The stochastic gravitational wave background (SGWB) can be observed in the nanohertz band using a pulsar timing array (PTA). Here, a computationally efficient state-space framework is developed for analysing SGWB data, in which the stochastic gravitational wave strain at Earth is tracked with a non-linear Kalman filter and separated simultaneously from intrinsic achromatic pulsar spin wandering. The filter is combined with a nested sampler to estimate the parameters of the model and to calculate a Bayes factor for selecting between models with and without a SGWB. The procedure extends previous state-space formulations of PTA data analysis applied to individually resolvable binary black hole sources. The performance of the new algorithm is tested on synthetic data from the first International PTA Mock Data Challenge. It is shown that the algorithm distinguishes a SGWB from pure noise for A(gw) >= 3x10(-14), where A(gw) denotes the standard normalization factor for a power spectral density with power-law exponent -13/3. Additional, systematic validation tests are also performed with synthetic data generated independently by adjusting the injected parameters to cover astrophysically plausible ranges. Full posterior distributions are recovered and tested for accuracy. The state-space procedure is memory-light and evaluates the likelihood for a standard-sized PTA data set in less than or similar to 10(-1) s without optimization on a standard central processing unit.

Continuous gravitational wave searches with terrestrial, long-baseline interferometers are hampered by long-lived, narrow band features in the power spectral density of the detector noise, known as lines. Candidate gravitational wave signals which overlap spectrally with known lines are typically vetoed. Here we demonstrate a line subtraction method based on adaptive noise cancellation, using a recursive least squares algorithm, a common approach in electrical engineering applications such as audio and biomedical signal processing. We validate the line subtraction method by combining it with a hidden Markov model (HMM), a standard continuous wave search tool, to detect an injected continuous wave signal with an unknown and randomly wandering frequency, which overlaps with the mains power line at 60 Hz in the Laser Interferometer Gravitational Wave Observatory. The performance of the line subtraction method is tested on a injected continuous wave signal obscured by (i) synthetic noise data with both Gaussian and non-Gaussian components, and (ii) real noise data obtained from the Laser Interferometer Gravitational Wave Observatory Livingston detector. In both cases, before applying the line subtraction method the HMM does not detect the injected continuous wave signal. After applying the line subtraction method the mains power line is suppressed by 20-40 dB, and the HMM detects the underlying signal, with a time- averaged root-mean-square error in the frequency estimate of similar to 0.05 Hz. The performance of the line subtraction method with respect to the characteristics of the 60 Hz line and the control parameters of the recursive least squares algorithm is quantified in terms of receiver operating characteristic curves.

Continuous nanohertz gravitational waves from individual supermassive black hole binaries may be detectable with pulsar timing arrays. A novel search strategy is developed, wherein intrinsic achromatic spin wandering is tracked simultaneously with the modulation induced by a single gravitational wave source in the pulse times of arrival. A two-step inference procedure is applied within a state-space framework, such that the modulation is tracked with a Kalman filter, which then provides a likelihood for nested sampling. The procedure estimates the static parameters in the problem, such as the sky position of the source, without fitting for ensemble-averaged statistics such as the power spectral density of the timing noise, and therefore complements traditional parameter estimation methods. It also returns the Bayes factor relating a model with a single gravitational wave source to one without, complementing traditional detection methods. It is shown via astrophysically representative software injections in Gaussian measurement noise that the procedure distinguishes a gravitational wave from pure noise down to a characteristic wave strain of h(0 )approximate to 2 x 10(-15). Full posterior distributions of model parameters are recovered and tested for accuracy. There is a bias of approximate to 0.3 rad in the marginalized one-dimensional posterior for the orbital inclination i, introduced by dropping the so-called pulsar terms. Smaller biases less than or similar to 10 are also observed in other static parameters.

Pulsar timing arrays (PTA) can detect continuous nanohertz gravitational waves (GW) emitted by individual supermassive black hole binaries. The data analysis procedure can be formulated within a time-domain, state-space framework, in which the radio timing observations are related to a temporal sequence of latent states, namely the intrinsic pulsar spin frequency. The achromatic wandering of the pulsar spin frequency is tracked using a Kalman filter concurrently with the pulse frequency modulation induced by a GW from a single source. The modulation is the sum of terms proportional to the GW strain at the Earth and at every pulsar in the array. Here, we generalize previous state-space formulations of the PTA problem to include the pulsar terms; that is, we copy the pulsar terms from traditional, non-state-space analyses over to the state-space framework. The performance of the generalized Kalman filter is tested using astrophysically representative software injections in Gaussian measurement noise. It is shown that including the pulsar terms corrects for previously identified biases in the parameter estimates (especially the sky position of the source) which also arise in traditional matched-filter analyses that exclude the pulsar terms. Additionally, including the pulsar terms decreases the minimum detectable strain by 14 percent. Overall, the study verifies that the pulsar terms do not raise any special extra impediments for the state-space framework, beyond those studied in traditional analyses. The inspiral-driven evolution of the wave frequency at the Earth and at the retarded time at every pulsar in the array is also investigated.

