Carles Fernández-Prades

Jordi Vilà-Valls
email: jordi.vilavalls@gipsa-lab.grenoble-inp.fr

BAYESIAN NONLINEAR FILTERING USING QUADRATURE AND CUBATURE RULES APPLIED TO SENSOR DATA FUSION FOR POSITIONING

Keywords: Bayesian estimation, Gauss-Hermite quadrature, cubature rules, nonlinear filtering, square-root filters, sensor data fusion, positioning

This paper shows the applicability of recently-developed Gaussian nonlinear filters to sensor data fusion for positioning purposes. After providing a brief review of Bayesian nonlinear filtering, we specially address squareroot, derivative-free algorithms based on the Gaussian assumption and approximation rules for numerical integration, namely the Gauss-Hermite quadrature rule and the cubature rule. Then, we propose a motion model based on the observations taken by an Inertial Measurement Unit, that takes into account its possibly biased behavior, and we show how heterogeneous sensors (using time-delay or received-signalstrength based ranging) can be combined in a recursive, online Bayesian estimation scheme. These algorithms show a dramatic performance improvement and better numerical stability when compared to typical nonlinear estimators such as the Extended Kalman Filter or the Unscented Kalman Filter, and require several orders of magnitude less computational load when compared to Sequential Monte Carlo methods, achieving a comparable degree of accuracy.

I. INTRODUCTION TO BAYESIAN FILTERING

The problem under study, concerns the derivation of efficient methods for online estimation and prediction (filtering), of the time-varying unknown state of a system (state could refer to position, velocity, attitude and so on), along with the continuous flow of information (observations) from the system. A positioning system, is nonlinear and dynamic by nature, and thus the objective is to recursively compute estimates of states at time index k, x k ∈ χ, given measurements y 1:k ∈ Y and measured inputs u k ∈ U, where χ, Y and U are their respective supports, not necessarily with the same dimension. State equation models the evolution of target states (on which measurements depend on), as a discretetime stochastic model. Following a first-order Markov chain approach, this can be modeled by

x k = f k-1 (x k-1 , u k-1 , v k-1 ), (1)
where f k-1 is a known, possibly nonlinear, function of the state x k , and v k-1 is referred to as process noise, which gathers any mismodeling effect or disturbances in the state characterization. The relation between the measurements and the state is modeled by

y k = h k (x k , u k , n k ) (2)
where h k is a known, possibly nonlinear, function and n k is referred to as measurement noise. The combination of (1) and (2) leads to a Markov chain whose state is not directly observable but indirectly inferred through measurements, an approach known as Hidden Markov Model. A generic recursive Bayesian filter is depicted in Algorithm 1, where E {•} stands for the statistical expectation operator. The state transition density p(x k |x k-1 , u k-1 ) is given by (1), and the old posterior density at time k -1, p(x k-1 |y 1:k-1 , u 1:k-1 ), is obtained in the previous update step. From ( 5) to [START_REF] Golub | Some modified matrix eigenvalue problems[END_REF] we have used p(

x k |y k-1 , u k-1 , u k ) = p(x k |y k-1 , u k-1
), which is equivalent to state that the input u k can be inferred using xk|k-1 . This allows the recursive relationship, since this a priori density is computed in the prediction step. The likelihood function p(y k |x k , u k ) is given by [START_REF] Arasaratnam | Cubature Kalman Filters[END_REF].

In this paper, we present the nonlinear Bayesian estimation problem, and we summarize the recently-developed Sigma-Point methods: the Quadrature Kalman Filter (QKF) [START_REF] Arasaratnam | Discretetime nonlinear filtering algorithms using gauss-hermite quadrature[END_REF] and the Cubature Kalman Filter (CKF) [START_REF] Arasaratnam | Cubature Kalman Filters[END_REF]. We propose a motion model, following a new approach, which includes the biases introduced by the Inertial Measurement Unit, into the state. We consider two different measurement models and we combine them into a nonlinear Bayesian filter. This approach can be useful when having different types of sensors for the same application. These methods, from the best of our knowledge, has never been used for data fusion for positioning.

II. GAUSSIAN NONLINEAR FILTERING

Algorithm 1 provides an estimation framework that is optimal in the Bayesian sense 1 and enjoys a recursive structure, which is a desirable property for online estimation. In the general case, however, this algorithm cannot be used in practice because the multi-dimensional integrals involved in ( 4) and ( 6) are intractable. For Markovian, nonlinear and non-Gaussian state-space models, Sequential Monte Carlo (SMC) methods [START_REF]Sequential Monte Carlo in Practice[END_REF] (also known as Particle Filters), provide a framework to obtain numerical approximations of such integrals by a stochastic sampling approach. This broad suitability comes at the expense of a high computational load, that makes this solution difficult to embed in digital light processors or real-time applications.

An alternative to the SMC methods, to solve the nonlinear filtering problem, are the so-called Sigma-Point Kalman filters (SPKF) [START_REF] Van Der Merwe | Sigma-Point Kalman filters for nonlinear estimation and sensor-fusion: Applications to integrated navigation[END_REF], [START_REF] Arasaratnam | Cubature Kalman Filters[END_REF]. The key assumption here, is to consider that both, process noise v k and measurement noise n k , are independent random additive processes, with Gaussian distribution. This leads to reckon the state transition density p(x k |x k-1 , u k-1 ) and the measurement likelihood function p(y k |x k , u k ) as Gaussian densities, which in turn reverts to a Gaussian posterior density p(x k |y 1:k , u 1:k ). If the state transition function f (•) and the measurement function h(•) are both linear, this approach leads to the Kalman filter. In case of dealing with slightly non-linear systems, we can resort to the Extended Kalman Filter (based on the linearization of these functions around the predicted and updated estimates), but this approach fails when nonlinearities are moderate or severe. As shown in [START_REF] Ito | Gaussian filters for nonlinear filtering problems[END_REF], for an arbitrary non-linear system and considering the Gaussian assumption, we can derive a recursive algorithm by equating the Bayesian formulae with respect to the first and second moments of the distributions, that is, means and covariances of the conditional densities involved. Since all the appearing multi-dimensional integrals are of the form

I = Z χ nonlinear function • Gaussian density • dx, (3)
the problem of Bayesian estimation under the Gaussian assumption reduces to a numerical evaluation of such integrals. Possible ways to approximate the definite integral of a function such as (3) are the Gauss-Hermite quadrature rules [START_REF] Golub | Some modified matrix eigenvalue problems[END_REF] or the cubature rules [START_REF] Stroud | Approximate calculation of multiple integrals[END_REF]. These approaches are based on a weighted sum of function values, at specified (i.e., deterministic) points within the domain of integration, as opposite to the stochastic sampling performed by SMC 1 The characterization of the posterior distribution allows us to compute the Minimum Mean-Squared Error (MMSE), the Maximum a Posteriori (MAP) or the median of the posterior (Minimax) estimators, addressing optimality in many senses.

