Finding a faint polarized signal in wide-band radio data

We develop two algorithms, based on maximum likelihood (ML) inference, for estimating the parameters of polarized radio sources which emit at a single rotation measure (RM), e.g., pulsars. These algorithms incorporate the flux density spectrum of the source, either a power law or a scaled version of the Stokes I spectrum, and a variation in sensitivity across the observing band. We quantify the detection significance and measurement uncertainties in the fitted parameters, and we derive weighted versions of the RM synthesis algorithm which, under certain conditions, maximize the likelihood. We use Monte Carlo simulations to compare injected and recovered source parameters for a range of signal-to-noise ratios, investigate the quality of standard methods for estimating measurement uncertainties, and search for statistical biases. These simulations consider one frequency band each for the Australia Telescope Compact Array (ATCA), the Square Kilometre Array (SKA), and the Low Frequency Array (LOFAR). We find that results obtained for one frequency band cannot be easily generalized, and that methods which were developed in the past for correcting bias in individual frequency channels do not apply to wide-band data sets. The standard method for estimating the measurement uncertainty in RM is not accurate for sources with non-zero spectral indices. Furthermore, dividing Stokes Q and U by Stokes I to correct for spectral index effects, in combination with RM synthesis, does not maximize the likelihood.