Vincent Savaux
email: vincent.savaux@supelec.fr

Moïse Djoko-Kouam
email: moise.djoko-kouam@ecam-rennes.fr

Yves Louët
email: yves.louet@supelec.fr

Alexandre Skrzypczak
email: alexandre.skrzypczak@zodiacaerospace.com

Effect of Polynomial Interpolations on the Estimation Performance of a Frequency-Selective Rayleigh Channel in OFDM Systems

Keywords: Bit error rate, Channel estimation, Interpolation, Mean square error, OFDM, Rayleigh channel

In this article, we provide an analytical expression of the mean square error (MSE) and the bit error rate (BER) lower bound of an orthogonal frequency division multiplexing (OFDM) signal transmission over a multipath Rayleigh channel considering estimation errors. For some pilot arrangements, an interpolation is required to perform the channel estimation. Due to their low complexity, polynomial based interpolations are usually applied at the receiver, which induces estimation and signal errors. Based on a statistical analysis of these errors, the exact MSE expression of the channel estimation is provided. Furthermore, with a geometrical study of the constellation, an analytical BER limit is derived. For a given channel, it is shown that the errors are perfectly characterized by the interpolation method and the frequency gap between the pilot tones. All the steps of the analytical developments are validated through simulations. The proposed analysis then predicts the performance of the receiver, thus enabling the latter to a priori select the interpolation method with minimum complexity, according to a given channel and a BER target.

Effect of Polynomial Interpolations on the Estimation

Performance of a Frequency-Selective Rayleigh Channel in OFDM Systems

I. INTRODUCTION

Orthogonal frequency division multiplexing (OFDM) has become a very popular modulation scheme, for its robustness against the frequency selective channels. In order to estimate the channel frequency response, some pilots are inserted in the OFDM frame. A wide range of pilot-symbol-aided channel estimation techniques are described in [START_REF] Coleri | A study of channel estimation in OFDM systems[END_REF]- [START_REF] Biguesh | Training-Based MIMO Channel Estimation: A Study of Estimator Tradeoffs and Optimal Training Signals[END_REF]. Among them, and according to the pilot distribution, an interpolation may be necessary to estimate the channel over the time-frequency lattice. For instance, a comb-type pilot scheme or an arrangement in staggered row are adapted to the time and frequency selective channel, as in the digital radio mondiale (DRM/DRM+) [START_REF] Etsi | Digital Radio Mondiale (DRM);System Specification[END_REF] or the digital video broadcasting (DVB) [START_REF]Digital video broadcasting (DVB) framing structure, channel coding and modulation for digital terrestrial television[END_REF] standard. These arrangements require an interpolation that may lead to estimation errors causing some decision errors and finally inducing a corrupted signal at the receiver. Consequently, the system performance, measured in bit error rate (BER) or mean square error (MSE) of estimation, reaches a lower limit even for a very low noise level.

The effects of the noise and the channel on the system performance have been studied for many years in the literature. In [START_REF] Alouini | A Unified Approach for Calculating Error Rates of Linearly Modulated Signals over Generalized Fading Channels[END_REF], the outcome of several channel models (e.g. Rayleigh, Rice, Nakagami) have been covered in order to express the theoretical symbol error rates (SER) of linear modulations in a single carrier context. The authors of [START_REF] Vitthaladevuni | A closed-form expression for the exact BER of generalized PAM and QAM constellations[END_REF] propose a closed-form of the exact BER expression for hierarchical modulations, and use the repetitive pattern of these kinds of constellations to derive a recursive algorithm for the exact BER computation [START_REF]A Recursive Algorithm for the Exact BER Computation of Generalized Hierarchical QAM Constellations[END_REF]. In [START_REF] Tang | Effect of Channel Estimation Error on M-QAM BER Performance in Rayleigh Fading[END_REF], the effects of the sinc channel estimation errors on the BER have been studied for a single carrier signal transmitted over a Rayleigh channel. This study has been extended in [START_REF] Cao | Exact Error-Rate Analysis of Diversity 16-QAM With Channel Estimation Error[END_REF], in which the spatial diversity is considered at the receiver.

More recently, in an OFDM context, the authors of [START_REF] Islam | Performance of Wireless OFDM System with LS-Interpolation-based Channel Estimation in Multi-path Fading Channel[END_REF] study the lower bound reached by the BER curves due to different interpolation methods and for different constellation sizes. However, it does not provide any analytical expression of this BER lower bound. In [START_REF] Krondorf | Symbol Error Rate of OFDM Systems with Carrier Frequency Offset and Channel Estimation Error in Frequency Selective Fading Channels[END_REF], [START_REF] Yih | Effects of Channel Estimation Error in the Presence of CFO on OFDM BER in Frequency-Selective Rayleigh Fading Channels[END_REF], an analytical expression of the SER and BER for a transmission in presence of channel frequency offset and channel estimation errors is derived for different modulation schemes. However, the developments are made by using the probability density functions (pdfs) given in [START_REF] Wilson | Probability Density Functions for Analyzing Multi-Amplitude Constellations in Rayleigh and Ricean Channels[END_REF], in which the channel coefficients and the error of estimation are supposed to be uncorrelated, which is not the case in practice.

In this article, we investigate the effect of interpolation methods based on low degrees polynomials, like the NN, linear or cubic interpolations [START_REF] Coleri | A study of channel estimation in OFDM systems[END_REF], [START_REF] Dong | Linear Interpolation in Pilot Symbol Assisted Channel Estimation for OFDM[END_REF], and that do not need physical or statistical channel properties as for the linear minimum mean square error (LMMSE) [START_REF] Edfors | OFDM Channel Estimation by Singular Value Decomposition[END_REF], [START_REF] Biguesh | Training-Based MIMO Channel Estimation: A Study of Estimator Tradeoffs and Optimal Training Signals[END_REF] or the maximum likelihood (ML) [START_REF] Kay | Fundamentals of Statistical Signal Processing : Estimation Theory[END_REF] techniques. As a matter of fact, these simple polynomial interpolation methods are the most used in practical receiver implementations, for they are much less complex than LMMSE or ML, although the latter are more accurate. Consequently, the analysis of their performance limits is of great interest. To this end, we characterize the system performance not only with a BER value as usually in the literature [START_REF] Tang | Effect of Channel Estimation Error on M-QAM BER Performance in Rayleigh Fading[END_REF], [START_REF] Cao | Exact Error-Rate Analysis of Diversity 16-QAM With Channel Estimation Error[END_REF], [START_REF] Krondorf | Symbol Error Rate of OFDM Systems with Carrier Frequency Offset and Channel Estimation Error in Frequency Selective Fading Channels[END_REF], [START_REF] Yih | Effects of Channel Estimation Error in the Presence of CFO on OFDM BER in Frequency-Selective Rayleigh Fading Channels[END_REF], but also with the MSE of the channel estimation performed by means of the least square (LS) method with interpolations, as it is done in [START_REF] Biguesh | Training-Based MIMO Channel Estimation: A Study of Estimator Tradeoffs and Optimal Training Signals[END_REF], [START_REF] Savaux | Minimum mean-square-error expression of LMMSE channel estimation in SISO OFDM systems[END_REF] for the LMMSE. As in [START_REF] Tang | Effect of Channel Estimation Error on M-QAM BER Performance in Rayleigh Fading[END_REF], [START_REF] Cao | Exact Error-Rate Analysis of Diversity 16-QAM With Channel Estimation Error[END_REF], we use a Rayleigh channel model with correlated interpolation errors, but we consider the transmission of an OFDM signal over a frequency selective channel. To provide an analytical BER lower bound expression, we propose to study the statistics of the estimation errors and the geometry of the constellation. From the statistics of the errors, we then derive an analytical expression of the MSE for the LS estimation with interpolation. A general expression of the errors is provided, and is then applied to the nearest neighbor (NN), the linear interpolations and the cubic interpolation methods.

The contributions of this article are the following ones: firstly, according to the channel parameters, we give an exact expression of the error statistics (variance, correlation coefficient, distribution) in function of the interpolation technique and the frequency gap between the pilot tones. Secondly, we propose an original approach for the exact BER floor calculation, by splitting the study into statistical and geometrical considerations. Thirdly, in addition to the BER analysis, the exact MSE of the LS estimation with interpolation is provided. This is an original performance study that highlights some interpolation properties, in particular regarding the interpolated noise. The proposed analysis can then be used at the receiver side to choose the less complex interpolation method for a BER target, and according to the channel.

The article is organized as follows: Section II describes the system model and gives the parameters used all along the paper. In Section III, the statistics of the errors are exactly expressed in function of the interpolation technique and the gap between the pilots. From these statistics, the mean square error of the estimations is analytically derived in Section IV, and then the exact BER floor value is given in Section V. The simulations results of Section VI validate the theoretical developments, and we draw conclusions and we give some prospects in Section VII.

II. SYSTEM MODEL

A. Background and Signal Model

We consider the transmission of OFDM symbols over a multipath Rayleigh channel. After the removal of the cyclic prefix (CP) and the Discrete Fourier Transform (DFT) of size M , we express the received OFDM symbol vector in the frequency domain by

U = CH + W, (1)
where

U = [U 0 , ..., U M -1 ] T , H = [H 0 , ..., H M -1 ] T and W = [W 0 , ..., W M -1 ] T denote the M × 1
complex vectors of the received signal, the time and frequency selective channel and the Gaussian white noise of variance σ 2 , respectively. The matrix C is the M ×M diagonal matrix of the transmitted signal containing the vector [C 0 , ..., C M -1 ]. Each C m is either a data element or a pilot, whose gain, phase and position are perfectly known at both transmitter and receiver sides. Since the channel is time-varying, let us assume OFDM symbols with a comb-type pilot arrangement [START_REF] Hsieh | Channel Estimation for OFDM Systems Based on Comb-Type Pilot Arrangement in Frequency Selective Fading Channels[END_REF], i.e. the pilot tones are evenly distributed in each symbol, with a frequency gap noted δ p . As reminded in [START_REF] Ozdemir | Channel Estimation for Wireless OFDM Systems[END_REF], in order to capture all the channel frequency variations, this frequency gap must respect the Nyquist sampling theorem δ p ≤ 1/(τ max ∆(f c )), where τ max is the maximum delay spread and ∆(f c ) the subcarrier spacing. As illustrated in Fig. 1, three values δ p ∈ {2, 3, 4} are considered, so as to generalize the further developments whatever δ p . For simulations purposes, each symbol has M = 241

subcarriers. The reason is that 241 = a × δ p + 1, where a is an integer. In order to only focus on the effect of the interpolation errors, the developments and the simulations are performed without considering channel coding. The frequency positions of the pilot tones are pointed out by the letter p, so as to write f = f p and m = p in the continuous and in the discrete formalism, respectively. We also define N p the number of pilot tones per OFDM symbol. In the rest of the article and without loss of generality, the pilot are supposed to have a normalized power, that is |C p | 2 = P = 1.

B. Channel Model

Since polynomial-based interpolations methods are usually expressed for continuous functions, some developments will be made in a continuous formalism. However, it will be shown that the developments remain valid in the discrete formalism. Thus, the Rayleigh channel frequency response will be equivalently expressed by

H(f ) = L-1 l=0 h l e -2jπf τl ⇐⇒ H m = L-1 l=0 h l e -2jπ β l m M , (2)
where β l = τ l /τ s are the sampled path delays, m = f M τ s the sampled frequency, with τ s the sampling time, L is the number of paths and h l are the zero-mean independent complex Gaussian path coefficients, according to the wide sense stationary uncorrelated scattering (WSSUS) model described in [START_REF] Bello | Characterization of randomly time-variant linear channels[END_REF]. We also define σ 2 l = E{|h l | 2 } the variance of the l th path.