Algorithm 1 Optimal recursive Bayesian filter structure

1: for k = 1 to ∞ do 2:
Prediction: estimate the predicted state

xk|k-1 = E {x k |y 1:k-1 , u 1:k-1 } = Z χ x k p(x k |y 1:k-1 , u 1:k-1 )dx k where: p(x k |y 1:k-1 , u 1:k-1 ) = Z χ p(x k , x k-1 |y 1:k-1 , u 1:k-1 )dx k-1 = Z χ p(x k |x k-1 , u k-1 )p(x k-1 |y 1:k-1 , u 1:k-1 )dx k-1 (4) 3:
Update: update the state estimation with the new measurement and input at time k

xk|k = E {x k |y 1:k , u 1:k } = Z χ x k p(x k |y 1:k , u 1:k )dx k
where, applying Bayes' rule:

p(x k |y 1:k , u 1:k ) = p(x k |y k , u k , y 1:k-1 , u 1:k-1 ) = p(y k |x k , u k )p(x k |u k , y 1:k-1 , u 1:k-1 ) p(y k |u k , y 1:k-1 , u 1:k-1 ) (5)
= p(y k |x k , u k )p(x k |y 1:k-1 , u 1:k-1 ) R χ p(y k |x k , u k )p(x k |y 1:k-1 , u 1:k-1 )dx k (6)
4: end for methods. A well known method within this family, is the Unscented Kalman Filter (UKF) [START_REF] Julier | A new extension of the Kalman filter to nonlinear systems[END_REF], that can be expressed as a particular case of a quadrature rule if parameters are tuned adequately [START_REF] Ito | Gaussian filters for nonlinear filtering problems[END_REF].

A further refinement of these schemes comes from the fact that, when we propagate the covariance matrix through a nonlinear function, the filter should preserve the properties of a covariance matrix, namely, its symmetry, and positivedefiniteness. In practice, however, due to lack of arithmetic precision, numerical errors may lead to a loss of these properties. To circumvent this problem, a square-root filter is introduced to propagate the square root of the covariance matrix instead of the covariance itself. Even more, it avoids the inversion of the updated covariance matrix, entailing additional computational saving. Although this idea is not new [START_REF] Potter | Statistical filtering of space navigation measurements[END_REF], it has been recently applied to the UKF [START_REF] Van Der Merwe | The squareroot unscented Kalman filter for state and parameter estimation[END_REF] (although it does not guarantee positive definiteness of the covariance matrix) and, more successfully, to the Square-Root Quadrature Kalman Filter (SQKF) [START_REF] Arasaratnam | Square-Root Quadrature Kalman Filtering[END_REF] and the Square-Root Cubature Kalman Filter (SCKF) [START_REF] Arasaratnam | Cubature Kalman Filters[END_REF].

The resulting procedure is sketched in Algorithm 2, where we present a general framework for square-root, derivativefree nonlinear Bayesian filtering. In steps 8 and 17, S = Tria (A) denotes a general triangularization algorithm (for instance, the QR decomposition), where A ∈ R p×q , p < q, Algorithm 2 Square-root, derivative-free nonlinear Kalman Filter Require: y1:K , u0:K , x0, Σx,0 = S x,0|0 S T x,0|0 , Σv,0, Σn,0.

Initialization: 1: Define sigma-points and weights {ξ i , ωi}i=1,...,L by using Algorithms 3, 4 or any other rule. 2: Set W = diag( √ ωi)

Tracking:

3: for k = 1 to ∞ do 4:
Time update:

5:
Evaluate the sigma points:

x i,k-1|k-1 = S x,k-1|k-1 ξ i + xk-1|k-1 , i = 1, ..., L. 6:
Evaluate the propagated sigma points:

xi,k|k-1 = f (x i,k-1|k-1 , u k-1 ). 7:
Estimate the predicted state:

xk|k-1 = P L i=1 ωi xi,k|k-1 . 8:
Estimate the square-root factor of the predicted error covariance:

S x,k|k-1 = Tria "h Xk|k-1 S Σ v,k-1 i"
, where:

S Σ v,k-1 is a square-root factor of Σ v,k-1 such that Σ v,k-1 = S Σ v,k-1 S T Σ v,k-1 , and
Xk|k-1 = ˆx 1,k|k-1 -xk|k-1 • • • xL,k|k-1 -xk|k-1 ˜W.

9:

Measurement update:

10:
Evaluate the sigma points:

x i,k|k-1 = S x,k|k-1 ξ i + xk|k-1 , i = 1, ..., L.
11:

Evaluate the propagated sigma points: ỹi,k|k-1 = h(x i,k|k-1 , u k ).

12:

Estimate the predicted measurement: ŷk|k-1 = L i=1 ω i ỹi,k|k-1 .

13:

Estimate the square-root of the innovation covariance matrix: S y,k|k-1 = Tria Y k|k-1 S Σ n,k , where:

S Σ n,k denotes a square-root factor of Σ n,k such that Σ n,k = S Σ n,k S T
Σ n,k , and

Y k|k-1 = ˆỹ 1,k|k-1 -ŷk|k-1 • • • ỹL,k|k-1 -ŷk|k-1 ˜W.

14:

Estimate the cross-covariance matrix

Σ xy,k|k-1 = X k|k-1 Y T k|k-1
, where:

X k|k-1 = ˆx1,k|k-1 -xk|k-1 • • • x L,k|k-1 -xk|k-1 ˜W.

15:

Estimate the Kalman gain

K k = Σ xy,k|k-1 /S T y,k|k-1 /S y,k|k-1 .
16:

Estimate the updated state

xk|k = xk|k-1 + K k y k -ŷk|k-1 . 17:
Estimate the square-root factor of the corresponding error covariance:

S x,k|k = Tria X k|k-1 -K k Y k|k-1 K k S Σ n,k .
18: end for and S is a lower triangular matrix.

In Algorithms 3 and 4, we present the way to compute the deterministic samples using, respectively, the Gauss-Hermite quadrature rule and the third-degree spherical-radial cubature rule. The symbol ⊗ is the Kronecker tensor product, and we Algorithm 3 Generation of Sigma-Points and weights for multi-dimensional Gauss-Hermite quadrature rule 1: Set the number of points per dimension α, and L = α dim(x) 2: Set Ji,i+1 = q i 2 , where i = 1, ..., (α -1) 3: Compute λi, the eigenvalues of J 4: Set ξ i = √ 2λi 5: Set ωi = (νi) 2 1 , where (νi)1 is the first element of the i-th normalized eigenvector of J 6: if dim(x) > 1 then have used the MATLAB notation for matrices: A 1:i,: refers to the first i rows of matrix A.

