Achraf Mallat

UWB Based Positioning in Multipath Channels: CRBs for AOA and for Hybrid TOA-AOA Based Methods

In this paper we consider UWB (ultra wide band) based positioning. We derive the CRBs (Cramer Rao bound) for AOA (angle of arrival) and for hybrid method using both AOA and TOA (time of arrival) information. Both non modulated and modulated IR-UWB (impulse-radio) waveforms are considered. The UWB channel is supposed to be of the multipath type. The AOA technique is investigated without NB (narrow band) approximation usually made. The analytical results obtained for the CRBs are illustrated by numerical results for typical scenarios.

UWB based positioning: Cramer Rao bound for Angle Of Arrival and comparison with Time Of Arrival

In this paper we consider the potential of UWB (ultra wide band) signals for positioning. We consider both TOA (time of arrival) and AOA (angle of arrival) based methods. For the AOA method we use a linear array of equi-spaced elements. We derive the CRB associated with this method, without using the monochromatic assumption. The CRBs of both methods are compared. The analytical results are illustrated for UWB signals having the spectral occupation fulfilling the FCC emission mask. We show that, for far field conditions and practical numbers of antenna array elements, TOA outperforms AOA. However the accuracy provided by AOA is shown to be very interesting

Statistics of the MLE and Approximate Upper and Lower Bounds—Part I: Application to TOA Estimation

In nonlinear deterministic parameter estimation, the maximum likelihood estimator (MLE) is unable to attain the Cramér-Rao lower bound at low and medium signal-to-noise ratios (SNRs) due the threshold and ambiguity phenomena. In order to evaluate the achieved mean-squared error (MSE) at those SNR levels, we propose new MSE approximations (MSEA) and an approximate upper bound by using the method of interval estimation (MIE). The mean and the distribution of the MLE are approximated as well. The MIE consists in splitting the a priori domain of the unknown parameter into intervals and computing the statistics of the estimator in each interval. Also, we derive an approximate lower bound (ALB) based on the Taylor series expansion of noise and an ALB family by employing the binary detection principle. The accuracy of the proposed MSEAs and the tightness of the derived approximate bounds are validated by considering the example of time-of-arrival estimation.

CRBs for UWB Multipath Channel Estimation: Impact of the Overlapping Between the MPCs on MPC Gain and TOA Estimation

In this paper we study the impact of the overlapping between neighboring MPCs (multipath component) on the performance of channel estimation. We consider IR-UWB (impulse radio) signals, and the IEEE802.15.3a and IEEE802.15.4a UWB channel models. We show that for a pulse width (PW) sufficiently smaller than the average MPC rate of arrival (ROA) the probability to have more than three overlapping MPCs is relatively small. We derive the CRBs (Cramer Rao bound) for the joint estimation of the MPC gain and TOA (time of arrival) in the case of up to three overlapping MPCs. We compute also the average CRBs. We show that the CRBs obtained by averaging more than 80% of possible cases of the channel, are very close to the bounds obtained under the non-overlapping assumption (NOLA).

Testbed for IR-UWB based ranging and positioning: Experimental performance and comparison to CRLBs

In this paper we describe a testbed for impulse radio ultra wideband based ranging and positioning. We show the characteristics of the generated, transmitted and received pulses. We consider both the maximum likelihood estimator and a threshold-based estimator for the estimation of the time of arrival. We measure the variances for ranging and positioning and compare them to the Cramer-Rao lower bounds for range and position estimation. We discuss the impact of both false multipath component detection and false sidelobe detection on the estimation accuracy. The obtained variances are close to the Cramer-Rao lower bounds when the mainlobe of the first multi-path component is detected. In realistic multipath environments we can use a threshold-based time of arrival estimator in order to detect the first multipath component which may be missed by the maximum likelihood estimator. We show also that the errors on positioning due to false multipath component and sidelobe detection can be highly mitigated by increasing the number of receivers. For a radiated energy of 8.1 pJ and a distance of 5 meters between the transmit and receive antennas, the obtained accuracy is in the order of one centimeter.