C. Channel Estimation

According to the pilot positions, the channel estimation is expressed by

Ĥ(f ) =        Ĥ(f ) LS = H(f ) + W (f ) C(f ) if f = f p Ĥ(f ) int + ( W (f ) C(f ) ) int if f = f p ⇐⇒ Ĥm =        ĤLS m = H m + Wm Cm if m = p Ĥint m + ( Wm Cm ) int if m = p , (3)
in the continuous and discrete formalisms, where Ĥ(f ) LS is the usual least square estimation [START_REF] Biguesh | Training-Based MIMO Channel Estimation: A Study of Estimator Tradeoffs and Optimal Training Signals[END_REF], [START_REF] Biguesh | Downlink Channel Estimation in Cellular Systems with Antenna Arrays at Base Stations Using Channel Probing with Feedback[END_REF], and the superscipt int denotes the interpolated values of the channel and the noise. Since the channel and the noise are uncorrelated, the interpolation of the sum of the channel and noise is the sum of the respective interpolations. Consequently, it is possible to study the statistics of the interpolation errors H m -Ĥint m independently of the interpolated noise ( Wm Cm ) int . The statistics of the noise and the channel estimation errors will be then used in order to derive the analytical expressions of the mean square error (MSE) and the bit error rate (BER) floor.

III. STATISTICS OF THE INTERPOLATION ERRORS

We propose to derive the analytical MSE and BER floor for three frequently used polynomial interpolations. To this end, the second-order statistics of the interpolation errors are required, in particular the variance of the error and the joint probability density function (pdf) of the channel and the error. In this section, we then statistically characterize the interpolation errors without taking into account the noise. We first give a general formulation of the statistical analysis, and we then apply it to the NN, the linear and the piecewise cubic interpolation methods.

A. General Formulation

From (3), for a given frequency f = f p and without taking into account the noise, the error induced by the interpolation, noted e h is defined by

e h = Ĥ(f ) int -H(f p ). (4)
Let us also define ξ = |e h | and the random Rayleigh process r = |H(f )|. Since the error in [START_REF] Biguesh | Training-Based MIMO Channel Estimation: A Study of Estimator Tradeoffs and Optimal Training Signals[END_REF] is function of the channel, it is assumed that it has the same distributions as this latter, i.e. e h is a complex Gaussian process and ξ a Rayleigh process, whose variance noted σ 2 ξ must be defined. This assumption will be verified afterward. From (4), we also deduce that the error depends on the frequency gap f -f p . In the discrete formalism, we denote by k the gap between a pilot p a given position m = p + k, as illustrated on Fig. 2. ξk . From Fig. 2, we observe that, by symmetry, ξ k has the same variance for k = 1 and k = 2 for δ p = 3, and in the same way, σ 2 ξ1 = σ 2 ξ3 for δ p = 4. We deduce the probability density function of the error

p ξ (ξ) =        ξ σ 2 ξ 1 e -ξ 2 2σ 2 ξ 1 , if δ p ∈ {2, 3} 2 3 ( ξ σ 2 ξ 1 e -ξ 2 2σ 2 ξ 1 ) + 1 3 ( ξ σ 2 ξ 2 e -ξ 2 2σ 2 ξ 2 ), if δ p = 4 . ( 5
)
The weights 2/3 and 1/3 in ( 5) correspond to the ratio between the symmetrical carriers having the same variances divided by the number of data-carriers in each discrete interval described in Fig. 2, for δ = 4. From (5), the general expression of p ξ (ξ) can be reasonably extrapolated (with an obvious recursion) whatever δ p > 2:

p ξ (ξ) =          1 δp-1 ξ σ 2 ξ δp /2 e -ξ 2 2σ 2 ξ δp/2 + 2 δp-1 δp/2-1 k=1 ξ σ 2 ξ k e -ξ 2 2σ 2 ξ k , if δ p is even 2 δp-1 (δp-1)/2 k=1 ξ σ 2 ξ k e -ξ 2 2σ 2 ξ k , if δ p is odd . ( 6
)
The limit δ p /2 -1 and (δ p -1)/2 in the sum are due to the fact that σ 2 ξk = σ 2 ξδ p -k . It is obvious from (4) that the channel frequency response and the interpolation error are correlated.

The model proposed in [START_REF] Wilson | Probability Density Functions for Analyzing Multi-Amplitude Constellations in Rayleigh and Ricean Channels[END_REF] is then not valid in this context. Since the channel r and the error ξ both follow a Rayleigh distribution, we know from [START_REF] Tang | Effect of Channel Estimation Error on M-QAM BER Performance in Rayleigh Fading[END_REF], [START_REF] Chau | On the Second-Order Statistics of Correlated Cascaded Rayleigh Fading Channels[END_REF], [START_REF] Simon | Probability Distributions Involving Gaussian Random Variables: A Handbook For Engineers and Scientists[END_REF] that the joint pdf of two correlated Rayleigh variables is

p r,ξ (r, ξ) = rξ σ 2 r σ 2 ξ (1 -ρ 2 rξ ) exp - σ 2 ξ r 2 + σ 2 r ξ 2 2σ 2 r σ 2 ξ (1 -ρ 2 rξ ) I 0 rξρ rξ σ r σ ξ (1 -ρ 2 rξ ) , (7)
where ρ rξ is the correlation coefficient between the channel and the error of estimation due to the interpolation, and I 0 is the modified Bessel function of first kind with order zero. As for the variance σ 2 ξ , ρ rξ depends on the interpolation method, and its expression will derived afterward. From ( 7), we also derive the conditional pdf p ξ|r,ξk (ξ|r, ξ k ) defined for a given value k by p ξ|r,ξk (ξ|r, ξ k ) = pr,ξ k (r,ξk) pr(r) . Similarly as ξ, the generalized expression of the conditional pdf is also a weighted sum of the conditional pdfs for different k values:

p ξ|r (ξ|r) =        1 δp-1 p ξ|r,ξδ p/2 (ξ|r, ξ δp/2 ) + 2 δp-1 δp/2-1 k=1 p ξ|r,ξk (ξ|r, ξ k ), if δ p is even 2 δp-1 (δp-1)/2 k=1 p ξ|r,ξk (ξ|r, ξ k ), if δ p is odd . (8)
This general statistics of the estimation errors are applied in the following, to the NN, the linear and the piecewise cubic interpolations.

B. Nearest Neighbor Interpolation

1) Reminder: Let us consider three consecutive frequency pilot positions f p-δp , f p and f p+δp , such as f p is not at the edge of the channel, i.e. p / ∈ {0, M -1}. From that, we denote f p -and

f p + the centers of the intervals [f p-δp , f p ] and [f p , f p+δp ] respectively. ∀f ∈ [f p -, f p + ],
the channel estimation made by a NN interpolation within this interval is simply written

Ĥ(f ) = Ĥ(f p ) LS = H(f p ) + W (f p ) C(f p ) , (9)
where Ĥ(f p ) LS is the LS estimation performed on the position

f p . If p = 0 or p = M -1, the NN interpolation is limited to the intervals [f 0 , f 0 + ] and [f M -1 -, f M -1 ],
respectively, but the expression (9) remains the same.

2) Variance of the Errors for the NN Interpolation:

By replacing H by its value given by (2), it yields

ξ N N = | L-1 l=0 h l e -2jπfpτl (1 -e -2jπ∆f τl )|, (10)
where

∆ f = f -f p . As e -2jπfpτl (1 -e -2jπ∆f τl ) is deterministic, we conclude that, since |H(f )|
follows a Rayleigh distribution, then ξ N N has the same distribution. From [START_REF] Tang | Effect of Channel Estimation Error on M-QAM BER Performance in Rayleigh Fading[END_REF], whatever p = 0, δ p , .., M -1 and whatever δ p , we now express ξ k,N N the discrete version of ξ N N in [START_REF] Tang | Effect of Channel Estimation Error on M-QAM BER Performance in Rayleigh Fading[END_REF]:

ξ k,N N = | L-1 l=0 h l e -2jπ pβ l M (1 -e -2jπ kβ l M )|. (11)
The variance of ξ k,N N , noted σ 2 ξk,NN , is expressed

σ 2 ξk,NN = E{e h,N N e * h,N N } = E L-1 l1=0 h l1 e -2jπ pβ l 1 M (1 -e -2jπ kβ l 1 M ) L-1 l2=0 h l2 e -2jπ pβ l 2 M (1 -e -2jπ kβ l 2 M ) * = E L-1 l=0 |h l | 2 (2 -e -2jπ kβ l M -e 2jπ kβ l M ) = L-1 l=0 σ 2 l (2 -2 cos(2π
kβ l M )). (12)
We observe in ( 12) that, since σ 2 ξk,NN depends on k, it consequently depends on the considered gap δ p . Thus, for δ p ∈ {2, 3}, |k| = 1, so σ 2 ξk,NN takes a sole value. For δ p = 4, |k| = 1 for two thirds of the estimated carriers in a discrete interval described in Fig. 2, and |k| = 2 for one third. Thus, we deduce the variance σ 2 ξNN of the error ξ N N (considered for all k):

σ 2 ξNN =                σ 2 ξ1,NN = L-1 l=0 σ 2 l (2 -2 cos(2π βl M )), if δ p ∈ {2, 3} 2 3 σ 2 ξ1,NN + 1 3 σ 2 ξ2,NN = 2 3 L-1 l=0 σ 2 l (2 -2 cos(2π βl M )) + 1 3 L-1 l=0 σ 2 l (2 -2 cos(2π 2βl M )), if δ p = 4, (13)
3) Correlation coefficient ρ rξ for the NN Interpolation: The correlation coefficient, noted ρ rξ,N N in the case of a NN interpolation, is expressed by

ρ rξ,N N = |E{H(f )e * h }| σ h σ ξNN = | L-1 l=0 E{|h l | 2 (e 2jπτl(fp-f ) -1)}| σ h σ ξNN = | L-1 l=0 σ 2 l (e -2jπτl∆f -1)| σ h σ ξNN , (14)
with σ h the variance of the channel. In the discrete domain, the error depends on the value of k, so we obtain:

ρ rξk,N N = | L-1 l=0 σ 2 l (e -2jπ β l k M -1)| σ h σ ξk,NN . (15)
C. Linear Interpolation 1) Reminder: In this section, as previously, a similar development is performed for the linear interpolation. We first remind the expression of the linear interpolation, for f ∈ [f p , f p+δp ], Ĥ(f ) is obtained by

Ĥ(f ) = ĤLS (f p ) + (f -f p ) ĤLS (f p+δp ) -ĤLS (f p ) f p+δp -f p , (16)
where ĤLS (f p ) is the LS estimation performed on the p th pilot tone. We remind that, in this section, we are only interested in the interpolation errors, so ( 16) is rewritten without the noise component:

Ĥ(f ) int = H(f p ) + (f -f p ) H(f p+δp ) -H(f p ) f p+δp -f p . ( 17
)
2) Variance of the Errors for the Linear Interpolation: For the linear interpolation, it is well known that the error between a given function g of class C 2 and its approximation ĝ by a linear function at a point x is given by

|g(x) -ĝ(x)| = 1 2 |(x p+δp -x)(x p -x)| × |g ′′ (y)|, (18)
where x p and x p+δp are two consecutive nodes such as ĝ(x p ) = g(x p ) and ĝ(x p+δp ) = g(x p+δp ), and y ∈ [x p , x p+δp ]. This expression comes from the Taylor's expansion of g(x) on x p and x p+δp .

