System equivalent flux density of a low-frequency polarimetric phased array interferometer
Authors: Sutinjo A. T. , Ung D. C. X. , Sokolowski M. , Kovaleva M. , McSweeney S. 
Published: February 2022

Aims.This paper extends the treatment of system equivalent flux density (SEFD), discussed in our earlier paper to interferometric phased array telescopes. The objective is to develop an SEFD formula involving only the most fundamental assumptions that is readily applicable to phased array interferometer radio observations. Our aim is to compare the resultant SEFD expression against the often-used root-mean-square (rms) SEFD approximation, ${\rm{SEFD}}_I^{{\rm{rms}}} = {1 \over 2}\sqrt {{\rm{SEFD}}_{XX}^{\rm{2}} + {\rm{SEFD}}_{YY}^{\rm{2}}} $, to study the inaccuracy of the SEFDIrms.Methods.We take into account all mutual coupling and noise coupling within an array environment (intra-array coupling). This intra-array noise coupling is included in the SEFD expression through the realized noise resistance of the array, which accounts for the system noise. No assumption is made regarding the polarization (or lack thereof) of the sky nor the orthogonality of the antenna elements. The fundamental noise assumption is that, in phasor representation, the real and imaginary components of a given noise source are independent and equally distributed (iid) with zero mean. Noise sources that are mutually correlated and non-iid among themselves are allowed, provided the real and imaginary components of each noise source are iid. The system noise is uncorrelated between array entities separated by a baseline distance, which in the case of the Murchison Widefield Array (MWA) is typically tens of wavelengths or greater. By comparing the resulting SEFD formula to the SEFDIrmsapproximation, we proved that SEFDIrmsalways underestimates the SEFD, which leads to an overestimation of array sensitivity.Results.We present the resulting SEFD formula that is generalized for the phased array, but has a similar form to the earlier result. Here, the physical meaning of the antenna lengths and the equivalent noise resistances have been generalized such that they are also valid in the array environment. The simulated SEFD was validated using MWA observation of a Hydra-A radio galaxy at 154.88 MHz. The observed SEFDXXand SEFDIare on average higher by 9% and 4%, respectively, while the observed SEFDYYis lower by 4% compared to simulated values for all pixels within the −12 dB beam width. The simulated and observed SEFD errors due to the rms SEFD approximation are nearly identical, with mean difference of images of virtually 0%. This result suggests that the derived SEFD expression, as well as the simulation approach, is correct and may be applied to any pointing. As a result, this method permits identification of phased array telescope pointing angles where the rms approximation underestimates SEFD (overestimates sensitivity). For example, for Hydra-A observation with beam pointing (Az, ZA) = (81°, 46°), the underestimation in SEFD calculation using the rms expression is 7% within the −3 dB beam width, but increases to 23% within the −12 dB beam width. At 199.68 MHz, for the simulated MWA pointing at (Az, ZA) = (45°, 56.96°), the underestimation reached 29% within the −3 dB beam width and 36% within the −12 dB beam width. This underestimation due to rms SEFD approximation at two different pointing angles and frequencies was expected and is consistent with the proof.