This paper presents a transformer-based doublyselective channel estimation (DSCE) network, conceptualized from the perspective of masked autoencoding, wherein pilot observations are regarded as the unmasked tokens and symbol observations as the masked tokens. To achieve robust channel estimation under varying pilot patterns, a relative positional encoding schema is adopted to incorporate the relative positions between tokens into the model, rendering the DSCE less sensitive to the absolute positions of pilot symbols. Additionally, the pilot augmentation strategy is implemented to simulate more diverse pilot patterns not presented in the training data, thereby mitigating overfitting and further improving the generalization ability of the model. The generalization and robustness of the proposed DSCE are validated through training and testing under various channel configurations. Comparative experiments demonstrate the superior performance of the proposed DSCE in terms of mean square error and bit error rate compared to traditional and deep learning-based methods, highlighting its state-of-the-art capabilities in channel estimation.

Orthogonal chirp division multiplexing (OCDM) emerges as a promising technique for multi-carrier communication systems in next-generation networks, offering better multipath diversity exploitation compared to orthogonal frequency division multiplexing (OFDM). In this paper, we study symbol-level precoding (SLP) design for reducing power consumption in the multiple-input multiple-output orthogonal chirp division multiplexing (MIMO-OCDM) systems through the idea of non-strict phase rotation constructive interference (non-strict CI), which leads to the proposed SLP methods. Our proposed methods minimize the total power of the transmitted signal on a symbol-by-symbol basis and address this optimization challenge by transforming it into a non-negative least-squares (NNLS) problem. Furthermore, frequency domain pre-equalization (pre-FDE) is incorporated into the SLP algorithm with non-strict CI, which not only enhances transmit power efficiency but also facilitates the acquisition of more optimal perturbation vectors. Numerical evaluations of the two SLP methods demonstrate substantial power savings, which are over 80.72% and 83.87%, respectively, compared with traditional MIMO-OCDM systems while simultaneously improving BER performance.

Orthogonal chirp division multiplexing (OCDM) is a promising technique for sixth-generation (6G), offering better resilience against multipath fading and interference compared to orthogonal frequency division multiplexing (OFDM). In this paper, widely linear minimum mean square error (WL-MMSE) detectors are introduced in multiple-input multiple-output orthogonal chirp division multiplexing (MIMO-OCDM) systems to deal with improper complex signals by minimizing noise and interference which outperform traditional MMSE detectors. Building on this, likelihood ascent search (LAS) detectors, known for their near-optimal performance with low complexity, are introduced. LAS detectors use iterative neighborhood search, initialized with MMSE and WL-MMSE outputs, to enhance detection performance by increasing likelihood and reducing error probability. The proposed detectors demonstrate over 5 dB SNR gain compared to MMSE in MIMO-OCDM systems and a 12 dB gain over MIMO-OFDM systems.

Non-orthogonal multiple access (NOMA) is a promising technology garnering significant attention among the Internet of Things (IoT) community due to its superior spectral efficiency. This work addresses the rate region boundary enhancement of downlink NOMA systems under imperfect successive interference cancellation (SIC) with advanced improper Gaussian signaling (IGS), which provides additional degrees of freedom for system design. We investigate a universal scenario in which two users adopt improper signaling and their transmit powers are optimized. We first formulate the achievable rate of both users in terms of the impropriety degree of the IGS. First, the analytical expressions for the best improper transmission are characterized by jointly optimizing the users’ power and the impropriety degree for the perfect SIC case. Then, a deep Q network (DQN)-based approach is provided to find the rate region of the IGS-aided NOMA system under imperfect SIC. Simulations presented for the downlink NOMA system support the analysis, illustrating that IGS can efficiently enhance the rate region of the NOMA system compared to proper signaling.

In this letter, we propose a widely linear momentum least mean squares (WLMLMS) algorithm by incorporating the momentum term into the widely linear framework. This approach effectively handles noncircular signals and achieves faster convergence compared to the conventional augmented complex LMS, with minimal additional computational cost. Since the lag term of momentum complicates the analysis, we perform the Schur-Cohn test in$z$domain, and the direct bound without mixing the momentum factor is derived, for the first time, which further simplifies the parameter selection. Our derived expressions elucidate that, although augmented statistics integrate into mean-square error (MSE) iterations, the stability threshold is predominantly influenced by the covariance of the input signal and the momentum factor. Additionally, the upper bound does not exhibit a monotonic decrease with the momentum factor. Finally, the theoretical findings of this study are corroborated through numerical simulations.