III. MOTION MODELS

The choosing of a motion model will depend on the targeted application, the availability of different technologies and the computational capability. In this paper, we propose a new motion model, inspired in the one presented in [START_REF] Godsill | Models and algorithms for tracking of maneuvering objects using variable rate particle filters[END_REF], based on an intrinsic coordinate system (i.e., the heading angle and the distance moved along the path), and focused on the thrusts applied when turning a vehicle. In an intrinsic coordinate system, applied forces can be represented relative to the heading of the object, rather than relative to the more standard Cartesian or polar fixed coordinate frame. Distance traveled along the path of motion is denoted s, while angle of the path relative to horizontal is denoted ψ. We assume that a piecewise constant thrust, relative to the direction of heading, is applied between any two times τ k and τ k+1 , with tangential component F T and perpendicular component F P . We also assume a damping term λ ds dt , applied in the opposite direction to the heading. Thus:

F T = λ ds dt + m d 2 s dt 2 , F P = m ds dt dψ dt ( 7
)
where m is the mass of the object and λ the coefficient of resistance. Accelerations tangential to and perpendicular to the motion are then given by standard dynamics of a point mass:

a T (t) = d 2 s dt 2 , a P (t) = ds dt dψ dt = 1 R ds dt 2 ( 8
)
where R is the instantaneous radius of curvature of the path. The tangential equation is readily integrated from time τ k to τ k+1 to give the speed v(t) along the path at time t = τ k + T :

v(τ k +T ) = v k+1 = 1 λ F T k -(F T k -λv(τ k )) e -T λ m (9)
Using ( 9), the perpendicular equation can be integrated from τ k to τ k + T , obtaining

ψ(τ k + T ) = ψ k+1 = ψ k + F P k F T k λT m -log v(τ k ) v(τ k + T ) (10)
The Cartesian position can be computed by an Euler approximation on a fine time grid, so the changes in x and y coordinates over a time interval T can be expressed as

x k+1 ≈ x k + v k+1 T cos(ψ k ) (11)
y k+1 ≈ x k + v k+1 T sin(ψ k ) (12)
Inertial Measurement Units (IMUs) are known to deliver biased estimations, and these biases are an underlying source of error. We propose to include those biases in the model:

ãT k = a T k + δa T k , ãP k = a P k + δa P k , ψk = ψ k + δψ k (13)
The state space is then defined as

x k = x k y k v k δa T k δa P k δψ k T (14)
and the input signal

u k = ãT k ãP k ψk T , (15)
and thus the state transition

x k = f (x k-1 , u k-1
) is completely defined by equations ( 7)-( 12), plus the assumptions

δa T k = δa T k-1 , δa P k = δa P k-1 , and δψ k = δψ k-1 .
An advantage of this model is that we can interpolate position, speed, and angle at any time between τ k and τ k+1 , by substituting T by ∆τ in the above equations (where 0 < ∆τ < T ), thus easing time alignment between inertial measurements and other measurements coming from wireless systems.

The reason for including the path angle bias δψ k and the acceleration biases δa T k and δa T k in the state vector, and the measured angle ψ k and accelerations a T k and a P k in the input signal is due to the well-known behavior of the gyroscopes and accelerometers of an IMU, and the high impact of such biases in dead reckoning methods used for position computation. Inertial measurement units are typically biased, and thus this effect is included in the model.

IV. MEASUREMENT MODEL

We assume a receiver equipped with a set of wireless system interfaces. Measurements taken from these systems (mainly time delay or received power strength) are nonlinearly related to the device position.

If we use technologies in which the natural measurement is time delay, it can be converted to range multiplying by the speed of light c, but taking into account the desynchronization between emitter and receiver (time stamp of measures has been omitted for clarity):

y = c (t Rx -t T x ) d +c(∆t Rx -∆t T x ) + e ( 16
)
where d = r " r T x x -r Rx x " 2 + " r T x y -r Rx y " 2 + " r T x z -r Rx z " 2
is the geometric distance, ∆t refers to the device's clock bias with respect to an agreed time framework, and e is a random variable modeling noise. In case of technologies in which we can easily have access to the received signal strength, we will assume an stochastic model for the strength loss:

y = P T x -PL (d 0 ) -10n log 10 d d 0 + ǫ, (17)
where P T x is the transmitted power, d 0 is the reference distance, PL (d 0 ) is the mean loss at d 0 , n is the slope of the loss (depending on the scenario) and ǫ is a random variable modeling noise.

V. COMPUTER SIMULATIONS

In order to provide illustrative numerical results, we have particularized the system model to the following network deployment. Let us assume a 2D field R, a square centered at (0, 0) with sides of 500 m length, where we have a total number of N p = 20 beacons of a power-based ranging technology (with the measurement function of equation ( 17)) and N t = 5 beacons of time-based ranging (equation ( 16)), all with known positions and a coverage of 30 m. We have assumed a random, uniformly distributed beacon deployment over R. The chosen values for the simulations have been P T x = 0 dBm, d 0 = 1 m, PL (d 0 ) = 30 dB, n = 3 (a typical value for urban area cellular radio) and ǫ is a random variable with a log-normal distribution with zero mean and σ ǫ = 3 dBm. The noise e is Gaussian with zero mean and σ e = 1 m. We have simulated We compare the results with a Particle Filter (PF) using a residual resampling step [START_REF]Sequential Monte Carlo in Practice[END_REF]. We note that we are using Fig. 2. RMSE for the estimation of the position, for the SQKF, the SCKF and the PF using 729 and 12 particles L = 2dim(x) = 12 deterministic samples for the SCKF, and L = 3 dim(x) = 729 deterministic samples for the SQKF, which considers 3 samples per dimension. To have a coherent comparison, we consider two PFs, using 12 and 729 particles, respectively. In figure 1, we plot a trajectory example in R, and the estimated trajectories using both methods. We can see that using both Sigma-point methods, and the PF with 729 particles, we obtain a correct estimation of the trajectory. Using the PF with 12 particles, we obtain a worse estimation, because the PF needs more particles to obtain good results. In figure 2, we plot the RMSE of the estimation of the position. We can see that the deterministic methods obtain better results than the PF. If we have no computational constraints, we will prefer the SQKF, otherwise, the SCKF will be the choice for real-time applications.

F T k ∼ N (µ T +m•δa T k , σ 2 T ) and F P k ∼ N (m • δa P k , σ 2

VI. CONCLUSIONS

This paper has presented the applicability of Gaussian nonlinear filters to the data fusion of heterogeneous sensors for positioning. We have proposed a state-space approach, which includes the biases introduced by the into the state, to compensate measurement errors. Our numerical results indicate that the square-root versions of both, the Gauss-Hermite quadrature Kalman filter and the thirddegree spherical-radial cubature Kalman filter, achieve good performance in scenarios where the square-root Unscented Kalman filter fails, while maintaining numerical stability. We have also compared the results with a SMC method, which give poorer performances while being computationally heavier. The quadrature filter attains a lower estimation error, while the cubature filter requires much less computational load, being the algorithm of choice for real-time implementations.

