352 research outputs found
Performance Bounds for Parameter Estimation under Misspecified Models: Fundamental findings and applications
Inferring information from a set of acquired data is the main objective of
any signal processing (SP) method. In particular, the common problem of
estimating the value of a vector of parameters from a set of noisy measurements
is at the core of a plethora of scientific and technological advances in the
last decades; for example, wireless communications, radar and sonar,
biomedicine, image processing, and seismology, just to name a few. Developing
an estimation algorithm often begins by assuming a statistical model for the
measured data, i.e. a probability density function (pdf) which if correct,
fully characterizes the behaviour of the collected data/measurements.
Experience with real data, however, often exposes the limitations of any
assumed data model since modelling errors at some level are always present.
Consequently, the true data model and the model assumed to derive the
estimation algorithm could differ. When this happens, the model is said to be
mismatched or misspecified. Therefore, understanding the possible performance
loss or regret that an estimation algorithm could experience under model
misspecification is of crucial importance for any SP practitioner. Further,
understanding the limits on the performance of any estimator subject to model
misspecification is of practical interest. Motivated by the widespread and
practical need to assess the performance of a mismatched estimator, the goal of
this paper is to help to bring attention to the main theoretical findings on
estimation theory, and in particular on lower bounds under model
misspecification, that have been published in the statistical and econometrical
literature in the last fifty years. Secondly, some applications are discussed
to illustrate the broad range of areas and problems to which this framework
extends, and consequently the numerous opportunities available for SP
researchers.Comment: To appear in the IEEE Signal Processing Magazin
A review of closed-form Cramér-Rao Bounds for DOA estimation in the presence of Gaussian noise under a unified framework
The Cramér-Rao Bound (CRB) for direction of arrival (DOA) estimation has been extensively studied over the past four decades, with a plethora of CRB expressions reported for various parametric models. In the literature, there are different methods to derive a closed-form CRB expression, but many derivations tend to involve intricate matrix manipulations which appear difficult to understand. Starting from the Slepian-Bangs formula and following the simplest derivation approach, this paper reviews a number of closed-form Gaussian CRB expressions for the DOA parameter under a unified framework, based on which all the specific CRB presentations can be derived concisely. The results cover three scenarios: narrowband complex circular signals, narrowband complex noncircular signals, and wideband signals. Three signal models are considered: the deterministic model, the stochastic Gaussian model, and the stochastic Gaussian model with the a priori knowledge that the sources are spatially uncorrelated. Moreover, three Gaussian noise models distinguished by the structure of the noise covariance matrix are concerned: spatially uncorrelated noise with unknown either identical or distinct variances at different sensors, and arbitrary unknown noise. In each scenario, a unified framework for the DOA-related block of the deterministic/stochastic CRB is developed, which encompasses one class of closed-form deterministic CRB expressions and two classes of stochastic ones under the three noise models. Comparisons among different CRBs across classes and scenarios are presented, yielding a series of equalities and inequalities which reflect the benchmark for the estimation efficiency under various situations. Furthermore, validity of all CRB expressions are examined, with some specific results for linear arrays provided, leading to several upper bounds on the number of resolvable Gaussian sources in the underdetermined case
Asymptotic normality and efficiency of the maximum likelihood estimator for the parameter of a ballistic random walk in a random environment
We consider a one dimensional ballistic random walk evolving in a parametric
independent and identically distributed random environment. We study the
asymptotic properties of the maximum likelihood estimator of the parameter
based on a single observation of the path till the time it reaches a distant
site. We prove an asymptotic normality result for this consistent estimator as
the distant site tends to infinity and establish that it achieves the
Cram\'er-Rao bound. We also explore in a simulation setting the numerical
behaviour of asymptotic confidence regions for the parameter value
Quantum estimation of coupled parameters and the role of entanglement
