Locally optimal invariant detector for testing equality of two power spectral densities

This work addresses the problem of determining whether two multivariate random time series have the same power spectral density (PSD), which has applications, for instance, in physical-layer security and cognitive radio. Remarkably, existing detectors for this problem do not usually provide any kind of optimality. Thus, we study here the existence under the Gaussian assumption of optimal invariant detectors for this problem, proving that the uniformly most powerful invariant test (UMPIT) does not exist. Thus, focusing on close hypotheses, we show that the locally most powerful invariant test (LMPIT) only exists for univariate time series. In the multivariate case, we prove that the LMPIT does not exist. However, this proof suggests two LMPIT-inspired detectors, one of which outperforms previously proposed approaches, as computer simulations show.This work was partly supported by the Spanish MINECO grants OTOSIS (TEC2013-41718-R), COMONSENS Network (TEC2015-69648-REDC) and KERMES Network (TEC2016-81900-REDT/AEI); by the Spanish MINECO and the European Commission (ERDF) grants ADVENTURE (TEC2015-69868-C2-1-R), WINTER (TEC2016-76409-C2-2-R), CARMEN (TEC2016-75067-C4-4-R) and CAIMAN (TEC2017-86921-C2-1-R and TEC2017-86921-C2-2-R); by the Comunidad de Madrid grant CASI-CAM-CM (S2013/ICE-2845); by the Xunta de Galicia and ERDF grants GRC2013/009, R2014/037 and ED431G/04 (Agrupación Estratéxica Consolidada de Galicia accreditation 2016-2019); by the SODERCAN and ERDF grant CAIMAN (12.JU01.64661); and by the Research Council of Norway grant FRIPRO TOPPFORSK (250910/F20)

Testing equality of multiple power spectral density matrices

This paper studies the existence of optimal invariant detectors for determining whether P multivariate processes have the same power spectral density. This problem finds application in multiple fields, including physical layer security and cognitive radio. For Gaussian observations, we prove that the optimal invariant detector, i.e., the uniformly most powerful invariant test, does not exist. Additionally, we consider the challenging case of close hypotheses, where we study the existence of the locally most powerful invariant test (LMPIT). The LMPIT is obtained in the closed form only for univariate signals. In the multivariate case, it is shown that the LMPIT does not exist. However, the corresponding proof naturally suggests an LMPIT-inspired detector that outperforms previously proposed detectors.Comunidad de Madrid | Ref. CASI-CAM-CM (S2013/ICE-2845)Research Council of Norway | Ref. FRIPRO TOPPFORSK (250910/F20)Ministerio de Economía y Competitividad | Ref. TEC2015-69648-REDCMinisterio de Economía y Competitividad | Ref. TEC2016-81900-REDT/AEIMinisterio de Economía y Competitividad | Ref. TEC2015-69868-C2-1-RAgencia Estatal de Investigación | Ref. TEC2016-76409-C2-2-RAgencia Estatal de Investigación | Ref. TEC2016-75067-C4-4-RAgencia Estatal de Investigación | Ref. TEC2017-86921-C2-1-RAgencia Estatal de Investigación | Ref. TEC2017-86921-C2-2-RXunta de Galicia | Ref. GRC2013/009Xunta de Galicia | Ref. GRC2013/009Xunta de Galicia | Ref. ED431G/0

Testing equality of multiple power spectral density matrices

This paper studies the existence of optimal invariant detectors for determining whether P multivariate processes have the same power spectral density. This problem finds application in multiple fields, including physical layer security and cognitive radio. For Gaussian observations, we prove that the optimal invariant detector, i.e., the uniformly most powerful invariant test, does not exist. Additionally, we consider the challenging case of close hypotheses, where we study the existence of the locally most powerful invariant test (LMPIT). The LMPIT is obtained in the closed form only for univariate signals. In the multivariate case, it is shown that the LMPIT does not exist. However, the corresponding proof naturally suggests an LMPIT-inspired detector that outperforms previously proposed detectors.This work was partly supported by the Spanish MINECO grants COMONSENS Network (TEC2015-69648-REDC) and KERMES Network (TEC2016-81900-REDT/AEI); by the Spanish MINECO and the European Commission (ERDF) grants ADVENTURE (TEC2015-69868-C2-1- R), WINTER (TEC2016-76409-C2-2-R), CARMEN (TEC2016-75067-C4-4- R) and CAIMAN (TEC2017-86921-C2-1-R and TEC2017-86921-C2-2-R); by the Comunidad de Madrid grant CASI-CAM-CM (S2013/ICE-2845); by the Xunta de Galicia and ERDF grants GRC2013/009, R2014/037 and ED431G/04 (Agrupacion Estratexica Consolidada de Galicia accred- ´ itation 2016-2019); by the SODERCAN and ERDF grant CAIMAN (12.JU01.64661); and by the Research Council of Norway grant FRIPRO TOPPFORSK (250910/F20). This paper was presented in part at the 2018 IEEE International Conference on Acoustics, Speech and Signal Processing

