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Single Channel Equivalent Point Processes of Poisson Networks With Multiple Channel Laws

We present the construction and characterization of novel single channel equivalent representations of a Poisson Point Process (PPP) whose points follow multiple channel laws with respect to the origin. Two variants are presented. The first, called the Type 1 Equivalent Process (EPP), allows for IID fading while maintaining equivalence in terms of path loss and fading with respect to a single channel law. The second, called the Type 2 EPP, also maintains equivalence with respect to a single channel law, but requires the fading distribution to be dependent on the underlying point process. The Type 1 EPP only exists in certain settings, while the Type 2 EPP always exists.