The modified GTCF method and its application to multi-branch multihop relaying systems with full selection diversity

A modification to the new transform method, the generalized transformed characteristic function (GTCF), is proposed to reduce its computational complexity. The GTCF and the modified GTCF (M-GTCF) use a new transform domain rather than the conventional moment generating function or characteristic function approaches. The M-GTCF gives exact solutions in single integral form for the probability density function (PDF) and the cumulative distribution function (CDF) of the end-to-end received signal-to-noise ratio (SNR) of multihop amplify-and-forward (AF) relaying systems. The M-GTCF method is applied to obtain exact, integral solutions for the average symbol error probability and the outage probability of multi-branch, multihop AF relaying systems with full selection diversity. The results are precise for any number of hops, N, any number of branches, L, and can accommodate any channel fading distribution. The GTCF method can provide exact solution for such systems. However, the M-GTCF method provides the first exact theoretical solution for such systems with complexity less than the GTCF method.