Multiuser and multipath interference mitigation in UWB TOA estimation

In this paper we consider the impact of MUI (multiuser interference) on coherent TOA (time of arrival) estimation. We consider TH-PPM (time-hopping pulse position modulation). We propose estimators based on the ML (maximum likelihood) criterion together with interference cancellation. We propose algorithms for SP (single-path) and MP (multipath) communication channels. We report simulation results and we conclude that the proposed estimators obtain smaller variance than the per-user ML estimator. Moreover, with well-chosen signature parameters, it is possible to reach the CRB (Cramèr-Rao bound) when multiple sources are transmitting in SP channels and values close to the CRB in MP channels. We also show that the interference cancellation technique makes the estimation robust against near-far situations.

Discrete fourier transform-based TOA estimation in UWB systems

In this paper, we propose two time of arrival estimators for ultra wideband signals based on the phase difference between the discrete Fourier transforms (DFT) of the transmitted and received signals. The first estimator is based on the slope of the unwrapped phase and the second one on the absolute unwrapped phase. We derive the statistics of the unwrapped phase. We show that slope-based estimation almost achieves asymptotically the baseband Cramer-Rao lower bound (CRLB), while the absolute-phase-based estimator achieves asymptotically the passband CRLB. We compare the proposed estimators to the time-domain maximum likelihood estimator (MLE). We show that the MLE achieves the CRLB faster than the DFT-based estimator, while the DFT-based estimator outperforms the MLE for low signal to noise ratios. We describe also how to use the proposed estimators in multipath UWB channels.

CRBS for the Joint Estimation of TOA and AOA in Wideband MISO and MIMO Systems: Comparison with SISO and SIMO Systems

We derive the CRBs (Cramer Rao bound) for the joint estimation of the TOA (time of arrival) and the AOA (angle of arrival) in wideband (WB) MISO (multiple input single output) and MIMO (multiple input multiple output) systems.We consider both cases of orthogonal and non-orthogonal transmitted signals. We compare the CRBs obtained in SISO (single input single output), SIMO (single input multiple output), MISO and MIMO systems under the assumption that the total transmitted energy is the same for all systems. We show that SIMO and MIMO are equivalent for TOA estimation, and MISO and SISO as well when the transmitted signals are orthogonal. For non-orthogonal signals, MIMO is better than SIMO, and MISO is better than SISO when the received signals are constructive. For AOA estimation, we show that MIMO is better than SIMO and SIMO is better than MISO when the transmitted signals are orthogonal. For non-orthogonal signals, MIMO is much better when the received signals are constructive. The CRBs obtained for non-orthogonal signals are very sensitive to the angle. Numerical results obtained in a typical scenario are provided.

Cramér-rao bound analysis of UWB based localization approaches

The problem of localizing one impulse radio (IR) Ultra-wideband (UWB) transmitter from the signals received at several anchors is assessed. In the direct position estimation (DPE) approach the target position is estimated in one step from the received signals from all anchors, while in two steps positioning (2SP) estimation, the times of arrival (TOAs) at different anchors are estimated in a first step, and the associated position estimate is found in a second step. This paper derives the Cramér-Rao Bound (CRB) for both DPE and 2SP in a single-path additive white Gaussian noise (AWGN) ultra-wideband (UWB) channel. In 2SP the amplitudes received at each anchor need to be estimated in order to generate the optimal weighting matrix. Considering the received amplitudes as a new unknown parameter for the 2SP approach gives rise to an expression relating the CRBs for DPE and 2SP. We find that DPE approach always outperforms 2SP approach. The derived bounds are compared with simulations using the corresponding Maximum Likelihood Estimators (MLE).

Ziv-Zakai lower bound for UWB based TOA estimation with unknown interference

This paper derives the Ziv-Zakai lower bound (ZZLB) for the time of arrival (TOA) estimation in the presence of one interfering pulse from which no a priori knowledge is available. The bound is obtained by including the interference in the system model but only the transmitted pulse as a candidate for the likelihood ratio (LR) test. A compact ZZLB expression that depends on the time delay and amplitude of the interference is obtained. We compare the performance of the first path maximum likelihood estimation (MLE) with the bound as a function of the relative distance between the first path and the interfering path.