Mihai I Stanciu

Stéphane Azou
email: azou@enib.fr

Emanuel Rȃdoi

Alexandru Şerbȃnescu

A Statistical Analysis of Multipath Interference for Impulse Radio UWB Systems

Keywords: Impulse Radio, Time-Hopping, Chip duration, Multipath Interference, Statistical Analysis, IEEE 802.15.4a, Channel Model

In this paper, we develop a statistical characterization of the multipath interference in an Impulse Radio (IR)-UWB system, considering the standardized IEEE 802.15.4a channel model. In such systems, the chip length has to be carefully tuned as all the propagation paths located beyond this limit can cause interframe/intersymbol interferences (IFI/ISI). Our approach aims at computing the probability density function (PDF) of the power of all multipath components with delays larger than the chip time, so as to prevent such interferences. Exact analytical expressions are derived first for the probability that the chip length falls into a particular cluster of the multipath propagation model and for the statistics of the number of paths spread over several contiguous clusters. A power delay profile (PDP) approximation is then used to evaluate the total interference power as the problem appears to be mathematically intractable. Using the proposed closed-form expressions, and assuming minimal prior information on the channel state, a rapid update of the chip time value is enabled so as to control the signal to interference plus noise ratio.

I. INTRODUCTION

Since its approval by the U.S. Federal Communications Commission (FCC) in 2002, UWB has given rise to considerable interest in wireless communications research community, due to its many attractive properties [START_REF] Arslan | Ultra Wideband Wireless Communication[END_REF]. Today, IR is considered as a promising technique from an industrial perspective [START_REF] Willig | Recent and Emerging Topics in Wireless Industrial Communications: A Selection[END_REF]; in particular, it is a main candidate solution for applications such as wireless sensor networks [START_REF] Zhang | UWB systems for wireless sensor networks[END_REF] due to its ability to provide joint data transmission and precise positioning [START_REF] Gezici | Localization via ultra-wideband radios: a look at positioning aspects of future sensor networks[END_REF]. However, a number of IR-UWB system design challenges remain to be solved to ensure a broader use of that technology in practice [START_REF] Lampe | Challenges and recent advances in IR-UWB system design[END_REF].

The very fine time resolution in IR transmission coupled with the rich multipath diversity of the UWB channels need a careful signal and architecture design to achieve good performances at reasonable complexity. The large path delay spread may require sophisticated signal processing algorithms at the receiver to cope with IFI/ISI. This can be avoided if we increase the minimum delay between two consecutive time hopping code values, that is to say if we decrease the chip rate (and thus the bit rate), or if we don't make use of consecutive code elements (corresponding to a time lag of one chip). The multiuser interference (MUI) has also to be taken into account in system design and some schemes limiting its impact on performances may be required.

Considering the chips as adjacent channels available for multiple users, the main cause of interference between channels is the time spread of the channel, if it exceeds the chip duration. In particular, it is shown in [START_REF] Hu | Accurate evaluation of multiple access performance in TH-PPM and TH-BPSK UWB systems[END_REF] that performance of correlator-based TH-PPM and TH-BPSK receivers is severely degraded by the MUI. Thus, given a particular channel profile, any information on the power of interfering multipath components (MPCs) can provide valuable assistance either for evaluating the signal to interference plus noise ratio (SINR) and the performance in terms of bit error rate, or for adapting the system parameters to the propagation conditions. This problem has been investigated only in a limited number of papers. Approximate analytical expressions for Signal to ISI power ratio was first derived in [START_REF] Piazzo | On the Inter-Symbol-Interference in several Ultra Wideband systems[END_REF] for various UWB transmission formats over two paths channels. In [START_REF] Deleuze | Inter-symbol/Inter-Frame Interference in Time-Hopping Ultra Wideband Impulse Radio System[END_REF], a closed-form expression of the ISI/IFI energy at the output of a rake receiver is stated by considering the IEEE 802. 15.3a statistical channel model [START_REF] Molisch | Ultra-Wide-Band Propagation Channels[END_REF] with one cluster. The influence of various parameters (time-hopping code, rake number of fingers, guard-time size) on the performance is conducted, hence enabling a pertinent design of the system. Even if IFI is an important issue, especially for high data rate UWB systems, [START_REF] Ahmed | Inter-Frame Interference in Time Hopping Impulse Radio Based UWB Systems for Coherent Receivers[END_REF] considers that its influence can eventually be neglected in BPSK or PPM time-hopping systems under certain conditions : a large number of frames per symbol is required and the considered scenario neglects ISI and multiple access (transmission of a single symbol for a single user). Gezici et al.

established in [15] that there is a tradeoff between the pulse combining gain N f and the pulse spreading gain N c = T f /T c to get a minimal bit errror probability in presence of timing jitter, for frequency-selective environments, as IFI is mitigated for larger values of N c , the effect of timing jitter being mitigated by increasing N f . A different approach is proposed in [START_REF]Statistical Analysis of the IEEE 802.15.4a UWB PHY over Multipath Channels[END_REF] to analyse the influence of channel and system parameters : by computing the first and second order moments of the received UWB pulses, the authors can express the intra-burst interference resulting from multipath propagation. In [START_REF] Rahman | Analysis and Comparison of Inter-Symbol/Frame Interference in Pulsed DS-and Hybrid DS/TH-UWB Communications[END_REF], the authors derive a few analytical expressions for IFI/ISI in pulsed direct sequence (DS) and hybrid DS/TH UWB communications, considering a frequency selective Nakagami fading channel. With standard Gaussian approximation of ISI/IFI, an expression of the error probability is given for a MRC rake combiner and the effects of various system parameters on the performances are studied.

In this paper, we seek to express the probability density function (PDF) of the power of the whole set of MPCs having delays exceeding the chip length, given a particular channel profile stemming from a prior identification step [START_REF] Venkatesh | Non-line-of-sight identification in ultra-wideband systems based on received signal statistics[END_REF]. We consider in particular the statistical channel model adopted by the IEEE 802.15.4a Task Group1 [START_REF] Molisch | A comprehensive standardized model for ultrawideband propagation channels[END_REF] to derive an analytical form of the PDF with some approximations. Proposed method may facilitate the characterization of a radio link in terms of SINR.

Our theoretical analysis involves the following intermediate steps to achieve the interference power characterization : first, we derive the probability that the chip length falls into a particular cluster of the multipath propagation model; then, the statistics of the number of paths in a cluster is expressed, which yields the statistics of the number of paths spread over several contiguous clusters; finally, we obtain an approximate relationship of the PDF of the power of all interfering MPCs using a simplified power delay profile expression. Due to dense multipath propagation and the non-stationary nature typical to UWB channels, the level of interference can change very quickly and an adaptation of some key parameters of the modulation may be required so as to control IFI/ISI (or multiuser interference) according to the environment in which the system is operating. Such possibility has been recently investigated in [START_REF] Molisch | Adaptive Frame Durations for Time-Hopped Impulse Radio Systems[END_REF], where the frame duration is periodically updated depending on the channel state information and SINR, which are measured using training sequences. The development proposed here can serve as a tool for a rapid performance assessment of a TH UWB radio link; as only a LOS/NLOS detection is required as prior information, our result could possibly be applied in a non-data aided scenario.

The paper is organized as follows. Section II briefly recalls some properties of the IEEE 802.15.4a propagation channel and specifies the problem of statistical analysis of the interference power. Then, in section III, we derive the statistics of the first cluster matching the chip length and the number of MPCs is investigated in section IV. A power delay approximation then follows in section V and a procedure for interference power estimation is developed in section VI. A few numerical results are then discussed before some conclusions to confirm the pertinence of our theoretical analysis.

