Orthogonal Space Research Articles

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

On Orthogonal Fuzzy Interpolative Contractions with Applications to Volterra Type Integral Equations and Fractional Differential Equations

In this paper, orthogonal fuzzy versions are reported for some celebrated iterative mappings. We provide various concrete conditions on the real valued functions J,S:(0,1]→(−∞,∞) for the existence of fixed-points of (J,S)-fuzzy interpolative contractions. This way, many fixed point theorems are developed in orthogonal fuzzy metric spaces. We apply the (J,S)-fuzzy version of Banach fixed point theorem to demonstrate the existence and uniqueness of the solution. These results are supported with several non-trivial examples and applications to Volterra-type integral equations and fractional differential equations.

A self-consistent extension of flamelet theory for partially premixed combustion

Orthogonal coordinate systems represent an attractive alternative for the formulation of two-dimensional composition space equations for partially premixed combustion: They avoid the need of closure models for a cross scalar dissipation rate and allow for a direct recovery of the corresponding flamelet equations in the asymptotic limits of non-premixed and premixed combustion. Despite these remarkable features, this kind of coordinate system still presents some important unsolved issues, which have limited their application so far. These difficulties are mainly associated with i) the lack of an appropriate formalism for the definition of two variables having orthogonal gradients in the entire flame domain and ii) the absence of corresponding closure models for the gradients of these variables in two-dimensional composition space. In the present work, it is shown how a Lagrangian interpretation of the flamelet derivative allows solving both problems. More specifically, after the mixture fraction, Z, is adopted as first coordinate, the proposed approach allows to derive i) two-dimensional composition space equations for all reactive scalars, ii) a transport equation for a modified reaction progress variable, φ, satisfying the desired orthogonality condition, ∇Z·∇φ=0, and iii) two-dimensional composition space equations for gZ=|∇Z| and gφ=|∇φ|. The obtained set of two-dimensional equations in orthogonal composition space is general and it can describe different flames of interest. In order to illustrate the capabilities of the formulation, the equations are then specialized for planar flames with unity Lewis number, obtaining in this way a solvable set of composition space equations in orthogonal coordinates. After an appropriate numerical approach is introduced, the resulting 2D equations are solved to analyze the interaction between premixed flamelets with a strain rate prescribed along the Z-dimension controlling the interaction. Both flame structures and budgets of the scalar gradient equations are studied and the results provide new insights into the physics of scalar gradients in two-dimensional composition space. Finally, conceivable coupling strategies of the present formulation with CFD codes for the simulation of turbulent flames are discussed.

A BER formula of OTFS modulation with message‐passing receiver

AbstractThe orthogonal time frequency space (OTFS) modulation is a promising multicarrier waveform which can have good performance in high‐mobility scenarios. Among various receivers designed for the OTFS modulation, the message‐passing (MP) receiver has received much attention which can achieve a good bit error rate (BER) performance. However, the BER performance of MP receiver is usually obtained through simulations due to the lack of corresponding BER formula in the literature. This work studies the BER formula of OTFS modulation with MP receiver over doubly selective fading channel. Specifically, equivalent signal models of the MP receiver are first defined and the expression of the corresponding signal‐noise‐ratio (SNR) is derived, then the probability distribution function of the SNR is estimated by assuming the distribution of SNR as a lognormal distribution, then the methods to estimate the parameters of the distribution function are proposed, and finally the formula for calculating the BER of the OTFS modulation with MP receiver is obtained. Simulation results verify that the proposed BER formula is accurate.

Predictive Precoder Design for OTFS-Enabled URLLC: A Deep Learning Approach

This paper investigates the orthogonal time frequency space (OTFS) transmission for enabling ultra-reliable low-latency communications (URLLC). To guarantee excellent reliability performance, pragmatic precoder design is an effective and indispensable solution. However, the design requires accurate instantaneous channel state information at the transmitter (ICSIT) which is not always available in practice. Motivated by this, we adopt a deep learning (DL) approach to exploit implicit features from estimated historical delay-Doppler domain channels (DDCs) to directly predict the precoder to be adopted in the next time frame for minimizing the frame error rate (FER), that can further improve the system reliability without the acquisition of ICSIT. To this end, we first establish a predictive transmission protocol and formulate a general problem for the precoder design where a closed-form theoretical FER expression is derived serving as the objective function to characterize the system reliability. Then, we propose a DL-based predictive precoder design framework which exploits an unsupervised learning mechanism to improve the practicability of the proposed scheme. As a realization of the proposed framework, we design a DDCs-aware convolutional long short-term memory (CLSTM) network for the precoder design, where both the convolutional neural network and LSTM modules are adopted to facilitate the spatial-temporal feature extraction from the estimated historical DDCs to further enhance the precoder performance. Simulation results demonstrate that the proposed scheme facilitates a flexible reliability-latency tradeoff and achieves an excellent FER performance that approaches the lower bound obtained by a genie-aided benchmark requiring perfect ICSI at both the transmitter and receiver.

