Revisiting the radio interferometer measurement equation. IV. A generalized tensor formalism

TL;DR

This paper introduces a tensor-based formalism for radio interferometry measurement equations, overcoming limitations of the traditional matrix approach, and enabling more comprehensive modeling of modern observational scenarios.

Contribution

It develops a generalized tensor formalism for the radio interferometer measurement equation, extending its applicability to complex scenarios like wide-field polarimetry and phased array feeds.

Findings

Tensor formalism describes beamforming, mutual coupling, and wide-field polarimetry.

Tensor equations reduce to 2x2 RIME under certain assumptions.

Proposes methods for practical implementation of tensor equations.

Abstract

The radio interferometer measurement equation (RIME), especially in its 2x2 form, has provided a comprehensive matrix-based formalism for describing classical radio interferometry and polarimetry, as shown in the previous three papers of this series. However, recent practical and theoretical developments, such as phased array feeds (PAFs), aperture arrays (AAs) and wide-field polarimetry, are exposing limitations of the formalism. This paper aims to develop a more general formalism that can be used to both clearly define the limitations of the matrix RIME, and to describe observational scenarios that lie outside these limitations. Some assumptions underlying the matrix RIME are explicated and analysed in detail. To this purpose, an array correlation matrix (ACM) formalism is explored. This proves of limited use; it is shown that matrix algebra is simply not a sufficiently flexible tool for the job. To overcome these limitations, a more general formalism based on tensors and the Einstein notation is proposed and explored both theoretically, and with a view to practical implementations. The tensor formalism elegantly yields generalized RIMEs describing beamforming, mutual coupling, and wide-field polarimetry in one equation. It is shown that under the explicated assumptions, tensor equations reduce to the 2x2 RIME. From a practical point of view, some methods for implementing tensor equations in an optimal way are proposed and analysed. The tensor RIME is a powerful means of describing observational scenarios not amenable to the matrix RIME. Even in cases where the latter remains applicable, the tensor formalism can be a valuable tool for understanding the limits of such applicability.