II. SYSTEM MODEL AND PROBLEM STATEMENT

We consider here a Time-Hopping Binary Pulse Amplitude Modulation (BPAM) format, with the following typical expression of the transmitted signal :

s tx (t) = E s N f ∞ p=-∞ d p/N f w tx (t -pT f -c p T c ) , (1)
where E s denotes the symbol energy and d p/N f ∈ {-1, 1} are the binary transmitted symbols, T f stands for the average period of the pulse train (also known as the frame time), the c p ∈ {0, 1, ..., N c -1} represent pseudo-random code elements required both for code division multiple access and spectral shaping purposes, the quantification of temporal hops being controlled by the chip length T c . The system is designed so that the unit energy pulse w tx (t) is confined within duration T c and we have T f = N c T c . The indoor propagation channel usually involves multiple reflections due to the objects in the vicinity of receiver and transmitter; a classical way to characterize such propagation conditions is to make use of the Saleh-Valenzuela (SV) channel model [START_REF] Molisch | Ultra-Wide-Band Propagation Channels[END_REF] whose basic assumption is that MPCs arrive in clusters, with complex variations of the received power due to path loss, large scale fading and small scale fading. The corresponding discrete-time impulse response is usually expressed as

h(t) = L l=0 K k=0 a k,l exp(jφ k,l )δ(t-T l -τ k,l ) (2)
where a k,l denotes the tap weight of the k-th component in the l-th cluster, τ k,l is the delay of the k-th MPC relative to the l-th cluster arrival time T l and the phases φ k,l being uniformly distributed in the range [0, 2π]; δ(t) is the Dirac delta function.

In case the maximum delay spread of the channel is larger than the chip time, the transmitted signal corresponding to one pulse may overlap with signals in some next frames, as shown in Fig. 1, thus causing IFI/ISI. Our objective is to derive a closed form expression of the power lying in time bins with delays beyond the chip length, with the aim of controlling the SINR.

As will be seen in the sequel, the problem rapidly becomes intractable due to the complexity of the mathematical relations involved; a few approximations will then be proposed, in such a way that the SINR is overestimated.

III. STATISTICS OF THE FIRST CLUSTER MATCHING THE CHIP LENGTH

In this section, we turn our attention to the probability P (T c ∈ C ) that the chip length "falls into the -th cluster" 2

C = [T -1 , T ), that is the probability that T -1 ≤ T c ≤ T .
To achieve this goal, it is required to compute the PDF

f (x) = x 0 f (x|T 0 ) f (T 0 ) dT 0 of the cluster arrival time T , supposing that f (T 0 ) = Λ 0 exp (-Λ 0 T 0 ), T 0 ≥ 0.
Using the expression T = j=1 ∆T j + T 0 and considering that distributions of the cluster arrival times are given by a Poisson processes with cluster arrival rate

Λ, i.e. f ∆T (x) = Λexp (-Λx) where ∆T = T -T -1 , we get f (x) = Λ 0 Λ exp (-Λx) ( -1)! x 0 (x -t) -1 exp (-(Λ 0 -Λ) t) dt. (3)
Then, by observing that integral of the form

I l,b (x) = x 0 (x -t) l exp (-bt) dt satisfies the recursive relation 3 I l,b (x) = x l b -l b I l-1,b (x) with b = Λ 0 -Λ, we can easily show that f (x) = Λ 0 Λ exp (-Λx) ( -1)! (-1) -1 ( -1)! (Λ 0 -Λ) (4)
2 In the sequel, the index will refer to the cluster containing the chip length value whereas index l will be used for any other cluster. 3 This relation results from the integration by parts

x 0 u v = [uv] x 0 -x 0 uv , with u = exp(-bt) and v = (x -t) l . × -1 i=0 (-bx) -1-i ( -1 -i)! -exp (-(Λ 0 -Λ)x)
Before computing the probability of interest P (T c ∈ C ), let us define z = T c , x = T -1 and y = ∆T to simplify the mathematical notations. Then, we can rewrite P (T c ∈ C ) as

P (z ∈ C ) = P (z -y < x ≤ z) (5)
= z 0 z z-y f (x) dx Λexp (-Λy) dy + ∞ z z 0 f (x) dx Λexp (-Λy) dy
To proceed further, it is required to express integral of the type

J m,Λ (T c ) = ∞ Tc x m exp (-Λx) dx and J m,Λ (T c ) = Tc 0 x m exp (-Λx) dx, where Λ ∈ R * + and m ∈ N. It can be easily verified that J m,Λ (T c )+J m,Λ (T c ) = m! Λ m+1 and J m,Λ (T c ) = exp(-ΛTc) Λ m+1 m p=0 m!(TcΛ) p p! .
Finally, from the relations above we obtain the following result after a few mathematical calculations :

P (T c ∈ C ) = Λ 0 Λ (Λ -Λ 0 ) +1 e -ΛTc R (-(Λ 0 -Λ) T c ) (6)
where R (u) stands for the rest of the -th order Taylor series expansion of the function e u , that is

R (u) = e u - i=0 u i i! (7)
Note that we assumed throughout the previous development that T c is larger than the first cluster delay T 0 . In case we have T c < T 0 , the interference corresponds to all the received pulse power. This event has the probability P ({T c < T 0 }) = e -Λ0Tc .

IV. ON THE NUMBER OF INTERFERING MPCS

This section is devoted to the estimation of the number of MPCs that can cause interference, that is the overall number of components located beyond the chip length. So, for any cluster index l ∈ N , we need to compute first the probability

P (n i ) for each subsequent cluster C i , l + 1 ≤ i ≤ n,
where n i denotes the number of MPCs belonging to the i-th cluster. The distribution associated to the whole set of interfering clusters will then be easily obtained in a second step.

In order to simplify our development, we will consider the approximation that there is no inter-cluster interference; hence the delay τ m,i of the m-th MPC relative to the i-th cluster arrival time T i-1 belongs to the interval

[T i-1 , T i ). A second
simplification is assumed for the ray arrival times; from [START_REF] Molisch | A comprehensive standardized model for ultrawideband propagation channels[END_REF] we know that they can be modelled with mixtures of two Poisson processes with mixture probability β and arrival rates {λ 1 , λ 2 } being determined experimentally for various environments.

As in the classical SV model, we will adopt a Poisson process for the ray arrival times, so that the distribution of the delay difference τ l+1,i -τ l,i of any two adjacent MPCs in cluster C i takes the form f (τ l+1,i -τ l,i ) = λexp(-λ(τ l+1,i -τ l,i )), where the value of the parameter λ is computed by minimizing the mean squared error between simplified and original models, for a given radio environment (known values for β, λ 1 , and λ 2 ).

