Estimation of Radio Channel Parameters

Kurzfassung Diese Dissertation behandelt die Schätzung der Modellparameter einer Momentanaufnahme des Mobilfunkkanals. Das besondere Augenmerk liegt zum einen auf der Entwicklung eines generischen Datenmodells für den gemessenen Funkkanal, welches für die hochauflösende Parameterschätzung geeignet ist. Der zweite Schwerpunkt dieser Arbeit ist die Entwicklung eines robusten Parameterschätzers für die Bestimmung der Parameter des entworfenen Modells aus Funkkanalmessdaten. Entsprechend dieser logischen Abfolge ist auch der Aufbau dieser Arbeit. Im ersten Teil wird ausgehend von einem aus der Literatur bekannten strahlenoptischen Modell eine algebraisch handhabbare Darstellung von beobachteten Wellenausbreitungspfaden entwickelt. Das mathematische Modell erlaubt die Beschreibung von SISO (single-input-single-output)- Übertragungssystemen, also von Systemen mit einer Sendeantenne und einer Empfangsantenne, als auch die Beschreibung von solchen Systemen mit mehreren Sende- und/oder Empfangsantennen. Diese Systeme werden im Allgemeinen auch als SIMO- (single-input-multiple-output), MISO- (multiple-input-single-output) oder MIMO-Systeme (multiple-input-multiple-output) bezeichnet. Im Gegensatz zu bekannten Konzepten enthält das entwickelte Modell keine Restriktionen bezüglich der modellierbaren Antennenarrayarchitekturen. Dies ist besonders wichtig in Hinblick auf die möglichst vollständige Erfassung der räumlichen Struktur des Funkkanals. Die Flexibilität des Modells ist eine Grundvoraussetzung für die optimale Anpassung der Antennenstruktur an die Messaufgabe. Eine solche angepasste Antennenarraystruktur ist zum Beispiel eine zylindrische Anordnung von Antennenelementen. Sie ist gut geeignet für die Erfassung der räumlichen Struktur des Funkkanals (Azimut und Elevation) in so genannten Outdoor- Funkszenarien. Weiterhin wird im ersten Teil eine neue Komponente des Funkkanaldatenmodells eingeführt, welche den Beitrag verteilter (diffuser) Streuungen zur Funkübertragung beschreibt. Die neue Modellkomponente spielt eine Schlüsselrolle bei der Entwicklung eines robusten Parameterschätzers im Hauptteil dieser Arbeit. Die fehlende Modellierung der verteilten Streuungen ist eine der Hauptursachen für die begrenzte Anwendbarkeit und die oft kritisierte fehlende Robustheit von hochauflösenden Funkkanalparameterschätzern, die in der Literatur etabliert sind. Das neue Datenmodell beschreibt die so genannten dominanten Ausbreitungspfade durch eine deterministische Abbildung der Pfadparameter auf den gemessenen Funkkanal. Der Beitrag der verteilten Streuungen wird mit Hilfe eines zirkularen mittelwertfreien Gaußschen Prozesses beschrieben. Die Modellparameter der verteilten Streuungen beschreiben dabei die Kovarianzmatrix dieses Prozesses. Basierend auf dem entwickelten Datenmodell wird im Anschluss kurz über aktuelle Konzepte für Funkkanalmessgeräte, so genannte Channel-Sounder, diskutiert. Im zweiten Teil dieser Arbeit werden in erster Linie Ausdrücke zur Bestimmung der erzielbaren Messgenauigkeit eines Channel-Sounders abgeleitet. Zu diesem Zweck wird die untere Schranke für die Varianz der geschätzten Modellparameter, das heißt der Messwerte, bestimmt. Als Grundlage für die Varianzabschätzung wird das aus der Parameterschätztheorie bekannte Konzept der Cramér-Rao-Schranke angewandt. Im Rahmen der Ableitung der Cramér-Rao-Schranke werden außerdem wichtige Gesichtspunkte für die Entwicklung eines effizienten Parameterschätzers diskutiert. Im dritten Teil der Arbeit wird ein Schätzer für die Bestimmung der Ausbreitungspfadparameter nach dem Maximum-Likelihood-Prinzip entworfen. Nach einer kurzen Übersicht über existierende Konzepte zur hochauflösenden Funkkanalparameterschätzung wird die vorliegende Schätzaufgabe analysiert und in Hinsicht ihres Typs klassifiziert. Unter der Voraussetzung, dass die Parameter der verteilten Streuungen bekannt sind, lässt sich zeigen, daß sich die Schätzung der Parameter der Ausbreitungspfade als ein nichtlineares gewichtetes kleinstes Fehlerquadratproblem auffassen lässt. Basierend auf dieser Erkenntnis wird ein generischer Algorithmus zur Bestimmung einer globalen Startlösung für die Parameter eines Ausbreitungspfades vorgeschlagen. Hierbei wird von dem Konzept der Structure-Least-Squares (SLS)-Probleme Gebrauch gemacht, um die Komplexität des Schätzproblems zu reduzieren. Im folgenden Teil