It is obvious that the channel frequency response (2) is C 2 on [f 0 , f M -1 ], so [START_REF] Hsieh | Channel Estimation for OFDM Systems Based on Comb-Type Pilot Arrangement in Frequency Selective Fading Channels[END_REF] can be used to determine the error of interpolation noted ξ li , on each interval [f p , f p+δp ]:

ξ li = |H(f ) -Ĥ(f )| = 1 2 |(f p+δp -f )(f p -f )| × |H ′′ (α)|, (19)
where α ∈ [f p , f p+δp ] and

H ′′ (f ) = -4π 2 L-1 l=1 h l τ 2 l e -2jπf τl . ( 20
)
It is straightforward from [START_REF] Bello | Characterization of randomly time-variant linear channels[END_REF] that if H(f ) has a Rayleigh distribution, then H ′′ (f ) follows a Rayleigh distribution as well. For further results, we deduce that, whatever n ∈ N, the n th derivative

H (n) (f )
follows a Rayleigh distribution. We then deduce the variance of H ′′ (f ):

σ 2 H ′′ = E{H ′′ (f )H ′′ (f ) * } = L-1 l=1 E{|4π 2 h l τ 2 l | 2 } = 16π 4 L-1 l=1 τ 4 l σ 2 l , (21)
and by inserting ( 21) into [START_REF] Ozdemir | Channel Estimation for Wireless OFDM Systems[END_REF], we deduce the variance of ξ li :

σ 2 ξli = 1 4 |(f p+δp -f )(f p -f )| 2 σ 2 H ′′ = |(f p+δp -f )(f p -f )| 2 4π 4 L-1 l=1 τ 4 l σ 2 l . (22)
In the discrete formalism, σ 2 ξli depends on the value k, and we note σ 2 ξk,li the variance given by

σ 2 ξk,li = | (δ p -k)k M 2 | 2 4π 4 L-1 l=1 β 4 l σ 2 l , (23)
for k = 1, .., δ p -1. One can notice if the channel tends to have flat fading (i.e. β l tends to zero), then σ 2 ξk,li logically tends to zero. Indeed, the linear interpolation of a constant function leads to an error equal to zero. The pdf of the error of the linear interpolation is denoted by p ξli (ξ). It has exactly the same formulation than the NN interpolation given in [START_REF] Etsi | Digital Radio Mondiale (DRM);System Specification[END_REF], by replacing σ 2 ξk,NN by σ 2 ξk,li . 3) Correlation coefficient ρ rξ for the Linear Interpolation: Since |H ′′ (f )| in (20) follows a Rayleigh distribution, the joint pdf p ξ,r (ξ, r) of the channel and the linear interpolation follows the distribution given by [START_REF] Alouini | A Unified Approach for Calculating Error Rates of Linearly Modulated Signals over Generalized Fading Channels[END_REF]. We now express the correlation coefficient ρ rξ,li between the channel and its linear interpolation error:

ρ rξ,li = |E{H(f )e * h }| σ h σ ξli = |(f p+δp -f )(f p -f )|2π 2 L-1 l=1 τ 2 l σ 2 l σ h σ ξli . ( 24
)
Like in the NN case, we notice from (24) that ρ rξ,li depends on f . We directly derive the discrete expression of the correlation coefficient depending on k and noted ρ rξk,li :

ρ rξk,li = | (δp-k)k M 2 |2π 2 L-1 l=1 β 2 l σ 2 l σ r σ ξk,li . (25)

D. Piecewise Cubic Interpolation 1) Reminder:

We now consider the piecewise cubic polynomial interpolation made with the Lagrange basis polynomials {L p } defined by

                         L p (f ) = f -fp+δ p fp-fp+δ p × f -fp+2δ p fp-fp+2δ p × f -fp+3δ p fp-fp+3δ p L p+δp (f ) = f -fp fp+δ p -fp × f -fp+2δ p fp+δ p -fp+2δ p × f -fp+3δ p fp+δ p -fp+3δ p L p+2δp (f ) = f -fp fp+2δ p -fp × f -fp+δ p fp+2δ p -fp+δ p × f -fp+3δ p fp+2δ p -fp+3δ p L p+3δp (f ) = f -fp fp+3δ p -fp × f -fp+δ p fp+3δ p -fp+δ p × f -fp+2δ p fp+3δ p -fp+2δ p . ( 26
)
The interpolated channel Ĥ(f ) build thanks to the Lagrange basis (26) and defined on [f p , f p+3δp ] is then given by

Ĥ(f ) = 3 n=0 H(f p + nδ p )L p+nδp (f ). ( 27
)
The whole estimated channel is then obtained by performing (27

) on each interval [f 0 , f 3δp ], [f 3δp , f 6δp ], ... , [f M -1-3δp , f M -1 ] extracted from [f 0 , f M -1 ].
2) Variance of the Errors for the Piecewise Cubic Interpolation: In the same way as the linear interpolation, it is known that, thanks to a Taylor's expansion of a function g of class C 4 , the error of interpolation in the case of a Lagrange cubic interpolation is given by

|g(x) -ĝ(x)| = 1 24 | 3 n=0 (x -x p+nδp )| × |g (4) (y)|, (28)
where g (4) is the fourth derivative of g and y is in [x p , x p+nδp ]. As we notice that the channel frequency response (2) is C 4 , we replace g(x) by H(f ) in (28) to get the expression of the error:

ξ cu = 1 24 | 3 n=0 (f -f p+nδp )| × |H (4) (α)|. (29)
From (29), we deduce that ξ cu follows a Rayleigh distribution and, by using the same development as from ( 19) to [START_REF] Chau | On the Second-Order Statistics of Correlated Cascaded Rayleigh Fading Channels[END_REF], we obtain the variance of the error σ 2 ξcu in the continuous formalism:

σ 2 ξ,cu = 1 24 2 | 3 n=0 (f -f p+nδp )| 2 × 2 8 π 8 L-1 l=1 σ 2 l τ 8 l . (30)
The discrete formalism slightly differs from the one described on Fig. 2 The discrete expression of the error variance is written

σ 2 ξk,cu = 1 24 2 | 3 n=0 (nδ p -k) M 4 | 2 × 2 8 π 8 L-1 l=1 σ 2 l β 8 l . (31)
We observe on Fig. 3 that the interval [[p, p + 3δ p ]] has a central symmetry, hence we deduce the general expression of the probability density of ξ cu as a sum of weighted Rayleigh distributions, by following the same reasoning leading to (6):

p ξcu (ξ cu ) =                1 3(δp-1) ξcu σ 2 ξ δp +δp /2,cu e - ξ 2 cu 2σ 2 ξ δp +δp/2,cu + 2 3(δp-1) (δp+δp/2)-1 k=1 k =δp ξcu σ 2 ξ k,cu e - ξ 2 cu 2σ 2 ξ k,cu , if δ p is even 2 3(δp-1) δp+(δp-1)/2 k=1 k =δp ξcu σ 2 ξ k,cu e - ξ 2 cu 2σ 2 ξ k,cu , if δ p is odd . ( 32
)
For each value k, the weight 2 3(δp-1) is the ratio between the number of symmetrical subcarriers on the number of interpolated subcarriers in [[p, p + 3δ p ]].

3) Correlation coefficient ρ rξ for the Piecewise Cubic Interpolation: Since |H (4) (f )| follows a Rayleigh distribution, and by analogy with (25), we directly deduce the expression of the correlation coefficient in the case of the cubic interpolation:

ρ rξk,cu = | 3 n=0 (nδ p -k)|2 4 π 4 L-1 l=1 σ 2 l β 4 l 24M 4 σ r σ ξcu,k . (33)
Finally, the generalized expression of the conditional pdf p ξk,cu|r (ξ k,cu |r) for the cubic interpolation is the one of (8) in which the weights are those of (32).

In this section, it has been shown that a general expression of the interpolation errors statistics can be analytically derived. We then applied the analysis to three interpolation methods. By extrapolation, we deduce that it is possible to express the statistical properties of the interpolation errors as soon as the interpolation has an analytical expression of its error. In the following, we use these results to predict the mean square error of the LS estimator when performed with an interpolation.

IV. MEAN SQUARE ERROR OF LS ESTIMATION WITH INTERPOLATION

In [START_REF] Savaux | Artificial channel aided LMMSE estimation for timefrequency selective channels in OFDM context[END_REF], it is assumed that the mean square error of the estimations performed by LS with interpolations cannot be analytically expressed. Thus, the different methods are compared by means of simulations. In this section, thanks to the previous results concerning the statistics of the interpolations errors, we show that a theoretical expression of the MSE can be derived. To this end, since the interpolation of the channel is independent from the one of the noise, we now study the statistics of

( Wm Cm ) int (3)
Ŵ int m = W p /C p . (34)
From (34), we simply deduce that the NN-interpolated noise has exactly the same statistics as the initial noise Ŵm , i.e. Ŵ int m ∼ N (0, σ 2 ). 2) Linear-Interpolated Noise: Using the linear interpolation [START_REF] Kay | Fundamentals of Statistical Signal Processing : Estimation Theory[END_REF] in the discrete formalism and without taking the channel into account, the expression of Ŵ int m with m = p + k is derived as:

Ŵ int m = W p /C p + k W p+1 /C p+δp -W p /C p δ p . ( 35
)
As whatever p = 0, δ p , .., M -1, we have E{W p } = 0, we directly deduce that E{ Ŵ int m } = 0. Since k appears in (35), the variance of the linear interpolated noise, noted σ 2 li,k , is a function of k such as

σ 2 li,k = E{| Ŵ int m | 2 } = E{| W p /C p (δ p -k) + kW p+1 /C p+δp δ p | 2 }. (36)
Reminding that the noise samples have the same variance and are uncorrelated, we get

σ 2 li,k = (δ p -k) 2 + k 2 δ 2 p σ 2 . ( 37
)
The total variance of Ŵ int m , noted σ 2 li is the weighted sum of σ 2 li,k :

σ 2 li =        1 δp-1 σ 2 li,δp/2 + 2 δp-1 δp/2-1 k=1 σ 2 li,k , if δ p is even 2 δp-1 (δp-1)/2 k=1 σ 2 li,k , if δ p is odd . ( 38
)
3) Piecewise Cubic-Interpolated Noise: By using the Lagrange cubic interpolation (27) applied to the noise samples, we obtain the variance of the interpolated noise σ 2 li,k as a function of k:

σ 2 cu,k = E{| Ŵ int m | 2 } = E{| 3 n=0 W (p + nδ p )L p+nδp (p + k)| 2 } = 3 n=0 L p+nδp (p + k) 2 σ 2 . ( 39
)
The total variance of Ŵ int m , noted σ 2 cu is the weighted sum of σ 2 cu,k :

σ 2 cu =              1 3(δp-1) σ 2 cu,δp+δp/2 + 2 3(δp-1) (δp+δp/2)-1 k=1 k =δp σ 2 cu,k , if δ p is even 2 3(δp-1) (δp+δp/2)-1 k=1 k =δp σ 2 cu,k , if δ p is odd . ( 40
)
In this first part, the interpolation errors and the interpolated noise have been statistically characterized. This can now be used to express the MSE of the LS with interpolations estimation.