We consider an instance of a radar jamming countermeasure problem, where the radar and the jammer have uncertainty about the radar environment (for instance, about the noise variance and the radar cross section variance), and they have to account for these uncertainties with statistical priors. The radar-jammer interaction is modelled as a two-player zero-sum repeated Bayesian game (also called an incomplete information game). We discuss the computation of the optimal strategies and present numerical results that illustrate the optimal strategies in an example.

The design of detectors for weak sinusoidal signals with random varying phase and frequency has been studied for many decades and continues to be a topic of significant theoretical and practical interest. For a random process, the likelihood ratio detector is known to have an estimator-correlator structure that requires generation of the causal minimum mean square error (MMSE) estimate of the random signal, which generally is very challenging for non-Gaussian signals. In this paper we present methods to project an infinite dimensional solution to the stochastic differential equations generating the required MMSE estimate onto a finite dimensional space, closely approximating the exact solution. Simulations of our proposed detectors are presented and compared with existing approaches, specifically optimized quadratic detectors and extended Kalman filter based methods.

Detecting random processes on a circle has been studied for many decades. The Neyman-Pearson detector, which evaluates the likelihood ratio, requires first the conditional mean estimate of the circle-valued signal given noisy measurements, which is then correlated with the measurements for detection. This is the estimator-correlator detector. However, generating the conditional mean estimate of the signal is very rarely solvable. In this paper, we propose an approximate estimator-correlator detector by estimating the truncated moments of the signal, with estimated signal substituted into the likelihood ratio. Instead of estimating the random phase, we estimate the complex circle-valued signal directly. The effectiveness of the proposed method in terms of estimation and detection is shown through numerical experiments, where the tracking accuracy and receiver operating curves, compared with the extended Kalman filter are shown under various process/measurement noise.

Filtering of circular data from noisy measurements is well known to be a hard problem because of the ambiguity of the wrapped phase and further complicated by the constraint that filtered states need to be on the circle. Probabilistic models focus on specifying the dynamics of the random phase and then estimate these states recursively using Bayes' formula, while deterministic approaches normally define a cost function containing the error between the states and measurements and then minimize it over all allowed state paths. In this paper, we construct a deterministic filter on the circle by minimizing the least square error based on Pontryagin's minimum principle, where the optimal state trajectory is described by a bilinear differential equation with deterministic optimal control input. The effectiveness of the proposed filter is shown through numerical experiments.

The problem of detecting a sinusoidal signal with randomly varying frequency has a long history. It is one of the core problems in signal processing, arising in many applications including, for example, underwater acoustic frequency line tracking, demodulation of FM radio communications, laser phase drift in optical communications and, recently, continuous gravitational wave astronomy. In this paper we describe a Markov Chain Monte Carlo based procedure to compute a specific detection posterior density. We demonstrate via simulation that our approach results in an up to 25 percent higher detection rate than Hidden Markov Model based solutions, which are generally considered to be the leading techniques for these problems.

Tracking on the rotation group is a key component of many modern systems for estimation and tracking of the orientation of rigid bodies. To address this problem, here, we describe a Bayesian algorithm that relies on directional measurements, acquired, for example, by an inertial measurements unit (IMU), for tracking on the special orthogonal (rotation) group. Its novelty lies in the use of maximum entropy distributions on these groups as models for the priors, and justifiable relaxation algorithms that permit the recursive implementation of such a model to provide a filter. We provide the solutions in a recursive closed form. In the two-dimensional case, the parameters of the prior and posterior distributions can be computed exactly and the solution has low complexity. Adoption of this approach eliminates the problem of angle wrapping. In higher dimensions, the exact solution cannot be computed, and it is necessary to make appropriate relaxations, which is done here. We demonstrate in simulations that, in contrast with some other approaches, our algorithm produces very accurate and statistically meaningful outputs. Pseudocode, specific to the IMU measurements is also provided.