dieses Abschnitts wird basierend auf aus der Literatur bekannten robusten numerischen Algorithmen ein Schätzer zur genauen Bestimmung der Ausbreitungspfadparameter abgeleitet. Im letzten Teil dieses Abschnitts wird die Anwendung unterraumbasierter Schätzer zur Bestimmung der Ausbreitungspfadparameter diskutiert. Es wird ein speichereffizienter Algorithmus zur Signalraumschätzung entwickelt. Dieser Algorithmus ist eine Grundvoraussetzung für die Anwendung von mehrdimensionalen Parameterschätzern wie zum Beispiel des R-D unitary ESPRIT (Estimation of Signal Parameters via Rotational Invariance Techniques) zur Bestimmung von Funkkanalparametern aus MIMO-Funkkanalmessungen. Traditionelle Verfahren zur Signalraumschätzung sind hier im Allgemeinen nicht anwendbar, da sie einen zu großen Speicheraufwand erfordern. Außerdem wird in diesem Teil gezeigt, dass ESPRIT-Algorithmen auch zur Parameterschätzung von Daten mit so genannter versteckter Rotations-Invarianzstruktur eingesetzt werden können. Als Beispiel wird ein ESPRIT-basierter Algorithmus zur Richtungsschätzung in Verbindung mit multibeam-Antennenarrays (CUBA) abgeleitet. Im letzten Teil dieser Arbeit wird ein Maximum-Likelihood-Schätzer für die neue Komponente des Funkkanals, welche die verteilten Streuungen beschreibt, entworfen. Ausgehend vom Konzept des iterativen Maximum-Likelihood-Schätzers wird ein Algorithmus entwickelt, der hinreichend geringe numerische Komplexität besitzt, so dass er praktisch anwendbar ist. In erster Linie wird dabei von der Toeplitzstruktur der zu schätzenden Kovarianzmatrix Gebrauch gemacht. Aufbauend auf dem Schätzer für die Parameter der Ausbreitungspfade und dem Schätzer für die Parameter der verteilten Streuungen wird ein Maximum-Likelihood-Schätzer entwickelt (RIMAX), der alle Parameter des in Teil I entwickelten Modells der Funkanalmessung im Verbund schätzt. Neben den geschätzten Parametern des Datenmodells liefert der Schätzer zusätzlich Zuverlässigkeitsinformationen. Diese werden unter anderem zur Bestimmung der Modellordnung, das heißt zur Bestimmung der Anzahl der dominanten Ausbreitungspfade, herangezogen. Außerdem stellen die Zuverlässigkeitsinformationen aber auch ein wichtiges Schätzergebnis dar. Die Zuverlässigkeitsinformationen machen die weitere Verarbeitung und Wertung der Messergebnisse möglich.The theme of this thesis is the estimation of model parameters of a radio channel snapshot. The main focus was the development of a general data model for the measured radio channel, suitable for both high resolution channel parameter estimation on the one hand, and the development of a robust parameter estimator for the parameters of the designed parametric radio channel model, in line with this logical work flow is this thesis. In the first part of this work an algebraic representation of observed propagation paths is developed using a ray-optical model known from literature. The algebraic framework is suitable for the description of SISO (single-input-single-output) radio transmission systems. A SISO system uses one antenna as the transmitter (Tx) and one antenna as the receiver (Rx). The derived expression for the propagation paths is also suitable to describe SIMO (single-input-multiple-output), MISO (multiple-input-single-output), and MIMO (multiple-input-multiple-output) radio channel measurements. In contrast to other models used for high resolution channel parameter estimation the derived model makes no restriction regarding the structure of the antenna array used throughout the measurement. This is important since the ultimate goal in radio channel sounding is the complete description of the spatial (angular) structure of the radio channel at Tx and Rx. The flexibility of the data model is a prerequisite for the optimisation of the antenna array structure with respect to the measurement task. Such an optimised antenna structure is a stacked uniform circular beam array, i.e., a cylindrical arrangement of antenna elements. This antenna array configuration is well suited for the measurement of the spatial structure of the radio channel at Tx and/or Rx in outdoor-scenarios. Furthermore, a new component of the radio channel model is introduced in the first part of this work. It describes the contribution of distributed (diffuse) scattering to the radio transmission. The new component is key for the development of a robust radio channel parameter estimator, which