The multiple-input multiple-output (MIMO) wiretap channel (WTC) serves as a fundamental model for exploring information-theoretic secrecy in wireless communication systems, involving a transmitter, a legitimate user, and an eavesdropper. This paper investigates the optimality of proper complex signals in complex WTCs. Our primary contribution lies in the derivation of a determinant inequality, which establishes that the secrecy rate of degraded complex MIMO WTCs is maximized when the signal is proper, meaning that its pseudo-covariance matrix is a zero matrix. Remarkably, we extend this result beyond the degraded scenario to the general complex WTC by leveraging a min-max reformulation of the secrecy capacity. Thus, we demonstrate that focusing on proper signals is sufficient when examining the secrecy capacity of the complex WTC. Overall, this work highlights the significance of the determinant inequality we derive and its implications for optimizing secrecy rates in the complex WTC.

In this paper, we study low-complexity linear precoding for downlink massive multiple-input multiple-output (MIMO) systems, exploiting a statistical method. In sharp contrast to traditional linear precoding algorithms, our proposed efficient randomized iterative precoding algorithm (ERIPA) not only avoids costly matrix inversion but also considers the complexity reduction of matrix multiplication involved, thus enabling more efficient linear precoding. Additionally, ERIPA is demonstrated to have both exponentially fast and global convergence, making it adaptable to various practical scenarios of massive MIMO. We also investigate the convergence phenomenon of ERIPA in relation to the selection of the sampling distribution during random iterations. After that, the concept of conditional sampling is introduced to ERIPA such that significant system potential can be beneficially exploited in terms of both precoding performance and computational complexity. Finally, simulation results regarding the downlink massive MIMO are presented to confirm the superiorities of the proposed ERIPA.

In this paper, we study the optimality of proper signals for wiretap channels (WTC). We characterize the secrecy capacity of WTC in both separate and augmented forms, and show that proper Gaussian signals can achieve the secrecy capacity for both single-input multiple-output (SIMO) and single-input single-output (SISO) WTC. We also use an accelerated difference of convex functions algorithm (ADCA) to verify our results.

In this paper, a new randomized iterative detection algorithm (NRIDA) is proposed for uplink large-scale MIMO systems, where the random iterations in it are designed to work for the detection model (denoted by$\mathbf {y}=\mathbf {Hx}+\mathbf {n}$) directly. Different from those traditional iterations designed for the linear system (denoted by$\mathbf {Ax}=\mathbf {b}$with$\mathbf {A}=\mathbf {H}^{H}\mathbf {H}$and$\mathbf {b}=\mathbf {H}^{H}\mathbf {y}$), we show that besides the complexity reduction about the matrix inversion, in the proposed NRIDA the computational complexity of matrix multiplication for the linear detection is also greatly reduced without any performance loss, thus leading to a much lower detection complexity. Meanwhile, according to convergence analysis, we demonstrate that the proposed NRIDA enjoys a globally exponential convergence performance, enabling it well suited to the various detection cases of interest. Besides, further complexity reduction and the choices of the sampling distribution in NRIDA are studied as well in full details. Moreover, in order to achieve a better detection trade-off between performance and complexity, we introduce the concept of the conditional sampling into NRIDA, which brings significant gains in both iteration convergence and efficiency. Finally, simulations with respect to the uplink large-scale MIMO detection are presented to illustrate the remarkable gains of the proposed NRIDA in both performance and complexity.

The secrecy capacity of the multiple-input multiple-output (MIMO) Gaussian cognitive interference channel with common, private and confidential messages is characterized in this paper. To show the achievability, we use jointly Gaussian auxiliary random variables to evaluate the existing single-letter description of the capacity region for the discrete memoryless channel. The converse part is established by invoking linear estimation theory and a new version of the extremal inequality. To resolve the Gaussian optimality problem in the extremal inequality, the perturbation framework is employed. Our results explore the connections between the MIMO Gaussian broadcast channel and its distributive antennas counterpart, i.e., the MIMO Gaussian interference channel.

