267 research outputs found
Optimized Quantification of Spin Relaxation Times in the Hybrid State
Purpose: The analysis of optimized spin ensemble trajectories for relaxometry
in the hybrid state.
Methods: First, we constructed visual representations to elucidate the
differential equation that governs spin dynamics in hybrid state. Subsequently,
numerical optimizations were performed to find spin ensemble trajectories that
minimize the Cram\'er-Rao bound for -encoding, -encoding, and their
weighted sum, respectively, followed by a comparison of the Cram\'er-Rao bounds
obtained with our optimized spin-trajectories, as well as Look-Locker and
multi-spin-echo methods. Finally, we experimentally tested our optimized spin
trajectories with in vivo scans of the human brain.
Results: After a nonrecurring inversion segment on the southern hemisphere of
the Bloch sphere, all optimized spin trajectories pursue repetitive loops on
the northern half of the sphere in which the beginning of the first and the end
of the last loop deviate from the others. The numerical results obtained in
this work align well with intuitive insights gleaned directly from the
governing equation. Our results suggest that hybrid-state sequences outperform
traditional methods. Moreover, hybrid-state sequences that balance - and
-encoding still result in near optimal signal-to-noise efficiency. Thus,
the second parameter can be encoded at virtually no extra cost.
Conclusion: We provide insights regarding the optimal encoding processes of
spin relaxation times in order to guide the design of robust and efficient
pulse sequences. We find that joint acquisitions of and in the
hybrid state are substantially more efficient than sequential encoding
techniques.Comment: 10 pages, 5 figure
Matched direction detectors and estimators for array processing with subspace steering vector uncertainties
In this paper, we consider the problem of estimating and detecting a signal whose associated spatial signature is known to lie in a given linear subspace but whose coordinates in this subspace are otherwise unknown, in the presence of subspace interference and broad-band noise. This situation arises when, on one hand, there exist uncertainties about the steering vector but, on the other hand, some knowledge about the steering vector errors is available. First, we derive the maximum-likelihood estimator (MLE) for the problem and compute the corresponding Cramer-Rao bound. Next, the maximum-likelihood estimates are used to derive a generalized likelihood ratio test (GLRT). The GLRT is compared and contrasted with the standard matched subspace detectors. The performances of the estimators and detectors are illustrated by means of numerical simulations
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
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
A Fresh Look at the Bayesian Bounds of the Weiss-Weinstein Family
International audienceMinimal bounds on the mean square error (MSE) are generally used in order to predict the best achievable performance of an estimator for a given observation model. In this paper, we are interested in the Bayesian bound of the WeissâWeinstein family. Among this family, we have Bayesian CramĂ©r-Rao bound, the BobrovskyâMayerWolfâZakaĂŻ bound, the Bayesian Bhattacharyya bound, the BobrovskyâZakaĂŻ bound, the ReuvenâMesser bound, and the WeissâWeinstein bound. We present a unification of all these minimal bounds based on a rewriting of the minimum mean square error estimator (MMSEE) and on a constrained optimization problem. With this approach, we obtain a useful theoretical framework to derive new Bayesian bounds. For that purpose, we propose two bounds. First, we propose a generalization of the Bayesian Bhattacharyya bound extending the works of Bobrovsky, MayerâWolf, and ZakaĂŻ. Second, we propose a bound based on the Bayesian Bhattacharyya bound and on the ReuvenâMesser bound, representing a generalization of these bounds. The proposed bound is the Bayesian extension of the deterministic Abel bound and is found to be tighter than the Bayesian Bhattacharyya bound, the ReuvenâMesser bound, the BobrovskyâZakaĂŻ bound, and the Bayesian CramĂ©râRao bound. We propose some closed-form expressions of these bounds for a general Gaussian observation model with parameterized mean. In order to illustrate our results, we present simulation results in the context of a spectral analysis problem
Intrinsic Sensitivity Limits for Multiparameter Quantum Metrology
The quantum Cramér-Rao bound is a cornerstone of modern quantum metrology, as it provides the ultimate precision in parameter estimation. In the multiparameter scenario, this bound becomes a matrix inequality, which can be cast to a scalar form with a properly chosen weight matrix. Multiparameter estimation thus elicits tradeoffs in the precision with which each parameter can be estimated. We show that, if the information is encoded in a unitary transformation, we can naturally choose the weight matrix as the metric tensor linked to the geometry of the underlying algebra su(n). This ensures an intrinsic bound that is independent of the choice of parametrization
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
Cram\'er-Rao bound-informed training of neural networks for quantitative MRI
Neural networks are increasingly used to estimate parameters in quantitative
MRI, in particular in magnetic resonance fingerprinting. Their advantages over
the gold standard non-linear least square fitting are their superior speed and
their immunity to the non-convexity of many fitting problems. We find, however,
that in heterogeneous parameter spaces, i.e. in spaces in which the variance of
the estimated parameters varies considerably, good performance is hard to
achieve and requires arduous tweaking of the loss function, hyper parameters,
and the distribution of the training data in parameter space. Here, we address
these issues with a theoretically well-founded loss function: the Cram\'er-Rao
bound (CRB) provides a theoretical lower bound for the variance of an unbiased
estimator and we propose to normalize the squared error with respective CRB.
With this normalization, we balance the contributions of hard-to-estimate and
not-so-hard-to-estimate parameters and areas in parameter space, and avoid a
dominance of the former in the overall training loss. Further, the CRB-based
loss function equals one for a maximally-efficient unbiased estimator, which we
consider the ideal estimator. Hence, the proposed CRB-based loss function
provides an absolute evaluation metric. We compare a network trained with the
CRB-based loss with a network trained with the commonly used means squared
error loss and demonstrate the advantages of the former in numerical, phantom,
and in vivo experiments.Comment: Xiaoxia Zhang, Quentin Duchemin, and Kangning Liu contributed equally
to this wor
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