7 :

 7 Set ζ 1,: = ξ i and ̟1,: = ωi 8: for m = 2 to dim(x) do 9: Set Ξ1:m-1,: = ζ ⊗ 11×α 10: Set Ξm,: = 11×α ⊗ ζ m-1,: 11: Set Ω1:m-1,: = ̟ ⊗ 11×α 12: Set Ωm,: = 11×α ⊗ ̟m-1,: 13: Set ζ = Ξ and ̟ = Ω 14: end for 15: ξ i = Ξ:,i, where i = 1, ..., L 16: ωi = Q dim(x) l=1 Ω l,i 17: end if Algorithm 4 Generation of Sigma-Points and weights for third-degree spherical-radial cubature rule 1: Set M = dim(x) and L = 2M . 2: Set the cubature points ξ i = √ M h IM×M -IM×M i i , where [•]i=1,...,L indicates the i-th column. 3: Set the cubature weights ωi = 1 2M , i = 1, ..., L.

  P ). The parameters of the motion model and the IMU are m = 500 kg, λ = 0.3, µ T = 200 N, σ T = 5000 N, σ P = 3000 N, and T = 1 s. Results are averaged over 500 independent computer simulations, each one with a different network deployment and target trajectory. Both filters have been initialized with x0 = [0 0 2 0.5 0.5 0.1] T , Σ x,0 = diag ([10 10 5 1 1 0.1]), Σ v,0 = diag ([1 1 1 0.1 0.1 0.1]), and Σ n,0 = diag σ ǫ I Np×Np σ e I Nt×Nt .

Fig. 1 .

 1 Fig. 1. Example of a true trajectory, and the estimate for a SQKF, a SCKF, and a PF using 12 and 729 particles

This work has been partially supported by the Spanish Science and Technology Commission: CENIT2007-2002 (TIMI) and TEC2008-02685/TEC (NARRA), and by the European Commission in the framework of the COST Action IC0803 (RFCSET) and the FP7 Network of Excellence in Wireless COMmunications NEWCOM++ (contract n. 216715).


I H Naqvi

A Khaleghi
email: ali.khaleghi@rr-research.no

G El Zein

Multiuser Time Reversal UWB Communication System: A Modified Transmission Approach

Keywords: Multiuser, time reversal (TR), dense multipath propagation channel, ultra-wideband (UWB), bit error rate (BER)

In this paper, ultra-wideband (UWB), time reversal (TR) communication is investigated by modifying the transmission prefilter. Mathematical expressions for received signal and the interference in the modified transmission scheme are derived. It is shown that the modified transmission approach reduces multi-user interference which eventually translates into a better bit error rate (BER) performance than simple TR multiuser scheme. Channel impulse responses (CIR) of a typical indoor channel are measured. In a multi-user scenario, both TR and the modified TR schemes are studied using the measured CIRs. It is shown that the proposed modified TR scheme outperforms the original TR scheme.

I. INTRODUCTION

Pulsed Ultra-Wideband (UWB) communication system can achieve high data rate due to its ultra large bandwidth. The ability of pulsed UWB to resolve individual multi-path components is exploited in the recent research for short range communication applications. However, large number of resolvable paths and low power limitations necessitate a complex receiver system. To collect the received signal energy, Rake receiver, transmit-reference or decision feedback autocorrelation receiver can be implied [START_REF] Hoctor | Delay-Hopped Transmitted-Reference RF Communications[END_REF]- [START_REF] Choi | Performance of Ultra-Wideband Communications with Suboptimal Receivers in Multipath Channels[END_REF]. Each technique has different difficulties and drawbacks. The characteristics of an Impulse radio and use of Rake receivers for signal detection has already been studied for communication in dense multipath environment [START_REF] Win | Impulse radio: How it works[END_REF], [START_REF] Win | Characterization of ultra-wide bandwidth wireless indoor channels: A communication-theoretic view[END_REF].

Time Reversal (TR) has been proposed as a technique to shift the design complexity from the receiver to the transmitter. Classically, TR has been applied in acoustics and under water communication applications [START_REF] Fink | Time-reversed acoustic[END_REF], [START_REF] Derode | Taking advantage of multiple scattering to communicate with timereversal antennas[END_REF], but recently, it has been widely studied for UWB communication [START_REF] Nguyen | A Time Reversal Transmission Approach for Multiuser UWB Communications[END_REF]- [START_REF] Khaleghi | Signal Frequency And Bandwidth Effects On The Performance Of UWB Time-Reversal Technique[END_REF]. The received signal in a TR system is considerably focused in spatial and temporal domains. As a result, the received power is concentrated within few taps and the effects of inter symbol interference (ISI) are greatly reduced. The receiver system becomes simpler than without TR and signal can be collected using a simple energy threshold detector [START_REF] Nguyen | A Time Reversal Transmission Approach for Multiuser UWB Communications[END_REF]. Spatial focusing enables low multi-user and co-channel interference in a multiuser and multi cell systems respectively. 1 Work supported by ANR project MIRTEC and French research ministry.

In [START_REF] Nguyen | A Time Reversal Transmission Approach for Multiuser UWB Communications[END_REF], TR filter was modified using circular shift operation in a multiuser scenario. Inherent limitations of the scheme were identified in [START_REF] Naqvi | Performance Enhancement of a Multiuser Time Reversal UWB Communication System[END_REF]. In this paper, a further modified transmission scheme for the TR system is proposed for a multiuser TR communication system. The proposed scheme addresses these limitations and gives a significantly better performance than the other schemes. The improvement in the performance increases with the increase in the number of users. Mathematical expressions for received signal and the interference of the proposed transmission scheme are derived. Experiments are performed in a typical indoor environment and results show that the proposed scheme outperforms the original TR scheme. Maximum number of simultaneous users increases with the proposed scheme. Expression for maximum number of simultaneous users is proposed.

The rest of the paper is organized as follows. A brief introduction of the TR scheme is presented in the Section II. The description of the proposed transmission scheme is presented in Section III. Experimental measurement setup, simulation results for the performance of the proposed transmission scheme are presented and analysed in Section IV. Finally, Section V concludes this paper. II. TIME REVERSAL Time reversal (TR) is a transmission scheme in which a time reversed channel impulse response (CIR) is used as a transmitter pre-filter. In a first step the CIR is estimated between an emitter and a receiver. The CIR is then flipped in time and transmitted in the same channel. The time reversed wave back propagates in the channel following the same paths as the CIR's ones but in the reverse order. Finally at the receiver, all the paths add up coherently in the time and spatial domains. For dense multipath propagation channels, strong temporal compression and high spatial focusing can be achieved with a focusing gain of about 8 dB [START_REF] Khaleghi | Demonstration of Time-Reversal in Indoor Ultra-Wideband Communication: Time Domain Measurement[END_REF]. For communication purposes, this gain improves, for instance, the transmission range. Inter Symbol Interference (ISI) effects are mitigated by temporal compression and multiuser interference is reduced due to spatial focusing. The received signal (y j (t)) at the intended receiver (j) can be mathematically represented as:

y j (t) = s(t) h ij (-t) h ij (t) = s(t) R auto ij (t) (1)
where h ij (t) is the CIR from the transmitting point (i) to an intended receiver (j), s(t) is the transmitted signal, denotes convolution product and R auto ij (t) is the autocorrelation of the CIR, h ij (t). The received signal at any non intended receiver (k) is written as:

y k (t) = s(t) h ij (-t) h ik (t) = s(t) R cross ikj (t) (2
) where h ik (t) is the CIR from the transmitting point (i) to an unintended receiver (k) and R cross ikj (t) is the cross-correlation of the CIR h ik (t) and h ij (t). If the channels are uncorrelated, then the signal transmitted for one receiver will act as a noise for a receiver at any other location. Thus, a secure communication is achieved with low probability of detection and low probability of interception.