For any cluster C i , it can be shown (see Appendix) that the probability P (n i ) that the number of MPCs in the cluster is

n i , is P (n i ) = λ ni Λ (λ + Λ) ni+1 , n i ≥ 0 (8)
Then, we can characterize the number of MPCs within clusters {C i ; i = , + 1, ..., L}, where L is the total number of clusters and supposing that T c ∈ C . This can be achieved by summing r + 1 indepedent and identically discrete random variables, each with probability [START_REF] Molisch | IEEE 802.15.4a channel model -final report[END_REF], where r = L -. The resulting probability P r (n) can be computed by recursion 4 , starting with P 1 (n) = n k=0 P 0 (k) P 0 (n -k) and P 0 (n) being expressed as in [START_REF] Molisch | IEEE 802.15.4a channel model -final report[END_REF] :

P r (n) = λ n (λ + Λ) n Λ (λ + Λ) r+1 (n + r)! n!r! . (9)
V. POWER DELAY PROFILE APPROXIMATION

The Power Delay Profile (PDP) is defined as the squared magnitude of the channel impulse response, averaged over the small-scale fading. In the frame of IEEE 802.15.4a it is expressed as

E |a k,l | 2 ∝ Ω l exp (-τ k,l /γ l ) (10)
where the integrated energy Ω l over the lth cluster follows an exponential decay and the intra-cluster decay time constant γ l depends linearly on the arrival time of the cluster. This model makes the closed-form derivation of the interfering power estimate intractable; so we will adopt the following first approximation for the PDP :

Ω(t) = Ω c exp - t Γ (11)
where t, Ω c and Γ denote the path delay, the integrated energy of the cluster and the intra-cluster decay time constant, respectively.

Concerning the small-scale fading, it can be shown that the i-th path relative to the l-th cluster has its power distributed as

f (x| Ω il ) = 1 Γ (m) m Ω il m x m-1 exp - mx Ω il , (12)
where Ω il stands for the PDP of the considered path, m is the Nakagami m-factor of the small-scale amplitude distribution and Γ (m) is the gamma function.

For CM1/CM2 channel models, the above relation can be simplified by considering a mean value5 m ∼ = 2 for the parameter m ; in this case Γ(m) = 1 and we get

f (x| Ω il ) = 2 Ω il 2 x.exp - 2x Ω il , (13)
which represents a Γ(α, θ) distribution with α = 2 and θ = Ω il 2 .

Now, we need to express the mean value of the PDP corresponding to the whole set of MPCs located beyond the chip length

:

Ω 0 = E [Ω Tc (τ i )] (14)
where Ω Tc (τ i ) = Ω (τ i ), for τ i ≥ T c and zero elsewhere, τ i denoting the path delay. We propose to derive an approximate expression of Ω 0 by picking n uniformly spaced samples of the original distribution [START_REF] Venkatesh | Non-line-of-sight identification in ultra-wideband systems based on received signal statistics[END_REF] in the interval [T c , T L ), where T L corresponds to the upper bound of the L-th cluster :

Ω 0 (L) = exp - T c Γ 1 n n-1 i=0 exp T c Γ exp - T N Γ i/n , (15)
which has the equivalent form

Ω 0 (L) = 1 n g L -g 0 (g L /g 0 ) 1/n -1 , (16)
where g 0 = exp (-T c /Γ) and g L = exp (-T L /Γ).

An additional simplification can be achieved for sufficiently large value of n by considering that

(g L /g 0 ) 1/n -1 / (1/n) ln (g L /g 0 ) (17)
Hence, we finally obtain the following approximation for the PDP :

Ω 0 (L) ∼ =            Γ [exp (-T c /Γ) -exp (-T L /Γ)] / (T L -T c ) , T L > T c , exp (-T c /Γ) , T L = T c . . ( 18
)
As can be seen in Fig. 2, considering [START_REF] Rahman | Analysis and Comparison of Inter-Symbol/Frame Interference in Pulsed DS-and Hybrid DS/TH-UWB Communications[END_REF] generally gives a good approximation of ( 16), for various values of L and T c , with an overestimation of the true value. Concerning the number of clusters, it can be assumed a Poisson distribution [9]

P L (L) = L L e -L L! , (19)
where L denotes the average number of clusters.

We know also that the last cluster delay has its PDF given by f (T L ) =

T L-1 L Λ L exp(-ΛT L ) (L-1)! , hence f T L (T L |T L ≥ T c ) =            T L-1 L exp (-ΛT L ) / ∞ Tc t L-1 exp (-Λt) dt , T L ≥ T c , 0, T L < T c . , (20)
From this last equation, we can then derive a closed form expression of the mean PDP, that will be used in the next section, devoted to interference power estimation : first, we derived in section III the probability that the chip length "falls into a particular cluster" of the multipath propagation model; -then, the statistics of the number of paths in any cluster was expressed in section IV, together with the statistics of the total number of paths located beyond the chip duration; -a simplified power delay profile expression has then be derived in section V.

Ω 0 (L) = E [Ω 0 (L)]
We are now going to derive an approximate relationship of the PDF of the power of all interfering MPCs. Our approach relies on the summation of a number n of r.v. each following the PDF [START_REF] Deleuze | Inter-symbol/Inter-Frame Interference in Time-Hopping Ultra Wideband Impulse Radio System[END_REF], with the factor θ = Ω 0 (L) computed as (21). Hence, for a given number of clusters N , the resulting PDF is expressed as, for x > 0,

g (x| Ω 0 (L)) = n=1 g (x| Ω 0 (L) , n) • P (n|L) (22)
where

g (x| Ω 0 (L) , n) = 2 Ω 0 (L) 2n x 2n-1 (2n -1)! • exp - 2x Ω 0 (L) (23)
and the number of paths with delays exceeding T c being determined as (under the assumption that each cluster contains at least one path)

P (n|L) = L-1 k=0 P (n|k, L) • P (k) , (24)
The above expression can be computed by taking advantage of previous developments ; for example, in case of a LOS environment (and considering that T 0 = 0), we get

P (n|k, L) = λ λ+Λ n-L+k Λ (λ+Λ) L-k × (n-1)! (n-L+k)!(L-k-1)! , ∀n ≥ 1 (25)
and

P (k) = (ΛT c ) k exp (-ΛT c ) k! . ( 26
)
To complete our analysis, we need also to consider the case x = 0, which results in a discrete part of the PDF :

g (x = 0| Ω 0 (L)) = P (n = 0|L) (27) = ∞ k=L P (k)
And finally, the interference power is obtained as

g (x) = ∞ L=1 g (x| Ω 0 (L)) P L (L) . (28)
No compact analytical expression of this PDF can be obtained due to the high complexity of the terms involved ; however, we can observe that the proposed result can easily be implemented to get an estimate at very limited computational cost. Note also that our development remains valid for both LOS and NLOS environments ; in the latter case, the probability that the chip length falls into the kth cluster has the alternative expression

P (k) = (ΛT c ) k+1 exp (-ΛT c ) (k + 1)! . ( 29
)
The proposed algorithm for interference power estimation is summarized in the form of a block diagram in Fig. (3).

Nb of clusters PDP Nb interfering paths

Interf. Power Estim.

Candidate chip time value

Channel Profile Id. As mentioned before, our theoretical analysis involves a few approximations to get the statistics of the interference power, due to the high complexity of some mathematical relations :

1) A Poisson process is adopted for modelling the ray inter-arrival times, instead of a mixture of two Poisson processes as recommended in the frame of the IEEE 802.15.4a channel model;

2) A simplified exponential expression [START_REF] Venkatesh | Non-line-of-sight identification in ultra-wideband systems based on received signal statistics[END_REF] is considered for the mean power of the different paths (PDP); we do not take into account the possible different values of the integrated energy and decay time constant for distinct clusters;

3) The computation of the mean PDP associated to MPCs located beyond the chip duration is achieved owing to a simple sampling scheme.

