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.

The rich information in the power spectral density (PSD) matrix of a received signal can be extracted in different ways for various purposes in signal processing. Care, however, must be taken when distances between PSD matrices are considered because these matrices form a Riemannian manifold in the signal space. In this paper, two Riemannian distances (RD) are introduced for the measurement of distances between PSD matrices on the manifold. The principle by which the geodesics on the manifold can be lifted to a Euclidean subspace isometric with the tangent space of the manifold is also explained and emphasized. This leads to the concept that any optimization involving the RD on the manifold can be equivalently performed in this Euclidean subspace. The application of this principle is illustrated by the development of iterative algorithms to find the Riemannian means and Riemannian medians according to the two RD. These distance measures are then applied to the detection of narrow-band sonar signals. Exploration of the translation of measure reference as well as the application of optimum weighting show substantial improvement in detection performance over classical detection method.

We consider the design of the precoder for a multi-input multi-output (MIMO) communication system equipped with a decision feedback equalizer (DFE) receiver. For such design problems, perfect knowledge of the channel state information (CSI) at both the transmitter and the receiver is usually required. However, in the environment of wireless communications, it is often difficult to provide sufficiently timely and accurate feedback of CSI from the receiver to the transmitter for such designs to be practically viable. In this paper, we consider the optimum design of a precoder for a wireless communication link having M transmitter antennas and N receiver antennas (M < N), in which the channels are assumed to be flat fading and may be correlated. We assume that full knowledge of CSI is available at the receiver. At the transmitter, however, only the first- and second-order statistics of the channels are available. Our goal here is to come up with an efficient design of the optimal precoder for such a MIMO system by minimizing the average arithmetic mean-squared error (MSE) of zero-forcing (ZF) decision feedback detection subject to a constraint on the total transmitting power. Applying some of the properties of the matrix parameters, this nonconvex optimization problem can be transformed into a convex geometrical programming problem which can then be efficiently solved using an interior point method. The performance of the MIMO system equipped with this optimum precoder and a ZF-DFE has also been found to be comparable, and in some cases, superior to that of V-BLAST which necessitates optimally ordered successive interference cancellation based on the largest postdetection signal-to-noise ratio (SNR). In terms of tradeoff between performance and implementation simplicity, our new system is certainly an attractive alternative.

The power spectral density (PSD) of a signal is often used as a feature for signal classification for which a distance measure must be chosen to compare the similarity between the signal features. We reason that PSD matrices have structural constraints and describe a manifold in the signal space. Thus, instead of the widely used Euclidean distance (ED), a more appropriate measure is the Riemannian distance (RD) on the manifold. Here, we develop a closed-form RD between two PSD matrices on the manifold and also an optimum weighting matrix for the purpose of signal classification. We then apply this new measure for electroencephalogram (EEG) classification for the determination of sleep states and the results are very encouraging.

The existing multiplexing techniques, such as TDM, FDM, and CDM, use different time slots, different frequency bands, and orthogonal codes, respectively, to distinguish each user. In this paper, a multiplexing model based on wavelet packets is presented. This model employs another kind of orthogonality -- the waveform orthogonality. In this approach, the channels are assigned to different users based on the mutual orthogonality of wavelet packets. After a brief review of the properties of wavelet packets, we can see that multiplexing based on the wavelet packets is a generalized form of TDM and FDM and it can be easily realized by the multiresolution algorithm developed for wavelet packets. In the multiplexing techniques, synchronization is an important problem. In this paper, the interference and cross- talk due to non-perfect synchronization between the receiver and transmitter are analyzed. It is demonstrated that with a suitable wavelet function, digital signals can be well recovered under noisy environments even when the synchronization is not accurate. The analysis on the synchronization also leads to the design of optimum wavelets for the application. Simulation results are presented and compared for the use of different wavelets.