B. Analytical Expression of the MSE of the LS Estimations Performed with Interpolation

The mean square error of a given interpolation is noted M SE int and is written

M SE int = 1 M E{|| Ĥ -H|| 2 }. ( 41
)
Each OFDM symbol being composed of pilot tones and data carriers, we split the development into two parts. On each pilot tone, the MSE has already been expressed in [START_REF] Biguesh | Downlink Channel Estimation in Cellular Systems with Antenna Arrays at Base Stations Using Channel Probing with Feedback[END_REF], [START_REF] Savaux | Artificial channel aided LMMSE estimation for timefrequency selective channels in OFDM context[END_REF]. In that case, the MSE, denoted by M SE p , is given by

M SE p = σ 2 P , (42)
with P the pilots power. Although in this article, it is assumed that P = 1, we keep the notation P in the further developments to provide a general expression of the MSE. On the data carriers m = p, the MSE is noted M SE d , and from (3), we obtain

M SE d = E | Ĥint m + W m C m int -H m | 2 , ( 43
)
and as the noise and the channel are uncorrelated, it yields

M SE d = E | Ĥint m -H m | 2 + E | W m C m int | 2 = σ 2 ξ + σ 2 int P , (44)
where σ 2 int is equal to σ 2 , σ 2 li (38) and σ 2 cu (40) for the NN, the linear and the piecewise cubic interpolations, respectively. Finally, the MSE is obtained by inserting ( 44) and ( 42) in (41):

M SE int = N p M σ 2 P + M -N p M (σ 2 ξ + σ 2 int P ), (45)
where N p is the number of pilot tones in an OFDM symbol. When performed on a preamble, we know from [START_REF] Biguesh | Downlink Channel Estimation in Cellular Systems with Antenna Arrays at Base Stations Using Channel Probing with Feedback[END_REF] that the MSE of LS tends to zero for high SNR values. We notice in (45) that M SE int tends to M -Np M σ 2 ξ for high SNR values. This is the result of the residual error due to the interpolations. Fig. 4 displays the pdfs of the interpolated noise for the three considered methods, assuming a frequency gap δ p = 2. The original noise has a variance σ 2 = 1, as depicted by its pdf. As expected, we observe that the NN-interpolated noise has the same variance as the original one, whereas the linear and cubic interpolations reduce the noise variance. We should also notice that the linear-interpolated noise has a lower variance than the cubic. Consequently, we deduce that the former interpolator makes the LS estimation less sensitive to the noise than the latter one. This observation will be confirmed in terms of performance in Section VI.

V. BIT ERROR RATE FLOOR OF LS ESTIMATION WITH INTERPOLATION

A. Geometrical Considerations 1) Reminder: In this section, the error floor reached by the BER curves for high SNR values is analytically evaluated when interpolation errors occur. In order to only consider the residual errors due to the interpolations, it is assumed high SNR values, so the noise is then neglected in this section. Since we focus on the effects of the interpolations errors on the BER, the geometry of the constellation has to be taken into account. In the following, a BPSK (C ∈ {-1, +1}) and a 4-QAM

(C ∈ { + -d + -jd}
) mapped with a Gray encoding are considered, with C a transmitted symbol and d a normalization coefficient.

At the receiver side, the channel is estimated by means of LS with interpolations. Then, the received symbol is estimated by a simple zero forcing equalizer. As the noise is supposed to be null, the estimated symbol Ĉ is written as follows

Ĉ = H(f ) H(f ) + e h C. (46)
For a BPSK, the decision domains are exactly symmetric for C = + -1. Let us then set C = +1 in the following developments. The probability of detection error, noted P BP SK e is then given by

P BP SK e = P r Re{ Ĉ} < 0|C = 1 . (47)
In the same way for a 4-QAM, and as mentioned in [START_REF] Yih | Effects of Channel Estimation Error in the Presence of CFO on OFDM BER in Frequency-Selective Rayleigh Fading Channels[END_REF], the decision domains are exactly symmetric with regard to the Q and I axes for the most significant bit (MSB) and for the least significant bit (LSB), respectively. Thus, let us set C = +d + jd in the following developments. In that case, the probability of error of detection, noted P 4QAM e is then given by

P 4QAM e = P r Re{ Ĉ} < 0|C = d + jd . (48)
For a given value |H(f )| = r, we denote the complex expressions of the channel H(f ) = re jΘH and the error e h = ξe jΘξ , and we define Θ = Θ H -Θ ξ . Let us now derive the analytical expression of (47) and (48).

2) BPSK: After some straightforward developments, the inequality in (47) yields

Re{ Ĉ} < 0 with C = 1 ⇔ Re{ H(f ) H(f ) + e h × 1} < 0 ⇔ r 2 + rξ cos(Θ) |H(f ) + e h | 2 < 0. (49)
Since |H(f ) + e h | 2 > 0, we limit the study to the numerator of (49). It can be seen that if ξ ≤ r, then ∀ Θ ∈ [0, 2π] the inequality cannot be valid because -1 ≤ cos(Θ) ≤ 1. Thus, P BP SK e = 0 as long as ξ ≤ r. For ξ ≥ r, the inequality (49) is valid for

Θ ∈ [-arccos(- r ξ ), arccos(- r ξ )]. ( 50
)
As Θ is defined in [0, 2π], we then express P BP SK e for ξ ≥ r as a function of r and ξ

P BP SK e = arccos(-r ξ ) -(-arccos(-r ξ )) 2π = 1 - arccos(-r ξ ) π , (51)
and then, finally, P BP SK e is expressed by

P BP SK e =        0 if ξ ≤ r 1 - arccos(-r ξ ) π if ξ ≥ r . ( 52
)
As arccos(1) π = 1, then P BP SK e is a continuous function on R + . From the conditional pdf p ξ|r (ξ|r),

we extract the conditional pdf of errors of estimation causing a wrong decision on the bits for a BPSK, that we denote Φ ξ|r,BP SK (ξ|r, Re{ Ĉ} = Re{C}):

Φ ξ|r,BP SK (ξ|r, Re{ Ĉ} = Re{C}) = P BP SK e p ξ|r (ξ|r).

Fig. 5 (a) displays the Rayleigh density p ξ|r (ξ|r) for r = 1, σ 2 ξ = 1, ρ rξ = 0 (i.e. p ξ (ξ)) and the extracted function Φ ξ|r,BP SK (ξ|r, Re{ Ĉ} = Re{C}), drawn by simulations and by means of (53).

The curves are obtained after the transmission of 10 6 bits.

It is verified that Φ ξ|r,BP SK (ξ|r, Re{ Ĉ} = Re{C}) = 0 for ξ ≤ r, with r = 1 in the simulation. Furthermore, the theoretical curve and the one drawn by simulations exactly match, which validates the previous developments.

3) 4-QAM Constellation: After some developments, the inequality Re{ Ĉ} < 0 in (48) yields

Re{ Ĉ} < 0 ⇔ Re{H(f )(H(f ) + e h ) * (1 + j)} < 0 ⇔ Re{(r 2 + rξe j(ΘH -Θξ) )(1 + j)} < 0 ⇔ r + ξ(cos(Θ) -sin(Θ)) < 0 (54)
If we set t = tan( Θ 2 ), we then get cos(Θ) = 1-t 2 1+t 2 and sin(Θ) = 2t 1+t 2 , and (54) yields

r(1 + t 2 ) + ξ(1 -t 2 -2t) 1 + t 2 < 0. (55)
Since 1 + t 2 > 0 whatever the value of t, we study the sign of the numerator in (55), and we denote Q the polynomial:

Q(t) = t 2 (r -ξ) -2ξt + r + ξ. ( 56
)
The solving of the inequality (55) is made by studying the convexity or concavity of Q and by looking for its roots. We finally obtain:

P 4QAM e =                0, if 0 ≤ ξ ≤ r √ 2 arctan(t + 2 )-arctan(t - 2 ) π , if r √ 2 ≤ ξ ≤ r 1 + arctan(t - 3 )-arctan(t + 3 ) π , if r ≤ ξ , ( 57
)
where t ± 2 and t ± 3 are expressed by

t ± 2 = 2ξ + - ∆ p 2(r -ξ)
, and

t ± 3 = 2ξ - + ∆ p 2(r -ξ) , (58)
with ∆ p = 4ξ 2 -4(r 2 -ξ 2 ) the discriminant of Q. It is straightforward to show that P 4QAM drawn with the same parameters as previously. Once more, it is verified that the theoretical curve and the one drawn by simulation perfectly match.

B. Analytical Expression of the BER Floor

The bit error rate floor, noted BER f loor , is the double integral of the conditional pdf Φ ξ|r,const (ξ|r, Re{ Ĉ} = Re{C}) multiplied by p r (r). Some simplifications lead to:

where P const e depends on the constellation (it is defined in (52) for the BPSK and in (57) for the 4-QAM).

VI. SIMULATIONS AND DISCUSSION

A. Simulations Parameters

In order to validate the previous developments by simulations, we do not limit us to a given standard, but we define two different general channels noted H (1) and H [START_REF] Dong | Linear Interpolation in Pilot Symbol Assisted Channel Estimation for OFDM[END_REF] . Their parameters (the delay and variance of each non-null path) are summed up in Tables I andII. It can be seen that the number of paths, the maximum delay and the variance of the channels differ from H (1) to H (2) . Firstly, we observe that the theoretical curves and the ones drawn by simulations perfectly match, which validates the theoretical results derived in this article. As expected, the MSE reaches an error floor that decreases when the polynomial order increases. Furthermore, these errors floors are higher for the channel H Secondly, from Figs. (c) and (d), we observe that the linear interpolation has a lower MSE than the cubic one for low SNR values, whereas the cubic interpolator has a better performance than linear one for higher SNR values. It is explained by the aforementioned observation in regard to the variance of the interpolated noise made in Section IV. Indeed, at low SNR values, the noise effect prevails on the interpolation errors, and since the linear-interpolated noise has a lower variance than the cubic-interpolated noise, linear interpolation has a lower MSE. On the contrary, for higher SNR, interpolation errors effect prevails on the noise, and since the cubic interpolation is more accurate than the linear one, then the former has a lower MSE than the latter. These results will be discussed in Section VI-D. It can be seen on Fig. 7 that, whatever the chosen set of parameters, the simulated curves reach a BER floor matching the analytical values. The very slight differences can be explained by the fact that the integral (62) is computed in a discrete formalism. However, we clearly observe that the BER lower bound decreases when the degree of the polynomial interpolator increases. Furthermore, the performance is more degraded in H (2) compared to H (1) , due to H (2) is more frequency selective than H (1) . These results then allow to validate the theoretical developments proposed in this article.

C. Bit Error Rate Low Bound

D. Discussion

In addition to be a theoretical analysis of the LS estimator performance in a frequency selective Rayleigh channel by means of polynomial interpolations, we can deduce a practical implementation from the proposed method. Indeed, an adaptive receiver (in terms of estimation technique) can be implemented, according to the trade-off between performance and complexity. It is known that the NN, the linear and the cubic interpolations require zero, two and four multiplications per interpolated value, respectively. Thus, for a given BER target, and given the frequency selectivity, the proposed method enables the receiver to choose the interpolation technique with minimum computation cost.

Moreover, as it has been noticed for low SNR values, it may appear advantageous to perform the estimation by means of the linear interpolation instead of the cubic one, while it is not a priori intuitive. Besides, this study can be applied in other domains, for which interpolations of random functions are required, as in image processing for instance.