A great number of Monte-Carlo simulations have been conducted using the Matlab implementation [START_REF] Molisch | IEEE 802.15.4a channel model -final report[END_REF] of the IEEE 802.15.4a channel model to verify the pertinence of our approach. A few illustrations are given hereafter to show the impact of various approximations on the statistics. The case of CM1 channel model is considered here, but it should be noticed that the same procedure could be applied in another radio environement. Firstly, we can examine the error resulting from the proposed uniform sampling for the computation of the mean PDP [START_REF] Rahman | Analysis and Comparison of Inter-Symbol/Frame Interference in Pulsed DS-and Hybrid DS/TH-UWB Communications[END_REF]. As can be seen in Fig. 4, the computed value of Ω 0 (L) tends to match the true value when the number of clusters L increases and the error decreases for larger chip length. Also, it can be clearly observed that our approximation yields an overestimation of the mean PDP, which is particularly important from the point of view of applications. We can now consider the PDF (24) of the number of paths with delays larger than the chip length, which plays a central role in estimating the interference power. As shown in Fig. 5, there is almost a perfect match between the values obtained through simulations and the values obtained by numerical evaluation of analytical expressions. Evidently, it can be seen that the numbers of MPCs increases with the number of clusters. In order to assess the goodness of fit of our statistical model of the number of paths with delays exceeding T c , we applied the one-sample Kolmogorov-Smirnov (K-S) test [START_REF] Soong | Fundamentals of probability and statistics for engineers[END_REF] which consists in measuring the deviation of the observed Cumulative Distribution Function (CDF) from the hypothesized CDF (corresponding to our model); the case of CM1 with T c = 50 was chosen as it corresponds to the most difficult case that we considered in our study. As can be clearly observed in Fig. 6, plotting the p-values of the K-S test against the number of clusters L (for a significance level α of 0.05), the hypothesis that the simulated data are drawn from the proposed model is adopted. Figure [START_REF] Molisch | Ultra-Wide-Band Propagation Channels[END_REF] illustrates the PDF g (x| Ω 0 (L)) resulting from summing n r.v., each one following [START_REF] Deleuze | Inter-symbol/Inter-Frame Interference in Time-Hopping Ultra Wideband Impulse Radio System[END_REF], once the mean PDP associated to interfering MPCs has been computed. Then, Fig. 8 shows the PDF of the interference power for two distinct chip time values. The difference between the "true" distribution (estimated through MC simulations) and the PDF derived from our method comes from the approximations 2) and 3) explained above. However, this limited statistical precision is not an obstacle for practical applications since a low error is noticed for the two first moments of the distribution (as will be shown in a next figure).

Additional simulations have been conducted in NLOS radio environments to evaluate the pertinence of the proposed algorithm.

For CM2/CM4 channels (NLOS residential/office), we obtained the PDFs depicted in Fig. 9 & 10, for the same candidate values of the chip time.

First and second order moments have been computed for various channel models and different T c (Fig. 11). Overall, the agreement is rather good between our model and the simulated data (provided that T c is not too small) and we can note that the two moments are in all cases overestimated; hence, the proposed approach can be very helpful for decreasing the SINR.

VIII. CONCLUSIONS

A novel approach for the statistical characterization of the multipath interference in IR-UWB systems has been developed in this paper. In the frame of the IEEE 802.15.4a channel model, we derived a theoretical analysis which aims to compute the power corresponding to all multipath components located beyond the chip length. Since the proposed approach requires minimal knowledge on the channel state, it can be helpful for real time adaptation of modulation parameters so as to limit the signal to interference plus noise ratio. Our method relies on the following key developments : first, we derived the probability that the chip length falls into a particular cluster of the multipath propagation model; then, the statistics of the number of paths spread over several contiguous clusters have been computed in closed-form; finally, we obtained an approximate relationship of the PDF of the power of all interfering MPCs using a simplified power delay profile expression. Numerous Monte-Carlo simulations have been carried out to verify the pertinence of our results; although these simulations revealed a significant gap between the true interference power distribution and that obtained through our approach, the first two computed moments are very close and upper-bound the true ones. This is a very useful result because they represent an information that helps us to control the SINR level and to ensure an effective functioning of the IR-UWB system in real-world scenarios. The proposed approach is rather general and could actually be implemented for another channel model (however, a specific study is required depending on the channel model considered). Different application scenarios of the proposed algorithm could be imagined; for example, key parameters associated to the channel could be identified based on some received pilot symbols statistics with a default chip time value known by the receiver. Data symbols could then be transmitted with an "optimized" chip time duration depending on the channel state information sent to the transmitter; a mapping between channel profiles and chip time value could be designed and implemented using look-up-tables.

APPENDIX

Appendix A -Proof of [START_REF] Molisch | IEEE 802.15.4a channel model -final report[END_REF] For any cluster C i , let us define the following notations: x = τ ni,i -τ 0,i , y = τ (ni+1),i -τ ni,i and z = ∆T i . The probability

which finally yields [START_REF] Molisch | IEEE 802.15.4a channel model -final report[END_REF].

Figure 1 .

 1 Figure 1. Interframe interference from the (p -1)th frame to the pth frame; in this example Nc = 3 and the two consecutive pulses have a relative delay equal to Tc.

FIGURE 2 .∞

 2 FIGURE 2. Proposed approximation of the PDP

FIGURE 3 .

 3 FIGURE 3. Block diagram of the proposed algorithm for interference power estimation

FIGURE 4 .

 4 FIGURE 4. Mean value of Ω 0 (L) for various chip length values (CM1 channel model ;10 4 MC simulations)

FIGURE 5 .

 5 FIGURE 5. PDF of the number of paths with delays exceeding the chip length Tc = 50 (CM1 channel model; 10 5 MC runs) : (a) L = 5 clusters (b) L = 10 clusters

FIGURE 6 .

 6 FIGURE 6. Kolmogorov-Smirnov test for evaluating the distance between our statistical model (24) and the distribution resulting from simulations.

FIGURE 7 .FIGURE 8 .

 78 FIGURE 7. PDF g (x| Ω 0 (L))for CM1 channel model, Tc = 50 : (a) L = 2 clusters (b) L = 4 clusters

FIGURE 9 .FIGURE 10 .

 910 FIGURE 9. PDF of the interference power g(x) for CM2 channel model (10 5 MC runs) : (a) Tc = 25 (b) Tc = 50

FIGURE 11 . 0 Pf

 110 FIGURE 11. Theoretical first and second order moments compared to simulated data, for different CM and Tc.

The interest to consider this model is that it is widely studied by the research community, due to its ability to reflect propagation phenomena with acceptable statistical precision. However, there is still ongoing researches to improve this model such as[START_REF] Haneda | Modeling the Frequency Dependence of Ultra-Wideband Spatio-Temporal Indoor Radio Channels[END_REF] where the frequency dependence of the multipath components is investigated. Extensions of the present work could then be considered for future research, depending on the evolution of standards.

Note that this probability depends on the number (r + 1) of clusters beyond C , with Tc ∈ C .

As the random variable m follows a lognormal distribution, the mean value is expressed as µm = exp(m 0 + m2 0 /2),where the parameters (m 0 , m0 ) have specified values depending on the considered environment[START_REF] Molisch | A comprehensive standardized model for ultrawideband propagation channels[END_REF].


