Pulsar timing analysis in the presence of correlated noise

Pulsar timing observations are usually analysed with least-square-fitting procedures under the assumption that the timing residuals are uncorrelated (statistically "white"). Pulsar observers are well aware that this assumption often breaks down and causes severe errors in estimating the parameters of the timing model and their uncertainties. Ad hoc methods for minimizing these errors have been developed, but we show that they are far from optimal. Compensation for temporal correlation can be done optimally if the covariance matrix of the residuals is known using a linear transformation that whitens both the residuals and the timing model. We adopt a transformation based on the Cholesky decomposition of the covariance matrix, but the transformation is not unique. We show how to estimate the covariance matrix with sufficient accuracy to optimize the pulsar timing analysis. We also show how to apply this procedure to estimate the spectrum of any time series with a steep red power-law spectrum, including those with irregular sampling and variable error bars, which are otherwise very difficult to analyse.Comment: Accepted by MNRA

Characterization of the Crab Pulsar's Timing Noise

We present a power spectral analysis of the Crab pulsar's timing noise, mainly using radio measurements from Jodrell Bank taken over the period 1982-1989. The power spectral analysis is complicated by nonuniform data sampling and the presence of a steep red power spectrum that can distort power spectra measurement by causing severe power ``leakage''. We develop a simple windowing method for computing red noise power spectra of uniformly sampled data sets and test it on Monte Carlo generated sample realizations of red power-law noise. We generalize time-domain methods of generating power-law red noise with even integer spectral indices to the case of noninteger spectral indices. The Jodrell Bank pulse phase residuals are dense and smooth enough that an interpolation onto a uniform time series is possible. A windowed power spectrum is computed revealing a periodic or nearly periodic component with a period of about 568 days and a 1/f^3 power-law noise component with a noise strength of 1.24 +/- 0.067 10^{-16} cycles^2/sec^2 over the analysis frequency range 0.003 - 0.1 cycles/day. This result deviates from past analyses which characterized the pulse phase timing residuals as either 1/f^4 power-law noise or a quasiperiodic process. The analysis was checked using the Deeter polynomial method of power spectrum estimation that was developed for the case of nonuniform sampling, but has lower spectral resolution. The timing noise is consistent with a torque noise spectrum rising with analysis frequency as f implying blue torque noise, a result not predicted by current models of pulsar timing noise. If the periodic or nearly periodic component is due to a binary companion, we find a companion mass > 3.2 Earth masses.Comment: 53 pages, 9 figures, submitted to MNRAS, abstract condense

Millisecond and Binary Pulsars as Nature's Frequency Standards. II. Effects of Low-Frequency Timing Noise on Residuals and Measured Parameters

Pulsars are the most stable natural frequency standards. They can be applied to a number of principal problems of modern astronomy and time-keeping metrology. The full exploration of pulsar properties requires obtaining unbiased estimates of the spin and orbital parameters. These estimates depend essentially on the random noise component being revealed in the residuals of time of arrivals (TOA). In the present paper, the influence of low-frequency ("red") timing noise with spectral indices from 1 to 6 on TOA residuals, variances, and covariances of estimates of measured parameters of single and binary pulsars are studied. In order to determine their functional dependence on time, an analytic technique of processing of observational data in time domain is developed which takes into account both stationary and non-stationary components of noise. Our analysis includes a simplified timing model of a binary pulsar in a circular orbit and procedure of estimation of pulsar parameters and residuals under the influence of red noise. We reconfirm that uncorrelated white noise of errors of measurements of TOA brings on gradually decreasing residuals, variances and covariances of all parameters. On the other hand, we show that any red noise causes the residuals, variances, and covariances of certain parameters to increase with time. Hence, the low frequency noise corrupts our observations and reduces experimental possibilities for better tests of General Relativity Theory. We also treat in detail the influence of a polynomial drift of noise on the residuals and fitting parameters. Results of the analitic analysis are used for discussion of a statistic describing stabilities of kinematic and dynamic pulsar time scales.Comment: 40 pages, 1 postscript figure, 1 picture, uses mn.sty, accepted to Mon. Not. Roy. Astron. So

Measuring the parameters of massive black hole binary systems with Pulsar Timing Array observations of gravitational waves

The observation of massive black hole binaries (MBHBs) with Pulsar Timing Arrays (PTAs) is one of the goals of gravitational wave astronomy in the coming years. Massive (>10^8 solar masses) and low-redshift (< 1.5) sources are expected to be individually resolved by up-coming PTAs, and our ability to use them as astrophysical probes will depend on the accuracy with which their parameters can be measured. In this paper we estimate the precision of such measurements using the Fisher-information-matrix formalism. We restrict to "monochromatic" sources. In this approximation, the system is described by seven parameters and we determine their expected statistical errors as a function of the number of pulsars in the array, the array sky coverage, and the signal-to-noise ratio (SNR) of the signal. At fixed SNR, the gravitational wave astronomy capability of a PTA is achieved with ~20 pulsars; adding more pulsars (up to 1000) to the array reduces the source error-box in the sky \Delta\Omega by a factor ~5 and has negligible consequences on the statistical errors on the other parameters. \Delta\Omega improves as 1/SNR^2 and the other parameters as 1/SNR. For a fiducial PTA of 100 pulsars uniformly distributed in the sky and a coherent SNR = 10, we find \Delta\Omega~40 deg^2, a fractional error on the signal amplitude of ~30% (which constraints only very poorly the chirp mass - luminosity distance combination M_c^{5/3}/D_L), and the source inclination and polarization angles are recovered at the ~0.3 rad level. The ongoing Parkes PTA is particularly sensitive to systems located in the southern hemisphere, where at SNR = 10 the source position can be determined with \Delta\Omega ~10 deg^2, but has poorer performance for sources in the northern hemisphere. (Abridged)Comment: 20 pages, 12 figures, 2 color figures, submitted to Phys. Rev.