VII. CONCLUSION

In this article, a theoretical study of the interpolation errors applied to the channel estimation has been presented. Firstly, a characterization of the errors statistics has been performed, and a geometrical analysis of the error effect on the constellation has been proposed. Then, an analytical expression of the MSE of the LS estimation performed with interpolation has been derived, just as the expression of the BER floor. For a given channel, we have shown that these values are functions of the frequency gap between the pilot tones δ p , the interpolation technique and the constellation size. The developments have been made for the polynomials of degree zero, one and three, and for the BPSK and the 4-QAM constellations as well. It has been shown that the theoretical results perfectly match the simulations.

This analysis can then be used to a priori predict the performance of the system. It differs from the existing works of the literature, because the errors and the channel are correlated, and the MSE and BER can be exactly characterized. Beyond the theoretical performance analysis, we have proposed a possible practical implementation, in which the method is used to adapt the receiver according to the trade-off between the expected performance and the complexity of the interpolation technique.

Fig. 1 .

 1 Fig. 1. Illustration of the pilot arrangements for different values of δp, the pilot tones are black and the data carriers are white.

Fig. 2 .

 2 Fig. 2. Discrete intervals for the nearest neighbor interpolation, for different values of δp.

  in the case of the piecewise cubic interpolation, since each interval of interpolation is composed of four pilots instead of two. Thus, let us now consider frequency positions m = p + k in the discrete interval [[p, p + 3δ p ]], as described on Fig. for δ p = 2.

Fig. 3 .

 3 Fig. 3. Discrete representation of an interval for the piecewise cubic interpolation, with δp = 2.

Fig. 4 .

 4 Fig. 4. Pdfs of the interpolated noise, for three interpolations methods, and for δp = 2, and σ 2 = 1.

Fig. 5 (

 5 Fig. 5 (b) depicts the Rayleigh density p ξ|r (ξ|r) and the function Φ ξ|r,4QAM (ξ|r, Re{ Ĉ} = Re{C})

Fig. 5 .

 5 Fig. 5. Extracted conditional pdfs, (a) for a BPSK and (b) for a 4-QAM.

Fig. 6

 6 Fig. 6 displays the MSE of the channel estimations performed with the NN, the linear (Figs. (a) and (b)) and the cubic (subfigures (c) and (d)) interpolations versus the SNR given in dB. The theoretical curves given by (45) are compared to the ones drawn by simulation. The curves are obtained by means of 2400 simulations runs.

( 2 )

 2 (Figs. (b) and (d)) than for the channel H (1) (Figs. (a) and (c)). This is in accordance with the theory, since H (2) is more frequency selective than H (1) .

Fig. 7

 7 Fig. 7 shows the simulated BER curves and the theoretical BER low bound values versus the SNR given in dB. Figs. 7 (a) -(b) display the BER curves versus the SNR for the NN interpolation, Figs. 7 (c) -(d) for the linear interpolation and Figs. 7 (e) -(f) for the cubic interpolation. In order to validate the proposed developments, the simulations are made for both channels H (1) and H (2) , and for δ p ∈ {2, 4}. The curves are obtained with 2 × 10 7 bits.

Fig. 7 .

 7 Fig. 7. BER of the channel estimation performed with (a) -(b) NN interpolation, (c) -(d) linear interpolation and (e) -(f) cubic interpolation, for δp ∈ {2, 4}, and for the channels H (1) and H (2) .

  separately. It is assumed that |C m | = 1, so ( Wm Cm ) int has the same statistics as W m , and we simply note Ŵ int m = ( Wm Cm ) int in the following.

A. Statistics of the

Interpolated Noise 1) NN-Interpolated Noise: Whatever the values of δ p and k, and for a fixed value p, the interpolated noise sample Ŵ int m such as with m = p + k, is expressed by

TABLE II CHANNEL

 II H (2) PARAMETERS.

		Channel H (2)
	paths l	0 1	2	3	4	5	6
	delays β l	0 4	9	14	16	18	21
	variance σ 2 l	1 0.5 0.4 0.3 0.3 0.2 0.1
	B. Mean Square Error


Meriem Mhedhbi
email: meriem.mhedhbi@univ-rennes1.fr

Stéphane Avrillon
email: stephane.avrillon@univ-rennes1.fr

Troels Pedersen
email: troels@es.aau.dk

Bernard Uguen
email: bernard.uguen@univ-rennes1.fr

Modelling of UWB Antenna Perturbed by Human Phantom in Spherical Harmonics Space

Keywords: UWB antenna, spherical harmonics, Body Area Networks, measurement

The spherical harmonics expansion is an efficient tool to represent the antenna radiation pattern, especially in physical simulator, where the knowledge of full antenna data is fundamental and computationally intensive as well. Thus, the SH become imperative and represent compactly the antenna by reducing the data into a small number of coefficients. In Body Area Network (BAN) context, the goal is to go further than a compact storage of antenna data. In fact, we aim to study the effect of human phantom presence on the antenna in terms of spherical harmonics coefficients. The proposed paper will present a simple SH coefficients model allowing to predict the antenna behaviour disrupted by a human phantom, based on the knowledge of the SH coefficient of antenna in free space and antenna-phantom distance.

I. INTRODUCTION

In the context of modeling and simulation of the radio channel experienced in a wireless body area network (WBAN), the knowledge of the antenna radiation pattern is important. Moreover, a nearby body affects the antenna pattern and should therefore be included in the simulation model. However, this poses a number of challenges.

Firstly, if relying directly on measurements of the antenna alone, the obtained radiation pattern should be adapted to include the presence of the human body. For this purpose, simple models describing the relation between simple parameters such as antenna-body distance and body size would be of importance. Secondly, if relying on measurements of the combined of an antenna and body yields a new data set for each pair of antennas and bodies. The potential large number of combinations of body parts, distances, and antennas, would require a large database of antenna measurement data, which is inconvenient in such a simulator. It is therefore natural to consider models in which the antenna pattern for each antenna body configuration can be represented by only few coefficients and where the pattern reconstruction is straightforward.

In this work we approach the antenna data storage problem by means of expansion of the pattern in spherical harmonics (SH) [START_REF] Roblin | Ultra compressed parametric modelling of uwb antenna measurements[END_REF], [START_REF] Burghelea | Vector spherical harmonics antenna description for ir-uwb ray tracing simulator[END_REF], [START_REF] Burghelea | Ultra-compact parametric modelling of three-dimensional antenna radiation pattern for impulse radio-ultra-wide band ray tracing[END_REF]. In the SH basis, the full antenna data can be compressed to only a few coefficients. Traditionally, vector spherical harmonic (VSH) expansion have been the most popular basis for representation of far-field antenna radiation patterns because it preserves inherently the transversal field. While this feature is attractive, the VSH basis has the drawback of more complicated basis functions burdening the reconstruction task. We follow the alternative approach proposed in [START_REF] Rahola | Modelling of radiation patterns using scalar spherical harmonics with vector coefficients[END_REF] of expanding the radiation pattern in terms of scalar spherical harmonics (SSHs).

In the present contribution, we study the effect of the human body presence close to the antenna in terms of SSH coefficients variation. In particular we focus on the change in the expansion coefficients as a function of phantom-antenna distance. We propose and validate a model for the SSH coefficients able to predict the observed variation of antenna radiation pattern.

II. MEASUREMENT AND POST-PROCESSING A. Measurement Setup

For the study, we designed a measurement campaign specifically to study the antenna response variation in the presence of a dielectric cylinder acting as a human phantom for different antenna/phantom distances (d). The antenna radiation pattern was measured in a SATIMO SG32 near-field measurement chamber at INSA Rennes. As depicted in Fig. 1, the presence of a human arm was mimicked by a phantom mounted in the vicinity of the antenna using a custom-made platform of dielectric foam with relative permittivity close to unity. The phantom consists of a nearly cylindrical plastic bottle with of radius 35 mm filled with MSL1800 phantom liquid. The size and shape of the phantom was chosen to represent a human arm, while still being lighted enough for the platform to support it. The bottle was fitted with a screw in the bottom to allow for mounting on the platform groove or holes. A detailed description of the measurement setup is available in [START_REF] Pedersen | Fp7 where2 deliverable d1[END_REF].

B. Extraction of Scalar spherical Harmonics Coefficients

The obtained data consists of a far-field pattern in Cartesian coordinates systems denoted by F c (θ, φ, f ) at co-elevation θ and azimuth φ, for each frequency f . We used a regular directional sampling grid given by:

(θ i , φ j ) = i N t , 2j N p π (1)
where 0 ≤ i ≤ N t , 0 ≤ j < N p , N t and N p are the numbers of θ and φ samples respectively. Following [START_REF] Rahola | Modelling of radiation patterns using scalar spherical harmonics with vector coefficients[END_REF], [START_REF] Chen | Radiation pattern analysis of arbitrary wire antennas using spherical mode expansions with vector coefficients[END_REF], the far-field radiation pattern represented in Cartesian systems and expanded in terms of SSHs with vector coefficients:

F c (θ, φ, f ) =   F x F y F x   = L l=0 l m=-l c lm (f ) Y m l (θ, φ) , (2)
where the SSH function of level l and mode m is defined by:

Y m l (θ, φ) = (2l + 1)(l -m)! 4π(l + m)! P m l (cos θ)e imφ (3)
with P m l denoting the associated Legendre polynomials of level l and mode m. We limit the expansion to a maximum level denoted by L.

The SSH vector coefficients are obtained via least squares as follows. At each directional grid point, the antenna response is a three-dimensional complex vector. We arranging these N t N p vectors as columns the matrix F c of dimension 3 × (N t N p ). With this definition, the expansion in (2) reads:

F c = CY (4)
where the entries of the (1 + L) 2 , N t × N p matrix Y, are the spherical harmonic functions evaluated at the sampling grid points. The matrix, C is of dimension a 3, (1 + L) 2 with the c lm as columns arranged accordingly. Estimates of the coefficients c lm are obtained in a straight forward manner by the lest square method

Ĉ = F c Y + , (5)
where Y + is the Moore-Penrose pseudo-inverse of Y.

III. DATA ANALYSIS AND OBSERVATIONS FROM MEASUREMENTS

In this section we are studying the effect of human phantom at different configurations in terms of antenna radiation pattern, Figure 2, and SH coefficients, Figure 3. Especially, the SH coefficients C (d, f ) are investigated in terms of number of coefficients and magnitude. In the following, most important observations,represent the key of the modelling task later,are presented.

A. Observation 1: Number of significant modes representing the antenna pattern

From Figure 2 we observe that antenna pattern is radiating mostly in the half space opposite to the phantom and converges to the free space case when d increases. This behaviour is expected considering the omnidirectional nature of the antenna. Besides, some ripples around the free space case are observed. Hence the pattern converge to the free space case by recovering the omnidirectional behaviour and oscillating around the free space case. These ripples might be the origin of the increase of modes number to represent the perturbed antenna shown in Figure 3. The increase of coefficient number might be explained differently. In fact, the number of coefficient depends on antenna dimensions [START_REF] Chen | Radiation pattern analysis of arbitrary wire antennas using spherical mode expansions with vector coefficients[END_REF], [START_REF] Hansen | Institution of Electrical Engineers[END_REF]. By introducing the phantom, an equivalent antenna (antenna+phantom) may be considered which dimensions increase with antenna/phantom distance.