Keywords: Modelling, ElecFET, microelectrode, pH-ChemFET, water electrolysis, hydrogen peroxide detection

This paper presents a COMSOL Multiphysics 2D axisymmetric model of an pH-ElecFET (pH-sensitive electrochemical field effect transistor) microsensor. This device combines an integrated microelectrode with a pH-sensitive chemical field effect transistor (pH-ChemFET). Thus, by triggering electrolysis phenomena owing to the integrated microelectrode, associated local pH variations in microvolumes are monitored thanks to the pH-ChemFET microdevice. Taking into account (electro) chemical reactions and diffusion phenomena in liquid phase, the proposed model points out the role of the ElecFET geometrical design (microelectrode width w, gate sensitive radius re and distance between the pH-ChemFET gate and the microelectrode d), as well as polarization parameters, (polarization voltage Vp and time tp), on the microsensor response. It is first applied to water electrolysis in order to validate pH impulsional variations in microvolume. Then, oxidation of hydrogen peroxide in phosphate buffer (PBS, pH0=7.2) solutions is studied, evidencing the H2O2 potentiometric detection in the [10-100mM] concentration range. This developed model paves new ways for sensor applications, opening several new opportunities for pH-ElecFET devices for H2O2-related enzymatic detection of biomolecules.

I. INTRODUCTION

n the last decade, the electrochemical microsensors have received an increasing interest in a wide range of applications such as clinical diagnostics, food analysis, environmental monitoring due to their low cost, simple operation, small size, and rapidity, sensitivity and real-time [START_REF] Zaidi | Recent developments in nanostructure based electrochemical glucose sensors[END_REF][START_REF] Chaiyo | Electrochemical sensors for the simultaneous determination of zinc, cadmium and lead using a Nafion/ionic liquid/graphene composite modified screen-printed carbon electrode[END_REF][START_REF] Vasiliev | Non-silicon MEMS platforms for gas sensors[END_REF]. The electrochemical sensors can be divided into three groups depending on the measured electrical signal [START_REF] Safavi | Fabrication of an Amperometric Sensor for Hydroxylamine Based on Silver Paste Nanocomposite Electrode[END_REF][START_REF] Sayyah | A High-Sensitivity Potentiometric Mercuric Ion Sensor Based on m-Toluidine Films[END_REF][START_REF] Chen | A Wide Range Conductometric pH Sensor Made With Titanium Dioxide / Multiwall Carbon Nanotube /Cellulose Hybrid Nanocomposite[END_REF]: amperometric, potentiometric, and conductometric. Even so, the combination of amperometric and potentiometric techniques is a very promising method in terms of detection [START_REF] Alberti | Potentiometric and amperometric gas sensors based on the protonic conduction of layered zirconium phosphates and phosphonates[END_REF][START_REF] Diallo | Development of pH-based ElecFET biosensors for lactate ion detection[END_REF][START_REF] Diallo | Modelling of impulsional pH variations using ChemFET-based microdevices: application to hydrogen peroxide detection[END_REF][START_REF] Ouyang | Performance of Amperometric and Potentiometric Hydrogen Sensors[END_REF]. Diallo et al. have developed an electrochemical field effect transistor (ElecFET) microsensor based on this technique [START_REF] Diallo | Development of pH-based ElecFET biosensors for lactate ion detection[END_REF][START_REF] Diallo | Modelling of impulsional pH variations using ChemFET-based microdevices: application to hydrogen peroxide detection[END_REF]. This device is achieved through the integration of a planar noble metal electrode around the dielectric gate area of a pH-sensitive ChemFET microdevice. By triggering pH-related electrochemical reactions thanks to the microelectrode polarization and by monitoring the so-N. Aoun and F. Echouchene are with the Laboratory of Electronics and Microelectronics, University of Monastir, 5000, Tunisia (e-mail: nejibaaoun@yahoo.com, frchouchene@yahoo.fr) A. K. Diallo, J. Launay and Temple-Boyer are with the University of Toulouse, UPS, LAAS, F-31400 Toulouse, France.

H. Belmabrouk is with the Department of Physics, College of Science AlZulfi, Majmaah University, Saudi Arabia. obtained pH variations thanks to the pH-ChemFET, electrolysis phenomena and pH measurement are closely embedded at the microscale, enabling new electrochemical detection potentialities. Among the ElecFET microdevice applications, the most frequent is the manufacturing of a pHrelated enzyme sensor. In this case, the ChemFET detects the pH change resulting from the enzymatic reaction in the membrane that covers the sensor [11]. The output voltage of the ChemFET controls the current flowing through the sensoractuator system [START_REF] Luo | Modelling of the migration effect occuring at an ISFET-based coulometric sensor-actuator system[END_REF]. ElecFET has been successfully used to determine acid or base concentration [13], to form the heart of a carbon dioxide sensor [14] and detect different biomolecules [START_REF] Diallo | Development of pH-based ElecFET biosensors for lactate ion detection[END_REF]11]. Various mathematical models of microsensors have been developed and successfully used to study and optimize analytical characteristics of microsensors [START_REF] Meena | Mathematical modeling of amperometric and potentiometric biosensors and system of non-linear equations-Homotopy perturbation approach[END_REF][START_REF] López-Gándaraa | Modeling and optimization of diffusive layers in potentiometric and amperometric electrochemical gas sensors[END_REF][START_REF] Morf | Theoretical treatment and numerical simulation of potentiometric and amperometric enzyme electrodes and of enzyme reactors Part 1: Steady-state concentration profiles, fluxes, and responses[END_REF][START_REF] Morf | Theoretical treatment and numerical simulation of potentiometric and amperometric enzyme electrodes and of enzyme reactors. Part 2: Time-dependent concentration profiles, fluxes, and responses[END_REF]. Meena and Rajendran [START_REF] Meena | Mathematical modeling of amperometric and potentiometric biosensors and system of non-linear equations-Homotopy perturbation approach[END_REF] have derived analytical expressions of concentration and current in order to describe and evaluate the performances of amperometric and potentiometric biosensors using homotopy perturbation method. A numerical study of potentiometric and amperometric electrochemical gas sensors based in a solid-state ion conducting electrolyte has presented by López-Gándaraa et al. [START_REF] López-Gándaraa | Modeling and optimization of diffusive layers in potentiometric and amperometric electrochemical gas sensors[END_REF] in order to optimize the diffuse layers covering one of their catalytic electrodes. The model describes the current-voltage characteristics in the system layer/electrode/electrolyte/electrode. A theoretical study and numerical simulation of potentiometric and amperometric enzyme electrodes and of enzyme reactors have been developed by Morf et al. [START_REF] Morf | Theoretical treatment and numerical simulation of potentiometric and amperometric enzyme electrodes and of enzyme reactors Part 1: Steady-state concentration profiles, fluxes, and responses[END_REF][START_REF] Morf | Theoretical treatment and numerical simulation of potentiometric and amperometric enzyme electrodes and of enzyme reactors. Part 2: Time-dependent concentration profiles, fluxes, and responses[END_REF]. The response characteristics of potentiometric and amperometric sensor systems, as well as the product release from enzyme reactors, are analyzed, and the influence of the relevant parameters on the steady-state response is demonstrated and discussed [START_REF] Morf | Theoretical treatment and numerical simulation of potentiometric and amperometric enzyme electrodes and of enzyme reactors Part 1: Steady-state concentration profiles, fluxes, and responses[END_REF].

In this paper, we apply the concept of the combination of amperometric and potentiometric techniques to simulate the ElecFET microdevice detection principles, in order to monitor water (H2O) electrolysis phenomena in a first stage and hydrogen peroxide (H 2 O 2 ) electrochemical detection in the second stage. We then focus on the study of the influences of the main parameters, i.e. (i) polarization voltage Vp and time tp on the integrated microelectrode, (ii) characteristic of the microelectrode width w, (iii) distance between the gate sensitive radius and the integrated microelectrode d and (iv) gate sensitive radius re.