OTFS Signaling for SCMA With Coordinated Multi-Point Vehicle Communications

This paper investigates an uplink coordinated multi-point (CoMP) coverage scenario, in which multiple mobile users are grouped for sparse code multiple access (SCMA), and served by the remote radio head (RRH) in front of them and the RRH behind them simultaneously. We apply orthogonal time frequency space (OTFS) modulation for each user to exploit the degrees of freedom arising from both the delay and Doppler domains. As the signals received by the RRHs in front of and behind the users experience respectively positive and negative Doppler frequency shifts, our proposed OTFS-based SCMA (OBSCMA) with CoMP system can effectively harvest extra Doppler and spatial diversity for better performance. Based on maximum likelihood (ML) detector, we analyze the single-user average bit error rate (ABER) bound as the benchmark of the ABER performance for our proposed OBSCMA with CoMP system. We also develop a customized Gaussian approximation with expectation propagation (GAEP) algorithm for multi-user detection and propose efficient algorithm structures for centralized and decentralized detectors. Our proposed OBSCMA with CoMP system leads to stronger performance than the existing solutions. The proposed centralized and decentralized detectors exhibit effective reception and robustness under channel state information uncertainty.

Performance of an OVT inverter reliant on the switching frequency of the filtering inverter

The paper presents an inverter based on the orthogonal space vectors theory (OVT inverter) and its performance. The OVT inverter consists of two constituent twolevel 3-phase inverters: the main inverter (MI) and the auxiliary one (AI). The MI is controlled in a simple way to generate the stepped output voltage and the AI works as an active filter limiting the higher harmonics in the OVT output voltage. The filtering process is formed by the use of the orthogonal space vector theory. The output voltage of the OVT inverter takes the shape of a stepped voltage comparable to the voltage generated by multilevel inverters. The AI operates as a very effective active power filter (APF) of the OVT inverter output voltage. The AI power is significantly lower in comparison to the MI power nevertheless its switching frequency affects the OVT performance. The article describes studies in which the impact of changes in the switching frequency filter inverter on the operation of the orthogonal inverter and the output voltage THD. Increasing the filter inverter switching frequency improved performance and reduced the transformer size, which acts as a summing node.

Output voltage control of the OVT inverter

The paper presents the possibility of the output voltage control of an OVT inverter. The idea of the OVT inverter construction is based on the orthogonal space vectors theory. The OVT inverter is built of two components two-level three-phase inverters: the main inverter (MI) and the auxiliary one (AI). The output voltage of the OVT inverter takes the shape of a stepped voltage analogous to the voltage generated by multilevel inverters. The paper demonstrates possible control methods of the output voltage frequency and amplitude. The results obtained during simulation studies prove that the output voltage essential parameters may be a function of the DC voltage and the control circuitry permits easy defining a selection of voltage/frequency characteristics.

Hamming Distance Optimized Underwater Acoustic OTFS-IM Systems.

The orthogonal time frequency space (OTFS) modulation technique can provide reliable communication in time-varying channels. Due to the dispersive characteristics of underwater acoustic channels, this paper proposes an OTFS-IM underwater acoustic communication system based on Hamming distance optimization to reduce the impact of dispersion in underwater acoustic communication. Firstly, the OTFS-IM underwater acoustic communication system is introduced, which introduces index modulation into the Delay-Doppler (DD) domain to make the OTFS system have stronger anti-Delay-Doppler capability. In contrast, since there is index sequence redundancy in a specific index combination, a Hamming distance optimization model is used to eliminate the redundant combination in the specific index combination sequence and further improve the bit error rate performance of the system. In addition, the Hamming distance optimized OTFS-IM underwater acoustic communication system is verified by simulation analysis. The results show that the proposed Hamming distance optimized OTFS-IM can achieve more reliable bit error rate performance.