This paper considers the channel tracking and angle of arrival (AoA) acquisition for millimeter wave (mmWave) massive multiple-input multiple-output (MIMO) systems, in which each antenna at the base station is equipped with low-resolution analog-to-digital converters (ADCs) to quantize the received signals. We utilize the beamspace Gauss-Markov model to capture the sparsity and temporal correlation of the time-varying mmWave channel, and an off-grid model is incorporated for an accurate AoA acquisition. Essentially, the beamspace channel tracking is a quantized sparse Bayesian learning problem, which is solved under the expectation maximization (EM) framework. We employ the variational inference to calculate the statistics in the expectation step. In this way, we propose a variational inference joint channel tracking and data detection (VIJ-CTDD) algorithm, in which the detected data symbols are reused to enhance the tracking without extra pilot overhead. Finally, extensive simulations validate the superiority of the proposed VIJ-CTDD algorithms over several existing works.

The affine combination of two complex-valued least-mean-squares filters (aff-CLMS) addresses the trade-off between fast convergence rate and small steady-state misadjustment error. However, a rigorous analysis of the aff-CLMS algorithm for second-order non-circular inputs is still under investigation. To this end, the focus in this letter is on the full mean-square analysis of the aff-CLMS algorithm, in which the transient analyses of the mixing parameter, as well as the standard and complementary weight-error covariance matrices, are completed. In addition, we derive the closed-form solutions of the steady-state weight-error power and its complementary version of the aff-CLMS. Finally, the effectiveness of the theoretical analysis is supported by computer simulations.

For the estimation of real-valued Gaussian signals, the unscented Kalman filter (UKF) can provide a state estimate with second-order accuracy. However, when a general complex-valued system is considered, a direct extension of UKF from the real domain to the complex domain is inadequate, since the complementary covariance information associated with general improper complex-valued signals has been systematically ignored. To this end, in this work, we propose a general complex-valued unscented Kalman filter (GCUKF) algorithm which can be applied for both proper and improper signals. This is achieved by first proposing a novel sigma points selection scheme for the general complex-valued case, followed by a modified state update method to fully utilize both the innovation and its conjugate. A rigorous MSE analysis illustrates the superiority of the proposed state update method, and simulations support the analysis.

The multivariate autoregressive (MVAR) model is widely used in describing the dynamics of nonlinear systems, in which the estimates of model parameters and underlying states can be achieved by dual extended Kalman filter (DEKF). However, when the measurements are corrupted by complicated non-Gaussian noises, the DEKF based on the minimum mean-square error (MMSE) criterion may provide biased estimates. In the present article, we develop a novel dual Kalman-type filter, referred to as DEKF under minimum error entropy (MEE) with fiducial points (MEEFs-DEKF) to deal with the non-Gaussian noises. First, the equivalent state-space model and parameter-space model are presented based on the MVAR model. Then, an optimality criterion based on MEE with fiducial points (MEEFs) is applied in the batch-mode regression equation to improve the robustness. Finally, a fixed-point iteration algorithm gives the posterior estimates of state and parameter. Simulation results confirm that the proposed MEEF-DEKF can achieve excellent performance in various noises with different distributions, especially in heavy-tailed and multimodal noises.

Non-orthogonal multiple access (NOMA) exhibits superiority in spectrum efficiency which is particularly essential in the Internet of Things (IoT) system involving massive number of device connections. This paper addresses the achievable rate improvement for the downlink NOMA system, in the context of imperfect successive interference cancellation (SIC), by means of the improper Gaussian signaling (IGS) technique. We investigate a basic scenario where the strong user transmits the conventional proper data, while the weak user adopts an improper signaling scheme. The users’ data rates are first formulated in terms of the impropriety degree of the IGS, under residual interference introduced by the imperfect SIC. In this way, analytical expressions for the best improper transmission can be characterized by jointly optimizing the user’s power and the impropriety degree, where their sufficient and necessary conditions are provided. When the strong user transmits with its maximum power, the IGS scheme always increases the achievable rate of the strong user while the weak user may also benefit. When the weak user transmits with its maximum power, such a scheme enables us optimize the achievable rate of the strong user under various levels of channel-to-noise ratios (CNR) and imperfect SIC. Finally, when both the users are imposed by quality of service (QoS) constraints, a Q-learning based solution is proposed to maximize their sum rate. Simulations on the downlink NOMA system support the analysis.

