Outage Performance Analysis of Improper Gaussian Signaling for Two-User Downlink NOMA Systems with Imperfect Successive Interference Cancellation.

The improper Gaussian signaling (IGS) technique can improve the achievable rate of an interference-limited network by fully exploiting the second-order statistics of complex signaling. This paper addresses the outage performance analysis of a two-user downlink non-orthogonal multiple access (NOMA) system using the IGS technique in the presence of imperfect successive interference cancellation (SIC). The strong channel user (SU) adopts the IGS, while the weak channel user (WU) adopts the traditional proper Gaussian signaling (PGS). Considering a practical scenario where the transmitter has obtained the statistics of the channel coefficients instead of the instantaneous channel state information (CSI), the expressions of the achievable rates of both users under residual interference due to imperfect SIC are derived, together with their outage probabilities, subject to predetermined target rates and channel statistics. Given a fixed transmit power of the WU, both the transmit power and the degree of impropriety of the SU are optimized to minimize the outage probability subject to the outage constraint of the WU. Numerical results are provided to assess the benefits of the proposed IGS-based downlink NOMA system, which are consistent with the analysis.

Optimization in Quaternion Dynamic Systems: Gradient, Hessian, and Learning Algorithms.

The optimization of real scalar functions of quaternion variables, such as the mean square error or array output power, underpins many practical applications. Solutions typically require the calculation of the gradient and Hessian. However, real functions of quaternion variables are essentially nonanalytic, which are prohibitive to the development of quaternion-valued learning systems. To address this issue, we propose new definitions of quaternion gradient and Hessian, based on the novel generalized Hamilton-real (GHR) calculus, thus making a possible efficient derivation of general optimization algorithms directly in the quaternion field, rather than using the isomorphism with the real domain, as is current practice. In addition, unlike the existing quaternion gradients, the GHR calculus allows for the product and chain rule, and for a one-to-one correspondence of the novel quaternion gradient and Hessian with their real counterparts. Properties of the quaternion gradient and Hessian relevant to numerical applications are also introduced, opening a new avenue of research in quaternion optimization and greatly simplified the derivations of learning algorithms. The proposed GHR calculus is shown to yield the same generic algorithm forms as the corresponding real- and complex-valued algorithms. Advantages of the proposed framework are illuminated over illustrative simulations in quaternion signal processing and neural networks.

Quaternion-valued echo state networks.

Quaternion-valued echo state networks (QESNs) are introduced to cater for 3-D and 4-D processes, such as those observed in the context of renewable energy (3-D wind modeling) and human centered computing (3-D inertial body sensors). The introduction of QESNs is made possible by the recent emergence of quaternion nonlinear activation functions with local analytic properties, required by nonlinear gradient descent training algorithms. To make QENSs second-order optimal for the generality of quaternion signals (both circular and noncircular), we employ augmented quaternion statistics to introduce widely linear QESNs. To that end, the standard widely linear model is modified so as to suit the properties of dynamical reservoir, typically realized by recurrent neural networks. This allows for a full exploitation of second-order information in the data, contained both in the covariance and pseudocovariances, and a rigorous account of second-order noncircularity (improperness), and the corresponding power mismatch and coupling between the data components. Simulations in the prediction setting on both benchmark circular and noncircular signals and on noncircular real-world 3-D body motion data support the analysis.

Performance Bounds of Quaternion Estimators.

The quaternion widely linear (WL) estimator has been recently introduced for optimal second-order modeling of the generality of quaternion data, both second-order circular (proper) and second-order noncircular (improper). Experimental evidence exists of its performance advantage over the conventional strictly linear (SL) as well as the semi-WL (SWL) estimators for improper data. However, rigorous theoretical and practical performance bounds are still missing in the literature, yet this is crucial for the development of quaternion valued learning systems for 3-D and 4-D data. To this end, based on the orthogonality principle, we introduce a rigorous closed-form solution to quantify the degree of performance benefits, in terms of the mean square error, obtained when using the WL models. The cases when the optimal WL estimation can simplify into the SWL or the SL estimation are also discussed.

An augmented echo state network for nonlinear adaptive filtering of complex noncircular signals.

A novel complex echo state network (ESN), utilizing full second-order statistical information in the complex domain, is introduced. This is achieved through the use of the so-called augmented complex statistics, thus making complex ESNs suitable for processing the generality of complex-valued signals, both second-order circular (proper) and noncircular (improper). Next, in order to deal with nonstationary processes with large nonlinear dynamics, a nonlinear readout layer is introduced and is further equipped with an adaptive amplitude of the nonlinearity. This combination of augmented complex statistics and enhanced adaptivity within ESNs also facilitates the processing of bivariate signals with strong component correlations. Simulations in the prediction setting on both circular and noncircular synthetic benchmark processes and real-world noncircular and nonstationary wind signals support the analysis.

Application of multivariate empirical mode decomposition for seizure detection in EEG signals.

We present a method for the analysis of electroencephalogram (EEG) signals which has the potential to distinguish between ictal and seizure-free intracranial EEG recordings. This is achieved by analyzing common frequency components in multichannel EEG recordings, using the multivariate empirical mode decomposition (MEMD) algorithm. The mean frequency of the signal is calculated by applying the Hilbert-Huang transform on the resulting intrinsic mode functions (IMFs). It has been shown that the mean frequency estimates for the ictal and seizure-free EEG recordings are statistically different, and hence, can serve as a test statistic to distinguish between the two classes of signals. Simulation results on real world EEG signals support the analysis and demonstrate the potential of the proposed scheme.