In the practical implementation of the TR system, the precoding filter is truncated in time to reduce the filter length and thus the system complexity. The truncated response is represented as h (-t). For data communication purpose, the transmitted symbols are modulated by binary pulse amplitude modulation (BPAM) scheme. The k th symbol, d k , of the symbol sequence is equal to 1 or -1 for the data bits 1 or 0 respectively. Therefore, the received signal at the intended receiver is written as:

y(t) = A k d k h (-t -k T s ) Transmitted RF signal h ( t) CIR +n(t) ≈ A k d k R hh (t -k T s ) + n(t) (3)
where A is a normalization factor and T s is the inter-symbol interval. n(t) is the noise and R hh (t) is the autocorrelation function of the CIR (h(t)). For the sake of simplicity, we have supposed that the T s = T sig , where T sig is the length of the measured time reversed CIR (h (-t)) in seconds. As the amplitude of the peak of the received signal is proportional to the energy of the transmitted signal ( h 2 dt), the truncation process decreases the peak of the received signal. Due to BPAM, the polarity of the received signal peaks depends on the transmitted data bit and is used for the detection of the data bits.

III. DESCRIPTION OF THE PROPOSED TRANSMISSION APPROACH

In a multi user TR system, multiple signals for different users are transmitted simultaneously. Let the measured and truncated CIR from the transmitter to the user i can be written as:

h i (-t) = L-1 m=0 a m δ(t -m τ s ) ( 4
)
where L is number of time reversal filter taps which corresponds to T sig = L τ s in time(seconds), a m is the associated amplitude and m τ s is the associated delay of the multi-path components. τ s is the time between two consecutive sample and depends on the sampling rate of the time reversal filter.

For instance, if the measured CIR is sampled with a sampling rate of 5 GS/s, the delay of τ s = 0.2 ns is obtained between two consecutive samples, therefore the number of taps (L) in filter having a length of 50 ns is L = 250 samples or taps. The transmitted signal for multi user TR can be written as:

T x (t) = k Nu-1 i=0 1 √ N u d ik A i h i (-t -kT s ) (5)
where d ik is the k th information bit of the i th user, N u is the total number of simultaneous user, T s is the inter symbol interval and A i is the normalization factor which can be written as:

A i = 1 h i (-t -kT s ) 2 ( 6
)
where . is the Frobenius norm operation. The term 1

Nu

insures that the signals for all the users are transmitted with a constant power. This is in contrast to [START_REF] Nguyen | A Time Reversal Transmission Approach for Multiuser UWB Communications[END_REF], where this normalization has not been carried out resulting in an unfair comparison of the performance with different number of users.

The received signal for the j th user and the k th symbol can be written as:

R x kj (t) = 1 √ N u d jk h j (t) h j (-t + jT u -kT s ) Signal + 1 √ N u Nu-1 i=0,i =j d ik h j (t) h i (-t + iT u -kT s ) Interf erence (7)
The responses of the modified filter are produced by shifting h (-t) to either left or right and then forcing the shifted part to zero so that the shifted signal can be packed in the same signal duration. Fig. 1 shows the pattern of the left or right shift for l = 1, 2 taps. As shown for the left shift of 1 tap, the last three taps are shifted to left by one tap and the slot for last tap is filled with zero. For the right shift of 1 tap, first three taps are shifted to right and then the slot for the first tap is filled with zero. With the process of filling the shifted taps with zeros, we get rid of the unwanted signal, which causes interference to the adjacent symbols (called Image in [START_REF] Naqvi | Performance Enhancement of a Multiuser Time Reversal UWB Communication System[END_REF]). As the taps which result in an undesired signal are forced to zero, the received peak signal increases for an equal transmitted power. If h (-t) is shifted left by T u = l τ s , the expression is given by:

lef t shif t(h (-t), T u ) = ĥ(-t + T u ) = [ L-l-1 i=0 a i+l δ(t -i τ s ) zeros(1, l)] (8)
where zeros(1, l) is a vector containing l number of zeros, l is the number of taps required to carry out a shift of T u seconds. Note that ĥ(-t + T u ) has L -l non zero taps. The transmitted signal with the modified transmission scheme can be thus written as:

T x (t) = k Nu-1 i=0 1 √ N u d ik A i ĥi (-t + iT u -kT s ) (9)
where A i = 1 ĥ(-t+iTu-kTs)

2 and the term 1 √ Nu insures that power transmitted for different number of users is constant. The received signal for the k th symbol of the j th user can be written as:

R x kj (t) = 1 √ N u d jk h j (t) ĥj (-t + jT u -kT s ) Signal + 1 √ N u Nu-1 i=0,i =j d ik h j (t) ĥi (-t + iT u -kT s ) Interf erence (10)

A. Interference Analysis of the Proposed Scheme

To analyze the worst case performance, it is assumed that the transmitter communicates with all users at the same time. Therefore, TR received signal in a multi user scenario consists of a sum of one autocorrelation function and N u -1 cross correlation functions. The N u -1 cross correlation functions cause multi user interference. The received signal peak is the sum of the square of the coefficients of CIR (from the properties of auto correlation function). The interference term at the time of the peak (t peak ) for the j th user in a simple TR scheme can be written as:

I j simple T R (t peak ) = Nu i=1,i =j L-1 m=0 b mi × a mj ( 11
)
where b mi are the coefficients of the time reversed CIR h i (-t + i T u -k T s ) (from 7)) specific to the user i, a mj are the coefficients of h j (t). As (11) uses simple TR transmission scheme, the high powered taps of h i (-t) and h j (-t) are approximately in the same time interval. Therefore, taps in the propagating channel containing more energy are multiplied by the taps of the transmitted signal with more energy resulting in a relatively higher interference.

