Orthogonal Space Research Articles

Sparse Linear Discriminant Analysis Based on lq Regularization

Linear discriminant analysis plays an important role in feature extraction, data dimensionality reduction, and classification. With the progress of science and technology, the data that need to be processed are becoming increasingly large. However, in high-dimensional situations, linear discriminant analysis faces two problems: the lack of interpretability of the projected data since they all involve all p features, which are linear combinations of all features, as well as the singularity problem of the within-class covariance matrix. There are three different arguments for linear discriminant analysis: multivariate Gaussian model, Fisher discrimination problem, and optimal scoring problem. To solve these two problems, this article establishes a model for solving the kth discriminant component, which first transforms the original model of Fisher discriminant problem in linear discriminant analysis by using a diagonal estimated matrix for the within-class variance in place of the original within-class covariance matrix, which overcomes the singularity problem of the matrix and projects it to an orthogonal projection space to remove its orthogonal constraints, and subsequently an lq norm regularization term is added to enhance its interpretability for the purpose of dimensionality reduction and classification. Finally, an iterative algorithm for solving the model and a convergence analysis are given, and it is proved that the sequence generated by the algorithm is descended and converges to a local minimum of the problem for any initial value.

Design and performance evaluation of a hybrid LiFi transceiver using OSTBC-based spatial multiplexing and transmit diversity

The need for data transmission systems that are high-speed, secure, and reliable increases as wireless communication technologies continue to advance rapidly. Light fidelity (LiFi) can meet this need through its several advantages, including high transmission rates, low electromagnetic interference, and indoor localization capabilities. This study proposes a hybrid LiFi transceiver, which utilizes LiFi technology using orthogonal space–time block coding (OSTBC), based on the spatial multiplexing (SM) and transmit diversity (Tx). The suggested hybrid LiFi transceiver aims to integrate the advantages of OSTBC, Tx diversity, and SM techniques to enhance the performance of the overall transceiver. The suggested transceiver plan to implement one of the diversity techniques, which are either Tx diversity or SM at a time. The selected technique will depends on the channel quality, optimizing the transceiver's throughput and ensuring effective utilization of the available spectrum. Through simulation results, this research aims to demonstrate that the proposed hybrid LiFi transceiver surpasses conventional LiFi transceivers that lack Tx diversity or SM. Anticipated outcomes of this research include superior data rates, enhanced reliability, and increased resilience against real channel impairments. These findings hold substantial promise for applications that requires high-speed and reliable communication, including smart homes, the Internet of Things, and forthcoming 5 G/6 G communication networks.

A Novel Joint Channel Estimation and Symbol Detection Receiver for Orthogonal Time Frequency Space in Vehicular Networks.

A vehicular network embodies a specialized variant of wireless network systems, characterized by its capability to facilitate inter-vehicular communication and connectivity with the encompassing infrastructure. With the rapid development of wireless communication technology, high-speed and reliable communication has become increasingly important in vehicular networks. It has been demonstrated that orthogonal time frequency space (OTFS) modulation proves effective in addressing the challenges posed by high-mobility environments, as it transforms the time-varying channels into the delay-Doppler domain. Motivated by this, in this paper, we focus on the theme of integrated sensing and communication (ISAC)-assisted OTFS receiver design, which aims to perform sensing channel estimation and communication symbol detection. Specifically, the estimation of the sensing channel is accomplished through the utilization of a deep residual denoising network (DRDN), while the communication symbol detection is performed by orthogonal approximate message passing (OAMP) processing. The numerical results demonstrate that the proposed ISAC system exhibits superior performance and robustness compared to traditional methods, with a lower complexity as well. The proposed system has great potential for future applications in wireless communication systems, especially in challenging scenarios with high mobility and interference.

Some recent and new fixed point results on orthogonal metric-like space

In this paper, we give some recent and new results for some contraction mappings on O−complete metric-like space and also we give illustrative examples. At the end, we give an application to show the existence of a solution of a differential equation.

