185 research outputs found
Maximum Likelihood-based Gridless DoA Estimation Using Structured Covariance Matrix Recovery and SBL with Grid Refinement
We consider the parametric data model employed in applications such as line
spectral estimation and direction-of-arrival estimation. We focus on the
stochastic maximum likelihood estimation (MLE) framework and offer approaches
to estimate the parameter of interest in a gridless manner, overcoming the
model complexities of the past. This progress is enabled by the modern trend of
reparameterization of the objective and exploiting the sparse Bayesian learning
(SBL) approach. The latter is shown to be a correlation-aware method, and for
the underlying problem it is identified as a grid-based technique for
recovering a structured covariance matrix of the measurements. For the case
when the structured matrix is expressible as a sampled Toeplitz matrix, such as
when measurements are sampled in time or space at regular intervals, additional
constraints and reparameterization of the SBL objective leads to the proposed
structured matrix recovery technique based on MLE. The proposed optimization
problem is non-convex, and we propose a majorization-minimization based
iterative procedure to estimate the structured matrix; each iteration solves a
semidefinite program. We recover the parameter of interest in a gridless manner
by appealing to the Caratheodory-Fejer result on decomposition of PSD Toeplitz
matrices. For the general case of irregularly spaced time or spatial samples,
we propose an iterative SBL procedure that refines grid points to increase
resolution near potential source locations, while maintaining a low per
iteration complexity. We provide numerical results to evaluate and compare the
performance of the proposed techniques with other gridless techniques, and the
CRB. The proposed correlation-aware approach is more robust to
environmental/system effects such as low number of snapshots, correlated
sources, small separation between source locations and improves sources
identifiability.Comment: Submitted to the IEEE Transactions on Signal Processing (Previous
submission date: 29-Oct-2021
Quantum metrology with full and fast quantum control
We establish general limits on how precise a parameter, e.g. frequency or the
strength of a magnetic field, can be estimated with the aid of full and fast
quantum control. We consider uncorrelated noisy evolutions of N qubits and show
that fast control allows to fully restore the Heisenberg scaling (~1/N^2) for
all rank-one Pauli noise except dephasing. For all other types of noise the
asymptotic quantum enhancement is unavoidably limited to a constant-factor
improvement over the standard quantum limit (~1/N) even when allowing for the
full power of fast control. The latter holds both in the single-shot and
infinitely-many repetitions scenarios. However, even in this case allowing for
fast quantum control helps to increase the improvement factor. Furthermore, for
frequency estimation with finite resource we show how a parallel scheme
utilizing any fixed number of entangled qubits but no fast quantum control can
be outperformed by a simple, easily implementable, sequential scheme which only
requires entanglement between one sensing and one auxiliary qubit.Comment: 17 pages, 7 figures, 6 appendice
Cramér-Rao bound for a mixture of real- and integer-valued parameter vectors and its application to the linear regression model
Performance lower bounds are known to be a fundamental design tool in parametric estimation theory. A plethora of deterministic bounds exist in the literature, ranging from the general Barankin bound to the well-known Cramér-Rao bound (CRB), the latter providing the optimal mean square error performance of locally unbiased estimators. In this contribution, we are interested in the estimation of mixed real- and integer-valued parameter vectors. We propose a closed-form lower bound expression leveraging on the general CRB formulation, being the limiting form of the McAulay-Seidman bound. Such formulation is the key point to take into account integer-valued parameters. As a particular case of the general form, we provide closed-form expressions for the Gaussian observation model. One noteworthy point is the as- sessment of the asymptotic efficiency of the maximum likelihood estimator for a linear regression model with mixed parameter vectors and known noise covariance matrix, thus complementing the rather rich literature on that topic. A representative carrier-phase based precise positioning example is provided to support the discussion and show the usefulness of the proposed lower bound
Exploiting Sparse Structures in Source Localization and Tracking
This thesis deals with the modeling of structured signals under different sparsity constraints. Many phenomena exhibit an inherent structure that may be exploited when setting up models, examples include audio waves, radar, sonar, and image objects. These structures allow us to model, identify, and classify the processes, enabling parameter estimation for, e.g., identification, localisation, and tracking.In this work, such structures are exploited, with the goal to achieve efficient localisation and tracking of a structured source signal. Specifically, two scenarios are considered. In papers A and B, the aim is to find a sparse subset of a structured signal such that the signal parameters and source locations maybe estimated in an optimal way. For the sparse subset selection, a combinatorial optimization problem is approximately solved by means of convex relaxation, with the results of allowing for different types of a priori information to be incorporated in the optimization. In paper C, a sparse subset of data is provided, and a generative model is used to find the location of an unknown number of jammers in a wireless network, with the jammers’ movement in the network being tracked as additional observations become available
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 fĂĽr den gemessenen Funkkanal,
welches für die hochauflösende Parameterschätzung geeignet ist. Der zweite
Schwerpunkt dieser Arbeit ist die Entwicklung eines robusten Parameterschätzers
fĂĽr 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)-
Ăśbertragungssystemen, 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 eingefĂĽhrt, welche den Beitrag verteilter (diffuser)
Streuungen zur FunkĂĽbertragung 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 fĂĽr 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 GauĂźschen 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 AusdrĂĽcke 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 fĂĽr 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 groĂźen Speicheraufwand erfordern. AuĂźerdem 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 heiĂźt zur
Bestimmung der Anzahl der dominanten Ausbreitungspfade, herangezogen. AuĂźerdem
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
New concepts in quantum-metrology: From coherent averaging to Hamiltonian extensions
This thesis is dedicated to the understanding of the metrology of quantum systems by using
the tools of quantum parameter estimation, in particular the quantum Fisher information (QFI).