A Low-Cost Robust Distributed Linearly Constrained Beamformer for Wireless Acoustic Sensor Networks with Arbitrary Topology

We propose a new robust distributed linearly constrained beamformer which utilizes a set of linear equality constraints to reduce the cross power spectral density matrix to a block-diagonal form. The proposed beamformer has a convenient objective function for use in arbitrary distributed network topologies while having identical performance to a centralized implementation. Moreover, the new optimization problem is robust to relative acoustic transfer function (RATF) estimation errors and to target activity detection (TAD) errors. Two variants of the proposed beamformer are presented and evaluated in the context of multi-microphone speech enhancement in a wireless acoustic sensor network, and are compared with other state-of-the-art distributed beamformers in terms of communication costs and robustness to RATF estimation errors and TAD errors

Quantum Measurement Theory in Gravitational-Wave Detectors

The fast progress in improving the sensitivity of the gravitational-wave (GW) detectors, we all have witnessed in the recent years, has propelled the scientific community to the point, when quantum behaviour of such immense measurement devices as kilometer-long interferometers starts to matter. The time, when their sensitivity will be mainly limited by the quantum noise of light is round the corner, and finding the ways to reduce it will become a necessity. Therefore, the primary goal we pursued in this review was to familiarize a broad spectrum of readers with the theory of quantum measurements in the very form it finds application in the area of gravitational-wave detection. We focus on how quantum noise arises in gravitational-wave interferometers and what limitations it imposes on the achievable sensitivity. We start from the very basic concepts and gradually advance to the general linear quantum measurement theory and its application to the calculation of quantum noise in the contemporary and planned interferometric detectors of gravitational radiation of the first and second generation. Special attention is paid to the concept of Standard Quantum Limit and the methods of its surmounting.Comment: 147 pages, 46 figures, 1 table. Published in Living Reviews in Relativit

Measurement uncertainty relations

Measurement uncertainty relations are quantitative bounds on the errors in an approximate joint measurement of two observables. They can be seen as a generalization of the error/disturbance tradeoff first discussed heuristically by Heisenberg. Here we prove such relations for the case of two canonically conjugate observables like position and momentum, and establish a close connection with the more familiar preparation uncertainty relations constraining the sharpness of the distributions of the two observables in the same state. Both sets of relations are generalized to means of order $\alpha$ rather than the usual quadratic means, and we show that the optimal constants are the same for preparation and for measurement uncertainty. The constants are determined numerically and compared with some bounds in the literature. In both cases the near-saturation of the inequalities entails that the state (resp. observable) is uniformly close to a minimizing one.Comment: This version 2 contains minor corrections and reformulation

Detecting a stochastic background of gravitational waves with non-standard polarizations

In this thesis work I consider the detection of a stochastic background of gravitational waves (SGWB) produced in the context of a generic theory of gravity. I will show in the second chapter that most theories admit solutions in terms of gravitational waves(GWs); these may differ in their propagation speed or, most relevant for the present work, in their polarization modes. For example, it is well-known that many theories of gravity, obtained for example as low-energy limits of string theories, predict a propagating scalar mode of polarization in addition to the usual tensor ones of General Relativity (GR). I will present a thorough discussion of the classification of GWs polarizations according to the E(2) scheme, which comprises the analysis of the non-vanishing components of the Riemann tensor as measured by a locally inertial observer, and thereafter the interaction of GWs with a detector. In the third chapter I present a comprehensive characterization of an SGWB with non-standard polarizations in terms of the detector response to it. Some considerations are made on the most general form that the corresponding signal may have according to alternative theories of gravity and the production mechanisms described before. It follows then the discussion about some “first order approximations” that will be useful for its study in these preliminary phases. The aim of the present work is to relax some of the usual constraints adopted in standard literature to include also the possibility of non-standard polarizations, and open up to a new more general class of possible SGWBs. In the second part of this chapter I construct and study an optimal detection algorithm for a generic SGWB. In particular, I will give importance to a procedure that is as less dependent as possible on the details of the model; for example, I begin without introducing any assumptions about the shape of the power spectrum densities of the stochastic signal. Only later they will be considered cases where it becomes necessary to include further assumptions, for example, in order to obtain some estimates on the parameters characterizing a certain model. This choice is motivated by the desire of understanding how much sensitivity is lost when a not well defined model is available, which is even more true when we extend the framework to include also alternative theories. In this sense, the present work is meant as an upgrade to those already present in literature and commonly adopted in the standard data analysis for the research of an SGWB. I will recover the known results from the literature adding only later some further assumptions. This treatment has some advantages over the standard one, in particular from a theoretical point of view. Finally, in Chapter 4, I make use of the proposed algorithm to study real data from Virgo and LIGO. The current upper limit on the intensity of the (standard) SGWB, published in 2009, is reconsidered. As it was reasonable to expect, it is not possible to improve this limit or, even more so, to perform a detection of a non-standard SGWB. Anyway, the upper limits on the non-standard polarization modes are computed and compared with the standard one. Also, several related quantities are computed and analysed from the point of view of the detection. The important news comes from the study of the predicted sensitivities that will be achieved by the advanced detectors with the scheduled upgrades (2015-2021). I will show that these sensitivities will become good enough to test several mechanisms of production of an SGWB, both of cosmological and of astrophysical origin, or at least to determine further upper limits on them. Therefore, we can expect that the tools provided by the study of GWs within an SGWB will become worth for testing alternative theories of gravity, as well as early Universe cosmological models and astrophysical ones