II. PRESENTATION OF THE SIMULATION MODEL

The modelling approach used takes into account the different chemical, electrochemical and physical phenomena occurring in the frame of the ElecFET detection principles:

▪ Oxido-reduction on the integrated microelectrode;

Finite-element simulations of the pH-ElecFET microsensors N. Aoun, F. Echouchene, A. K. Diallo, J. Launay, P. Temple-Boyer and H. Belmabrouk I ▪ Diffusion phenomena; ▪ (H3O + /H2O) and (H2O/OH -) and acid-base analysis of aqueous solutions at pH0=7.2. The ElecFET detection properties were thus modeled by studying the diffusion phenomena of the main chemical species in the electrolyte, focusing on the H3O + /OH -waterbased ions, and finally by analyzing the pH detection properties of the silicon nitride Si3N4 ChemFET gate.

A. Geometry of ElecFET: comparison between real and simulated model

Since their fabrication process used silicon-based microtechnologies and therefore photolithography, the real ElecFET microdevices are based on a rectangular concentric geometry [START_REF] Diallo | Modelling of impulsional pH variations using ChemFET-based microdevices: application to hydrogen peroxide detection[END_REF]. In order to decrease calculation times, the simulated model was simplified by using r-z cylindrical geometry, leading to a 2D axisymmetric geometry (Fig. 1).

The geometrical parameters are: the gate sensitive radius re, the distance between the sensitive gate and the electrode d, and w the microelectrode ring width. The depth of the sensitive gate ze is 0.5µm and the height of the modeled geometry is h=300 µm. The standard values of the parameters (re, d, w) are (10µm, 50µm, 100 µm). At the end of the paper these values will be modified in order to enhance the performance of pH-ElecFET. The dimensions of electrolyte domain are indicated in Fig. 1.

B. Modelling of the electrochemical reactions (H2O &H2O2)

The ElecFET concept was initially used for the monitoring of water (H2O) hydrolysis and secondly of the hydrogen peroxide (H2O2) oxidation in water based solutions:

       4e O O 4H O 6H : E 2 3 2 0 p V (1)        4OH 2H 4e O 4H : E 2 2 0 p V (2) 2 3 2 2 2 1 O 2e O 2H O 2H O H : E         p V ( 3
)
where E 0 + , E 0 -, and E 1 + are equilibrium potentials of the O2/H2O, the H2/H2O and O2/H2O2 redox couple, respectively. The values of the constants are: E 0 + = 1.2V, E 0 -= -0.8V [9] and E 1 + = 0.7V [START_REF] Diallo | Development of pH-based ElecFET biosensors for lactate ion detection[END_REF].

C. Modelling of the diffusion phenomena in water

Diffusion phenomena of the most influential chemical species, i.e. H3O + and OH -ions in case of H2O detection and H3O + , OH -and H2O2 species in case of hydrogen peroxide detection, into water, have been modeled using the Fick law:

i i i i i i R z c r c r r c D t c             ) 1 ( 2 2 2 2 (4)
where t denotes the time, ci is the concentration of the studied chemical species, Di is its diffusion coefficient and Ri is an additional term accounting for the production/consumption rate of the ion species through a possible chemical reaction. The reaction term can be deduced using the water protolysis reaction: [START_REF] Sayyah | A High-Sensitivity Potentiometric Mercuric Ion Sensor Based on m-Toluidine Films[END_REF] where kf and kb are the forward and backward reaction kinetic rates, respectively.

One can thus express the reaction for the production of proton and hydroxide ions as [START_REF] Pardon | Modeling and simulation of electrostatically gated nanochannels[END_REF]:

  ] OH ][ O H [ ] OH [ ] O H [ ] O H [ 3 3 2 3             e f f b OH O H K k k k R R (6)
where Ke = kb[H20]/kf is the equilibrium constant of water.

The forward rate constant kf of the water protolysis reaction is estimated by values taken from literature [START_REF] Moritsuka | The pH change after HCl titration into resting and stimulated saliva for a buffering capacity test[END_REF] while the backward rate constants kb is obtained from where Ke denotes the water dissociation constant.

Buffer solution is of great importance in biosensor applications. Experimentally, the buffer concentration could be optimized. So, a lot of experiments have been reported on the buffer properties which enhance the response and the stability of the biosensor [START_REF] Saeedfar | Potentiometric Urea Biosensor Based on an Immobilised Fullerene-Urease Bio-Conjugate[END_REF][START_REF] Alqasaimeh | A Large Response Range Reflectometric Urea Biosensor Made from Silica-Gel Nanoparticles[END_REF]. In our kinetic model the chemical effect of the buffer solution was neglected and was not taken into account. In the frame of a kinetic approach, the influences of acid/bases other than the water-based ones are limited by diffusion phenomena (Eq. 4). In another way, we have been interested only on the protonation reaction because, as a solvent, water is in excess in solution. As a result, we can assume that the buffer solution does not have an effect on the biosensor response which can be expressed with the chemical effect modeled with physical equations.

D. Initial and boundary conditions

Boundary conditions are crucial for a correct description of phenomena with partial differential equations. For our model, several important conditions have to be set. The initial condition related to mass transport of species is given by: kf kb

                               h 0 300µm 0 50 0 ; , 10
H3O + + OH - 2H2O
On the axis of symmetry, the boundary condition reads:

0 ) , 0 (     t ,z r r c i (8)
At the upper interface of the electrolyte (z=h) and the lower interface (z=0 or ze), the interface is assumed to be impermeable: In order to do not disturb the diffusion phenomena operating from the surface of the microelectrode, the outer surface of the device is assumed to be at a constant concentration:

) , , 300 (

0 , i i c t z µm r c   (10)
An oxido-reduction reaction takes place at the electrode for t<tp. Thus, the boundary condition reads:

w ; = ) ( ,         d r r d r i F N t t , z=z r, z c D e e j j j i p e i i  ( 11
)
where Nj is the number of electron transferred for reaction "j", i,j is the stoichiometric coefficient, F is Faraday's constant and tp is the polarization time. ij denotes the current density given from the kinetic equations for the electrochemical reactions at the electrode surface based on the Butler-Volmer expression [START_REF] Temple-Boyer | Study of front side connected chemical field effect transistor for water analysis[END_REF]:

                                       i i j j c j qi ref i i j j a j pi ref i i ref j j RT F c c RT F c c i i ) exp( ) exp( , , , , , , ; 0 
   (12)
where i0j,ref is the exchange current density due to reaction "j" at the reference concentrations in A/cm 2 ,ci is the concentration of species "i" adjacent to the surface of electrode in mol/cm 3 , ci,ref is the reference concentration of species "i" in mol/cm 3 , αaj is the anodic transfer coefficient for reaction "j", αc,j is the cathodic transfer coefficient for reaction "j", pi, j=i,j is the anodic reaction order of species "i" in reaction "j", qi, j=-i,j is the cathodic reaction order of species "i" in reaction "j", R is the gas constant, T is the temperature and ηj is the overpotential of reaction "j" in volts (V), and it is measured with respect to a reference electrode of a given kind in a solution at the reference concentrations.

The overpotential for electrochemical reaction "j", (η) in Eq. 12 is given by:

j p V 0, j E -   ( 13
)
where Vp is the applied potential surface in V, E0,j is the equilibrium potential for reaction "j".

Total current is obtained by numerical integration of the local current density over the entire ElecFET area. It is given by:

  A j dA i t I ) ( ( 14
)
where A is the surface area of the electrode.

E. Numerical method

The governing equations with initial and boundary conditions are solved numerically using the finite element software COMSOL Multiphysics 4.3b [START_REF]Comsol multiphysics finite element analysis software[END_REF]. An electroanalysis module has been chosen for solving the mass transport of diluted species in electrolytes using the diffusion equation.