To calculate the interference of the modified transmission scheme, we assume that the transmitted signal for the intended receiver is not shifted. Then by using [START_REF] Kyritsi | One-bit Time Reversal for WLAN Applications[END_REF] and [START_REF] Akogun | Demonstrating time reversal in ultra-wideband communications using time domain measurements[END_REF], interference at the peak of the intended signal is then written as:

I j mod.T R (t peak ) = Nu-1 i=0,i =j L-l-1 m=0 b (m+l) i × a mj ( 12
)
where b mi and a mj are the same coefficients used in [START_REF] Akogun | Demonstrating time reversal in ultra-wideband communications using time domain measurements[END_REF]. Here the interference is the product of L -l coefficients (as l taps are forced to zero). Furthermore, [START_REF] Nguyen | The Potential Use of Time Reversal Technique in Multiple Elements Antenna Systems[END_REF] suggests that in case of modified TR scheme, the taps in the propagating channel containing more energy are multiplied by the taps of the transmitted signal with less energy and vice versa. Therefore, interference is reduced with the new proposed transmission scheme. This reduced interference helps to improve the bit error rate (BER) performance of the system.

B. Effects of Shift on Received Signal Peak

In a TR communication system, as the received signal is the auto correlation function of the CIR, the received signal peak is the sum of the square of the coefficients of CIR. Neglecting the interference and the noise, the received signal peak with the modified TR scheme for the k th symbol of the j th user is written as:

R x modT R (t peak jk ) = d jk A j ( L-lj -1 m=0 a 2 m ) (13)
where a i are the coefficients of taps of ĥ(-t + j T b -kT s ) and l j (j T b τs ) are the number of taps required for a shift of j T b . The received signal peak depends on the energy contents of (L -l j ) filter coefficients. Thus, the amplitude of the received peak decreases by the sum of the square coefficients in the shifted part of the transmitted signal. Therefore with the new modulation scheme, the received signal peak reduces in proportion to the energy of the shifted part of the transmitted signal.

Fig. 2 shows the peak power of the received signal peak for the shifted signals normalized to the received peak with no shift. The shift percentage corresponds to the percentage of the total length of the transmitted signal. A set of 243 measured CIRs are used for the simulation. Experimental setup and the measurement procedure are explained in Section IV. The loss of the received peak power for transmitted signals corresponding to individual CIRs is represented by the dots and the dashed line is the mean of power loss. To calculate the maximum number of simultaneous users that a system can support, we must take the decision in accordance to the threshold (say 3 dB) which can vary for different applications. For a 3 dB threshold, our system can support a shift percentage 2). Thus, the number of users with the proposed scheme can be written as:

N mod.T R u = 0.95 × L δ = 0.95 × T sig ∆T R (14)
where . denotes the floor operator, L is the total number of taps in the transmitted signal, δ is the shift percentage between two simultaneous users, T sig is the channel length in s and ∆T R is shift separation between two users in s.

For the same threshold, the previously proposed scheme in [START_REF] Nguyen | A Time Reversal Transmission Approach for Multiuser UWB Communications[END_REF] can only support a shift of 0.75L which is contrary to their claim of 100% shift (as power loss with circular shift operation was not considered by the authors). The power loss for left shift is lesser than the power loss for the right shift as the energy contained in the shifted parts of the right shift is greater than the energy contained in the shifted parts of the left shift. Although a combination of right and left shifts can be used for the communication, for the sake of simplicity we have only used left shift. In the rest of the paper unless otherwise mentioned, a shift is meant to be a left shift. Power spectral density (PSD) of the transmitted signal of a TR communication system takes into account the effects of the antennas and the propagation channel including the path loss. In contrast to the pulse signal, the spectrum of a TR signal has a descending shape with increasing the frequency because the higher frequency components experience a greater path loss as compared to the lower frequency components in the spectrum. Fig. 3 shows the PSD plots of the transmitted signal with simple TR and modified TR schemes where a left shift of 0.20N taps is carried out for both modified TR scheme. The plots of both schemes have a descending shape. Maximum spectral power is experienced at the same frequency. Therefore, both the signals must be attenuated with the same factor in order to respect the UWB spectral mask proposed by FCC. Frequency selectivity of the transmitted signals is similar for the two schemes. In short, the both schemes have resembling spectral properties.

IV. EXPERIMENTAL SETUP AND SIMULATION RESULTS

Experiments are performed in a typical indoor environment. The environment is an office space of 14 m × 8 m in the

A. BER Performance

In the proposed transmission scheme, one user is separated with the other by a shift of a fixed number of taps. This separation is named as δ, which is a percentage of total number of taps in the transmitted signal. Signal for User 1 is transmitted without any shift. As discussed in Section III-A, that interference between users is greatly reduced with the proposed modulation scheme. To study the impact of the reduced interference, we evaluate the BER performance with the proposed scheme using left shift for 5, 10 and 15 simultaneous user for δ = 0.05 L.

From the measured CIRs, we generate almost 35 × (35 -N u -1) combinations for simulating different number of simultaneous users (N u ). For every combination of simultaneous users, 10000 symbols are transmitted which makes it sufficient for statistical analysis. The measured CIR is truncated for 90% energy contained in the CIR. Thus, the transmitted symbol has a length of 55 ns and a per user bit rate of 18 M bps. Perfect synchronization and no ISI effects are assumed. Signal to noise ratio (SNR) is varied by varying the noise variance, as:

SN R = P j /σ 2 noise ( 15
)
where P j is the power of the received signal at its peak and σ 2 noise is the noise variance. Bipolar pulse amplitude modulation (BPAM) is used for these simulations. The received signal y j (t) is sampled at its peak and is detected based on 1

Institute of Electronics and Telecommunications of Rennes

Detected bit = 1 if y j (t peak ) ≥ 0 0 if y j (t peak ) < 0 (16)
Fig. 4a-c shows the BER performance of the simple TR, TR with circular shift operation and the modified TR scheme for 5, 10 and 15 simultaneous users. The modified TR scheme outperforms the other two schemes specially for higher number of simultaneous users [START_REF] Kyritsi | One-bit Time Reversal for WLAN Applications[END_REF][START_REF] Khaleghi | Demonstration of Time-Reversal in Indoor Ultra-Wideband Communication: Time Domain Measurement[END_REF]. For instance for 10 simultaneous users, the modified TR scheme results in a 1.4 dB better performance than the TR with circular shift for a BER of 10 -4 . The simple TR scheme has already reached a plateau. To perform an analysis in the presence of extreme multi user interference, BER performance is studied for 15 simultaneous users. Fig. 4d compares the performance of the three schemes for 15 simultaneous users. The modified TR scheme gives significantly better performance than the other two schemes. The improvement is in the order of 4.5 dB or more.

If a system has large number of users, the users experiencing higher shift percentages will give poorer performance than the users experiencing lower shift percentages. To have a consistent system, we propose to rotate the shift percentages for different users so that no user is subjected to permanent high shift percentage.

