A new mathematical interpretation of the FSAN crosstalk-summing method

It was recently shown that the Full Service Access Network (FSAN) method for computing the power spectral density (PSD) of the crosstalk noise produced by a mixture of digital subscriber line (DSL) disturbers may be considered a particular case of a general lower bound —obtained using Minkowski’s Inequality—of the sum of the power spectra of the noises generated by the individual interferer classes. Such result was deemed important because it gave certain mathematical validation to the FSAN method, which currently lacks a theoretical foundation. The main contribution of the present paper is a proof that the FSAN method is equivalent to the Lp norm of a function suitably defined in a finite counting measure space, for a particular value of p. This result is regarded as relevant to the search of a mathematical basis for the FSAN method because it brings forward a new perspective for the theoretical analysis of the technique. Several results obtained with the help of this formulation are presented, including (1) a proof that the FSAN method always produces results smaller than or equal to the ones obtained by the simple sum of the crosstalk PSDs of the individual interferer classes and (2) a proof that the Lp norm with p → ∞ may be used to estimate the PSD of the crosstalk produced by the dominant disturber or disturber class at a particular frequency. The meaning of these results is illustrated with two examples.