Telecommunications with a relativistic spacecraft optimized via the Karhunen-Loève Transform (KLT)

The subject of telecommunications between the Earth and a spacecraft moving at a relativistic speed has received little attention so far. This is probably because the best thoughts of scientists have usually been devoted to the relativistic spacecraft propulsion problems instead. In this paper we would like to give the basic equations for relativistic telecommunications in the form of the relativistic Karhunen-Loève Transform (KLT), investigated by this author for over 15 years (Maccone, 1994). Essentially, the KLT is something superior than the Fourier Transform (FT) inasmuch as: 1) The KLT applies to any non-stationary stochastic process (input noisy signal), whereas the FT rigorously applies to stationary processes only (the latter result is usually referred to as the “Wiener-Khintchine Theorem”); 2) The KLT applies to any background noise distribution (i.e. to coloured noise in general) whereas the FT rigorously applies only when the background noise is white over the observing bandwidth; 3) The KLT is a more general filtering and compressing tool since it is a statistical procedure, rather than a deterministic procedure like the FT. This, unfortunately, makes the maths behind the KLT less easy to handle particularly in the case of the relativistic signals considered here. The KLT is a way of optimizing the signal processing of a given noisy signal by projecting the noisy signal itself onto the set of orthonormal basis functions spanned by the eigenfunctions of the autocorrelation of the noisy signal. Thus, the key problem in computing the KLT of a noisy signal is the computation of the eigenvalues and eigenfunctions of the autocorrelation of the noisy signal. For the special case of the Brownian motion (i.e. the basic Gaussian noisy signal) it can be proved that the KLT eigenfunctions are just sines, i.e. the KLT is the same as the FT. Let us now bring relativity into the KLT picture (this paper is confined to special relativity; general relativity can be KLT-studied also, but the calculations are, of course, even more difficult). Also, only rectilinear motions will be considered here. So, if one considers a source in relativistic motion, then the noisy signal undergoes a time-rescaling that depends on the type of relativistic motion. This author has demonstrated (Maccone, 1994) that the eigenfunctions of the time-rescaled, relativistic Brownian motion are Bessel functions of the first kind, and their eigenvalues are the zeros of such Bessel functions. In addition, it is shown here that explicit formulas for the KLT signal processing can be found for the particularly important cases of the noisy signals received on Earth from a relativistic spacecraft whose motion is either: 1) uniform: or 2) uniformly accelerated; or 3) arbitrarily accelerated (the most general case of arbitrary, rectilinear motion).