Publication# A Novel Maximum Likelihood Estimation Of Superimposed Exponential Signals In Noise And Ultra-Wideband Application

Abstract

We pose the estimation of the parameters of multiple superimposed exponential signals in additive Gaussian noise problem as a Maximum Likelihood (ML) estimation problem. The ML problem is very non linear and hard to solve. Some previous works focused on finding alternative estimation procedures, for example by denoising. In contrast, we tackle the ML estimation problem directly. First, we use the same transformation as the first step of Iterative Quadratic Maximum Likelihood (IQML) and transform the ML problem into another optimization problem that gets rid of the amplitude coefficients. Second, we solve the remaining optimization problem with a gradient descent approach (“pseudo-quadratic maximum likelihood”). We also use this algorithm for Ultra-Wideband channel estimation and estimate ranging in non-line of sight environment.

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John Farserotu, Jean-Yves Le Boudec, Hai Zhan

We propose a high-resolution ranging algorithm for impulse radio (IR) ultra-WideBand (UWB) communication systems in additive white Gaussian noise. We formulate the ranging problem as a maximum- likelihood (ML) estimation problem for the channel delays and amplitudes at the receiver. Then we translate the obtained delay estimates into an estimate of the distance. The ML estimation problem is a non-linear problem and is hard to solve. Some previous works focus on finding alternative estimation procedures, for example by denoising. In contrast, we tackle the ML estimation problem directly. First, we use the same transformation as the first step of Iterative quadratic maximum likelihood (IQML) and we transform the ML problem into another optimization problem that avoids the estimation of the amplitude coefficients. Second, we solve the remaining optimization problem with a gradient descent approach (pseudo-quadratic maximum likelihood (PQML) algorithm). To demonstrate the good performance of the proposed estimator, we present the numerical evaluations under the IEEE 802.15.4a channel model. We show that our algorithm performs significantly better than previously published heuristics. We also derive a reduced complexity version of the algorithm algorithm, which will be implemented on the Xinlix field-programmable gate array (FPGA) board in the future. We test the approach in a real weak line of sight (LOS) propagation environment and obtained good accuracy for the ranging.

John Farserotu, Jean-Yves Le Boudec, Hai Zhan

We propose a high resolution ranging algorithm for unsynchronized impulse radio Ultra-wideband (UWB) communication systems in gaussian noise. We pose the ranging problem as a Maximum Likelihood (ML) estimation problem for the channel delays and attenuations and phase offset at receiver. Then we translate the obtained delay estimates into an estimate of the distance. The ML problem is very non linear and hard to solve. Some previous works focused on finding alternative estimation procedures, for example by denoising. In contrast, we tackle the ML estimation problem directly. First, we use the same transformation as the first step of Iterative Quadratic Maximum Likelihood (IQML) and transform the ML problem into another optimization problem that gets rid of the amplitude coefficients. Second, we solve the remaining optimization problem with a gradient descent approach (“pseudo-quadratic maximum likelihood”). We show that our algorithm performs significantly better than previously published heuristics. We tested the approach on a real non-line of sight system and obtained good accuracy.

2007In this thesis, we focus on Impulse Radio (IR) Ultra-WideBand (UWB) ranging and positioning techniques under indoor propagation environments. IR-UWB, a new carrierless communication scheme using impulses, is a candidate technology for future communication, ranging and positioning applications. Recent progress on both the technical and regulatory side of this technology has made this possible [1][2][3]. The fine time resolution of UWB signals has created a vision of novel ranging and positioning applications to augment existing narrowband systems operating in dense multipath environments [4][5][6]. We propose a high-resolution IR-UWB ranging algorithm based on Maximum Likelihood (ML) when the noise is additive Gaussian noise or multi-user interference. First, we pose the ranging problem as an ML estimation problem for the channel delays and their amplitudes at the receiver. We evaluate the ranging by translating the received delay estimates into an estimate of the distance. Then, we use the same transformation as the first step of Iterative Quadratic Maximum Likelihood (IQML) and we transform the ML problem into another optimization problem that avoids the estimation of the amplitude coefficients. We solve the remaining optimization problem with a gradient descent approach (Pseudo-Quadratic Maximum Likelihood (PQML) algorithm). Most previous works assume the distribution of the targeted distance is a uniform distribution (for example [7], [8] and [9]). In contrast to the previous works, we propose that the distribution of the targeted distance, which is not necessarily a uniform distribution, should depend on the geometry of the indoor environments of interest. We propose a Bayesian detection algorithm where the prior distribution of the channel follows the IEEE 802.15.4a channel model to identify whether the received signal is Line-Of-Sight (LOS) signal or a Non-Line-Of-Sight (NLOS) signal. If it is a LOS signal, we use a Bayesian estimation approach to estimate the joint posterior probability density function (pdf) of the channel and the targeted distance. We use this pdf function to estimate the ranging with Minimum Mean Square Error Estimator (MMSE). For computing the joint posterior pdf of the channel and the targeted distance, we derive a novel algorithm based on Sampling and Importance Resampling (SIMR) and Expectation Maximum (EM) techniques. Furthermore, we propose a reduced-complexity architecture of an IR-UWB ranging system by using our proposed algorithms. We also implement the Bayesian ranging algorithm on the Xinlix ML410 Field-Programmable Gate Array (FPGA) board. The used FPGA resources show that the Bayesian ranging algorithm is a low-complexity algorithm. We derive a novel Ziv-Zakai lower bound for IR-UWB ranging error that depends on the geometry of the indoor environments of interest. In contrast to the work in [7], we do not introduce, during our derivation process, any approximation for our log-likelihood function. Therefore, we obtain a more accurate Ziv-Zakai lower bound for the IR-UWB ranging error with IEEE 802.15.4a channel models. Based on the geometry of the indoor environments of interest, we can find the "best" position of the base station that provides the lowest Ziv-Zakai lower bound of the IR-UWB ranging error. Our bound can also be used in real environments with the channel measurements from real environments. We also propose that the distribution of the targeted position should depend on the geometry of the indoor environments of interest. We propose a novel one-step approach that estimates the position and channels directly and jointly from the received signals of the used base stations. We use a Bayesian approach where the prior distribution of the channels follows the IEEE 802.15.4a channel model to estimate the joint posterior pdf of the channels, the targeted position and the transmit time. One application of the joint posterior pdf of the channels, the targeted position and the transmit time is the determination of the position with classical posterior estimator (such as MMSE). For computing the joint posterior pdf of the channels, the targeted position and the transmit time, we derived an algorithm that is based on SIMR and EM techniques. We derive Ziv-Zakai position error lower bounds for a one-step positioning scheme, a Time Of Arrival (TOA)-based positioning scheme and a Time Difference Of Arrival (TDOA)-based positioning scheme. As our derived bounds depend on the geometry of the indoor environments of interest, our bounds can also be used in real environments with the channel measurements from real environments. Multi-User Interference (MUI) statistical models for IR-UWB systems can be important in providing an accurate estimate of the channel state. As such, it can have a major impact on the overall system performance. In the literature, MUI in the time domain is often approximated with Middleton class A and Gaussian Mixture Models. We use measurements from an indoor IR-UWB testbed to assess the validity of these models. We analyzed the statistical properties of IR-UWB MUI with the measurements from real indoor environments.