The Varied Avatars of Time-delay Interferometry

Time-Delay Interferometry (TDI) is the data processing technique that cancels the large laser phase fluctuations affecting the one-way Doppler measurements made by unequal-arm space-based gravitational wave interferometers. By taking finite linear combinations of properly time-shifted Doppler measurements, laser phase fluctuations are removed at any time t and gravitational wave signals can be studied at a requisite level of sensitivity. In the past, other approaches to this problem have been proposed. Recently, matrix based approaches have been put forward; two such approaches are by Vallisneri et al. and Tinto, Dhurandhar and Joshi. In this paper we establish a close relationship between these approaches. In fact we show that the matrices involved in defining the operators in the two approaches exhibit an isomorphism and therefore in both approaches one is dealing with matrix representations of the time-delay operators.