In this paper, the random iterative method is introduced to massive multiple-input multiple-output (MIMO) systems for the efficient downlink linear precoding. By adopting the random sampling into the traditional iterative methods, the matrix inversion within the linear precoding schemes can be approximated statistically, which not only achieves a faster exponential convergence with low complexity but also experiences a global convergence without suffering from the various convergence requirements. Specifically, based on the random iterative method, the randomized iterative precoding algorithm (RIPA) is firstly proposed and we show its approximation error decays exponentially and globally along with the number of iterations. Then, with respect to the derived convergence rate, the concept of conditional sampling is introduced, so that further optimization and enhancement are carried out to improve both the convergence and the efficiency of the randomized iterations. After that, based on the equivalent iteration transformation, the modified randomized iterative precoding algorithm (MRIPA) is presented, which achieves a better precoding performance with low-complexity for various scenarios of massive MIMO. Finally, simulation results based on downlink precoding in massive MIMO systems are given to show the system gains of RIPA and MRIPA in terms of performance and complexity.

Along with current multi-scale based detectors, Feature Aggregation and Enhancement (FAE) modules have shown superior performance gains for cutting-edge object detection. However, these hand-crafted FAE modules show inconsistent improvements on face detection, which is mainly due to the significant distribution difference between its training and applying corpus, i.e. COCO vs. WIDER Face. To tackle this problem, we essentially analyse the effect of data distribution, and consequently propose to search an effective FAE architecture, termed AutoFAE by a differentiable architecture search, which outperforms all existing FAE modules in face detection with a considerable margin. Upon the found AutoFAE and existing backbones, a supernet is further built and trained, which automatically obtains a family of detectors under the different complexity constraints. Extensive experiments conducted on popular benchmarks, i.e. WIDER Face and FDDB, demonstrate the state-of-the-art performance-efficiency trade-off for the proposed automatic and scalable face detector (ASFD) family. In particular, our strong ASFD-D6 outperforms the best competitor with AP 96.7/96.2/92.1 on WIDER Face test, and the lightweight ASFD-D0 costs about 3.1 ms, i.e. more than 320 FPS, on the V100 GPU with VGA-resolution images.

Deep learning based image denoising methods have been extensively investigated. In this paper, attention mechanism enhanced kernel prediction networks (AME-KPNs) are proposed for burst image denoising, in which, nearly cost-free attention modules are adopted to first refine the feature maps and to further make a full use of the inter-frame and intra-frame redundancies within the whole image burst. The proposed AME-KPNs output per-pixel spatially-adaptive kernels, residual maps and corresponding weight maps, in which, the predicted kernels roughly restore clean pixels at their corresponding locations via an adaptive convolution operation, and subsequently, residuals are weighted and summed to compensate the limited receptive field of predicted kernels. Simulations and real-world experiments are conducted to illustrate the robustness of the proposed AME-KPNs in burst image denoising.

Although the widely linear minimum mean-square error (WLMMSE) receiver has been widely applied in multiple-input-multiple-output (MIMO) systems, there has been no theoretical analysis to quantify its distribution of signal-to-interference-plus-noise ratio (SINR) in arbitrary fading environments. In this paper, the closed-from expression of SINR at the output of the WLMMSE detection is presented, which, in its essence, can be interpreted as the sum of a series of gamma distributed random variables. The general probability density function of SINR is derived for the first time, which is explicitly expressed in terms of the confluent Lauricella hypergeometric function. Simulations on MIMO transmission systems over Rayleigh fading channels support the analytic results.