Gravitational waves from coalescing neutron stars encode information about nuclear matter at extreme densities, inaccessible by laboratory experiments. The late inspiral is influenced by the presence of tides, which depend on the neutron star equation of state. Neutron star mergers are expected to often produce rapidly-rotating remnant neutron stars that emit gravitational waves. These will provide clues to the extremely hot post-merger environment. This signature of nuclear matter in gravitational waves contains most information in the 2-4 kHz frequency band, which is outside of the most sensitive band of current detectors. We present the design concept and science case for a neutron star extreme matter observatory (NEMO): a gravitational-wave interferometer optimized to study nuclear physics with merging neutron stars. The concept uses high circulating laser power, quantum squeezing and a detector topology specifically designed to achieve the high-frequency sensitivity necessary to probe nuclear matter using gravitational waves. Above one kHz, the proposed strain sensitivity is comparable to full third-generation detectors at a fraction of the cost. Such sensitivity changes expected event rates for detection of post-merger remnants from approximately one per few decades with two A+ detectors to a few per year, and potentially allows for the first gravitational-wave observations of supernovae, isolated neutron stars, and other exotica.

We investigate the challenging problem of integrating detection, signal processing, target tracking, and adaptive waveform scheduling with lookahead in urban terrain. We propose a closed-loop active sensing system to address this problem by exploiting three distinct levels of diversity: (1) spatial diversity through the use of coordinated multistatic radars; (2) waveform diversity by adaptively scheduling the transmitted waveform; and (3) motion model diversity by using a bank of parallel filters matched to different motion models. Specifically, at every radar scan, the waveform that yields the minimum trace of the one-step-ahead error covariance matrix is transmitted; the received signal goes through a matched-filter, and curve fitting is used to extract range and range-rate measurements that feed the LMIPDA-VSIMM algorithm for data association and filtering. Monte Carlo simulations demonstrate the effectiveness of the proposed system in an urban scenario contaminated by dense and uneven clutter, strong multipath, and limited line-of-sight.

Core-collapse supernovae (CCSNe) are a potential source for ground-based gravitational wave detectors, as their predicted emission peaks in the detectors' frequency band. Typical searches for gravitational wave bursts reconstruct signals using wavelets. However, as CCSN signals contain multiple complex features in the time-frequency domain, these techniques often struggle to reconstruct the entire signal. An alternative method developed in recent years involves applying principal component analysis (PCA) to a set of simulated CCSN models. This technique enables model selection between astrophysical CCSN models as well as waveform reconstruction. However, PCA faces its own difficulties, such as being unable to reconstruct signals longer than the simulations; many CCSN simulations are stopped before the emission peaks due to insufficient computational resources. In this study, we show how combining PCA with dynamic time warping improves the reconstruction of CCSN gravitational wave signals in Gaussian noise characteristic of Advanced LIGO at design sensitivity. For the waveforms used in this analysis, we find that the number of PCs needed to represent 90% of the data is reduced from nine to four by applying dynamic time warping and that the match between the original and reconstructed waveforms improves for signal-to-noise ratios in the range [0, 50].

Work of Pezeshki, Calderbank and others [1, 2] has shown that choice of transmission order of Golay complementary waveforms in radar pulse trains can significantly improve Doppler resilience in detecting nonzero Doppler targets. For multiple targets a Weighted average Doppler (WD) algorithm [3] based on our earlier work has been shown to provide improvements. In this paper, we demonstrate that the Weighted Sidelobe Minimization procedure, developed in our recent work that combines the outputs of WD and the existing Binominal Design algorithms for ordering transmitted complementary waveform sequences using a point-wise minimization process (PMP), is also effective for the elimination of false target returns arising from sea clutter. A detailed analysis based on a Swerling II target model in terms of target detection probability is presented to validate the use of PMP. Our validation is strengthened by numerical simulations for both fixed and randomized scenarios. Crown Copyright (C) 2017 Published by Elsevier B.V. All rights reserved.

Sonobuoy fields, comprising a network of transmitters and receivers, are commonly deployed to find and track underwater targets. For a given environment and sonobuoy field layout, the performance of such a field depends on the scheduling, that is, deciding which source should transmit, and which from a library of available waveforms should be transmitted at any given time. In this paper, we propose a novel scheduling framework based on multi-objective optimization. Specifically, we pose the two tasks of the sonobuoy field-tracking and searching-as separate, competing, objective functions. Using this framework, we propose a characterization of scheduling based on Pareto optimality. This characterization describes the trade-off between the search-track objectives and is demonstrated on realistic multistatic sonobuoy simulations.