is derived in the main part of this work. The ignorance of the contribution of distributed scattering to radio propagation is one of the main reasons why high-resolution radio channel parameter estimators fail in practice. Since the underlying data model is wrong the estimators produce erroneous results. The improved model describes the so called dominant propagation paths by a deterministic mapping of the propagation path parameters to the channel observation. The contribution of the distributed scattering is modelled as a zero-mean circular Gaussian process. The parameters of the distributed scattering process determine the structure of the covariance matrix of this process. Based on this data model current concepts for radio channel sounding devices are discussed. In the second part of this work expressions for the accuracy achievable by a radio channel sounder are derived. To this end the lower bound on the variance of the measurements i.e. the parameter estimates is derived. As a basis for this evaluation the concept of the Cramér-Rao lower bound is employed. On the way to the Cramér-Rao lower bound for all channel model parameters, important issues for the development of an appropriate parameter estimator are discussed. Among other things the coupling of model parameters is also discussed. In the third part of this thesis, an estimator, for the propagation path parameters is derived. For the estimator the 'maximum-likelihood' approach is employed. After a short overview of existing high-resolution channel parameter estimators the estimation problem is classified. It is shown, that the estimation of the parameters of the propagation paths can be understood as a nonlinear weighted least squares problem, provided the parameters of the distributed scattering process are known. Based on this observation a general algorithm for the estimation of raw parameters for the observed propagation paths is developed. The algorithm uses the concept of structured-least-squares (SLS) and compressed maximum likelihood to reduce the numerical complexity of the estimation problem. A robust estimator for the precise estimation of the propagation path parameters is derived. The estimator is based on concepts well known from nonlinear local optimisation theory. In the last part of this chapter the application of subspace based parameter estimation algorithms for path parameter estimation is discussed. A memory efficient estimator for the signal subspace needed by, e.g., R-D unitary ESPRIT is derived. This algorithm is a prerequisite for the application of signal subspace based algorithms to MIMO-channel sounding measurements. Standard algorithms for signal subspace estimation (economy size SVD, singular value decomposition) are not suitable since they require an amount of memory which is too large. Furthermore, it is shown that ESPRIT (Estimation of Signal Parameters via Rotational Invariance Techniques) based algorithms can also be employed for parameter estimation from data having hidden rotation invariance structure. As an example an ESPRIT algorithm for angle estimation using circular uniform beam arrays (circular multi-beam antennas) is derived. In the final part of this work a maximum likelihood estimator for the new component of the channel model is developed. Starting with the concept of iterative maximum likelihood estimation, an algorithm is developed having a low computational complexity. The low complexity of the algorithm is achieved by exploiting the Toeplitz-structure of the covariance matrix to estimate. Using the estimator for the (concentrated, dominant, specular-alike) propagation paths and the parametric estimator for the covariance matrix of the process describing the distributed diffuse scattering a joint estimator for all channel parameter is derived (RIMAX). The estimator is a 'maximum likelihood' estimator and uses the genuine SAGE concept to reduce the computational complexity. The estimator provides additional information about the reliability of the estimated channel parameters. This reliability information is used to determine an appropriate model for the observation. Furthermore, the reliability information i.e. the estimate of the covariance matrix of all parameter estimates is also an important parameter estimation result. This information is a prerequisite for further processing and evaluation of the measured channel parameters