The quantum Cramer-Rao bound places a limit on the mean square error of a parameter estimation procedure, and its numerical value is determined by the quantum Fisher information. For single parameters, this leads to the well- known Heisenberg limit that surpasses the classical shot-noise limit. When estimating multiple parameters, the situation is more complicated and the quantum Cramer-Rao bound is generally not attainable. In such cases, the use of entanglement typically still offers an enhancement in precision. Here, we demonstrate that entanglement is detrimental when estimating some nuisance parameters. In general, we find that the estimation of coupled parameters does not benefit from either classical or quantum correlations. We illustrate this effect in a practical application for optical gyroscopes
Estimation of Radio Channel Parameters
Kurzfassung
Diese Dissertation behandelt die SchÀtzung der Modellparameter einer
Momentanaufnahme des Mobilfunkkanals. Das besondere Augenmerk liegt zum einen
auf der Entwicklung eines generischen Datenmodells fĂŒr den gemessenen Funkkanal,
welches fĂŒr die hochauflösende ParameterschĂ€tzung geeignet ist. Der zweite
Schwerpunkt dieser Arbeit ist die Entwicklung eines robusten ParameterschÀtzers
fĂŒr 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)-
Ăbertragungssystemen, 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 enthĂ€lt das entwickelte Modell keine Restriktionen bezĂŒglich
der modellierbaren Antennenarrayarchitekturen. Dies ist besonders wichtig in
Hinblick auf die möglichst vollstÀndige Erfassung der rÀumlichen Struktur des
Funkkanals. Die FlexibilitĂ€t des Modells ist eine Grundvoraussetzung fĂŒr die
optimale Anpassung der Antennenstruktur an die Messaufgabe. Eine solche
angepasste Antennenarraystruktur ist zum Beispiel eine zylindrische Anordnung
von Antennenelementen. Sie ist gut geeignet fĂŒr die Erfassung der rĂ€umlichen
Struktur des Funkkanals (Azimut und Elevation) in so genannten Outdoor-
Funkszenarien. Weiterhin wird im ersten Teil eine neue Komponente des
Funkkanaldatenmodells eingefĂŒhrt, welche den Beitrag verteilter (diffuser)
Streuungen zur FunkĂŒbertragung beschreibt. Die neue Modellkomponente spielt eine
SchlĂŒsselrolle bei der Entwicklung eines robusten ParameterschĂ€tzers im
Hauptteil dieser Arbeit. Die fehlende Modellierung der verteilten Streuungen ist
eine der Hauptursachen fĂŒr die begrenzte Anwendbarkeit und die oft kritisierte
fehlende Robustheit von hochauflösenden FunkkanalparameterschÀtzern, 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 GauĂschen Prozesses
beschrieben. Die Modellparameter der verteilten Streuungen beschreiben dabei die
Kovarianzmatrix dieses Prozesses. Basierend auf dem entwickelten Datenmodell
wird im Anschluss kurz ĂŒber aktuelle Konzepte fĂŒr FunkkanalmessgerĂ€te, so
genannte Channel-Sounder, diskutiert.
Im zweiten Teil dieser Arbeit werden in erster Linie AusdrĂŒcke zur Bestimmung
der erzielbaren Messgenauigkeit eines Channel-Sounders abgeleitet. Zu diesem
Zweck wird die untere Schranke fĂŒr die Varianz der geschĂ€tzten Modellparameter,
das heiĂt der Messwerte, bestimmt. Als Grundlage fĂŒr die VarianzabschĂ€tzung wird
das aus der ParameterschÀtztheorie bekannte Konzept der Cramér-Rao-Schranke
angewandt. Im Rahmen der Ableitung der CramĂ©r-Rao-Schranke werden auĂerdem
wichtige Gesichtspunkte fĂŒr die Entwicklung eines effizienten ParameterschĂ€tzers
diskutiert.
Im dritten Teil der Arbeit wird ein SchĂ€tzer fĂŒr die Bestimmung der
Ausbreitungspfadparameter nach dem Maximum-Likelihood-Prinzip entworfen. Nach
einer kurzen Ăbersicht ĂŒber existierende Konzepte zur hochauflösenden
FunkkanalparameterschÀtzung wird die vorliegende SchÀtzaufgabe analysiert und in
Hinsicht ihres Typs klassifiziert. Unter der Voraussetzung, dass die Parameter
der verteilten Streuungen bekannt sind, lÀsst sich zeigen, daà sich die
SchÀtzung der Parameter der Ausbreitungspfade als ein nichtlineares gewichtetes
kleinstes Fehlerquadratproblem auffassen lÀsst. Basierend auf dieser Erkenntnis
wird ein generischer Algorithmus zur Bestimmung einer globalen Startlösung fĂŒr
die Parameter eines Ausbreitungspfades vorgeschlagen. Hierbei wird von dem
Konzept der Structure-Least-Squares (SLS)-Probleme Gebrauch gemacht, um die
KomplexitÀt des SchÀtzproblems zu reduzieren. Im folgenden Teil dieses
Abschnitts wird basierend auf aus der Literatur bekannten robusten numerischen
Algorithmen ein SchÀtzer zur genauen Bestimmung der Ausbreitungspfadparameter
abgeleitet. Im letzten Teil dieses Abschnitts wird die Anwendung
unterraumbasierter SchÀtzer zur Bestimmung der Ausbreitungspfadparameter
diskutiert. Es wird ein speichereffizienter Algorithmus zur SignalraumschÀtzung
entwickelt. Dieser Algorithmus ist eine Grundvoraussetzung fĂŒr die Anwendung von
mehrdimensionalen ParameterschÀtzern 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 SignalraumschÀtzung sind hier im Allgemeinen nicht anwendbar, da
sie einen zu groĂen Speicheraufwand erfordern. AuĂerdem wird in diesem Teil
gezeigt, dass ESPRIT-Algorithmen auch zur ParameterschÀtzung von Daten mit so
genannter versteckter Rotations-Invarianzstruktur eingesetzt werden können. Als
Beispiel wird ein ESPRIT-basierter Algorithmus zur RichtungsschÀtzung in
Verbindung mit multibeam-Antennenarrays (CUBA) abgeleitet.