Random Access in Massive MIMO by Exploiting Timing Offsets and Excess Antennas

Massive MIMO systems, where base stations are equipped with hundreds of antennas, are an attractive way to handle the rapid growth of data traffic. As the number of user equipments (UEs) increases, the initial access and handover in contemporary networks will be flooded by user collisions. In this paper, a random access protocol is proposed that resolves collisions and performs timing estimation by simply utilizing the large number of antennas envisioned in Massive MIMO networks. UEs entering the network perform spreading in both time and frequency domains, and their timing offsets are estimated at the base station in closed-form using a subspace decomposition approach. This information is used to compute channel estimates that are subsequently employed by the base station to communicate with the detected UEs. The favorable propagation conditions of Massive MIMO suppress interference among UEs whereas the inherent timing misalignments improve the detection capabilities of the protocol. Numerical results are used to validate the performance of the proposed procedure in cellular networks under uncorrelated and correlated fading channels. With $2.5\times10^3$ UEs that may simultaneously become active with probability 1\% and a total of $16$ frequency-time codes (in a given random access block), it turns out that, with $100$ antennas, the proposed procedure successfully detects a given UE with probability 75\% while providing reliable timing estimates.Comment: 30 pages, 6 figures, 1 table, submitted to Transactions on Communication

Parameter Estimation from Time-Series Data with Correlated Errors: A Wavelet-Based Method and its Application to Transit Light Curves

We consider the problem of fitting a parametric model to time-series data that are afflicted by correlated noise. The noise is represented by a sum of two stationary Gaussian processes: one that is uncorrelated in time, and another that has a power spectral density varying as $1/f^\gamma$. We present an accurate and fast [O(N)] algorithm for parameter estimation based on computing the likelihood in a wavelet basis. The method is illustrated and tested using simulated time-series photometry of exoplanetary transits, with particular attention to estimating the midtransit time. We compare our method to two other methods that have been used in the literature, the time-averaging method and the residual-permutation method. For noise processes that obey our assumptions, the algorithm presented here gives more accurate results for midtransit times and truer estimates of their uncertainties.Comment: Accepted in ApJ. Illustrative code may be found at http://www.mit.edu/~carterja/code/ . 17 page

TEE, a simple estimator for the precision of eclipse and transit minimum times

Context: Transit or eclipse timing variations have proven to be a valuable tool in exoplanet research. However, no simple way to estimate the potential precision of such timing measures has been presented yet, nor are guidelines available regarding the relation between timing errors and sampling rate. Aims: A `timing error estimator' (TEE) equation is presented that requires only basic transit parameters as input. With the TEE, it is straightforward to estimate timing precisions both for actual data as well as for future instruments, such as the TESS and PLATO space missions. Methods: A derivation of the timing error based on a trapezoidal transit shape is given. We also verify the TEE on realistically modeled transits using Monte Carlo simulations and determine its validity range, exploring in particular the interplay between ingress/egress times and sampling rates. Results: The simulations show that the TEE gives timing errors very close to the correct value, as long as the temporal sampling is faster than transit ingress/egress durations and transits with very low S/N are avoided. Conclusions: The TEE is a useful tool to estimate eclipse or transit timing errors in actual and future data-sets. In combination with an equation to estimate period errors (Deeg 2015), predictions for the ephemeris precision of long-coverage observations are possible as well. The tests for the TEE's validity-range led also to implications for instrumental design: Temporal sampling has to be faster than transit in- or egress durations, or a loss in timing-precision will occur. An application to the TESS mission shows that transits close to its detection limit will have timing uncertainties that exceed 1 hour within a few months after their acquisition. Prompt follow-up observations will be needed to avoid a `loosing' of their ephemeris.Comment: Accepted by A&A. Version 2 with updated timing uncertainties of TESS mission due to correction of a figure in Sullivan et al. 201

Space-Time diversity for NLOS mitigation in TDOA-based positioning systems

This paper studies the potential impact of using space-Time information in the mitigation of the Non-LineOf-Sight condition in mobile subscriber's positioning systems. First of all, this work discusses the positioning problem based on measures of Time Differences Of Arrival departing from a more exact characterization of the signal statistics and including some geometrical restrictions to achieve an improved accurate. Furthermore, a novel approach that integrates signal propagation characteristics to information provided by a suitable timing estimation model based on Cramer Rao Bound for a Rayleigh-fading channel, when antenna arrays are used at the receiver and when a set ofchannel vector estimates are available, has been introduced to study the positive benefits of space-Time diversity. These approaches are evaluated within a realistic simulation scenario.Peer ReviewedPostprint (published version