l'amplitude (ou l'intensité), mais ne contiennent pas la phase. Comme les méthodes d'inversion directes usuelles (MUSIC [START_REF] Ammari | An Introduction to Mathematics of Emerging Biomedical Imaging[END_REF], backpropagation method, Kirchhoff migration) reposent sur la connaissance de la phase, le principal intérêt et l'originalité de notre méthode concerne ce point.

Pour ajouter de l'information, de façon non restrictive par rapport aux applications, nous supposons que l'on peut prendre les mesures sur une autre droite, parallèle à la première. Ainsi, pour chaque angle d'éclairage, nous disposons de deux mesures d'amplitude prises dans la même direction. En approximation haute fréquence, le passage d'une mesure d'amplitude à l'autre s'obtient par une équation de transport ; ceci cache une certaine notion de phase dans le rapport entre les deux mesures. L'idée est de trouver les petits disques, qui, éclairés, rétrodiffusent une onde dont la phase permet de ≪ transporter ≫ les premières mesures vers les deuxièmes. En plus d'utiliser directement les mesures, cette idée permet d'exploiter l'information cachée au sujet de la phase. Cette méthode peut être formulée à l'aide d'un coût que l'on cherche à rendre petit. La ≪ validité ≫ d'un point est alors testé par la valeur de la dérivée topologique du coût : un point est bien placé par rapport à l'objet recherché si cette dérivée est très négative. Il s'agit ainsi d'une approche par analyse asymptotique topologique, dans le même esprit que celle conduite dans [START_REF] Auroux | Image processing by topological asymptotic expansion[END_REF][START_REF] Ammari | Imaging schemes for perfectly conducting cracks[END_REF].

∇ • 1 µ * 1 1 D + 1 µ 0 1 1 R 2 \D ∇u + ω 2 (ε * 1 1 D + ε 0 1 1 R 2 \D )u = 0, avec une condition de radiation sortante sur l'onde diffractée u dif [D](η, x) := u[D](η, x) -u inc (η, x), x ∈ R 2 \ D.
Numériquement, ce problème est résolu par la méthode des équations intégrales, comme suggéré dans [START_REF] Bellet | Modèle électromagnétique d'objet dissimulé[END_REF].

1.2 Problème inverse. On fixe deux droites de mesures parallèles en champ lointain : Γ 0 :

x 2 = γ 0 et Γ 1 : x 2 = γ 1 ,
A 1 [D ε (z)](x η ) = A 0 [D * ](p) γ 0 -z 2 γ 1 -z 2 1/2 1 -ε t η 4 |x η -z| |p η -z| + O ε 2 où x η = z + γ 1 -z 2 η 2 η = Γ 1 ∩ (z, η), et t η = -γ 1 -γ 0 η 2
est le temps de trajet.

Paramétrons par x = (x 1 , γ 1 ) :

η(x) = z-x |z-x| , p(x) = z + γ 0 -z 2 γ 1 -z 2 (x -z), et posons a 0 (x, z) := A 0 [D * ](p(x)) γ 0 -z 2 γ 1 -z 2 1/2 , a 1 (x, z) := -1 4 a 0 (x, z) γ 1 -γ 0 γ 0 -z 2
(z) = 1 a 0 (•, z) -A 1 [D * ] a 0 (•, z) -A 1 [D * ], a 1 (•, z) .

1 Position du problème 1 . 1

 11 Problème direct. On considére l'espace libre R 2 avec permittivité ε 0 > 0 et perméabilité µ 0 > 0. On ajoute dans ce milieu une inclusion D de permittivité ε * > 0 et perméabilité µ * > 0. D est supposée près de l'origine. On éclaire D par un laser, i.e., une onde incidente plane d'angle unitaire η, de pulsation ω > 0 : u inc (η, x) = e ik 0 η•x . Le nombre d'onde associé est k 0 := ω √ ε 0 µ 0 , et la longueur d'onde du laser est λ := 2π k 0 . Le nombre d'onde associé à D est k * := ω √ ε * µ * . L'onde qui se propage est alors u[D](η, x) = u(x), solution de :