Adverse impacts of in-phase and quadrature-phase (I/Q) imbalance in both the transmitter (Tx) and receiver (Rx) are quantified for the generalized frequency division multiplexing (GFDM) based transmission over frequency selective fading channels. To this end, we first equip the standard signal-to-interference-plus-noise (SINR) performance evaluation with the ability to consider second-order noncircular (improper) signals, and thus precisely evaluate performance deterioration caused by I/Q distortions over the in-phase (I) and quadrature-phase (Q) channels of a transmission system. Next, we propose a novel means to evaluate the individual SINR contributions from both the channels of GFDM, and hence, provide more meaningful insights into the underlying wireless transmission in the presence of complex non-circularity. This is accompanied by an account of complete augmented second-order statistics of I/Q imbalanced GFDM waveforms which caters for various sources of complex improperness. Simulations in the GFDM system setting support our analysis.

This study develops a pseudo-energy-conservation relation to analyze the complementary mean-square performance of a complex-valued normalized least mean-square (CNLMS) algorithm. Based on this relation, we derive a recursion describing the transient complementary mean-square deviation (CMSD) behavior of the CNLMS. Closed-form expressions to predict the steady-state CMSD and complementary excess mean-square error conjugate (CEMSEc) are obtained. Because this method inherits the advantages of the energy-conservation method, our results do not restrict the distribution of input signals. Numerical simulations validate our theoretical analysis results.

Direct-conversion receivers (DCRs) have been adopted in wideband communication systems owing to their simple structure and low cost, however, their operation is affected by amplitude and phase mismatches between their analog inphase (I) and quadrature (Q) branches, as well as the discrepancy of low-pass filter coefficients between these two channels. In this paper, a blind adaptive frequency-dependent I/Q imbalance compensator is proposed, which exploits the complex properness (second-order circularity) of ideal constellation mappings to provide more enhanced insight into the problem setting within the proposed compensator. This serves as a basis for a novel full second-order performance assessment framework, which is established through a joint consideration of the weight error covariance and complementary covariance in both the transient and steady-state stages. This conjoint analysis is further shown to facilitate accurate quantification of the overall mirror-frequency interference attenuation capability of the proposed compensator. Simulation results in an orthogonal frequency division multiplexing (OFDM) transmission system demonstrate the excellent performance of the proposed compensator.

This paper presents a new course design to enhance students' interest in our graduate curriculum of morden digital signal processing (MDSP). The familiar radio frequency identification (RFID) technology is introduced to reveal the fundamentals of the least-mean-square (LMS) algorithm. Students are encouraged to encode their customised data into their respective RFID tags, and to use the software-defined radio (SDR) platform to monitor the inventory process integrated within RFID communication systems. After acquiring the digital signal from the SDR hardware, they need to perform adaptive channel estimation and signal detection to identify the customised data. By applying appropriate mathematical tools in the Matlab programming environment to solve the signal detection problem in two different communication scenarios, students reported to have had a deeper understanding of the signal processing concepts of adaptive filtering techniques in a self-directed manner.

Quaternion least-mean-quare (QLMS) algorithm has been thoroughly investigated in the past for the adaptive filtering of 3D and 4D signals. However, QLMS is restricted to one-channel processing of such 3D and 4D processes. To address this shortcoming, we proposed the multichannel QLMS so that we can exploit the additional information from the multitude of 3D-4D vector sensors. For rigour, we provide the stability bound of the multichannel QLMS and demonstrate its robustness against noise. Simulations on real world 4D signals support the analysis.

Complex least mean square (CLMS) based adaptive computation of discrete orthogonal transforms has been extensively investigated in the literature. However, all of these results provide only a means for the calculation of either forward orthogonal transforms or their inverse orthogonal transforms, separately. In this work, a way to simultaneously calculate the discrete Fourier transform (DFT) and the inverse DFT (IDFT) is established via the widely linear (WL) signal processing framework. We show that by appropriately selecting the input vector and adaptation speed of the widely linear complex least mean square (WL-CLMS), the resulting spectrum analyzer is capable of simultaneously performing DFT and IDFT of the signal to be Fourier analyzed in both the block-based and online manners.