B. Observation 2: Omnidirectional mode 0 energy variation Mode 0 energy E 0 refers to energy provided by coefficients C l0 , where 0 ≤ l ≤ L. Figure 4 shows the evolution of E 0 for the different measured d. This mode energy takes it is minimum value when the phantom is too close to the antenna and increases with d. As the phantom approaches the antenna the pattern is becomes increasingly directional, and since mode 0 coefficients represent the omnidirectional contribution which is required by increasing d, E 0 is increasing as well.

C. Observation 3: Coefficient correlation functions

As presented in previous section (Fig. 3) the presence of the phantom induces the appearance of new coefficients. In order to achieve the modelling task, the origin of the new coefficients needs to be identified. Therefore the correlation functions, equation 7, are studied in order to identify a possible relation between free space and perturbed antenna coefficients. number of peaks in CCF is higher than 2 and even more increases with the distance(same behaviour of modes number), we assume that there is a strong relation between the new coefficients and free space coefficients.

k = m (2L + 1 -m) 2 + l (6) R (n) = (1+L)(2+L)/2 k=0 C f s (k) C d (k + n) (7

IV. PROPOSED MODEL STRUCTURE

A SH coefficient model able to anticipate the pattern of a disrupted antenna by a human phantom is proposed in this part. The main goal of this model is to make the channel modelling task in physical simulators easier. Based on both knowledge of antenna SH coefficients in free space and the antenna-phantom distance, the challenge is to generate the SH coefficients of the perturbed antenna (Figure 6). The presence of the phantom induces the appearance of the new coefficients (modes), therefore the modelling task treat separately both types of coefficient (free space and new coefficients). However, before coefficients modelling, the number of modes M should be identified. Based on observation 1, M varies proportionally with the distance, and using a stair case fitting M is determined, Figure7. Another parameter should be introduced M 0 which is the maximum mode representing the free space antenna. We notice that for a pure omnidirectional pattern M 0 = 0, but for measured antennas higher modes M 0 = [1, 2] can be observed. It depends on antenna pattern and measurement conditions. b) m ≤ M 0 : The Spherical harmonics coefficient C lm where m ≤ M 0 represents the free space antenna coefficients whose behaviour is generally omnidirectional.The perturbation of these coefficients decrease with the antenna/phantom distance d. Based on observation 2 representing the variation of mode 0 energy (which is generally the most significant modes) we assume that the perturbation caused by the phantom is multiplicative and varies linearly with the distance. c) m > M 0 : The Spherical harmonics coefficient C lm where m > M 0 represents the coefficients that appear due to the phantom presence. This subtask of coefficient modelling is an extrapolation since these coefficient do not exist in free space. To be able to predict their behaviour we studied the cross correlation functions. Based on observation 3 which reveals a strong correlation between free space modes and new modes we propose that these coefficients represent a mitigated and shifted copy from the highest mode (M 0 ) representing free space antenna.

Based on observations and data analysis presented in previous, the proposed model structure is the following:

C lm (f, d) = A m (d) C f s lm (f ) , m ≤ M 0 A m (d) C f s lM0 (f ) , m > M 0 ( 8
)
where d is antenna/phantom distance,f is the frequency, C lm (f, d) are disrupted antenna SH coefficients, C f s lm (f ) are free space antenna coefficients, and A m (d) represents the

A m (d) = α m (d) exp (jφ m (d)) (9)
where α m (d) represents the real perturbation which varies linearly with the distance d (observation 2) and φ m (d) is a shift introduced introduced on disturbed coefficient.

The real perturbation α m (d) varies linearly with the distances, this choice is based on observation where the linear approximation seems to fit well the variation. But the structure depends on the modes:

α m (d) =    a m d + b m , m ≤ M 0 (a m d + b m ) m -M 0 , m > M 0 (10)
The free space coefficients (m ≤ M 0 ) depend only on the distance d which means the recovery of the free space structure and behaviour. The new coeffiecients (m > M 0 ) depend on the mode m and the distance d. In fact, the number of cofficient increases with the distance but not their values or effect which clear in Figure 3

φ m (d) = m (c m d + d m ) (11)
V. RESULTS

A. Model fitting

In this subsection the modelling results are presented for measured antenna Thomson 1 [START_REF] Gueguen | A low cost uwb printed dipole antenna with high performance[END_REF], in Figure 8, used to elaborate the model. These results are also obtained using numerical values in Table 1:

As the numerical values shows, α m (m ≤ M 0 ) increases with the distance which proves the fact that the antenna recover the behaviour in free space while α m (m < M 0 ) is decreasing with d because the energy is splitted over a higher number of coefficients (modes) but their impact is reduced. The value of the shift φ m is decreasing with respect to the distance because it has in impact on the antenna radiation direction. Besides observations show that the value of φ m depends on m and at a fixed distance, it increase with modes.

The Figure 11 shows the modelling results in terms antenna pattern in the plane θ = π 2 at different frequencies and antenna/phantom distances.

Thus, as the results show, the model is able to generate the global behaviour of perturbed antenna by human phantom. In fact, the half space radiation is controlled and maximum and minimum radiation directions as well and in terms of relative error (equation 12) averaged over the frequency bandwidth the results remain sufficient (around 10%, Figure 10).

= E m -E 2 E 2 (12)
where E m is the field obtained from modeled coefficient and E is the measured field.

The proposed model is frequency independent (we assume that the frequency dependency is in the coefficients themselves), level independent (because of the choice of measurement set: the phantom is parallel to z axis so the symmetry over φ is affected which means the modes) and linear (simple)is able to generate, with a sufficient reconstruction results, the set of disturbed antenna SH coefficient based on the knowledge of the free space coefficient and the antenna/phantom distance.

B. Model Validation

The same measurement campaign was carried out with another fractal structure antenna, Figure 8, in order to validate the model. In using the same model parameters the results seems to be very sufficient.

VI. CONCLUSION

The paper presents a simple model based on spherical harmonics coefficients to represent an antenna pattern, allowing to predict a disrupted antenna behaviour by a human phantom with respect to the antenna-phantom distance. This model is going to be integrated in a channel simulator where the body is approximated by dielectric cylinders animated thanks to motion capture files. The purpose being to have fast access to a representative perturbed antenna pattern with full polarization and ultra wide bandwidth. The dependency of the model parameters with constitutive parameters of the dielectric cylinder and cylinder dimension have still to be investigated. This work has demonstrated the following practical interesting fact that simple modification of the non perturbed antenna coefficient can reproduce reasonably accurate reconstruction of the antenna in the presence of a dielectric cylinder. In the future work, this model will be generalized and the different parameters should take into account the volume and the dimensions of the phantom.

Fig. 1 .

 1 Fig. 1. Measurement Platform

Fig. 2 .

 2 Fig. 2.Antenna radiation pattern at 3.1 GHz for different phantom configurations (x, y) plane.

Fig. 3 .

 3 Fig. 3. Magnitude of SSH coefficients ( c lm ) at 3.1 GHz in free space and with phantom at different distances

Fig. 4 .

 4 Fig. 4. Mode 0 energy w.r.t antenna/phantom distance

  The coefficients are indexed according to two integers l and m and to compute correlation functions we aggregate into one single index k. The equation 6 sequence the coefficients as follows (gathering all modes for different levels): m = [0, 0, ..., 0, 1, 1, ..., 1, ..., L, -L, ..., -1, -1, ..., -1] l = [0, 1, ...., L, 1, 2, ..., L, ..., L, L, ....., 1, 2, ....., L]

Fig. 5 .

 5 Fig. 5. Auto correlation coefficient function (top), cross correlation function between coefficient in free space and in phantom presence (bottom)

Fig. 6 .

 6 Fig. 6. Model Scheme

Fig. 7 .

 7 Fig. 7. Number of modes w.r.t antenna/phantom distance d

Fig. 8 .

 8 Fig. 8. Thomson 1 (left), fractal structure (right) antennas Parameters Values (m ≤ M 0 ) Values (m > M 0 ) am 0.002 -0.002 bm 0.55 1.55 cm 0 -0.006 dm 0 1.22

  (coefficients values) and Figure 5 (correlation functions). d) φ m (d): Regarding the shift φ m (d), it represents the shift between the different modes and follows a linear variation and depends on d and the mode m:

Fig. 9 .

 9 Fig. 9. Antenna pattern comparison (Thomson 1 antenna )

Fig. 10 .

 10 Fig. 10. Relative error (Thomson 1 antenna)

Fig. 11 .

 11 Fig. 11. Antenna pattern comparison (fractal structure antenna )

TABLE I

 I

	MODEL PARAMETERS NUMERICAL VALUES
	perturbation:

ACKNOWLEDGMENT

This work has been carried out in the frame of the COR-MORAN project, which is funded by the French National Research Agency (ANR) under the contract number ANR-11-INFR-010 and the framework of the ICT project ICT-248894 WHERE2, which is partly funded by the European Union.


The presented work addresses a comparison between measured and simulated data in an Indoor On-body channel for motion capture scenario. Measured data used in this work come from a large dataset of BAN channel open data, shared by the NICTA laboratory. The simulated data are produced using the PyLayers physical simulation platform based on graph based ray tracing techniques.

Introduction

In the last years the interest in Body Area Networks (BAN) is increasing and occupying many researchers and industrials mostly because of anticipated huge applications around the upcoming Internet of Things and nomadic connectivity.

In general, in the context of communication systems conception and design, the study of the propagation channel characteristics is fundamental. This is especially true, in the WBAN context, where the channel is considerably affected by the anthropoid nature of the motion.

Any propagation channel study begins with measurements for extraction of specific statistical models with the depth required by the targeted use case. Besides, it is also important to build ADAP (As Deterministic As Possible) simulators to address the very large space of particular situations defined and parameterized by data, which are becoming, over time, more and more available at the application level. The desirable balance between deterministic and statistical description of the channel being strongly dependent on the context, e.g very different whether only localization or only communications is involved. Moreover, any design of a simulation strategy has to do compromises between computation time and level of physical description, mostly when dealing with upper layer abstractions. In that very purpose, the presented work is aiming to investigate how the anthropoid nature of the subject motion is "etched" into the multi link indoor WBAN radio channel and how good (or bad) deterministic simulations compare with real data. Thus, in this work a PyLayers [START_REF]PyLayers: Python Indoor Simulator for Mobile Localization[END_REF], [START_REF] Amiot | Design of a Simulation Platform Joining Site Specific Radio Propagation and Human Mobility for Localization Applications[END_REF], [START_REF] Amiot | Pylayers : An indoor propagation tool for studying localization in wban context[END_REF], on-body channel motion capture based multi link simulation, has been compared to the corresponding scenario extracted from a sample of the measurement database carried out and shared by the NICTA laboratory, [START_REF]Nicta body area network radio channel data[END_REF] [START_REF] Smith | First-and second-order statistical characterizations of the dynamic body area propagation channel of various bandwidths[END_REF].

The measurements campaign is presented in section 2. The section 3 recalls the used simulation setup. Finally, in the section 4 a comparison between simulation and measurement results is presented and discussed.

Measurement Setup

A measurement campaign was carried out by the NICTA laboratory [START_REF] Smith | First-and second-order statistical characterizations of the dynamic body area propagation channel of various bandwidths[END_REF] aiming to study the different BAN channels (On-body, Off-body...) in several configurations (standing, walking, running...) at different frequencies (400 MHz, 900 MHz and 2400 MHz) in indoor environment.

In this work, only the data relative to On-body links has been used. The measurement environment is an indoor office presented in Figure 1. The human subject is moving on a treadmill. In this work, the walking scenario at the speed of 3 km/h during 60seconds has been selected for comparison.

