Abstract
Determining tractable analytical expressions for the probability density functions (PDFs) of correlated Rayleigh random variables (i.e., the multivariate Rayleigh joint PDF) is a longstanding problem in wireless communications. Recent results have reported simpler forms for the 3-D and 4-D Rayleigh joint PDFs. Yet, the new solutions are still mathematically complex. New representations for the general 3-D Rayleigh and the general 4-D Rayleigh fading PDFs are derived. An advantage of the new representations is that only a single-integral computation is needed to compute the 3-D Rayleigh PDF and only double integral computation is needed for the 4-D Rayleigh PDF, where the integrals are proper and definite on $[0,\pi )$ and ${[0,2\pi )}$ , respectively. The new expressions for the 3-D and 4-D Rayleigh PDFs are used to determine several special, “best” placements in the 2-D plane for three and four antennas. Closed-form solutions are found for three antennas with two or three independent antenna pairs, and four antennas with four or five independent antenna pairs.
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