Fixed Point Results via Orthogonal (α − 𝔶 − 𝔾)-Contraction in Orthogonal Complete Metric Space

In this publication, we establish a suitable symmetry structure for orthogonal (α−y−G)-contractive mappings and prove fixed point results for an orthogonal (α−y−G)-contractive via orthogonal metric spaces. We give an application to strengthen our main results from the existing literature to prove the existence of a unique analytical solution to the differential equation by converting it into an integral equation through fixed point analysis.

Orthogonal neutrosophic 2-metric spaces

In this study, we introduce the notion of an orthogonal neutrosophic 2-metric space and prove the common fixed-point theorem on an orthogonal neutrosophic 2-metric space. From the obtained results, we give an example to support our results.

DL-Based OTFS Signal Detection in Presence of Hardware Impairments

Orthogonal time frequency space (OTFS) modulation is an emerging technique for next-generation communication due to its robustness to the doubly dispersive channels under high mobility scenarios. We have designed and analyzed a deep learning (DL)-based OTFS system (DL-OTFS) in the presence of hardware impairments (HI) such as in-phase and quadrature-phase (IQ) component mismatch and DC offset. Further, data augmentation is also considered for the proposed DL-OTFS to enhance the system performance. Numerical results show that the DL-OTFS model can efficiently learn the input and output relation and leads to improved bit error rate (BER) performance than the conventional message passing and minimum mean square error (MMSE)-based receiver with and without HI.

A Low-Complexity OTFS Channel Estimation Method for Fractional Delay-Doppler Scenarios

We propose a low-complexity channel estimation method for Orthogonal Time Frequency Space (OTFS) modulation based systems. Two important aspects of OTFS modulation, namely, (i) fine delay-Doppler (DD) resolution which results in path separation and, (ii) the separability of the effective DD domain channel gain expression into delay domain and Doppler domain terms, are used to propose an iterative channel estimation method which separately estimates the delay, Doppler shift and the channel gain for each channel path. The proposed estimation does not require matrix inversion and therefore has lower complexity when compared to other known methods. The proposed method is shown to achieve a normalized mean square estimation error (NMSE) performance better than other methods.

Channel Estimator and Nonlinear Detector for mmWave Beamformed OTFS Systems in High Mobility Scenarios

Real-time data-intensive applications in highly mobile environments with high data rates and low latency are one of the key requirements for beyond fifth-generation (5 G) communications. Towards this end, a combination of the millimeter wave (mmWave) band with large available spectrum and orthogonal time-frequency space (OTFS) modulation, which offers robust performance in high mobility scenarios, can be envisioned as a promising solution. However, employing OTFS in the mmWave frequency band is challenging in presence of nonlinear distortions attributed to radio frequency (RF) power amplifier (PA) and complex equalization involved in OTFS receiver in the delay-Doppler domain. The nonlinearity effect of PA is inevitable in mmWave systems due to the high operating frequency and large bandwidth. The nonlinear distortions induce interference and make the target distribution analytically intractable. In this paper, we propose a receiver design for mmWave OTFS systems that aims to address the challenges posed by the nonlinear distortions of the RF PA. The proposed receiver design leverages the use of a large-scale antenna array to decouple the received signal into multiple branches through beamforming. Specifically, we first derive the input-output relation in the delay-Doppler domain with nonlinearity. Then, a combination of maximal-ratio combining (MRC) and iterative particle filter (PF)-based nonlinear detector is proposed to address the issue. We also design a pilot pattern considering the additional interference from the PA impairment. Furthermore, we reformulate the threshold-based channel estimation to incorporate the nonlinearly distorted pilot in its estimation process. Extensive numerical results demonstrate the proposed algorithm's reliability and superiority over existing digital predistorters (DPD) and conventional linear detectors.