Arbitrarily Strong Utility-Privacy Tradeoff in Multi-Agent Systems

Each agent in a network makes a local observation that is linearly related to a set of public and private parameters. The agents send their observations to a fusion center to allow it to estimate the public parameters. To prevent leakage of the private parameters, each agent first sanitizes its local observation using a local privacy mechanism before transmitting it to the fusion center. We investigate the utility-privacy tradeoff in terms of the Cram\'er-Rao lower bounds for estimating the public and private parameters. We study the class of privacy mechanisms given by linear compression and noise perturbation, and derive necessary and sufficient conditions for achieving arbitrarily strong utility-privacy tradeoff in a multi-agent system for both the cases where prior information is available and unavailable, respectively. We also provide a method to find the maximum estimation privacy achievable without compromising the utility and propose an alternating algorithm to optimize the utility-privacy tradeoff in the case where arbitrarily strong utility-privacy tradeoff is not achievable

A Sketching Framework for Reduced Data Transfer in Photon Counting Lidar

Single-photon lidar has become a prominent tool for depth imaging in recent years. At the core of the technique, the depth of a target is measured by constructing a histogram of time delays between emitted light pulses and detected photon arrivals. A major data processing bottleneck arises on the device when either the number of photons per pixel is large or the resolution of the time stamp is fine, as both the space requirement and the complexity of the image reconstruction algorithms scale with these parameters. We solve this limiting bottleneck of existing lidar techniques by sampling the characteristic function of the time of flight (ToF) model to build a compressive statistic, a so-called sketch of the time delay distribution, which is sufficient to infer the spatial distance and intensity of the object. The size of the sketch scales with the degrees of freedom of the ToF model (number of objects) and not, fundamentally, with the number of photons or the time stamp resolution. Moreover, the sketch is highly amenable for on-chip online processing. We show theoretically that the loss of information for compression is controlled and the mean squared error of the inference quickly converges towards the optimal Cram\'er-Rao bound (i.e. no loss of information) for modest sketch sizes. The proposed compressed single-photon lidar framework is tested and evaluated on real life datasets of complex scenes where it is shown that a compression rate of up-to 150 is achievable in practice without sacrificing the overall resolution of the reconstructed image.Comment: 16 pages, 20 figure

Observability Gramian for Bayesian Inference in Nonlinear Systems With Its Industrial Application

In this letter, we present a novel (empirical) observability Gramian for nonlinear stochastic systems in the light of Bayesian inference. First, we define our observability Gramian, which we refer to as the estimability Gramian, based on the relation to the so-called Bayesian Fisher Information Matrix for initial state estimation. Then, we study the fundamental properties of an empirical version of the estimability Gramian. The practical usefulness of the proposed framework is examined through its application to a parameter and initial state estimation in a natural gas engine cylinder

A New Compact Delay, Doppler Stretch and Phase Estimation CRB with a Band-Limited Signal for Generic Remote Sensing Applications

Since time-delay, Doppler effect and phase estimation are fundamental tasks in a plethora of engineering fields, tractable lower performance bounds for this problem are key tools of broad interest for a large variety of remote sensing applications. In the large sample regime and/or the high signal-to-noise ratio regime of the Gaussian conditional signal model, the Cramér–Rao bound (CRB) provides an accurate lower bound in the mean square error sense. In this contribution, we introduce firstly a new compact CRB expression for the joint time-delay and Doppler stretch estimation, considering a generic delayed and dilated band-limited signal. This generalizes known results for both wideband signals and the standard narrowband signal model where the Doppler effect on the band-limited baseband signal is not considered and amounts to a frequency shift. General compact closed-form CRB expressions for the amplitude and phase are also provided. These compact CRBs are expressed in terms of the baseband signal samples, making them especially easy to use whatever the baseband signal considered, therefore being valid for a variety of remote sensors. The new CRB expressions are validated in a positioning case study, both using synthetic and real data. These results show that the maximum likelihood estimator converges to the CRB at high signal-to-noise ratios, which confirms the exactness of the CRB. The CRB is further validated by comparing the ambiguity function and its 2nd order Taylor expansion where the perfect match also proves its exactness

Noise-Enhanced and Human Visual System-Driven Image Processing: Algorithms and Performance Limits

