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On the equivalence of two recent control over network results

In a recent work we considered the infimal signal-to-noise (SNR) ratio for stabilisability for an additive coloured Gaussian noise (ACGN) channel with bandwidth limitation. For an additive white Gaussian noise (AWGN) channel and a minimum phase plant with m unstable poles pi, i = 1, ⋯, m (each with multiplicity ni) such result should coincide with Σmi=1 2niRe {pi}, a result independently developed by other authors. However, this is not obvious and what observed for just one unstable real pole p in our recent previous work, is instead the factor 2p times a sum of alternating binomial coefficients. Here we study the properties of the finite Blaschke product containing the plant unstable poles and then explicitly prove the link between our recent result and the result independently developed by other authors, all for an AWGN channel and a minimum phase unstable plant. As a third contribution in this note, we prove the alternating binomial coefficients identity observed for the first time in our previous work (to the best knowledge of the author).