For the On-body measurement the devices are located on the different limbs (right/left wrists and ankles) and the trunk (front/back chest and left/right hip). This configuration, illustrated in Figure 2 allows the study of on-body dynamic and static channel characterization. Regarding the frequency, the channels was measured at different frequencies bands. In this work, the simulation and comparison are performed at f = 2.4 GHz. At this frequency the used antenna, Octane BW-2400-2500, is approximately omni-directional. The antenna were worn such as the E-plane is perpendicular to the environment floor.

3 Simulation Setup

Layout and propagation environment

The PyLayers simulation work-flow starts by defining the propagation environment. Here, the layout is a 3 meters × 5.5 meters room where the human subject is moving on a treadmill. Notice that in the presented results only the layout has been modeled but not the furniture present in the room as, for instance, the treadmill which may have had some effect on the propagation channel (see e.g fig 1(d) of [START_REF] Smith | First-and second-order statistical characterizations of the dynamic body area propagation channel of various bandwidths[END_REF]).

Large scale and body scale mobility

PyLayers simulator has been designed to handle large scale mobility [START_REF] Amiot | Pylayers : An indoor propagation tool for studying localization in wban context[END_REF], i.e, the mobility of an agent, considered as summarized to its center of gravity. In this paper this feature is not fully exploited because the center of gravity of the subject is assumed steady above the treadmill. After describing precisely the position and orientation of each device on the body, the trajectory is centered to simulate the walking on treadmill trajectory during the same 60 s measurement time and at speed = 3 km/h.

In order to simulate the BAN channel, the human motion has to be superimpose over the introduced. This has been done using the motion capture files C3D (Coordinates 3D) of a walking sequence [START_REF]Cmu graphics lab motion capture database[END_REF]. The corresponding body sequence (PyLayers topos concept) is presented in Figure 3. This walking sequence is generic and do not correspond to the actual motion of the subject during the measurement. This is a source of deviation between the simulated and measured data. Further work should investigate such comparison in exploiting the actual motion capture of the subject under test. The data extracted from these files is plugged in the PyLayers simulator and synchronized with the large scale mobility. In the current case the trajectory is limited to the center of gravity of the body. Then, the same devices configuration as measurements, in Figure 2, is set up.

Perturbed Antenna Model

The antenna modeling is key in the presented approach. Aiming the comparison with the Octane BW-2400-2500 antenna used in measurements a patch antenna radiating at f = 2.4GHz with similar radiation pattern has first been modeled using HFSS simulator, Figure 4. The antenna pattern is expanded in terms of spherical harmonics coefficients [START_REF] Rahola | Modelling of radiation patterns using scalar spherical harmonics with vector coefficients[END_REF], [START_REF] Burghelea | Ultra-compact parametric modelling of threedimensional antenna radiation pattern for impulse radio-ultra-wide band ray tracing[END_REF]. This approach reduces the amount of data required for describing the full antenna pattern. This has been the chosen manner for describing the antenna pattern in the PyLayers simulation platform.

The ray tracing is used to take into account the interaction on walls of the environment. This is much more questionable to apply the ray tracing for the On-body channel. There is current investigation in that direction [START_REF] Plouhinec | 3d utd modeling of a measured antenna disturbed by a dielectric circular cylinder in wban context[END_REF], but this is not this approach which has been chosen here.

The direct link between the different nodes on the body is taken into account by a proper modification of the antenna radiation pattern in the spherical harmonics space. We introduce a perturbation on the antenna, modeling the human presence. This approach is detailed in [START_REF] Mhedhbi | Modelling of UWB antenna perturbed by human phantom in spherical harmonics space[END_REF]. The free space antenna spherical harmonics coefficients are modified directly to generate the coefficients representing the perturbed antenna by the human body presence. By resorting to this approach, we avoid the ray tracing on the body and the body effect is taken into account into the antenna pattern itself. The Figure 5 shows the perturbed antenna outcome obtained from the perturbed model. In this figure the antenna has been placed at a distance of 5 mm from a 70 mm diameter cylinder. One all the aforementioned elements are set up, a ray tracing simulation is done for each link of interest and for each time-stamp.

Simulations vs Measurements Comparison

This section presents a preliminary comparison in terms of path loss amplitude of a multi links time series and the corresponding cumulative distribution functions (CDF) both for the measurements and simulations of the above described walking scenario. The Figures 6 and7 show the variation of the measured and simulated path loss during a walking sequence in indoor environment for the different on-body links. The Figures 8 and9 represent the corresponding CDF evaluated over 60 s of walking sequence. The link legend and the color are shared between all figures 6, 7 and 8 and 9 . Those 4 Figures show that the simulation are quite close to the measurement data at least on a relative scale. The breadth of the shadowing is clearly underestimated in the simulation. The current interpretation of this effect is that the trunk shadowing is underestimated by the perturbed antenna model which has been used. The dependency of the model with the radius of the equivalent cylinder associated to the trunk is clearly not properly handled and should be calibrated. This point has to be investigated more, hopefully the NICTA dataset is perfectly suited for this calibration task. Notice that in the presented measurement/simulation comparison there was no ad hoc effort for having an absolute fitting of the channel gain.

Several interesting aspects can be underlined. The link characteristic (reliable or not ...) and variation during the walking motion is well recovered. As it is observed in the measurement data, the best link is right hip to right wrist (purple curve in Figure 6). Interestingly the shape of the channel gain is very well retrieved with a peak appearing when both link antennas are in close vicinity just before digging in the shadow region. This shape is probably very dependent from one subject to an other depending on the subject gait. Further statistical analysis on that point would probably be of interest. This observed relatively good agreement in the multi link structure argues for the relevance of the approach which is potentially a good compromise between efficient implementation and physical realism. The front/chest to back/chest link which is static and intrinsically NLOS is well retrieved and of course those results would be completely impossible to model if the antenna patterns were kept unperturbed. Interestingly, the adopted approach exploits actual data from the knowable free space, far field antenna pattern. This means that the proposed approach could benchmark different antenna design in complex WBAN scenario, ceteris paribus.

However, other links are not well retrieved, for example the right hip to chest link. In fact, the simulated path loss is overestimated compared to the measurements. This link is more reliable in simulation because of the antenna perturbation model which introduce, here, the perturbation of the member on which the antenna is placed. This might be admitted for the antenna on the trunk but it is not the case of the antenna on the wrist which is perturbed by the trunk too.

Conclusions

This paper has presented a comparison between real data from a WBAN walking experiment and the simulation from the site specific simulator PyLayers. The results are in quite good agreement if we consider that the used human motion corresponds in each case to different subject. The relative level of the different link is quite well retrieved. It has to be noticed that PyLayers simulator can provide a much richer information about the channel as e.g angular and delay structure of the channel over ultra wide bandwidth. This comparison confirms a well admitted fact that the very nature of the multi link variation is strongly related to the human motion which can be captured and digitized for semi deterministic simulation. The motion captured is a key technology for decoding the rich information carried out by the multi-link variation observed in the BAN.

In this comparison the large scale motion of the problem has been removed because the subject is staying at the same place. The quality of the obtained results is strongly dependent on the manner the radiation pattern is handled. This is the key factor behind the presented comparison. This first validation paves the way for investigations of far more sophisticated scenarios as those which are currently investigated in the French ANR CORMORAN project as indoor group navigation.

In the future work we aim to simulate other UWB BAN channel, for instance, Body-to-Body and Body-to-infrastructure channels. Further step is to produce improved antenna model describing better the trunk perturbation and to introduce the statistic model described in [START_REF] Ouvry | Delay dispersion of the on-body dynamic channel[END_REF] for the on body channel because the current implemented approach cannot give access to the correct CIR delays.

Figure 1 :Figure 2 :

 12 Figure 1: On-Body Experimental treadmill environment, including subject location (from [5])

Figure 3 :

 3 Figure 3: Motion capture sequence from[START_REF]Cmu graphics lab motion capture database[END_REF]

Figure 4 :

 4 Figure 4: Simulated antenna in free space

Figure 5 :

 5 Figure 5: Simulated antenna perturbed by the model

Figure 6 :

 6 Figure 6: Path loss for walking sequence from measurements

Figure 7 :

 7 Figure 7: Pathloss for walking sequence from PyLayers simulation

Figure 8 :Figure 9 :

 89 Figure 8: CDF of the path loss from NICTA measurements

Acknowledgment

This work has been carried out in the frame of the CORMORAN project, which is funded by the French National Research Agency (ANR) under the contract number ANR-11-INFR-010.


Meriem Mhedhbi

Nicolas Amiot

Stéphane Avrillon

Bernard Uguen

Human Body Perturbed Antenna Integration In Indoor Propagation Simulator

Keywords: UWB antenna modeling, Spherical Harmonics, Physical Simulator, WBAN, Motion Capture, Radio Channel Simulation, Site specific modeling

This paper presents first results of simulation of a radio link between 2 antennas mounted on a the body. The body mobility is handled at two levels, the trajectory level and the body mobility level. A perturbed antenna model is used to acknowledge the effect of the body directly in the radiation pattern. Aggregated channel impulse response simulated in an Indoor environment is presented. The presented results validate the interest of the proposed approach for further integration with higher layer simulation.

Introduction

In the context of communication systems conception and design, the knowledge of the channel is always of importance. Especially, Body Area Networks (BAN) channels, since the transmission channel depends not only on the environment but it is also strongly affected by the body and its effect on body mounted antennas. In this work, for ease of the exposition, we focus on a simple single agent, single link. However the presented tool has been designed to handle Body Area network (BAN) for multi-agent and multi-links situations. This requirement being necessary for addressing the location related scenario defined in CORMORAN project : Large Scale Individual Motion Capture (LSIMC) and Coordinated Group Navigation (CGN) [START_REF] Derrico | Jihad Hamie. D1.1 applications scenarios, system requirements and prior models[END_REF].

The BAN channel is manifold as it may consist of different kind of links : On-body and Offbody, the later including body to infrastructure and body to body links. The used simulator implements classical ray tracing techniques [2] and handles the agents mobility in a specific layout, as well as the body mobility using realistic body abstraction obtained from motion capture files. The antennas are fully described in a compact form using spherical harmonics expansions as described in [START_REF] Rahola | Modelling of radiation patterns using scalar spherical harmonics with vector coefficients[END_REF].

The chosen approach has two levels. A first level acknowledges the motion of the center of gravity of the body along realistic indoor trajectories using a Discrete Event Simulator (DES) and a second level superimposes the body mobility on the trajectory. All the link of interest for the scenario can be calculated and stored for further post processing. To this end, a perturbed antenna model assessing the pattern behavior when antenna is placed around the human body is presented. This approach assumes that the effect of the body presence is being taken into account in the antenna radiation pattern, avoiding to manage a body description into the ray tracing. The paper is organized as follows: Section 1 describes human body mobility model, based on motion capture files, as well as the dedicated antenna model, Section 2 presents first integration simulations results and a comparison between situations using a body-perturbed and a non body-perturbed antenna.

Human Mobility and Antenna Model Integration

In this section, the Body Area Network channels simulation is presented including a description of how are handled the human body mobility and the antenna model.