This dissertation investigates the problem of image processing based on stochastic resonance (SR) noise and human visual system (HVS) properties, where several novel frameworks and algorithms for object detection in images, image enhancement and image segmentation as well as the method to estimate the performance limit of image segmentation algorithms are developed. Object detection in images is a fundamental problem whose goal is to make a decision if the object of interest is present or absent in a given image. We develop a framework and algorithm to enhance the detection performance of suboptimal detectors using SR noise, where we add a suitable dose of noise into the original image data and obtain the performance improvement. Micro-calcification detection is employed in this dissertation as an illustrative example. The comparative experiments with a large number of images verify the efficiency of the presented approach. Image enhancement plays an important role and is widely used in various vision tasks. We develop two image enhancement approaches. One is based on SR noise, HVS-driven image quality evaluation metrics and the constrained multi-objective optimization (MOO) technique, which aims at refining the existing suboptimal image enhancement methods. Another is based on the selective enhancement framework, under which we develop several image enhancement algorithms. The two approaches are applied to many low quality images, and they outperform many existing enhancement algorithms. Image segmentation is critical to image analysis. We present two segmentation algorithms driven by HVS properties, where we incorporate the human visual perception factors into the segmentation procedure and encode the prior expectation on the segmentation results into the objective functions through Markov random fields (MRF). Our experimental results show that the presented algorithms achieve higher segmentation accuracy than many representative segmentation and clustering algorithms available in the literature. Performance limit, or performance bound, is very useful to evaluate different image segmentation algorithms and to analyze the segmentability of the given image content. We formulate image segmentation as a parameter estimation problem and derive a lower bound on the segmentation error, i.e., the mean square error (MSE) of the pixel labels considered in our work, using a modified Cramér-Rao bound (CRB). The derivation is based on the biased estimator assumption, whose reasonability is verified in this dissertation. Experimental results demonstrate the validity of the derived bound

Caractérisation des performances minimales d'estimation pour des modèles d'observations non-standards

In the parametric estimation context, estimators performances can be characterized, inter alia, by the mean square error and the resolution limit. The first quantities the accuracy of estimated values and the second defines the ability of the estimator to allow a correct resolvability. This thesis deals first with the prediction the "optimal" MSE by using lower bounds in the hybrid estimation context (i.e. when the parameter vector contains both random and non-random parameters), second with the extension of Cramér-Rao bounds for non-standard estimation problems and finally to the characterization of estimators resolution. This manuscript is then divided into three parts :First, we fill some lacks of hybrid lower bound on the MSE by using two existing Bayesian lower bounds: the Weiss-Weinstein bound and a particular form of Ziv-Zakai family lower bounds. We show that these extended lower bounds are tighter than the existing hybrid lower bounds in order to predict the optimal MSE.Second, we extend Cramer-Rao lower bounds for uncommon estimation contexts. Precisely: (i) Where the non-random parameters are subject to equality constraints (linear or nonlinear). (ii) For discrete-time filtering problems when the evolution of states are defined by a Markov chain. (iii) When the observation model differs to the real data distribution.Finally, we study the resolution of the estimators when their probability distributions are known. This approach is an extension of the work of Oh and Kashyap and the work of Clark to multi-dimensional parameters estimation problems.Dans le contexte de l'estimation paramétrique, les performances d'un estimateur peuvent être caractérisées, entre autre, par son erreur quadratique moyenne (EQM) et sa résolution limite. La première quantifie la précision des valeurs estimées et la seconde définit la capacité de l'estimateur à séparer plusieurs paramètres. Cette thèse s'intéresse d'abord à la prédiction de l'EQM "optimale" à l'aide des bornes inférieures pour des problèmes d'estimation simultanée de paramètres aléatoires et non-aléatoires (estimation hybride), puis à l'extension des bornes de Cramér-Rao pour des modèles d'observation moins standards. Enfin, la caractérisation des estimateurs en termes de résolution limite est également étudiée. Ce manuscrit est donc divisé en trois parties :Premièrement, nous complétons les résultats de littérature sur les bornes hybrides en utilisant deux bornes bayésiennes : la borne de Weiss-Weinstein et une forme particulière de la famille de bornes de Ziv-Zakaï. Nous montrons que ces bornes "étendues" sont plus précises pour la prédiction de l'EQM optimale par rapport à celles existantes dans la littérature.Deuxièmement, nous proposons des bornes de type Cramér-Rao pour des contextes d'estimation moins usuels, c'est-à-dire : (i) Lorsque les paramètres non-aléatoires sont soumis à des contraintes d'égalité linéaires ou non-linéaires (estimation sous contraintes). (ii) Pour des problèmes de filtrage à temps discret où l'évolution des états (paramètres) est régit par une chaîne de Markov. (iii) Lorsque la loi des observations est différente de la distribution réelle des données.Enfin, nous étudions la résolution et la précision des estimateurs en proposant un critère basé directement sur la distribution des estimées. Cette approche est une extension des travaux de Oh et Kashyap et de Clark pour des problèmes d'estimation de paramètres multidimensionnels