Channel Estimation and Turbo Equalization for Coded OTFS and OFDM: A Comparison

In this letter, we study joint channel estimation and turbo equalization for coded orthogonal time frequency space (OTFS) and orthogonal frequency division multiplexing (OFDM). To the best of our knowledge, this is the first study that compares performance of OTFS and OFDM coded systems using the state-of-the-art message passing (MP) and soft minimum mean square error (MMSE) equalizers, under imperfect channel estimation. We show that the commonly used threshold-based channel estimator incurs noticeable performance loss for OTFS. Hence, we propose a basis expansion model (BEM) channel estimation technique for OTFS, which improves performance of the threshold-based channel estimator and yields a superior performance to OFDM when using small modulations of BPSK and QPSK. Our work also reveals that OTFS can perform inferior to OFDM as the modulation order increases. This is due to the increased 2D detection complexity of OTFS.

Expectation Propagation Aided Model Driven Learning for OTFS Signal Detection

Orthogonal time frequency space (OTFS) modulation, which modulates constellation symbols in the delay-Doppler (DD) domain, is capable of outperforming conventional multi-carrier schemes in the context of high mobility communication scenarios. However, due to its special structure, it poses serious challenge to its signal detection, especially in a rich-scattering environment. To address this issue, in this paper, an expectation propagation (EP) aided model-driven deep learning framework is proposed for OTFS signal detection. Specifically, each iteration of the EP algorithm is unfolded into a layer-wise network and trainable parameters are added to this network to accelerate the convergence of the algorithm, as well as improve the detection performance. Simulation results show that the proposed EP aided model-driven network is capable of providing significant performance gain over conventional EP counterparts.

Hybrid Feature-Learning-Based PSO-PCA Feature Engineering Approach for Blood Cancer Classification.

Acute lymphoblastic leukemia (ALL) is a lethal blood cancer that is characterized by an abnormal increased number of immature lymphocytes in the blood or bone marrow. For effective treatment of ALL, early assessment of the disease is essential. Manual examination of stained blood smear images is current practice for initially screening ALL. This practice is time-consuming and error-prone. In order to effectively diagnose ALL, numerous deep-learning-based computer vision systems have been developed for detecting ALL in blood peripheral images (BPIs). Such systems extract a huge number of image features and use them to perform the classification task. The extracted features may contain irrelevant or redundant features that could reduce classification accuracy and increase the running time of the classifier. Feature selection is considered an effective tool to mitigate the curse of the dimensionality problem and alleviate its corresponding shortcomings. One of the most effective dimensionality-reduction tools is principal component analysis (PCA), which maps input features into an orthogonal space and extracts the features that convey the highest variability from the data. Other feature selection approaches utilize evolutionary computation (EC) to search the feature space and localize optimal features. To profit from both feature selection approaches in improving the classification performance of ALL, in this study, a new hybrid deep-learning-based feature engineering approach is proposed. The introduced approach integrates the powerful capability of PCA and particle swarm optimization (PSO) approaches in selecting informative features from BPI mages with the power of pre-trained CNNs of feature extraction. Image features are first extracted through the feature-transfer capability of the GoogleNet convolutional neural network (CNN). PCA is utilized to generate a feature set of the principal components that covers 95% of the variability in the data. In parallel, bio-inspired particle swarm optimization is used to search for the optimal image features. The PCA and PSO-derived feature sets are then integrated to develop a hybrid set of features that are then used to train a Bayesian-based optimized support vector machine (SVM) and subspace discriminant ensemble-learning (SDEL) classifiers. The obtained results show improved classification performance for the ML classifiers trained by the proposed hybrid feature set over the original PCA, PSO, and all extracted feature sets for ALL multi-class classification. The Bayesian-optimized SVM trained with the proposed hybrid PCA-PSO feature set achieves the highest classification accuracy of 97.4%. The classification performance of the proposed feature engineering approach competes with the state of the art.

Superimposed Perfect Binary Array-Aided Channel Estimation for OTFS Systems.