Modeling of Human Body Mobility

Different ways can be used to model the human body mobility. Traditionally, bio-mechanical models are used to represent the human mobility in a walking situation [START_REF] Hamie | Joint motion capture and navigation in heterogeneous body area networks with distance estimation over neighborhood graph[END_REF]. An alternative way is to make use of data coming from motion capture. Among the different existing motion capture data file formats, the C3D (Coordinates 3D) format is used here due to its compliance with most of motion capture systems. Among the data stored in C3D files are the whole 3D coordinates of a fixed number of points, representing human body sensors at very specific conventional location on the body at each time stamp. A 3D frame consists of a set of those 3D points. For example Figure 1.1 represents a 41 points' frame. The main advantage of using motion capture data is that it enables to investigate a large variety of motion situations, not only limited to walking scenario.

The data extracted from C3D files are then used to build a human body cylindric model. The C3D data are filtered and a limited number of points is used 16 instead of 41 for creating a simplified cylindric model. This model being a good trade off between fidelity of the representation and simplicity of the implementation. Similar approaches are proposed in [START_REF] Iswandi | The utilization of body skeleton model for modeling the dynamic ban channels[END_REF]. The body trajectory is obtain from DES force based multi agent simulator which can produce an arbitrary number of realistic human trajectories into the Layout. Once the trajectory has been calculated, the body is inserted at the good location in time and space. Once properly positioned it remains to handle the body mounted radiating element positioning and orientation in the global scene. This is done through the introduction of a chain of time dependent (except the global one) coordinates systems relating one to another via specific linear transformations (translation or rotation). The important aspect being to properly ponder the rays with the appropriate complex gain from the antenna pattern. The defined chain of coordinate systems is as followed:

GCS → CCS(t) → DCS(t) → ACS(t) (1)
Where

• GCS refers to the global coordinates system. The Layout and the rays from the ray tracing are expressed in this GCS.

• CCS refers to Cylinder Coordinates System: each cylinder of the body model is associated with a dedicated CCS. A CCS is obtained from the large scale trajectory and the body frame corresponding to a given time on the body trajectory.

• DCS refers to Device Coordinates System : this CS is a rotated and/or translated version of the CCS. The origin of this coordinate system is the device position. A device is non ambiguously placed on a body cylinder via 3 parameters (l, h, cylId). 0 < l < 1 is a parameterization of the length of the cylinder,h is the height of the device w.r.t to the cylinder of Id CylId.

• ACS refers to Antenna Coordinate System: it is an arbitrarily rotated version of the DCS that defines the antenna local coordinates system. The radiation pattern of the antenna is known in this ACS. One device could have theoretically several different ACS. This possible feature is not exploited in the following.

The Figure 1.1 shows the C3D frame (a), the body cylindric model built using C3D data (b), the body and all the CCS (c), and the body with the 2 selected DCS (d).

Antenna Model

In the context of body area network radio channel simulation and modeling the knowledge of the antenna radiation pattern is fundamental. The human body disturbs in a considerable way the antenna pattern. One rigorous way to introduce the body effect in the physical simulator would be to apply an electromagnetic solver. However, for the purpose of higher level simulation we are addressing, this approach is too computationally intensive. As a compromise alternative, we proposed to account for the body effect in the antenna pattern itself in using an equivalent antenna obtained from the unperturbed free space antenna and parameters depending mostly of the distance of the antenna from the body.

To study the effect of a human phantom on a UWB antenna a measurement campaign was carried out to study the effect of the human presence on antenna, where the antenna is measured in free space and placed around a cylindric human phantom at different distances [START_REF] Pedersen | Fp7 where2 deliverable d1[END_REF].

It is well known that a natural and compact representation of the antenna pattern is based on spherical harmonic (SH) basis expansion. According to [START_REF] Rahola | Modelling of radiation patterns using scalar spherical harmonics with vector coefficients[END_REF], [START_REF] Chen | Radiation pattern analysis of arbitrary wire antennas using spherical mode expansions with vector coefficients[END_REF], the far-field radiation pattern F can be expanded in terms of scalar spherical harmonics with vector coefficients:

F (θ , φ , f , d) = L ∑ l=0 l ∑ m=-l c lm ( f , d)Y m l (θ , φ ), (2)
where Y m l (θ , φ ) are the SH functions and c lm are SH coefficients. Using spherical harmonics (SH) expansion the antenna data is reduced to few SH coefficients. Exploiting the above spherical harmonics decomposition we have proposed a model which predicts in the spherical harmonics coefficients domain the perturbation of the complex pattern caused by the human presence. It is obtained from experimental data and consists in a simple linear model relating SH coefficients of a disrupted antenna to free space coefficients and antenna-phantom distance:

C(d) = A(d) C fs ( 3
)
Where d is the antenna-phantom distance, C fs are SH coefficients of the antenna in free space, C(d) denotes the perturbed version of SH coefficients and A(d) is the linear model vector of parameters representing the perturbation. The Figure 1.2 shows the "Thomson" antenna patterns in the plane (x, y) at f = 4GHz in free space (green) and around the human phantom (red) cases. We observe the perturbation brought out by the human phantom presence which is placed at the distance d = 30mm from the antenna. This model is described with more details in [START_REF] Mhedhbi | Modelling of UWB antenna perturbed by human phantom in spherical harmonics space[END_REF]:

This model has been integrated in the indoor propagation simulator to simulate and model realistic BAN channel by taking into account the antenna perturbation.

Simulation and Results

In this section, we consider the link hip to right wrist. Comparisons between using a perturbed antenna and a free space antenna are presented. The simulation is made in a defined layout and for a specific trajectory in a walking scenario shown in figure 2.1. There are 3 different regimes in this trajectory. The agent starts to move in a room, then goes into a long corridor, At each instant the rays between the transmitter and the receiver are identified and evaluated. The antennas are applied with their proper orientation and the transmission channel impulse response h (t, τ) is evaluated. The two pattern which has been used are presented in Figure 1.2.

The pattern used for the perturbed antenna is obtained using a phantom modeling of the antenna in the vicinity of a human arm. This same pattern has also been used for modeling of the antenna mounted on the hip which is of course less justified. However, for the purpose of the illustration of the approach this is not an issue. Any more realistic antenna pattern coming from a rigorous simulation or measurement could be used instead if more realism is needed.

The Figure 2.2 represents the body cylindric model with antenna placed on the body. The chosen pattern radiates mostly toward the half space opposite to the body members (arms in this case). In fact, by using these antennas the human body presence is taken into account without requiring to actually represent the body in the global scene. Notice that this is an approximation which do not capture everything from the propagation phenomena appearing on the body.

Figure 2.2: Body cylindric model with perturbed antenna

The channel impulse response is evaluated over a bandwith from 3GHz to 6GHz allowing UWB waveform simulation. The time varying channel impulse response is expressed as:

h (t, τ) = K(t) ∑ k=0 α k (t)δ (τ -τ k (t)) (4)
where K(t) is the number of paths, α k (t) and τ k are the amplitude and the arrival time of the k ith path.

The situation we are studying is an on-body channel, in the sense that transmitter and receiver are placed on the body. However the part of the channel we are considering is obtained via off-body interactions. In this example the pure on body contribution is not fully taken into account especially in the delay domain. There exist a delay gap in the delay domain between the direct path and the first path coming to off body contribution (reflection on the environment). In this section the preliminary results obtained for human body mobility and antenna model obtained with the WBAN physical simulator are presented as well as a comparison between aggregated infinite bandwidth CIR obtained using a perturbed antenna and a free space antenna.

The Figure 2.3 shows the different paths amplitude and delays while the agent is traveling during the trajectory using free space antenna (right) and using a perturbed antenna (left). It shows the different regimes in the trajectory and how the channel structure varies with the environment. We notice that the global channel structure is nearly the same. However, we observe a dissimilarity in terms of paths amplitude since they are weighted differently according to the used antenna.

At the beginning of the trajectory during the 2 first seconds, the agent is walking in a small room the channel structure do not exceed 50ns which corresponds to the vertical band on the left. Then the agent goes out from the room, almost in the middle of the corridor, the two reflection at both end of the corridor are almost superimposed. Then the agent goes on the right until time 3s, then the delay from the reflection from corridor right wall is decreasing while delay from the left wall is increasing. Finally, the agent reach another room and the multi-path structure observed during the first second is re-established.

The major effect of the body perturbed antenna is observed in to two specific region. The first one is the corridor region where the rays weights difference is clear and it illustrates well the body shadowing of one reflected path coming from the back. Thus, this simple approach is in fact able to capture correctly what would be observed in practice.

The second region and the most important difference is when zooming and observing the first paths affect by the body presence during 3s which corresponds to approximately to 3 human steps in Figure 2. [START_REF] Hamie | Joint motion capture and navigation in heterogeneous body area networks with distance estimation over neighborhood graph[END_REF]. It shows that the first paths amplitude do not vary in considerable manner when using the free space antenna. The observed variation is due to distance variation between the receiver and the transmitter, relative to the arm movement in a walking scenario, not the human body shadowing which is quite well reproduced with the perturbed antenna case where the line of sight, non line of sight and even the transition region are observed. This result might be well observed when representing the total received power P r at each instant given by:

P r (t) = K(t) ∑ k=0 |α k (t)| 2 (5)
where K(t) is the number of paths at t and α k (t) is the k ith path amplitude. The Figure 2.5 shows explicitly the human trunk shadowing in hip to right wrist link in a walking scenario and confirms the paths amplitude results. It shows that the total received power variation when using free space antenna is around 5dB which do not consider the human body presence, it only a distance variation effect. However, with perturbed antenna P r variation is around 15dB which is more realistic in BAN channels. This variation in received power values and even behavior are coherent with measurement results in [START_REF] Ouvry | Delay dispersion of the on-body dynamic channel[END_REF] which confirm that the shadowing variation in almost same condition (hip to right wrist link, walking scenario) but in anechoic chamber is around 10dB.

Conclusion and perspectives

This paper has presented a full simulation of a radio link between two antennas mounted on the body integrating realistic large scale mobility and body mobility. An original technical choice for the integration of perturbed antenna model by a human phantom in the physical simulator PyLayers has been presented. The results show that the proposed approach provides realistic range in terms of received power and provides realistic radio observable variation.These observables can be always produced with ground truth values of the position and orientation of radiating element on the devices. Further step is to produce improved antenna describing better the trunk perturbation and to introduce the statistic model described in [START_REF] Ouvry | Delay dispersion of the on-body dynamic channel[END_REF] for the on body channel because the current implemented approach cannot give access to the correct CIR delays. An other immediate perspective is to study the body to body channel in realistic indoor human motion and exploitation of produced t races for evaluating various MAC layer scheme suited for BAN localization scenario of the CORMORAN project.

Figure 1 . 1 :

 11 Figure 1.1: C3D frame (a) and body cylindric model(b) with 11 CCS (c) and 2 DCS (d)

Figure 1

 1 Figure 1.2: Perturbed (red) and free space (green) antenna pattern at f = 4GHz in the plane (x, y)

Figure 2 . 1 :

 21 Figure 2.1: Simulation Layout and Agent 10 s trajectory from one room to another

Figure 2 . 3 :

 23 Figure 2.3: Paths amplitude during trajectory time using perturbed antenna (left) and free space antenna (right).

Figure 2 . 4 :

 24 Figure 2.4: First Path amplitude during trajectory time using perturbed antenna (left) and free space antenna (right).

Figure 2 . 5 :

 25 Figure 2.5: Shadowing for the link hip to right wrist for a walking sequence

Acknowledgment

This work has been carried out in the frame of the CORMORAN project, which is funded by the French National Research Agency (ANR) under the contract number ANR-11-INFR-010.