The overall computational domain is discretized using an unstructured triangular mesh Fig. 2. It is noticed that the region nearby the reaction surface is refined with a better mesh quality.

For the time discretization we have employed First-order backward differentiation formula (BDF), with the time steps controlled by the numerical solver during the computations. Total current, given by Eq. ( 14), is performed using a fourth order integration method.

III. RESULTS AND DISCUSSION

To describe the pH-ChemFET response, we can use the threshold voltage variation. The pH-ElecFET threshold voltage variation is related to the pH at the silicon nitride Si3N4 surface according to the following equation [START_REF] Moritsuka | The pH change after HCl titration into resting and stimulated saliva for a buffering capacity test[END_REF]:

) pH - ) (pH(0, s ) ( pzc 0 0 t V t V T T    (15)
where pH(0,t) is the pH at the Si3N4 sensitive surface when the diffusion phenomena "steady state" is reached and VT0 is a constant parameter depending on the SiO2/Si3N4 pH-ChemFET technology [START_REF] Bousse | Operation of chemically sensitive field effect sensors as a function of the insulator-electrolyte interface[END_REF], pH pzc is the point zero of charge which is estimated around 4 for Si3N4 [START_REF] Temple-Boyer | Study of front side connected chemical field effect transistor for water analysis[END_REF][START_REF] Poghossian | Determination of the pHpzc of insulators surface from capacitance-voltage characteristics of MIS and EIS structures[END_REF]). In the following, since the VT0 value is only related to the pH-ChemFET technological fabrication, it is of no influence concerning the ElecFET detection properties and it will not be taken into account, i.e. it will be chosen equal to zero [START_REF] Temple-Boyer | Modelling of EnFETs for the creatinine detection[END_REF][START_REF] Sant | On-line monitoring of urea using enzymatic field effect transistors[END_REF]. s0 is an ideal sensitivity of ISFET based pH sensor when pH is measured by the means of Nernst potential given by the following equation [ where k=1.38×10 -23 J/K is the Boltzmann constant, T=300K is temperature and q=1.6×10 -19 C the elementary charge.

A. Mesh sensitivity

To ensure that the convergence has been obtained and the numerical results are independent of the mesh size, Fig. 3 shows the pH local at z=150 μm of the electrolyte for several mesh grids. There was no significant difference between the curves obtained using these grids (6818, 13643, 27719, 53534 elements) and we may conclude that the numerical convergence has been reached with all the grids. In the results section, all simulations were done with a total elements number 13645 [START_REF] Selmi | Flow Confinement Enhancement of Heterogeneous Immunoassays in Microfluidics[END_REF][START_REF] Nama | Numerical study of acoustophoretic motion of particles in a PDMS microchannel driven by surface acoustic waves[END_REF].

B. Study of the most influential parameters

The kinetic parameters and the constants used in this simulation are listed in Table I and Table II [START_REF] Diallo | Modelling of impulsional pH variations using ChemFET-based microdevices: application to hydrogen peroxide detection[END_REF][START_REF] Bard | Electrochemical methods: fundamental and applications[END_REF][START_REF] Moore | Basic Physical Chemistry[END_REF]. ElecFET detection/transduction principles were first studied for water electrolysis monitoring H2O in aqueous solutions (pH0=7.2). Then, the ElecFET detection properties were tested for the hydrogen peroxide H2O2 also in aqueous solutions (pH0=7.2) solutions containing different stabilized concentrations [H2O2] ranging from 10 to 100mM because of their properties in terms of activity, stability and cost. According to theoretical equations, the most influential parameters are the polarization voltage Vp, the time of polarization tp, the concentration [H2O2], the distance between the gate sensitive radius and the microelectrode (parameter d), the surface of the microelectrode (associated with the parameter w) and gate sensitive radius (parameter re).

All series of simulation have been devoted to modelling the sensor response, i.e. the dependence of the threshold voltage versus time. The curves of this threshold voltage, measured at pH0=7.2, are shown in Figs. 456789.

Case1: water electrolysis monitoring:

Before testing the sensor performance for the H2O2 detection, one should ensure that the sensor is able to provide a response in the case of water electrolysis.

Fig. 4 presents the temporal evolution of the threshold voltage VT in the case of H2O redox phenomena for different polarization voltages while maintaining a constant the polarization time (tp = 5s). Since Vp is positive (respectively negative), oxidation (respectively reduction) of H2O takes place near the platinum microelectrode.

Therefore, a production of hydronium H3O + (respectively hydroxide OH -) ions, so for a local pH changes sharply and finally the threshold voltage VT increases (respectively decreases). In this case, phenomena occur at higher than 1.2V (positive polarization) and/or lower than -0.8V (negative polarization) polarization voltages Vp according to the associated equilibrium potential values E 0 + and E 1 + . The amplitude of the peak is so much greater than the value of the applied potential increases. When the polarization is interrupted, the ElecFET microsensor response follows a return to equilibrium of the system because of the diffusion laws.

Subsequently, the influence of polarization time tp was studied. Fig. 5 illustrates the temporal variations threshold voltage for different polarization times (tp = 5, 10, 10 and3 s) for two values of the polarization voltage on the integrated microelectrode (Vp = 1.4 or -1V). In agreement with the Equation ( 11), the polarization time increase is responsible for a local pH decrease and therefore a pH-ChemFET threshold voltage increases (Eq. 15). Nevertheless, threshold voltage variations are lower and tend to reach saturation. Fig. 4 and Fig. 5 show clearly that the microsensor has a response in the case of water electrolysis. Therefore, we can continue our analysis and assess the performance of the sensor for H2O2 detection.

Case2: hydrogen peroxide detection:

The ElecFET concept was studied by taking into account the H2O2 oxidation on the integrated microelectrode. According to the oxidation reaction (Eq.3), in presence of hydrogen peroxide H2O2, a positive polarization Vp on the platinum microelectrode produces hydronium H3O + ions. Therefore, the local pH decreases around the sensitive area of the microsensor and finally the threshold voltage VT increases. For this purpose, the polarization voltage Vp was chosen less than 𝐸 0 + to avoid any interference with water hydrolysis.Since the 𝐸 1 + is roughly equal to 0.7V, a similar polarization voltage Vp was applied on the platinum microelectrode at the different polarization times (tp = 5, 10, 15, 20, 30, 40 and 60s) while keeping constant [H2O2] concentration at 50mM in a solution of pH0=7.2, with a constant polarization voltage (Fig. 6). VT tends toward increasingly positive values (i.e local pH decreases). As previously for the water electrolysis, the amplitude of the peaks increases with tp. The maximum threshold voltage increases steeply over a short period of time and thereafter reaches a plateau value. It is clear that the influence of polarization time appears only in transient state. The equilibrium state is reached for tp=10s.

In the same way, Fig. 7 represents the H2O2-ElecFET response as a function of time for different polarization voltages Vp. This figure has been obtained with [H2O2] = 50mM in a solution of pH0=7.2 and tp = 30s. We found that the asymptotic value of threshold voltage is essentially linearly related to the polarization voltage Vp. As soon as the polarization is interrupted, the ElecFET microsensor response tends to turn to back equilibrium of the system because of diffusion phenomenon.In the case of hydrogen peroxide detection, the Butler-Volmer theory emphasizes on the influence of the [H2O2] concentration of H3O + ion production kinetics (Eq. 3). phenomenon is related to the previously discussed ElecFET detection principles [START_REF] Moore | Basic Physical Chemistry[END_REF].