Orthogonal time-frequency space (OTFS) modulation outperforms orthogonal frequency-division multiplexing in high-mobility scenarios through better channel estimation. Current superimposed pilot (SP)-based channel estimation improves the spectral efficiency (SE) when compared to that of the traditional embedded pilot (EP) method. However, it requires an additional non-superimposed EP delay-Doppler frame to estimate the delay-Doppler taps for the following SP-aided frames. To handle this problem, we propose a channel estimation method with high SE, which superimposes the perfect binary array (PBA) on data symbols as the pilot. Utilizing the perfect autocorrelation of PBA, channel estimation is performed based on a linear search to find the correlation peaks, which include both delay-Doppler tap information and complex channel gain in the same superimposed PBA frame. Furthermore, the optimal power ratio of the PBA is then derived by maximizing the signal-to-interference-plus-noise ratio (SINR) to optimize the SE of the proposed system. The simulation results demonstrate that the proposed method can achieve a similar channel estimation performance to the existing EP method while significantly improving the SE.

Timing synchronization based on Radon–Wigner transform of chirp signals for OTFS systems

Accurate timing synchronization is fundamental for the orthogonal time frequency space (OTFS) system to work properly, but the severe Doppler effect makes it a challenging task in high-mobility environments and high carrier frequency systems. To tackle this challenge, we propose a timing synchronization algorithm based on the Radon-Wigner transform (RWT) of chirp signals for OTFS modulation systems in high-mobility environments, which makes full use of the anti-Doppler performance of chirp signals and the energy aggregation characteristics of RWT to achieve accurate timing synchronization in high-Doppler conditions. Firstly, a training sequence consisting of a dual-chirp signal is transformed from the time domain to the delay-Doppler domain and then superimposed on the user data at the transmitter side. Subsequently, the timing synchronization of the OTFS system is implemented by detecting the peak value of the chirp signal after RWT and computing the difference in peak position between the transmitted and received chirp signals. Simulation results demonstrate that the proposed scheme offers a superior performance gain in low signal-to-noise ratio and high-Doppler conditions.

Data-Aided CSI Estimation Using Affine-Precoded Superimposed Pilots in Orthogonal Time Frequency Space Modulated MIMO Systems

An orthogonal affine-precoded superimposed pilot (AP-SIP)-based architecture is developed for the cyclic prefix (CP)-aided single input single output (SISO) and multiple input multiple output (MIMO) orthogonal time frequency space (OTFS) systems relying on arbitrary transmitter-receiver (Tx-Rx) pulse shaping. The data and pilot symbol matrices are affine-precoded and superimposed in the delay Doppler (DD)-domain followed by the development of an end-to-end DD-domain relationship for the input-output symbols. At the receiver, the decoupled pilot and data symbol are extracted by employing orthogonal precoder matrices, which eliminates the mutual interference. Furthermore, a novel pilot-aided Bayesian learning (PA-BL) technique is conceived for the channel state information (CSI) estimation of SISO OTFS systems based on the expectation-maximization (EM) technique. Subsequently, a data-aided Bayesian learning (DA-BL)-based joint CSI estimation and data detection technique is proposed, which beneficially harnesses the estimated data symbols for improved CSI estimation. In this scenario our sophisticated data detection rule also integrates the CSI uncertainty of channel estimation into our the linear minimum mean square error (LMMSE) detectors. The AP-SIP framework is also extended to MIMO OTFS systems, wherein the DD-domain input matrix is affine-precoded for each transmit antenna (TA). Then an EM algorithm-based PA-BL scheme is derived for simultaneous row-group sparse CSI estimation for this system, followed also by our data-aided DA-BL scheme that performs joint CSI estimation and data detection. Moreover, the Bayesian Cramer-Rao bounds (BCRBs) are also derived for both SISO as well as MIMO OTFS systems. Finally, simulation results are presented for characterizing the performance of the proposed CSI estimation techniques in a range of typical settings along with their bit error rate (BER) performance in comparison to an ideal system having perfect CSI.

