Hidden Frequency Estimation with Data Tapers

Detecting and estimating hidden frequencies have long been recognized as an important problem in time series. This paper studies the asymptotic theory for two methods of high-precision estimation of hidden frequencies (secondary analysis method and maximum periodogram method) under the premise of using a data taper. In ordinary situations, a data taper may reduce the estimation precision slightly. However, when there are high peaks in thespectral density of the noise or other strong hidden periodicities with frequencies close to the hidden frequency of interest, the procedures of detection of the existence and the estimation for the hidden frequency of interest fail if data are non-tapered whereas they may work well if the data are tapered. The theoretical results are verified by some simulated examples

Burg algorithm for enhancing measurement performance in wavelength scanning interferometry

Wavelength scanning interferometry (WSI) is a technique for measuring surface topography that is capable of resolving step discontinuities and does not require any mechanical movement of the apparatus or measurand, allowing measurement times to be reduced substantially in comparison to related techniques. The axial (height) resolution and measurement range in WSI depends in part on the algorithm used to evaluate the spectral interferograms. Previously reported Fourier transform based methods have a number of limitations which is in part due to the short data lengths obtained. This paper compares the performance auto-regressive model based techniques for frequency estimation in WSI. Specifically, the Burg method is compared with established Fourier transform based approaches using both simulation and experimental data taken from a WSI measurement of a step-height sample

A tapering window for time-domain templates and simulated signals in the detection of gravitational waves from coalescing compact binaries

Inspiral signals from binary black holes, in particular those with masses in the range 10M_\odot \lsim M \lsim 1000 M_\odot, may last for only a few cycles within a detector's most sensitive frequency band. The spectrum of a square-windowed time-domain signal could contain unwanted power that can cause problems in gravitational wave data analysis, particularly when the waveforms are of short duration. There may be leakage of power into frequency bins where no such power is expected, causing an excess of false alarms. We present a method of tapering the time-domain waveforms that significantly reduces unwanted leakage of power, leading to a spectrum that agrees very well with that of a long duration signal. Our tapered window also decreases the false alarms caused by instrumental and environmental transients that are picked up by templates with spurious signal power. The suppression of background is an important goal in noise-dominated searches and can lead to an improvement in the detection efficiency of the search algorithms

Visibility based angular power spectrum estimation in low frequency radio interferometric observations

We present two estimators to quantify the angular power spectrum of the sky signal directly from the visibilities measured in radio interferometric observations. This is relevant for both the foregrounds and the cosmological 21-cm signal buried therein. The discussion here is restricted to the Galactic synchrotron radiation, the most dominant foreground component after point source removal. Our theoretical analysis is validated using simulations at 150 MHz, mainly for GMRT and also briefly for LOFAR. The Bare Estimator uses pairwise correlations of the measured visibilities, while the Tapered Gridded Estimator uses the visibilities after gridding in the uv plane. The former is very precise, but computationally expensive for large data. The latter has a lower precision, but takes less computation time which is proportional to the data volume. The latter also allows tapering of the sky response leading to sidelobe suppression, an useful ingredient for foreground removal. Both estimators avoid the positive bias that arises due to the system noise. We consider amplitude and phase errors of the gain, and the w-term as possible sources of errors . We find that the estimated angular power spectrum is exponentially sensitive to the variance of the phase errors but insensitive to amplitude errors. The statistical uncertainties of the estimators are affected by both amplitude and phase errors. The w-term does not have a significant effect at the angular scales of our interest. We propose the Tapered Gridded Estimator as an effective tool to observationally quantify both foregrounds and the cosmological 21-cm signal.Comment: 20 pages, 15 figures, 1 table.One typo corrected in Fig.13. Accepted for publication in MNRA

Non-stationary log-periodogram regression

We study asymptotic properties of the log-periodogram semiparametric estimate of the memory parameter d for non-stationary (d>=1/2) time series with Gaussian increments, extending the results of Robinson (1995) for stationary and invertible Gaussian processes. We generalize the definition of the memory parameter d for non-stationary processes in terms of the (successively) differentiated series. We obtain that the log-periodogram estimate is asymptotically normal for dE[1/2, 3/4) and still consistent for dE[1/2, 1). We show that with adequate data tapers, a modified estimate is consistent and asymptotically normal distributed for any d, including both non-stationary and non-invertible processes. The estimates are invariant to the presence of certain deterministic trends, without any need of estimation.Publicad

Trimming and tapering semi-parametric estimates in asymmetric long memory time series

This paper considers semi-parametric frequency domain inference for seasonal or cyclical time series with asymmetric long memory properties. It is shown that tapering the data reduces the bias caused by the asymmetry of the spectral density at the cyclical frequency. We provide a joint treatment of different tapering schemes and of the log-periodogram regression and Gaussian semi-parametric estimates of the memory parameters. Tapering allows for a less restrictive trimming of frequencies for the analysis of the asymptotic properties of both estimates when allowing for asymmetries. Simple rules for inference are feasible thanks to tapering and their validity in finite samples is investigated in a simulation exercise and for an empirical example.Publicad

Gaussian semi-parametric estimation of fractional cointegration

We analyse consistent estimation of the memory parameters of a nonstationary fractionally cointegrated vector time series. Assuming that the cointegrating relationship has substantially less memory than the observed series, we show that a multi-variate Gaussian semi-parametric estimate, based on initial consistent estimates and possibly tapered observations, is asymptotically normal. The estimates of the memory parameters can rely either on original (for stationary errors) or on differenced residuals (for nonstationary errors) assuming only a convergence rate for a preliminary slope estimate. If this rate is fast enough, semi-parametric memory estimates are not affected by the use of residuals and retain the same asymptotic distribution as if the true cointegrating relationship were known. Only local conditions on the spectral densities around zero frequency for linear processes are assumed. We concentrate on a bivariate system but discuss multi-variate generalizations and show the performance of the estimates with simulated and real data.Publicad