Joint phase and envelope densities of a higher order

In dealing with narrowband noise processes the density p(R₁ ,R₂,Δ) in which R₁ and R₂ are envelope samples and Δ is the phase difference can play an important role in determining system design features such as diversity performance, crossing rates, error probabilities, optimum processing and frequency or phase distributions. Standard texts on signal detection or radar usually include a discussion of p(R) and, less often, of p(R₁ ,R₂) but the number of instances in which p(R₁ ,R₂,Δ) can be specified is limited, and here known results are touched upon while a general form for this joint density is developed. The work extends an earlier treatment of p(R₁ ,R₂,Δ) and includes some important observations regarding the method used here and a SIRP (spherically invariant random process) approach which has been widely proposed for the analysis of correlated radar returns. Two differing families of non-Gaussian processes are used to illustrate the working and a number of densities for the jointly correlated pair R₁ and R₂, and the phase difference Δ are given, so extending the pool of such results available to the analyst. The approach used is heuristic and although the positivity of p(Δ) is not proven outright, experience and the cases illustrated point to this requirement being met