| A multi-hop, amplify-and-forward (AF), cooperative diversity system with K relays is studied. An accurate approximate expression for the symbol-error-rate (SER) is derived for the multi-hop system. Also, a lower bound for the outage probability of the system, that is tight throughout nearly the entire signal-to-noise ratio (SNR) range, is presented. Neither an SER expression nor an outage probability expression had previously been reported in the literature for the multi-hop system. To assist in the derivation of the SER expression, the cumulative density function (CDF), probability density function (PDF), and moment generating function (MGF) are found for the random variable (RV), Z = X · Y/(X + Y + c), where X and Y are RVs which have PDFs that are sums of terms of the form xne −βx. It is shown that with the CDF, PDF, and MGF of this type of RV, it is possible to derive an expression for the SER of the multi-hop system for several important scenarios with respect to what type of fading is present in the channels of the system. To assist in the derivation of the lower bound of the outage probability, the CDF is found for an interesting new RV, presented in a recursive formula, that is used to represent the upper bound of the instantaneous end-to-end SNR of the multi-hop system. These mathematical results are useful beyond the scope of the multi-hop system researched in this thesis. Also, many of the results found in this thesis for the previously-scarcely-studied multi-hop sytem are shown to be generalizations of results that had been found for the previously-often-studied two-hop, AF, cooperative diversity system with K relays. |