Golay complementary waveforms can, in theory, yield radar returns of high range resolution with essentially zero sidelobes. In practice, when deployed conventionally, while high signal-to-noise ratios can be achieved for static target detection, significant range sidelobes are generated by target returns of nonzero Doppler causing unreliable detection. We consider signal processing techniques using Golay complementary waveforms to improve radar detection performance in scenarios involving multiple nonzero Doppler targets. A signal processing procedure based on an existing, so called, Binomial Design algorithm that alters the transmission order of Golay complementary waveforms and weights the returns is proposed in an attempt to achieve an enhanced illumination performance. The procedure applies one of three proposed waveform transmission ordering algorithms, followed by a pointwise nonlinear processor combining the outputs of the Binomial Design algorithm and one of the ordering algorithms. The computational complexity of the Binomial Design algorithm and the three ordering algorithms are compared, and a statistical analysis of the performance of the pointwise nonlinear processing is given. Estimation of the areas in the Delay–Doppler map occupied by significant range sidelobes for given targets are also discussed. Numerical simulations for the comparison of the performances of the Binomial Design algorithm and the three ordering algorithms are presented for both fixed and randomized target locations. The simulation results demonstrate that the proposed signal processing procedure has a better detection performance in terms of lower sidelobes and higher Doppler resolution in the presence of multiple nonzero Doppler targets compared to existing methods.

We consider optimal Bayesian detection of a slowly varying tone of unknown amplitude in situations characterized by very low signal-to-noise ratio (SNR) and a large number of measurements, as found in certain gravitational wave and passive sonar problems. We use a hidden Markov model (HMM) framework but, unlike typical HMM-based frequency line tracking methods, we develop a true track-before-detect algorithm, which does not threshold the blocked Fourier data and only considers frequency trails that have phase continuity across all HMM steps. We model the frequency and phase evolution as a phase-wrapped Ornstein-Uhlenbeck process. The resulting optimal detector is computationally efficient. The detectability improvement arising from phase continuity is characterized via comparative simulation for a mock, simplified gravitational wave search problem.

Sonobuoy fields, consisting of many distributed emitter and receiver sonar sensors on buoys, are used to seek and track underwater targets in a defined search area. A sensor scheduling algorithm is required in order to optimise tracking performance by selecting which emitter sonobuoy should transmit in each time interval, and which waveform it should use. In this paper we describe a new long term sensor scheduling algorithm for sonobuoy fields, called the continuous probability states algorithm. This algorithm reduces the scheduling search space by keeping track of the probability that a target is undetected, rather than modelling all possible detection outcomes, which reduces the computation complexity of the algorithm. It is shown that this approach results in high quality tracking for multiple targets in a simulated sonobuoy field.

Sonobuoy fields, consisting of a large network of emitter and receiver sonar sensors on buoys, are increasingly being used for detection and tracking of underwater targets in a defined maritime area. This study presents a Gaussian mixture version of a multitarget–multisensor (MS) Bayesian‐type tracker developed specifically for multistatic sonobuoy fields. Its foundation is the optimal Bayesian MS filter for a single target in clutter. The multi target feature is incorporated using the linear‐multitarget paradigm, which is a fast and accurate approximation assuming the density of underwater targets is low. Reliable track initiation and false track discrimination for low signal‐to‐noise ratio targets are achieved using the amplitude feature of reported detections. The developed tracker is capable of processing measurements with different modalities, depending on the transmitted signal waveform. It is integrated and tested within a realistic multistatic sonar emulator developed by DST Group.

Gravitational wave searches for continuous-wave signals from neutron stars are especially challenging when the star's spin frequency is unknown a priori from electromagnetic observations and wanders stochastically under the action of internal (e.g., superfluid or magnetospheric) or external (e.g., accretion) torques. It is shown that frequency tracking by hidden Markov model (HMM) methods can be combined with existing maximum likelihood coherent matched filters like the F-statistic to surmount some of the challenges raised by spin wandering. Specifically, it is found that, for an isolated, biaxial rotor whose spin frequency walks randomly, HMM tracking of the F-statistic output from coherent segments with duration T-drift = 10 d over a total observation time of T-obs = 1 yr can detect signals with wave strains h(0) > 2 x 10(-26) at a noise level characteristic of the Advanced Laser Interferometer Gravitational Wave Observatory (Advanced LIGO). For a biaxial rotor with randomly walking spin in a binary orbit, whose orbital period and semimajor axis are known approximately from electromagnetic observations, HMM tracking of the Bessel-weighted F-statistic output can detect signals with h(0) > 8 x 10(-26). An efficient, recursive, HMM solver based on the Viterbi algorithm is demonstrated, which requires similar to 10(3) CPU hours for a typical, broadband (0.5-kHz) search for the low-mass x-ray binary Scorpius X-1, including generation of the relevant F-statistic input. In a "realistic" observational scenario, Viterbi tracking successfully detects 41 out of 50 synthetic signals without spin wandering in stage I of the Scorpius X-1 Mock Data Challenge convened by the LIGO Scientific Collaboration down to a wave strain of h(0) = 1.1 x 10(-25), recovering the frequency with a root-mean-square accuracy of <= 4.3 x 10(-3) Hz.