  où les constantes γ 0 , γ 1 > 0 sont grandes par rapport à la longueur d'onde, avec γ 0γ 1 raisonnable (ni trop grand, ni trop petit). En éclairant l'objet D avec l'angle x/ |x|, l'amplitude rétrodiffusée enx est A[D](x) = |u dif [D](x/ |x| , x)|, notée A 0 [D] et A 1 [D] respectivement sur Γ 0 et Γ 1 .On suppose qu'un objet D * est situé à proximité de l'origine, avec pour paramètres électromagnétiques ε * > 0 et µ * > 0 connus. On suppose que l'on mesure A 0 [D * ] et A 1 [D * ]. Le problème inverse que l'on souhaite résoudre consiste à retrouver l'objet D * à partir des mesures, en inversant A[D * ] par A, i.e. en résolvant A[D] = A[D * ], sur Γ 0 ∪ Γ 1 . 2 Méthode de dérivée topologique 2.1 Démarche. Par approximation haute fréquence, on approche l'amplitude A 1 [D * ] en transportant l'amplitude A 0 [D * ], avec une certaine phase ϕ[D * ] : si A satisfait l'équation de transport : ∇A • ∇ϕ[D * ] + 1 2 A∆ϕ[D * ] = 0, avec A = A 0 [D * ] sur Γ 0 , alors la trace de A sur Γ 1 approche A 1 [D * ]. D'autre part, en transportant A 0 [D * ], avec la phase connue ϕ[D ε (z)] issue de l'onde diffractée par un disque D ε (z) de centre z et de petit rayon ε > 0 : ∇A • ∇ϕ[D ε (z)] + 1 2 A∆ϕ[D ε (z)] = 0, avec A = A 0 [D * ] sur Γ 0 , alors on obtient sur Γ 1 une amplitude A 1 [D ε (z)]. On a alors l'intuition que z est un point bien positionné par rapport à l'objet D * lorsque l'amplitude transportée A 1 [D ε (z)] est proche de l'amplitude mesurée A 1 [D * ]. L'idée originale de reconstruction que nous proposons consiste à chercher les z qui rendent petite la quantité A 1 [D ε (z)] -A 1 [D * ] . Nous obtenons une expression asymptotique de A 1 [D ε (z)] par la méthode des caractéristiques. Puis un développement asymptotique de A 1 [D ε (z)] -A 1 [D * ] permet d'identifier une dérivée topologique à rendre très négative pour estimer D * . 2.2 Dérivée topologique. La résolution de l'équation de transport par la méthode des caractéristiques conduit au théorème suivant. Théorème 2.1 Pour tout p ∈ Γ 0 , en posant η = z-p |z-p| l'angle d'éclairage, le transport de la mesure A 0 [D * ](p) vers Γ 1 , le long de la direction de rétropropagation (z, η), avec phase ϕ[D ε (z)](η, •), conduit à l'amplitude :

1

  |x-z| . Pour un domaine D, notons A 1 [D] l'amplitude issue du transport de A 0 [D * ] sur Γ 1 , dans la direction de rétropropagation, avec la phase ϕ[D] de l'onde diffractée par D. Le théorème précédent justifie le résultat asymptotique suivant : Théorème 2.2 (Dérivée topologique) La dérivée topologique de l'applica-tion D -→ A 1 [D] -A 1 [D * ] , évaluée en z, dans la direction le disque unité, est : d

2. 3

 3 Méthode d'inversion directe. Si le petit disque D ε (z) est bien placé, i.e. ≪ près ≫ de D * alors la phase ϕ[D ε (z)] approche la phase ϕ[D * ], et donc A 1 [D ε (z)] approche A 1 [D * ]. L'approche originale que nous proposons pour reconstruire la forme D* consiste à minimiser D → A 1 [D] -A 1 [D * ] ,parmi les petits disques. Le développement asymptotique du théorème :A 1 [D ε (z)] -A 1 [D * ] = a 0 (•, z) -A 1 [D * ] + εd(z) + O ε 2 , suggère que A 1 [D ε (z)] -A 1 [D * ]est petit lorsque la dérivée topologique d(z) est négative et très grande en valeur absolue. Pour estimer D * , on trace ainsi les lignes de niveau de z -→ d(z), puis on en choisit une de niveau d 0 arbitraire. Celle-ci peut servir de point départ pour un algorithme itératif.

Remerciements

Les auteurs remercient très chaleureusement Habib Ammari pour ses conseils et suggestions.