Matrix representation of time-delay interferometry

Time-Delay Interferometry (TDI) is the data processing technique that cancels large laser phase fluctuations affecting one-way Doppler measurements made by unequal-arm space-based gravitational wave interferometers. By taking finite linear combinations of properly time-shifted measurements, are removed at any time $t$ and signals can be studied a requisite level precision. In this article we show delay operators used in TDI represented as matrices acting on arrays associated with noises measurements. The matrix formulation nothing but group theoretic representation (ring homomorphism) earlier approach involving time-delay so principle same. It shown homomorphism valid generally cover all situations interest. To understand potential advantages brings, care must taken analyst to account for light travel times when linearly relating noises. This especially important view future projects envisaged. We results cancellation an arbitrary only combining number samples measured around $t$.