Our first project deals with a specific protocol of quantum enhanced measurement known
as coherent averaging [Braun and Martin, 2011]. This protocol is based on a star topology, with
one central object, the so-called quantum bus, connected to N extra subsystems, called probes.
For the estimation of a parameter characteristic of the interaction between the quantum bus
and the probes, coherent averaging leads to a Heisenberg limited (HL) scaling for the QFI (QFI
proportional to N 2 ). Importantly this HL scaling can be obtained while starting with a separable
state. This provides an advantage as generally one needs to use entangled states to achieve
this scaling. Another important aspect in coherent averaging is the possibility to obtain the HL
scaling by performing a measurement on the quantum bus only. These results were obtained using
perturbation theory in the regime of weak interactions.
In this thesis we go one step further in the study of the coherent averaging protocol. We extend
the formalism of perturbation theory to encompass the possibility of estimating any parameter, in
the regimes of strong and weak interactions. To illustrate the validity of our results, we introduce
two models as examples for a coherent averaging scheme. In these models both the quantum bus
and all the probes are qubits. In the ZZXX model, the free Hamiltonians do not commute with
the interaction Hamiltonians and we have to rely on numerics to find non-perturbative solutions
.In the ZZZZ model the free evolution Hamiltonians commute with the interaction Hamiltonians
and we can find the exact solution analytically.
Perturbation theory shows that in the strong interaction regime and starting with a separable
state, we can estimate the parameter of the free evolution of the probes with a HL scaling if the
free Hamiltonians do not commute with the interaction Hamiltonians. This is confirmed by the
non-perturbative numerical results for the ZZXX model. In the weak interaction regime we only
obtain a standard quantum limit (SQL) scaling for the parameter of the free evolution of the
probes (QFI proportional to N ). When one has only access to the quantum bus, we show that the
HL scaling found using the perturbation theory does not necessarily survive outside the regime
of validity of the perturbation. This is especially the case as N becomes large. It is shown by
comparing the exact analytical result to the perturbative result with the ZZZZ model. The same
behaviour is observed with the ZZXX model using the non-perturbative numerical results.
In our second project we investigate the estimation of the depolarizing channel and the
phase-flip channel under non-ideal conditions. It is known that using an ancilla can lead to an
improvement of the channel QFI (QFI maximized over input states feeding the channel) even
if we act with the identity on the ancilla. This method is known as channel extension. In all
generality the maximal channel QFI can be obtained using an ancilla whose Hilbert space has the
same dimension as the dimension of the Hilbert space of the original system. In this ideal scenario
using multiple ancillas — or one ancilla with a larger Hilbert space dimension — is useless.
To go beyond this ideal result we take into account the possibility of loosing either the probe
or a finite number of ancillas. The input states considered are GHZ and W states with n + 1
qubits (the probe plus n ancillas). We show that for any channel, when the probe is lost then
all the information is lost, and the use of ancillas cannot help. For the phase-flip channel the
introduction of ancillas never improves the channel QFI and ancillas are useless.
For the depolarizing channel the maximal channel QFI can be reached using one ancilla and
feeding the extended channel with a Bell state, but if the ancilla is lost then all the advantage
is lost. We show that the GHZ states do not help to fight the loss of ancillas: If one ancilla or
more are lost all the advantage provided by the use of ancillas is lost. More interestingly, we show
that the W states with more than one ancilla are robust against loss. For a given number of lost
ancillas, there always exists an initial number of ancillas for which a W state provides a higher
QFI than the one obtained without ancillas.
Our last project is about Hamiltonian parameter estimation for arbitrary Hamiltonians.
It is known that channel extension does not help for unitary channels. Instead we apply the
idea of extension to the Hamiltonian itself and not to the channel. This is done by adding
to the Hamiltonian an extra term, which is independent of the parameter and which possibly
encompasses interactions with an ancilla. We call this technique Hamiltonian extension. We show
that for arbitrary Hamiltonians there exists an upper bound to the channel QFI that is in general
not saturated. This result is known in the context of non-linear metrology. Here we show explicitly
the conditions to saturate the bound.
We provide two methods for Hamiltonian extensions, called signal flooding and Hamiltonian
subtraction, that allow one to saturate the upper bound for any Hamiltonian. We also introduce a
third method which does not saturate the upper bound but provides the possibility to restore the
quadratic time scaling in the channel QFI when the original Hamiltonian leads only to a periodic
time scaling of the channel QFI.
We finally show how these methods work using two different examples. We study the estimation
of the strength of a magnetic field using a NV center, and show how using signal flooding we
saturate the channel QFI. We also consider the estimation of a direction of a magnetic field using
a spin-1. We show how using signal flooding or Hamiltonian subtraction we saturate the channel
QFI. We also show how by adding an arbitrary magnetic field we restore the quadratic time
scaling in the channel QFI. Eventually we explain how coherent averaging can be scrutinized in
the formalism of Hamiltonian extensions
Self-Localization of Ad-Hoc Arrays Using Time Difference of Arrivals
This work was supported by the U.K. Engineering and Physical Sciences Research Council (EPSRC) under Grant EP/K007491/1
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