Detecting a stochastic background of gravitational waves with non-standard polarizations

In this thesis work I consider the detection of a stochastic background of gravitational waves (SGWB) produced in the context of a generic theory of gravity. I will show in the second chapter that most theories admit solutions in terms of gravitational waves(GWs); these may differ in their propagation speed or, most relevant for the present work, in their polarization modes. For example, it is well-known that many theories of gravity, obtained for example as low-energy limits of string theories, predict a propagating scalar mode of polarization in addition to the usual tensor ones of General Relativity (GR). I will present a thorough discussion of the classification of GWs polarizations according to the E(2) scheme, which comprises the analysis of the non-vanishing components of the Riemann tensor as measured by a locally inertial observer, and thereafter the interaction of GWs with a detector. In the third chapter I present a comprehensive characterization of an SGWB with non-standard polarizations in terms of the detector response to it. Some considerations are made on the most general form that the corresponding signal may have according to alternative theories of gravity and the production mechanisms described before. It follows then the discussion about some “first order approximations” that will be useful for its study in these preliminary phases. The aim of the present work is to relax some of the usual constraints adopted in standard literature to include also the possibility of non-standard polarizations, and open up to a new more general class of possible SGWBs. In the second part of this chapter I construct and study an optimal detection algorithm for a generic SGWB. In particular, I will give importance to a procedure that is as less dependent as possible on the details of the model; for example, I begin without introducing any assumptions about the shape of the power spectrum densities of the stochastic signal. Only later they will be considered cases where it becomes necessary to include further assumptions, for example, in order to obtain some estimates on the parameters characterizing a certain model. This choice is motivated by the desire of understanding how much sensitivity is lost when a not well defined model is available, which is even more true when we extend the framework to include also alternative theories. In this sense, the present work is meant as an upgrade to those already present in literature and commonly adopted in the standard data analysis for the research of an SGWB. I will recover the known results from the literature adding only later some further assumptions. This treatment has some advantages over the standard one, in particular from a theoretical point of view. Finally, in Chapter 4, I make use of the proposed algorithm to study real data from Virgo and LIGO. The current upper limit on the intensity of the (standard) SGWB, published in 2009, is reconsidered. As it was reasonable to expect, it is not possible to improve this limit or, even more so, to perform a detection of a non-standard SGWB. Anyway, the upper limits on the non-standard polarization modes are computed and compared with the standard one. Also, several related quantities are computed and analysed from the point of view of the detection. The important news comes from the study of the predicted sensitivities that will be achieved by the advanced detectors with the scheduled upgrades (2015-2021). I will show that these sensitivities will become good enough to test several mechanisms of production of an SGWB, both of cosmological and of astrophysical origin, or at least to determine further upper limits on them. Therefore, we can expect that the tools provided by the study of GWs within an SGWB will become worth for testing alternative theories of gravity, as well as early Universe cosmological models and astrophysical ones

COrE (Cosmic Origins Explorer) A White Paper

COrE (Cosmic Origins Explorer) is a fourth-generation full-sky, microwave-band satellite recently proposed to ESA within Cosmic Vision 2015-2025. COrE will provide maps of the microwave sky in polarization and temperature in 15 frequency bands, ranging from 45 GHz to 795 GHz, with an angular resolution ranging from 23 arcmin (45 GHz) and 1.3 arcmin (795 GHz) and sensitivities roughly 10 to 30 times better than PLANCK (depending on the frequency channel). The COrE mission will lead to breakthrough science in a wide range of areas, ranging from primordial cosmology to galactic and extragalactic science. COrE is designed to detect the primordial gravitational waves generated during the epoch of cosmic inflation at more than $3\sigma$ for $r=(T/S)>=10^{-3}$. It will also measure the CMB gravitational lensing deflection power spectrum to the cosmic variance limit on all linear scales, allowing us to probe absolute neutrino masses better than laboratory experiments and down to plausible values suggested by the neutrino oscillation data. COrE will also search for primordial non-Gaussianity with significant improvements over Planck in its ability to constrain the shape (and amplitude) of non-Gaussianity. In the areas of galactic and extragalactic science, in its highest frequency channels COrE will provide maps of the galactic polarized dust emission allowing us to map the galactic magnetic field in areas of diffuse emission not otherwise accessible to probe the initial conditions for star formation. COrE will also map the galactic synchrotron emission thirty times better than PLANCK. This White Paper reviews the COrE science program, our simulations on foreground subtraction, and the proposed instrumental configuration.Comment: 90 pages Latex 15 figures (revised 28 April 2011, references added, minor errors corrected