John T. Flåm

On MMSE Estimation: A Linear Model Under Gaussian Mixture Statistics

In a Bayesian linear model, suppose observation y = Hx + n stems from independent inputs x and n which are Gaussian mixture (GM) distributed. With known matrix H, the minimum mean square error (MMSE) estimator for x , has analytical form. However, its performance measure, the MMSE itself, has no such closed form. Because existing Bayesian MMSE bounds prove to have limited practical value under these settings, we instead seek analytical bounds for the MMSE, both upper and lower. This paper provides such bounds, and relates them to the signal-to-noise-ratio (SNR).

Gaussian mixture modeling for source localization

Exploiting prior knowledge, we use Bayesian estimation to localize a source heard by a fixed sensor network. The method has two main aspects: Firstly, the probability density function (PDF) of a function of the source location is approximated by a Gaussian mixture model (GMM). This approximation can theoretically be made arbitrarily accurate, and allows a closed form minimum mean square error (MMSE) estimator for that function. Secondly, the source location is retrieved by minimizing the Euclidean distance between the function and its MMSE estimate using a gradient method. Our method avoids the issues of a numerical MMSE estimator but shows comparable accuracy.

Using a Sensor Network to Localize a Source under Spatially Correlated Shadowing

This paper considers the use of a sensor network to estimate the position of a transmitting radio based on the received signal strength at the sensors. A generic path loss model which includes the effects of spatially correlated shadowing is assumed. A weighted likelihood (WL) estimator is proposed, which can be seen as a simplified minimum mean square error (MMSE) estimator. This estimator can be used for localizing a source in a static scenario or it can provide the initial position estimate of a tracking algorithm. The performance of the WL estimator is simulated, and robustness to erroneous assumptions about path loss exponent, shadowing variance and correlation distance is demonstrated.

On localisation accuracy inside the human abdomen region

In this work, localisation of a source within an absorbing medium is considered. By an absorbing medium, the authors mean an environment where the signal power decays exponentially with distance. The authors assume that the source is heard by nearby sensors when transmitting and its position shall be estimated based on the received signal strength (RSS) by these sensors. Under these assumptions, the focus is to determine the Cramer–Rao lower bound (CRLB). Thus, the goal is to derive the theoretical performance limit for an optimal estimator, and to study the feasibility of RSS-based localisation in an absorbing environment and specifically in human abdominal region. The authors demonstrate that the CRLB greatly depends on the shadowing conditions, and also on the relative positions of the sensors and the source. Although the obtained results are quite general, the motivating application is localisation of capsule endoscope in human abdominal region. The authors find that the RSS-based method can reach the needed accuracy for localising a capsule endoscope.

On change detection in a Kalman filter based tracking problem

Abstract Objective This work considers detecting an additive abrupt state change in a tracking process. It is assumed that the tracking is done by a Kalman filter and that the abrupt change takes place after the steady-state behavior of the filter is reached. Result The effect of the additive change on the innovation process is expressed in closed form, and we show that the optimal detection method depends on the available information, contained in the change vector. Method We take a Bayesian perspective and show that prior knowledge on the nature of the change can be used to significantly improve the detection performance. Result Specifically, we show that performance of such a detector coincides with that of a matched filter when the variance (uncertainty) of the change tends to zero, and it coincides with that of an energy detector when the variance tends to infinity. Conclusion Finally we conclude that utilizing the derived closed form improves the detection performance for abrupt changes for Kalman filter based tracking problems. In addition, it is concluded that incorporating prior knowledge can improve the detection performance only if the prior variance is less than a certain amount.

Pilot design for MIMO channel estimation: An alternative to the Kronecker structure assumption

This work seeks to design a pilot signal, under a power constraint, such that the channel can be estimated with minimum mean square error. The procedure we derive does not assume Kronecker structure on the underlying covariance matrices, and the pilot signal is obtained in three main steps. Firstly, we solve a relaxed convex version of the original minimization problem. Secondly, its solution is projected onto the feasible set. Thirdly we use the projected solution as starting point for an augmented Lagrangian method. Numerical experiments indicate that this procedure may produce pilot signals that are far better than those obtained under the Kronecker structure assumption.

The Linear Model Under Mixed Gaussian Inputs: Designing the Transfer Matrix

Suppose a linear model y=Hx+n, where inputs x,n are independent Gaussian mixtures. The problem is to design the transfer matrix H so as to minimize the mean square error (MSE) when estimating x from y. This problem has important applications, but faces at least three hurdles. Firstly, even for a fixed H, the minimum MSE (MMSE) has no analytical form. Secondly, the MMSE is generally not convex in H. Thirdly, derivatives of the MMSE w.r.t. H are hard to obtain. This paper casts the problem as a stochastic program and invokes gradient methods. The study is motivated by two applications in signal processing. One concerns the choice of error-reducing precoders; the other deals with selection of pilot matrices for channel estimation. In either setting, our numerical results indicate improved estimation accuracy-markedly better than those obtained by optimal design based on standard linear estimators. Some implications of the non-convexities of the MMSE are noteworthy, yet, to our knowledge, not well known. For example, there are cases in which more pilot power is detrimental for channel estimation. This paper explains why.

On optimal detection for matrix multiplicative data hiding

This paper analyzes a multiplicative data hiding scheme, where the watermark bits are embedded within frames of a Gaussian host signal by two different, but arbitrary, embedding matrices. A closed form expression for the bit error rate (BER) of the optimal detector is derived when the frame sizes tend to infinity. Furthermore, a structure is proposed for the optimal detector which divides the detection process into two main blocks: host signal estimation and decision making. The proposed structure preserves optimality, and allows for a great deal of flexibility: The estimator can be selected according to the a priori knowledge about host signal. For example, if the host signal is an Auto-Regressive (AR) process, we argue that a Kalman filter may serve as the estimator. Compared to a direct implementation of the Neyman-Pearson detector, this approach results in significantly reduced complexity while keeping optimal performance.

On localizing a capsule endoscope using magnetic sensors

In this work, localizing a capsule endoscope within the gastrointestinal tract is addressed. It is assumed that the capsule is equipped with a magnet, and that a magnetic sensor network measures the flux from this magnet. We assume no prior knowledge on the source location, and that the measurements collected by the sensors are corrupted by thermal Gaussian noise only. Under these assumptions, we focus on determining the Cramer-Rao Lower Bound (CRLB) for the location of the endoscope. Thus, we are not studying specific estimators, but rather the theoretical performance of an optimal one. It is demonstrated that the CRLB is a function of the distance and angle between the sensor network and the magnet. By studying the CRLB with respect to different sensor array constellations, we are able to indicate favorable constellations.

On the CRLB for source localization in a lossy environment

In this work, localization of a source within a lossy medium is considered. By a lossy medium we mean an environment where the signal power decays exponentially with distance. We assume no prior knowledge on the source location, but that the source is heard by nearby sensors when transmitting. The source position shall be estimated based on the power received by these sensors. Under these assumptions, our focus is to determine the Cramer-Rao Lower Bound (CRLB). Thus, we are not studying specific estimators, but rather the (theoretical) performance of an optimal one. We demonstrate that the CRLB greatly depends on the shadowing conditions, and also on the relative positions of the sensors and the source. This spatial variability of the CRLB is used to discuss favorable positioning of the sensors.