Ergodic Channel Capacities for the Amplify-and-Forward Half-Duplex Cooperative Systems

In this paper, the ergodic channel capacities are established for the amplify-and-forward (AF) half-duplex cooperative systems, which consist of a source node, a destination, and multiple-relay nodes. The relay nodes assist the transmission from the source node to the destination. Since the channel matrices for the cooperative systems involve product of Gaussian random variables, which are no longer Gaussian, the approach in obtaining the ergodic channel capacity for conventional multiple-input-multiple-output (MIMO) Gaussian channels (Telatar, 1999) is not applicable. By using a novel approach, we have arrived at the following conclusions about the ergodic channel capacities for the AF cooperative systems. For the single antenna AF relay systems in which all nodes are equipped with one antenna, the optimal covariance matrix of the input signals to achieve the ergodic channel capacity is diagonal, and the diagonal elements are obtained by solving optimization problems of multidimensional integrals. These diagonal entries are not all equal even if all the channel gains in the cooperative systems are independent and identically distributed (i.i.d.) Gaussian with unit variance. Therefore, a white input signal for the AF cooperative system may not achieve the ergodic channel capacity of the system. This is in direct contrast to the case of conventional multiple-input-single-output (MISO) systems having i.i.d. Gaussian channel gains of unit variance, in which case, the ergodic capacity is achieved if the input covariance matrix is a scaled identity matrix. For the MIMO relay system in which the nodes have multiple antennas, the input covariance matrix to achieve the ergodic capacity is block diagonal and each block diagonalizes the autocorrelation of channel matrix from the source to the destination. This is different from the case of conventional MIMO systems, where the input covariance matrix to achieve the ergodic channel capacity diagonalizes the channel autocorrelation matrix of the MIMO system (Tulina, Lozano, and Verdu, 2006). If the channel gains in conventional MIMO systems are correlated Gaussian random variables, the input covariance matrix is a full matrix, not block diagonal; and if the channel gains are i.i.d. Gaussians, the optimal input covariance matrix is a scaled identity. The observations obtained in this paper reveal useful insights of how the AF cooperative systems ldquomimicrdquo the conventional MISO and MIMO systems from the ergodic channel capacity perspective.