V. CONCLUSION

In this paper, a modified transmission scheme for a multi user time reversal system is proposed. With the help of mathematical derivations, it is shown that the interference in the modified TR scheme is reduced compared to simple TR scheme. Limitations of the proposed scheme are studied and an expression for maximum number of simultaneous users is proposed. It is shown that the modified TR scheme outperforms simple TR and TR with circular shift scheme specially at higher number of simultaneous users. For instance for 15 simultaneous users, the modified TR scheme improves the performance in the order of 4.5 dB or more for a constant BER.

Fig. 1 .

 1 Fig. 1. Pattern for the left and right shift

Fig. 2 .

 2 Fig. 2. Received signal peak power with left and right shifts normalized to the peak with no shift

Fig. 3 .

 3 Fig. 3. PSD of transmitted signal with simple TR and modified TR schemes

Fig. 4 .

 4 Fig. 4. BER performance with 5, 10, and 20 simultaneous users with a) simple TR b) TR with circular shift c) modified TR scheme d) 15 simultaneous users with all three schemes for δ = 0.05 L taps


I H Naqvi

G El Zein

Time Domain Measurements for a Time Reversal SIMO System in Reverberation Chamber and in an Indoor Environment

Keywords: Time reversal (TR), ultra-wideband (UWB), focusing gain, TR peak performance

Time domain measurements are conducted for ultrawideband (UWB) signals in a reverberation chamber (RC) and in a typical indoor environment for a single input multiple output (SIMO) time reversal (TR) system. Channel estimation is carried out using an Arbitrary Waveform Generator (AWG) and a high speed Digital Storage Oscilloscope (DSO) for all transmitter receiver links. The measured channel impulse responses (CIR) of all the links are added and re-transmitted from the transmitting antenna. Upon reaching the receiving antennas, the received signals are added to form the eventual received signal. Different TR characteristics i.e. TR peak performance, TR focusing gain, average power increase, signal to side lobe ratio (SSR) and RMS delay spread are analyzed and compared to that of a single input single output (SISO) TR system.

I. INTRODUCTION

Pulsed Ultra-wide band (UWB) is a communication technique in which high data rate can be achieved by making use of ultra large bandwidth. The ability of pulsed UWB to resolve individual multipath components is exploited in the recent research for short range communication applications. However, the large number of resolvable paths and the low power limitations necessitate a complex receiver system. To collect the received signal energy, Rake receiver, transmitreference or differential scheme and the decision feedback autocorrelation receiver can be implied [START_REF] Hoctor | Delay-Hopped Transmitted-Reference RF Communications[END_REF]- [START_REF] Choi | Performance of Ultra-Wideband Communications with Suboptimal Receivers in Multipath Channels[END_REF]. Each technique has its own difficulties and drawbacks. The characteristics of an Impulse radio and use of Rake receivers for signal detection has already been studied for communication in dense multipath environment [START_REF] Win | Impulse radio: How it works[END_REF], [START_REF] Win | Characterization of ultra-wide bandwidth wireless indoor channels: A communication-theoretic view[END_REF]. Time reversal (TR) has been proposed as a technique to shift the design complexity from the receiver to the transmitter. Classically, TR has been applied in acoustics and under water communication applications [START_REF] Fink | Time-reversed acoustic[END_REF]- [START_REF] Derode | Taking advantage of multiple scattering to communicate with timereversal antennas[END_REF] but in recent times, it has been widely studied for UWB communication. In a single input single output (SISO) TR scheme, the time-reversed channel impulse response (CIR) is used as a transmitter pre-filter. It results in a received signal which is considerably focused both in spatial and temporal domains. Owning to temporal focusing, the received power is concentrated within a few taps and the effects of inter symbol interference (ISI) are greatly reduced. The task of equalizer design becomes much simpler than without focusing. The receiver system becomes simpler and significant signal energy can be collected using simple energy threshold detectors [START_REF] Nguyen | A Time Reversal Transmission Approach for Multiuser UWB Communications[END_REF]. Another important advantage of TR is the spatial focusing. It enables very low co channel interference in a multi cell system, resulting in a very effective use of bandwidth in the overall network. It is because of these characteristics that TR is gaining more and more attention for communication in UWB [START_REF] Kyritsi | MISO time reversal and delay-spread compression for FWA channels at 5 GHz[END_REF]- [START_REF] Naqvi | Performance Enhancement of a Multiuser Time Reversal UWB Communication System[END_REF].

In [START_REF] Khaleghi | Demonstration of Time-Reversal in Indoor Ultra-Wideband Communication: Time Domain Measurement[END_REF], the authors have presented the first time domain measurements for an indoor environment. In this paper the, to the best of the authors knowledge, first ever time domain SIMO measurements are presented in reverberation chamber (RC) and in an indoor environment. A very narrow pulse is transmitted from an antenna toward multiple receiving antennas. The CIRs for one transmitting antenna and multiple receive antennas is measured and added together. The resulting signal is reversed in time and retransmitted from the same transmitting antenna and toward the same set of multiple antennas. The received signals from multiple antennas are again added together. TR characteristics like, focusing gain, peak to side lobe ratio, delay spread and average power augmentation are assessed. It is shown that with SIMO configuration a focusing gain of the order of 40dB is achieved in a RC environment.

The rest of the paper is organized as follows. First, a brief introduction of TR is presented in section II. Experimental measurement setup and results are presented and analyzed in Section III. Finally, Section IV concludes this paper.

II. TIME REVERSAL

TR is essentially a pre-Rake scheme in which time reversed channel impulse responses (CIR) are used as transmitter prefilter. The signal (after being pre-filtered) propagates in an invariant channel following the same paths and results in coherently adding all the received signals in the delay and spatial domains. With this technique, a focusing gain in the order of 8dB for indoor propagation channel, strong temporal compression and spatial focusing (depending on the signal band width) are observed [START_REF] Kyritsi | One-bit Time Reversal for WLAN Applications[END_REF]. The received signal quality is improved by the focusing gain, ISI effects are mitigated by temporal compression and multiuser interference is reduced due to spatial focusing. The received signal at the intended receiver (j) can be mathematically represented as:

s(t) h ij (-t) * h ij (t) = s(t) R auto ij (t) (1)
where h ij (t) is the CIR from the transmitting point to an intended receiver, s(t) is the transmitted signal, denotes convolution product and (.) * means the complex conjugate of the function and R auto ij (t) is the autocorrelation of the CIR between the transmitting antenna (i) and receiving antenna (j). The received signal at any non intended receiver (k) is:

s(t) h ij (-t) * h ik (t) = s(t) R cross ikj (t) (2)
where h ik (t) is the CIR from the transmitting point to an unintended receiver and R cross ikj (t) is the cross-correlation of the CIR h ik (t) and the time reversed complex conjugated version of the transmitted signal h ij (-t) * . If the channels are uncorrelated, then the signal transmitted for one receiver will act as a noise for a receiver at any other location. This results in a secure communication with low probability of detect and low probability of intercept. If there are N r receiving antennas and one transmitting antenna, the received signal by the jth receiving antenna is:

y j (t) = s j (t) R auto ij (t) Signal(j) + Nr k=1;k =j s k (t) R cross ikj (t)
Interf erence(j)

(

+ n j (t)

Noise(j)
where s j (t) and s k (t) are the transmitted signals intended for the jth user and the kth user respectively. If the channels are uncorrelated, the Interference part in (3) will be negligible, enabling an interference free communication with different users.