Contraction of Locally Differentially Private Mechanisms

We investigate the contraction properties of locally differentially private mechanisms. More specifically, we derive tight upper bounds on the divergence between $P\mathsf{K}$ and $Q\mathsf{K}$ output distributions of an $\varepsilon$-LDP mechanism $\mathsf{K}$ in terms of a divergence between the corresponding input distributions $P$ and $Q$, respectively. Our first main technical result presents a sharp upper bound on the $\chi^2$-divergence $\chi^2(P\mathsf{K}\|Q\mathsf{K})$ in terms of $\chi^2(P\|Q)$ and $\varepsilon$. We also show that the same result holds for a large family of divergences, including KL-divergence and squared Hellinger distance. The second main technical result gives an upper bound on $\chi^2(P\mathsf{K}\|Q\mathsf{K})$ in terms of total variation distance $\mathsf{TV}(P, Q)$ and $\varepsilon$. We then utilize these bounds to establish locally private versions of the van Trees inequality, Le Cam's, Assouad's, and the mutual information methods, which are powerful tools for bounding minimax estimation risks. These results are shown to lead to better privacy analyses than the state-of-the-arts in several statistical problems such as entropy and discrete distribution estimation, non-parametric density estimation, and hypothesis testing

Timing estimation and resynchronization for amplify-and-forward communication systems

This paper proposes a general framework to effectively estimate the unknown timing and channel parameters, as well as design efficient timing resynchronization algorithms for asynchronous amplify-and-forward (AF) cooperative communication systems. In order to obtain reliable timing and channel parameters, a least squares (LS) estimator is proposed for initial estimation and an iterative maximum-likelihood (ML) estimator is derived to refine the LS estimates. Furthermore, a timing and channel uncertainty analysis based on the CramrRao bounds (CRB) is presented to provide insights into the system uncertainties resulted from estimation. Using the parameter estimates and uncertainty information in our analysis, timing resynchronization algorithms that are robust to estimation errors are designed jointly at the relays and the destination. The proposed framework is developed for different AF systems with varying degrees of timing misalignment and channel uncertainties and is numerically shown to provide excellent performances that approach the synchronized case with perfect channel information. © 2006 IEEE.published_or_final_versio

Performance Analysis of Bistatic Radar and Optimization methodology in Multistatic Radar System.

This work deals with the problem of calculating the Cramér-Rao lower bounds (CRLBs) for bistatic radar channels. To this purpose we exploited the relation between the Ambiguity Function (AF) and the CRLB. The bistatic CRLBs are analyzed and compared to the monostatic counterparts as a function of the bistatic geometric parameters. In the bistatic case both geometry factors and transmitted waveforms play an important role in the shape of the AF, and therefore in the estimation accuracy of the target range and velocity. In particular, the CRLBs depend on the target direction of arrival, the bistatic baseline length, and the distance between the target and the receiver. The CRLBs are then used to select the “optimum” bistatic channel (or set of channels) for the tracking of a radar target moving along a trajectory in a multistatic scenario and for design weighting coefficients for the multistatic detection process. This work also deals with the calculation of the Posterior Cramér-Rao Lower Bound (PCRLB) for sequential target state estimators for a bistatic tracking problem. In the context of tracking, the PCRLB provides a powerful tool, enabling one to determine a lower bound on the optimal achievable accuracy of target state estimation. The bistatic PCRLBs are analyzed and compared to the monostatic counterparts for a fixed target trajectory. Two different kinematic models are analyzed: constant velocity and constant acceleration. The derived bounds are also valid when the target trajectory is characterized by the combination of these two motions