Fig. 8 inset presents the evolution of the asymptotic threshold voltage 𝑉 𝑇 𝑆𝑎𝑡 versus the concentration [H2O2] and at Vp=0.7V and tp=30s. As illustrated in Fig. 8 inset, the simulation shows a linear response through a large range of [H2O2] concentration, as well as a very insensitive behavior. The influence of the microelectrode positions, the surface of the microelectrode and the gate sensitive radius on the terms were also simulated.

The results of these simulations are shown in Fig. 8. It is worth noting that in Fig. 9, each of the three curves includes the results of simulations for Vp= 0.7 V, tp = 30s and [H2O2] = 50mM.

In Fig. 9a, the change of the threshold voltage is imposed by the variation of the microelectrode width (w), from 50 to 200 µm with a step of 50µm. In this case, the maximal variation 𝑉 𝑇 𝑆𝑎𝑡 was increased during the simulation of this stage. Far from the microelectrode (d = 20, 50, 80, 100 and 150 µm), once hydronium ions are created, they take longtime to diffuse into the solution causing a reduction in their number at very large distances leading to low response of the sensor as illustrated in Fig. 9b.

Finally, Fig. 9c shows the threshold voltage variation with time at various gate sensitive radius (re =5, 10, 15, 20 and 25µm). When the gate sensitive radius increases, the 𝑉 𝑇 𝑆𝑎𝑡 value also decreases slightly. This proves that the ElecFET detection principles depend mainly on the spatial integration of both microelectrode and pH-ChemFET microdevices.

C. Comparison with experimental results

This section presents a comparative study of experimental results and those obtained by simulation (Fig. 8) in the case of hydrogen peroxide detection. It is worth noting that our ChemFET-meter measurement interface was used to measure the pH-ChemFET gate-source voltage VGS, also called output voltage Vout, enabling the local pH monitoring in the microelectrode surroundings depending on the polarization conditions. Vout shift provides an identical threshold voltage shift and therefore thinking in term of Vout or threshold voltage is the same. Fig. 10 shows the evolution of the output voltage versus time at various [H2O2] concentrations, while inset represents the maximum of Vout versus [H2O2] concentrations. We can see that the curve of sensor response for experimental studies shows similar shape as in theoretical studies. As the [H2O2] concentrations increases, the sensor response increases too, with Vout shifted towards positive values.

Our numerical results show that the amplitude of the impulsional voltage variations increases with the [H2O2] concentration in solution, and saturates for the highest values. This saturation phenomenon is related to the ElecFET detection principles as previously mentioned: in the presence of hydrogen peroxide, a positive bias on the platinum microelectrode produces hydronium H3O + ions. Therefore the local pH decreases and finally the pH-ChemFET-meter output voltage Vout increases. When the [H2O2] concentration increases, the H3O + ions production also increases. Nevertheless, the H3O + ion increase is limited by diffusion and electromigration phenomena, leading to a new chemical equilibrium at the microscale. Hydrogen peroxide H2O2 potentiometric detection was finally demonstrated in the [10-100mM] range.

The response from our simulation is not in the same range of the value obtained from the experience. This is due to the effect of the buffer properties. Indeed, Chong et al. have shown that when concentration of the PBS solution decreases, the ionic strength of the PBS solution also decreases causing the drain current Id to decrease. The dependence of the Id to the pH value of the testing pH buffer solutions is related to the variation of the threshold voltage [START_REF] Soon Weng | The Effect of Phosphate Buffer Solution (PBS) Concentration on the Ion Sensitive Field-Effect Transistor (ISFET) Detection[END_REF]. In the future work, we intend to take into account the effect of the buffer.

IV. CONCLUSION

In this study, we have investigated the modelling of the ElecFET microdevice, taking into account the mass transport for different species, electrochemical reactions on the microelectrode and acid/basic reactions for couples (H3O + /H2O) and (H2O/OH -). The pH-ElecFET techniques were used for monitoring the water-based electrolysis by pH-ChemFET and extended to the hydrogen peroxide in order to obtain the threshold voltage (i.e local pH) variation. The influence of the ElecFET geometrical design, i.e. characteristic width of the microelectrode w, gate sensitive radius re and distance between the gate sensitive radius and the microelectrode d, as well as polarization parameters, i.e. polarization voltage Vp and time tp, on its response behavior was studied. So, the obtaining of pH impulsional variations in microvolumes was clarified and the potentiometric detection of hydrogen peroxide H2O2 was evidenced in the [10-100mM] concentration range. In the near future, we intend to model the ElecFET device for the enzymatic detection of lactate and glucose in cylindrical coordinates.

Fig. 1 .

 1 Fig. 1.Cross-section on the ElecFET device in r-z cylindrical coordinates.

Fig. 2 .

 2 Fig. 2. Two-dimensional unstructured mesh with triangular elements.

Fig. 4 .

 4 Fig. 4.Temporal variations of the pH-ChemFET threshold voltage for different polarization voltages.

Fig. 8

 8 Fig. 8 shows temporal variations of the pH-ChemFET threshold voltage for different H2O2 concentrations. Simulation results prove that the amplitude of the threshold voltage increases with the [H2O2] concentration in solution, and saturates for the highest values. This saturation

Fig. 7 .

 7 Fig. 7.Temporal variations of the pH-ChemFET threshold voltage for the different polarization voltages at [H2O2] =50mM, tp=30s, d=50µm, w=100µm and re=10µm and (inset) ElecFET response Vp.

Fig. 6 .Fig. 5 .

 65 Fig. 6.Temporal variations of the pH-ChemFET threshold voltage for the different polarization times at [H2O2] =50 mM, Vp =0.7V, d=50µm, w=100µm and re=10µm and (inset) ElecFET response versus tp.

Fig. 9 .

 9 Fig. 9.Temporal variations of the pH-ChemFET threshold voltage for various: (a) surface of the microelectrode at d=50µm and re=10µm (b) positions of the microelectrode at w=100µm and r e =10µm, and (c) gate sensitive radius at d=50µm and w=100µm and to all at Vp=0.7V, tp=30 s, [H2O2] =50 mM.

Fig. 8 .

 8 Fig. 8.Temporal variations of the pH -ChemFET threshold voltage for the different H2O2 concentrations at 0.7V, tp=30s, d=50µm, w=100µm and re=10µm and (inset) H2O2 response of the ElecFET microdevice (Vp =0.7V)

Fig. 10 .

 10 Fig. 10. Detection of impulsional pH variations related to the H2O2 electrolysis in phosphate buffer solutions: influence of the H2O2 concentration and (inset) ElecFET response for the H2O2 detection.

TABLE I PARAMETERS

 I

								USED IN THE SIMULATIONS
								Reaction 1	Reaction 2	Reaction 3
								i0j,ref (A/m 2 )	1.115×10 -6	10 -6	9.64×10 -6
								E0,j(V)	1.2	-0.8	0.7
								αa,j	0.5	0.5	0.5
								αc,j	0.5	0.5	0.5
								H3O+	+1	+1
								OH -	+1
								H2O2	-1
								Nj	4	4	2
		7,4
		7,2
		7,0
		6,8	3569 elements
			6818 elements
		6,6	13643 elements
			27719 elements
	pH	6,4	53534 elements
		6,2
		6,0
		5,8
		5,6
		5,4
		0	50	100	150	200	250	300
				r	(µm)

Fig. 3. pH at z = 150 μm of the electrolyte for several mesh grids.