Widely linear (WL) models have been demonstrated to be superior to conventional strictly linear models for the estimation of noncircular complex and quaternion signals. Existing studies on their performance bounds focus on the analysis of mean square error (MSE). However, the single degree of freedom within standard MSE allows for only the minimization of error power, with no means to understand how the error contribution is distributed across the data channels. To this end, we introduce novel complex and quaternion valued complementary quadratic cost functions for complex and quaternion signal estimation, which are extensions of the recently proposed complementary MSE metric. It is shown that for WL minimum MSE estimation and least squares regression, the complementary cost function and the standard cost function attain the same stationary point. We also show that for the former, the stationary point is a saddle point. This novel finding provides insight into the performance of complex and quaternion WL estimators, and offers a rigorous foundation for further developments in this field.

Generalized frequency division multiplexing (GFDM) has become one of the most important waveform candidates for beyond 5G (B5G) communications. However, physical distortions, such as in-phase and quadrature (I/Q) imbalance caused by the imperfections of radio frequency (RF) components within direct-conversion transceivers (DCTs), may cause severe performance degradation in GFDM-based wireless systems. To this end, we first conduct a rigorous sum rate analysis to quantify the impact of I/Q imbalance in both the transmitter and the receiver on the GFDM wireless transmission. An efficient I/Q imbalance compensation scheme is next proposed based on pilots; this is achieved through a nonlinear least squares analysis of the joint channel and I/Q imbalance estimation, and a simple symbol detection procedure. For rigor, the Cramer-Rao lower bounds for both the I/Q imbalance parameters and the channel coefficients are also derived. The simulation results illustrate that the mean square error performance of the proposed estimator closely approaches the corresponding CRLB over static frequency selective channels, thus significantly reducing the sensitivity of GFDM DCTs to physical I/Q impairments.

An augmented nonlinear complex LMS (ANCLMS) algorithm is proposed to adaptively mitigate both the linear and nonlinear self-interference (SI) components in a full-duplex direct-conversion transceiver (DCT). A data prewhitening scheme, which exploits the known SI signal distributions, is also adopted to accelerate the convergence. Theoretical mean and mean square performance evaluations of the proposed SI canceller are performed and fully support the proposed approach. Computer simulations on wireless local area network (WLAN) standard compliant waveforms in practical full-duplex (FD) direct-conversion transceiver settings support the analysis.

Real world complex-valued signals typically exhibit rotation-dependent distributions (noncircularity), and significant performance gains in learning algorithms can be obtained by accounting for information beyond the standard second-order noncircularity (impropriety). To this end, we introduce a new closed form definition of complex correntropy which is general enough to cater for both circular and noncircular distributions in complex data, and serves as a basis for a novel cost function for widely linear adaptive filtering, termed the maximum improper complex corren-tropy criterion (MICCC). A stochastic gradient adaptive filtering algorithm is developed based on the MICCC, and its standard and complementary convergence and stability analyses are conducted with respect to both the circularity of the estimation error and the kernel size in the underlying Parzen estimator. Performance advantages over the strictly linear correntropy algorithm (MCCC) and the mean square error based complex least mean square (CLMS) and augmented CLMS (ACLMS) are demonstrated through analysis and simulations.

A resource effective extension to the class of distributed real-time diffusion Kalman filters is proposed. The proposed scheme removes the need to share measurement variables explicitly, by sharing only the state estimates and state error covariance matrices which implicitly contain the information about the measurements, observations matrices, and noise covariance matrices. The proposed distributed Kalman filter is quiet general, and its performance is comparable to that of existing diffusion based schemes, while having lower communication and computational requirements per-iteration compared to current distributed Kalman filtering algorithms.

A novel physical insight is provided into the behavior and performance of the augmented complex least mean square (ACLMS) algorithm for widely linear adaptive estimation of general second-order noncircular (improper) Gaussian signals, whereby the off-diagonal elements of the covariance and complementary covariance matrices are nonzero. This is achieved through a novel complementary mean square analysis, a counterpart to the standard mean square analysis, and which focuses on the behavior of the complementary second-order statistics of the output error and the augmented weight error vector. We next establish the effect of the degree of input noncircularity on the evolution for these two key parameters that govern the ACLMS. Both transient and steady-state performances are addressed and a stability bound on the step-size for their convergence is established. Simulations in the system identification setting support the analysis.

