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Due to its conceptual simplicity and its proven effectiveness in real-time detection and removal of radio frequency interference (RFI) from radio astronomy data, the Spectral Kurtosis (SK) estimator is likely to become a standard tool of a new generation of radio telescopes. However, the SK estimator in its original form must be developed from instantaneous power spectral density (PSD) estimates, and hence cannot be employed as an RFI excision tool downstream of the data pipeline in existing instruments where any time averaging is performed. In this letter, we develop a generalized estimator with wider applicability for both instantaneous and averaged spectral data, which extends its practical use to a much larger pool of radio instruments.
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We obtain analytical approximations for the expectation and variance of the Spectral Kurtosis estimator in the case of Gaussian and coherent transient time domain signals mixed with a quasi-stationary Gaussian background, which are suitable for practical estimations of their signal-to-noise ratio and duty-cycle relative to the instrumental integration time. We validate these analytical approximations by means of numerical simulations and demonstrate that such estimates are affected by statistical uncertainties that, for a suitable choice of the integration time, may not exceed a few percent. Based on these analytical results, we suggest a multiscale Spectral Kurtosis spectrometer design optimized for real-time detection of transient signals, automatic discrimination based on their statistical signature, and measurement of their properties.
We describe in general terms the practical use in astronomy of a higher-order statistical quantity called Spectral Kurtosis (SK), and describe the first implementation of SK-enabled firmware in the F-engine (Fourier transform-engine) of a digital FX correlator for Expanded Owens Valley Solar Array (EOVSA). The development of the theory for SK is summarized, leading to an expression for generalized SK that is applicable to both SK spectrometers and those not specifically designed for SK. We also give the means for computing both the SK estimator and thresholds for its application as a discriminator of RFI contamination. Tests of the performance of EOVSA as an SK spectrometer are shown to agree precisely with theoretical expectations, and the methods for configuring the correlator for correct SK operation are described.
Detection of the cosmological neutral hydrogen signal from the Epoch of Reionization, and estimation of its basic physical parameters, is the principal scientific aim of many current low-frequency radio telescopes. Here we describe the Cosmological HI Power Spectrum Estimator (CHIPS), an algorithm developed and implemented with data from the Murchison Widefield Array (MWA), to compute the two-dimensional and spherically-averaged power spectrum of brightness temperature fluctuations. The principal motivations for CHIPS are the application of realistic instrumental and foreground models to form the optimal estimator, thereby maximising the likelihood of unbiased signal estimation, and allowing a full covariant understanding of the outputs. CHIPS employs an inverse-covariance weighting of the data through the maximum likelihood estimator, thereby allowing use of the full parameter space for signal estimation (foreground suppression). We describe the motivation for the algorithm, implementation, application to real and simulated data, and early outputs. Upon application to a set of 3 hours of data, we set a 2$sigma$ upper limit on the EoR dimensionless power at $k=0.05$~h.Mpc$^{-1}$ of $Delta_k^2<7.6times{10^4}$~mK$^2$ in the redshift range $z=[6.2-6.6]$, consistent with previous estimates.
The problem of astrometry is revisited from the perspective of analyzing the attainability of well-known performance limits (the Cramer-Rao bound) for the estimation of the relative position of light-emitting (usually point-like) sources on a CCD-like detector using commonly adopted estimators such as the weighted least squares and the maximum likelihood. Novel technical results are presented to determine the performance of an estimator that corresponds to the solution of an optimization problem in the context of astrometry. Using these results we are able to place stringent bounds on the bias and the variance of the estimators in close form as a function of the data. We confirm these results through comparisons to numerical simulations under a broad range of realistic observing conditions. The maximum likelihood and the weighted least square estimators are analyzed. We confirm the sub-optimality of the weighted least squares scheme from medium to high signal-to-noise found in an earlier study for the (unweighted) least squares method. We find that the maximum likelihood estimator achieves optimal performance limits across a wide range of relevant observational conditions. Furthermore, from our results, we provide concrete insights for adopting an adaptive weighted least square estimator that can be regarded as a computationally efficient alternative to the optimal maximum likelihood solution. We provide, for the first time, close-form analytical expressions that bound the bias and the variance of the weighted least square and maximum likelihood implicit estimators for astrometry using a Poisson-driven detector. These expressions can be used to formally assess the precision attainable by these estimators in comparison with the minimum variance bound.
We propose a generalized version of the Dantzig selector. We show that it satisfies sparsity oracle inequalities in prediction and estimation. We consider then the particular case of high-dimensional linear regression model selection with the Huber loss function. In this case we derive the sup-norm convergence rate and the sign concentration property of the Dantzig estimators under a mutual coherence assumption on the dictionary.
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