To assess the performance of a SIMO TR system, some TR characteristics are defined. TR peak performance improvement of a SIMO system is defined as the ratio of a TR received peak with SIMO system to the TR received peak with SISO system. Focusing gain (FG) of a TR system is defined as the ratio of the strongest tap power of the received signal in TR scheme to the strongest tap power of the pulse system:

F G = 20log 10 max|y j (t)| max|h ij (t)| (4)
The average received power with the TR scheme increases as compared to the pulsed system for the same value of the total transmitted energy. In this paper the comparison of the average power increase of the SIMO and SISO TR systems is made. Signal to side lobe ratio (SSR) is defined as ratio of the power of the first to second strongest tap in a TR received signal:

SSR = 20log 10 max|y j (t)| max|y j (t)| t =t peak ( 5
)
where t peak is the time for strongest tap. Another important TR characteristic is the instantaneous RMS delay spread (σ t ). It can be calculated by the first and the second moment of the measured CIR or the measured TR response:

σ τ = N l=1 P DP (l)τ 2 l N l=1 P DP (l) - N l=1 P DP (l)τ l N l=1 P DP (l) 2 (6)
where P DP (l) = |h ij (l)| 2 , τ l is the excess time delay and N is the total length of the PDP. RMS delay spread is considered a metric for temporal compression in TR systems. In this paper we study the RMS delay spread ratio of a pulse UWB system and a TR UWB system. The delay spread ratio (ds/ds T R ) must be as high as possible to have good temporal compression where ds and ds T R are the RMS delay spreads of a pulse UWB system and a TR UWB communication system respectively.

III. EXPERIMENTAL SETUP AND SIMULATION RESULTS

Experiments are performed in RC and typical indoor environments. RC is a metallic chamber of dimensions 8.7m × 3.7m × 2.9m present inside IETR laboratory whose layout is shown in Fig. 1. For an indoor channel, an experimental setup is established in a modern laboratory building in IETR having the plan shown in Fig. 1. The indoor environment is an office space of 14m × 8m. All rooms are furnished with office equipments: tables, PCs and seats. RC increases the wave reflections in the environment and allows accomplishing a non line-of-sight (NLOS) propagation scenario. Measurement setup is illustrated in Fig. 2. One log periodic antenna is used as a transmitter and two conical mono-pole antennas are used as the receiver. The height of the transmitting antenna is 1.5m from the ground and that of receiving antennas is 1m from the ground. The distance between the transmit and receive antennas is 4.5m and 9m in RC and in indoor respectively. The channel sounding pulse and the time reversed CIR are generated through the arbitrary waveform generator (AWG 7052) having a maximum sampling rate of 5 GS/s. The output signal of AWG is amplified to a level to drive the power amplifier (P.A), Mini-Circuits ZHL-42. The given P.A provides a constant gain of 30dB and covers the frequency range of 700-4200 MHz. The output power is limited to avoid the interference to unauthorized bands and to generate sufficient signal-to-noise ratio (SNR) at the receiver. The receiver is a Digital Storage Oscilloscope (Tektronix DSO 6124C) with band width of 12 G Hz and a maximum sampling rate of 40 GS/s. DSO captures the CIR of the channel as well as the TR response. DSO is operated in average mode so that 8 samples are taken and averaged together. A set of 100 measurements is taken in RC and a set of 1000 measurements is taken in the indoor environment. Thus the results presented in this paper are average of 100 and 1000 measurements in RC and the indoor environment.

A. SIMO in Reverberation Chamber

RC is a metallic chamber. It generates large amount of multipaths and is thus a perfect environment for a TR system. SISO and SIMO experiments are carried out in RC and the comparison is summarized in Table I. Fig. 3a shows power delay profile (PDP) of CIR with SISO configuration in RC. Large number of multipaths can be observed. Fig. 3b shows TR response in RC for both SISO and SIMO configurations for an equal value of transmitted energy. Almost all the multipaths are coherently added and the resulting signal has very high FG and SSR. Furthermore, it can be seen that SIMO TR has a better TR peak performance compared to SISO TR. Table I presents different characteristics of SIMO and SISO TR in RC environment. It shows that FG, average power increase, SSR and TR peak performance normalized to the SISO TR system improves significantly with SIMO TR. For instance, it can be seen that TR peak with SIMO TR is 4.84 dB more than SISO TR for same transmitted energy. 4a and5a show the measured SISO CIR in an indoor environment for LOS and NLOS configurations respectively. NLOS configuration has got larger number of multipaths and is thus better adopted for a TR system. Figs. 4b and 5b show the measured TR response for both SISO and SIMO configurations in LOS and NLOS configurations respectively. Total transmitted energy for all the experiments is kept same. Table II presents different characteristics of SIMO and SISO TR in an indoor environment for LOS and NLOS configurations. In case of LOS configuration, it is quite obvious that the TR peak performance is significantly better in the case of SIMO TR. For two receiving antennas, an improvement of 3.48 dB is observed. All other TR properties also improve except for average increase power. In the case of NLOS configuration, we observe even better TR peak performance. This time we achieve an improvement of 4.29 dB with SIMO configuration. The other factors also improve but SSR remains same. Thus SIMO NLOS system has the best TR performance amongst all systems. All TR properties improve significantly except for SSR which remains almost unchanged for both SISO and SIMO systems.

Fig. 1 .

 1 Fig. 1. Measurement environment layout

Fig. 3 .

 3 Fig. 3. In reverberation chamber a) SISO CIR b) TR response for SISO and SIMO configurations

Fig. 4 .Fig. 5 .

 45 Fig. 4. In LOS configuration a) Indoor CIR for SISO b) Indoor TR response for SISO and SIMO TR

ACKNOWLEDGMENT

This work was partially supported by ANR Project MIRTEC and French Ministry of Research.

IV. CONCLUSION

In this paper, time domain measurements are presented for a SIMO TR system in RC and in indoor environments. It is shown that SIMO TR improves different TR characteristics e.g. focusing gain, increased average power, signal to side lobe ratio, TR peak performance and RMS delay spread. In RC, SIMO and SISO TR measurements give a very high focusing gain and SSR. For instance, the focusing gain for SIMO TR in RC is 28.5 dB. TR peak performance of SIMO TR is 4.84 dB better than peak performance of SISO TR. In indoor, SIMO TR has a lot better performance than SISO TR especially for NLOS configuration. The TR peak performance of SIMO TR is 3.48 and 4.29 dB better than SISO TR for LOS and NLOS configurations respectively. Thus amongst all configurations, SIMO NLOS TR system gives the best performance.