Analyses and designs in amplify-and-forward half-duplex cooperative systems

In this thesis, amplify-and-forward (AF) half-duplex relay systems are studied in the following three aspects: ergodic channel capacity, pairwise error probability and optimal precoder designs. The results obtained are compared with the case of conventional Multiple Input and Multiple Output (MIMO) systems in a way that reveals useful insights on how the relay systems "mimic" MIMO systems.; Firstly, ergodic channel capacities are established for two types of relay system: single antenna relay systems and MIMO relay systems. The channel matrices for these systems involve the product of Gaussian random variables or matrices, which renders the channels no longer Gaussian. Thus, the approach for obtaining the ergodic capacity for conventional MIMO Gaussian channels [1] is not applicable. A novel approach is developed in deriving the ergodic capacities of the relay systems. For the single antenna relay systems, it is shown that the optimal input covariance matrix to achieve ergodic capacity is diagonal for both the single and multiple relay systems. The diagonal entries are obtained by solving the optimization problem that involves multiple-dimensional integrals. Unfortunately, it appears that there is no analytic solution to this problem, but some insight has been obtained. In particular, the diagonal entries are not all equal even if all channel gains in the relay systems are Gaussian independent and identically distributed (i.i.d.). This is in direct contrast to the case of conventional MISO systems having i.i.d. Gaussian channel gains, where the ergodic capacity is achieved if the input covariance matrix is a scaled identity. Approximations for the ergodic capacity and the optimal power loading matrix are derived for relay systems with i.i.d. channels at high and low Signal to Noise Ratios (SNRs). For the approximations of the ergodic capacity to be achieved, the amplification coefficients at relay nodes must be equal to 0 at high SNR and to their upper limits at low SNR. For the MIMO relay system, the optimal input covariance matrix to achieve the ergodic capacity is shown to be block diagonal and each block diagonalizes the autocorrelation of channel matrix from the source node to the destination. This is also different from that for conventional MIMO systems with correlated channel gains.; To characterize the error performance, asymptotically exact expressions for the pairwise error probability (PEP) are derived for single and multiple relay systems. It is found that the PEP for the relay systems is not simply an exponential function of SNR as in the case of conventional MIMO systems, rather, it involves the logarithm of the SNR. The term diversity gain function is introduced. The coding gain is found to be proportional to the determinant of autocorrelation of the coding error matrices, which is similar to the case of conventional MIMO systems. The amplification coefficients at relay nodes must not be zero if the full diversity gain function is achieved.; Unitary precoders are designed to improve the performance of the relay systems without reducing the ergodic capacity. Two design criteria are suggested by the derived expressions for the error probability, namely, a rank criterion to achieve full diversity gain function, and a coding gain criterion to obtain the greatest advantage of coding gain. Sufficient conditions to guarantee the rank criterion for both single and multiple relay systems are obtained by exploiting the properties of cyclotomic rings. Optimal coding gain precoders are then derived analytically for QAM signals with the help of Farey sequence in number theory. It is also observed that a space-time block code providing optimal coding gain for conventional MIMO systems does not necessary offer optimal coding gain for the relay system. Simulations show that proposed precoders not only significantly improve the system performance, but also outperform some existing designs.