OTFS Waveform Based on 3-D Signal Constellation for Time-Variant Channels

Orthogonal time frequency space (OTFS) is a new waveform modulated in the delay-Doppler domain that can achieve better communication performance than the conventional waveform, such as orthogonal frequency division multiplexing (OFDM), in high-mobility scenarios. In this letter, we introduce the three-dimensional (3-D) mapper into the conventional OTFS modulation scheme to further reduce the bit error rate (BER) at higher modulation orders. Moreover, the minimum Euclidean distance of the 3-D signal constellations and detector complexity analysis of the proposed 3D-OTFS are presented. The simulation results demonstrate that the proposed 3D-OTFS system outperforms the conventional OTFS system in terms of BER performance for higher modulation orders in time-variant channels, such as the Extended Vehicular A (EVA) channel and the Extended Pedestrian A (EPA) channel. Furthermore, performance comparisons between the proposed 3D-OTFS and 3D-OFDM in different time-variant channels show that the proposed 3D-OTFS is better suited for high-mobility communication scenarios.

Graph Neural Network Assisted Efficient Signal Detection for OTFS Systems

In this letter, an efficient graph neural network (GNN) assisted detector is conceived for the orthogonal time frequency space (OTFS) system. Specifically, the transmit symbols are viewed as the nodes of GNN, obtaining the detection results through aggregation, update, and output modules. Firstly, the aggregation module is employed to weigh the connections between adjacent nodes. Subsequently, the update module amends node features according to the calculated connection value and the node’s information. Finally, after a certain number of iterations, the output module classifies nodes relying on the final features to realize signal detection. Simulation results confirm that the proposed GNN-assisted detector outperforms the latest intelligent detector by around 1 dB.

Twisted GGP problems and conjectures

In a series of three earlier papers, we considered a family of restriction problems for classical groups (over local and global fields) and proposed precise answers to these problems using the local and global Langlands correspondence. These restriction problems were formulated in terms of a pair $W \subset V$ of orthogonal, Hermitian, symplectic, or skew-Hermitian spaces. In this paper, we consider a twisted variant of these conjectures in one particular case: that of a pair of skew-Hermitian spaces $W = V$.

Joint channel estimation and data detection for high rate orthogonal time frequency space systems

SummaryThis paper proposes a new pilot pattern in the delay‐Doppler (DD) domain for the orthogonal time frequency space (OTFS) system. In contrast to the embedded‐pilot schemes, guard intervals are not used so as increase the spectral efficiency. Also, compared to the superimposed design where data symbols and pilots are arranged on the entire DD grid, in the proposed rearrangement, the number of pilots used is only spread over a subgrid of the DD grid. Hence, the interference of pilots with data symbols is reduced. Afterwards, an algorithm for channel estimation (CE) and symbol detection in the DD domain benefiting from the sparsity of the DD channel is designed. The sparse CE step is formulated as a specific marginalization of the maximum a posteriori (MAP) criterion by providing a Bayesian approach via the mean‐field approximation that involves the variational Bayesian expectation maximization (VB‐EM) algorithm. Detection of data symbols is done using a low complexity MP algorithm. We also propose an interference cancellation (IC) scheme to mitigate contamination of data by pilots that is run after each CE step. To achieve a high CE accuracy, based on the mean mutual incoherence property (MIP), a pilot optimization problem for OTFS is formulated and develop a simulated annealing‐based algorithm to solve it. Finally, simulation results show that the proposed scheme achieves a good compromise between spectral efficiency, complexity, and performance in terms of bit error rate (BER) and normalized mean square error (NMSE) when compared to literature benchmarks.

On Optimal Convergence Rates for Discrete Minimizers of the Gross–Pitaevskii Energy in Localized Orthogonal Decomposition Spaces

On Optimal Convergence Rates for Discrete Minimizers of the Gross–Pitaevskii Energy in Localized Orthogonal Decomposition Spaces