The problem of off-nominal frequency estimation in unbalanced three-phase power systems is addressed from the frequency domain perspective. It is first established that the original Smart discrete Fourier transform (SDFT) technique designed for real-valued single-phase voltage can be extended to deal with complex-valued αβ transformed voltage. By observing that the underlying time series relationship among the consecutive DFT fundamental components employed by SDFT technique does not hold when noise or unexpected higher order harmonics are present, resulting in suboptimal estimation performances, the least squares framework is then built upon the underlying relationship among the consecutive DFT fundamental components to minimise the mean square model error. The benefits of the proposed LS-SDFT over the time-domain widely linear least squares (WL-LS) frequency estimator are verified by simulations for unbalanced power system conditions in the presence of noise and higher order harmonic pollution, as well as for real-world measurements.

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 (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.

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.

We introduce a novel and unifying framework for the calculation of gradients of both quaternion holomorphic functions and nonholomorphic real functions of quaternion variables. This is achieved by considering the isomorphism between the quaternion domain H and the bivariate complex domain C×C, and by exploiting complex calculus to simplify the quaternion gradient calculation. The validation of the proposed HC calculus is performed against the existing HR calculus, and its convenience is illustrated in the context of gradient-based quaternion optimisation as well as in adaptive learning systems. Quaternion adaptive filtering algorithms and a dynamical perceptron update are next derived based on the bivariate complex representation of quaternions and the HC calculus. Simulations on both synthetic and real-world multidimensional signals support the analysis.

A novel method to evaluate the statistical significance differences between the groups of coma and brain death patients is presented. This is achieved based on the electroencephalogram (EEG) and by using a collaborative filtering structure with the least mean square (LMS) and least mean phase (LMP) adaptive filters. By virtue of a complex-valued representation of pair-wise EEG signals, the evolution of the mixing parameter is used as an indicator of the fundamental amplitude-phase relationships of EEG recordings. Simulations illustrate the suitability of this approach to differentiate between the coma and quasi-brain-death states.

A new framework for the estimation of the instantaneous frequency in a three-phase power system is proposed. It is first illustrated that the complex-valued signal, obtained by the αβ transformation of three-phase power signals under unbalanced voltage sag conditions, is second order noncircular, for which standard complex adaptive estimators are suboptimal. To cater for second order noncircularity, an adaptive widely linear estimator based on the augmented complex least mean square (ACLMS) algorithm is proposed, and the analysis shows that this allows for optimal linear adaptive estimation for the generality of system conditions (both balanced and unbalanced). The enhanced robustness over the standard CLMS is illustrated by simulations on both synthetic and real-world voltage sags.

An adaptive diffusion augmented complex least mean square (D-ACLMS) algorithm for collaborative processing of the generality of complex signals over distributed networks is proposed. The algorithm enables the estimation of both second order circular (proper) and noncircular (improper) signals within a unified framework of augmented complex statistics. The analysis shows that the performance advantage of the widely linear D-ACLMS over the strictly linear D-CLMS increases with the degree of noncircularity while maintaining similar performance for proper data. Simulations on both synthetic benchmark and real world noncircular data support the approach.

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.

An original experimental design is combined with a novel signal processing approach so as to provide cognitive clues in the study of auditory scene analysis and in the design of auditory brain computer interfaces. Volunteers attended a single auditory stimulus in a perceptually complex auditory environment of speech and music, wherein the experiment aim was to estimate the attended stimulus from recorded electroencephalogram (EEG). Unlike previous studies, the complex nature of the auditory environment does not allow for straightforward analysis that exploits convenient properties of the stimuli. To provide insight, synchronised neuronal activity was analysed within a novel signal processing framework that models energy and phase dynamics independently using empirical mode decomposition. By design, the proposed approach caters for higher order information and is suitable for nonstationary data, both critical properties in the analysis of cognitive activity. The proposed methodology achieved a median classification accuracy of 71% in a series of selective attention experiments with several volunteers.