Im letzten Teil dieser Arbeit wird ein Maximum-Likelihood-SchĂ€tzer fĂŒr die neue
Komponente des Funkkanals, welche die verteilten Streuungen beschreibt,
entworfen. Ausgehend vom Konzept des iterativen Maximum-Likelihood-SchÀtzers
wird ein Algorithmus entwickelt, der hinreichend geringe numerische KomplexitÀt
besitzt, so dass er praktisch anwendbar ist. In erster Linie wird dabei von der
Toeplitzstruktur der zu schÀtzenden Kovarianzmatrix Gebrauch gemacht. Aufbauend
auf dem SchĂ€tzer fĂŒr die Parameter der Ausbreitungspfade und dem SchĂ€tzer fĂŒr
die Parameter der verteilten Streuungen wird ein Maximum-Likelihood-SchÀtzer
entwickelt (RIMAX), der alle Parameter des in Teil I entwickelten Modells der
Funkanalmessung im Verbund schÀtzt. Neben den geschÀtzten Parametern des
Datenmodells liefert der SchÀtzer zusÀtzlich ZuverlÀssigkeitsinformationen.
Diese werden unter anderem zur Bestimmung der Modellordnung, das heiĂt zur
Bestimmung der Anzahl der dominanten Ausbreitungspfade, herangezogen. AuĂerdem
stellen die ZuverlÀssigkeitsinformationen aber auch ein wichtiges SchÀtzergebnis
dar. Die ZuverlÀssigkeitsinformationen 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
Information Geometric Approach to Bayesian Lower Error Bounds
Information geometry describes a framework where probability densities can be
viewed as differential geometry structures. This approach has shown that the
geometry in the space of probability distributions that are parameterized by
their covariance matrix is linked to the fundamentals concepts of estimation
theory. In particular, prior work proposes a Riemannian metric - the distance
between the parameterized probability distributions - that is equivalent to the
Fisher Information Matrix, and helpful in obtaining the deterministic
Cram\'{e}r-Rao lower bound (CRLB). Recent work in this framework has led to
establishing links with several practical applications. However, classical CRLB
is useful only for unbiased estimators and inaccurately predicts the mean
square error in low signal-to-noise (SNR) scenarios. In this paper, we propose
a general Riemannian metric that, at once, is used to obtain both Bayesian CRLB
and deterministic CRLB along with their vector parameter extensions. We also
extend our results to the Barankin bound, thereby enhancing their applicability
to low SNR situations.Comment: 5 page
Bounds for maximum likelihood regular and non-regular DoA estimation in K-distributed noise
We consider the problem of estimating the direction of arrival of a signal embedded in -distributed noise, when secondary data which contains noise only are assumed to be available. Based upon a recent formula of the Fisher information matrix (FIM) for complex elliptically distributed data, we provide a simple expression of the FIM with the two data sets framework. In the specific case of -distributed noise, we show that, under certain conditions, the FIM for the deterministic part of the model can be unbounded, while the FIM for the covariance part of the model is always bounded. In the general case of elliptical distributions, we provide a sufficient condition for unboundedness of the FIM. Accurate approximations of the FIM for -distributed noise are also derived when it is bounded. Additionally, the maximum likelihood estimator of the signal DoA and an approximated version are derived, assuming known covariance matrix: the latter is then estimated from secondary data using a conventional regularization technique. When the FIM is unbounded, an analysis of the estimators reveals a rate of convergence much faster than the usual . Simulations illustrate the different behaviors of the estimators, depending on the FIM being bounded or not
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