Development of Fixed Point Results for αΓ-F-Fuzzy Contraction Mappings with Applications

This manuscript contains several fixed point results for αΓ-F-fuzzy contractive mappings in the framework of orthogonal fuzzy metric spaces. The symmetric property guarantees that the distance function is consistent and does not favour any one direction in orthogonal fuzzy metric spaces. No matter how the points are arranged, it enables a fair assessment of the separations between all of them. In fixed point results, the symmetry condition is preserved for several types of contractive self-mappings. Moreover, we provide several non-trivial examples to show the validity of our main results. Furthermore, we solve non-linear fractional differential equations, the Atangana–Baleanu fractional integral operator and Fredholm integral equations by utilizing our main results.

On orthogonal polar spaces

Let P be a non-degenerate polar space. In [6], we introduced an intrinsic parameter of P, called the anisotropic gap, defined as the least upper bound of the lengths of the well-ordered chains of subspaces of P containing a frame; when P is orthogonal, we also defined two other parameters of P, called the elliptic and parabolic gap, both related to the universal embedding of P. In this paper, assuming that P is an orthogonal polar space, we prove that the elliptic and parabolic gaps can be described as intrinsic invariants of P without directly appealing to the embedding.

Controlled Bond Expansion for Density Matrix Renormalization Group Ground State Search at Single-Site Costs

DMRG ground state search algorithms employing symmetries must be able to expand virtual bond spaces by adding or changing symmetry sectors if these lower the energy. Traditional single-site DMRG does not allow bond expansion; two-site DMRG does, but at much higher computational costs. We present a controlled bond expansion (CBE) algorithm that yields two-site accuracy and convergence per sweep, at single-site costs. Given a matrix product state $\mathrm{\ensuremath{\Psi}}$ defining a variational space, CBE identifies parts of the orthogonal space carrying significant weight in $H\mathrm{\ensuremath{\Psi}}$ and expands bonds to include only these. CBE-DMRG uses no mixing parameters and is fully variational. Using CBE-DMRG, we show that the Kondo-Heisenberg model on a width 4 cylinder features two distinct phases differing in their Fermi surface volumes.

INVESTIGATION OF THE SIBERIAN FEDERAL DISTRICT OF THE RUSSIAN FEDERATION SOCIO-ECONOMIC DEVELOPMENT LEVEL FOR 2018-2021 USING THE METHODS OF MULTIVARIATE DATA ANALYSIS

The paper analyzes the socio-economic development of the regions dynamics level of the Siberian Federal District of the Russian Federation for the period from 2018 to 2021. The basis of the study was the Federal State Statistics Service for monitoring the socio-economic situation in the regions of the Russian Federation. The following indicators are considered: average monthly nominal accrued wages of employees (rubles) per year, per capita monetary income of the population (rubles) on average per year, labor force (thousand people) on average per year, number of unemployed (thousand people) on average per year, the volume of investments in fixed assets (million rubles) per year, retail trade turnover (million rubles) per year, goods of own production were shipped, works and services performed on their own per year (excluding VAT, excises and similar mandatory payments) million rubles. The methodology used includes: bringing monetary indicators for the period under review to the prices of 2021 in accordance with the deflator indices of the gross domestic product of the Russian Federation;
 reduction of the original set of non-orthogonal variables using factor analysis to a small-sized orthogonal factor space; determination of the Russian Federation federal districts centroids by the center of mass method for each year of the period under review; analysis of the Siberian Federal District internal structure dynamics and its position.

Spatial-Index Modulation Based Orthogonal Time Frequency Space System in Vehicular Networks

In this paper, a spatial-index modulation (SIM) based orthogonal time frequency space (OTFS) system, named SIM-OTFS, is proposed to enhance the effectiveness and the reliability of high mobility vehicular networks in intelligent transportation systems. The SIM-OTFS system adopts a three dimensional index modulation (IM) technique which utilizes the transmit antenna, delay, and Doppler indexes in the space and delay-Doppler domains, respectively, to achieve a higher transmission rate. Considering the characteristics of vehicular networks, we first present the SIM-OTFS system design and the corresponding signal processing. Then, we derive the average bit error rate (ABER) upper bound of the SIM-OTFS system by the union bound theory. Moreover, the diversity, the coding gain, and the complexity of the SIM-OTFS system are further investigated. Numerical results verify the theoretical analysis of the ABER and the diversity of the SIM-OTFS system, which shows the superiority of the SIM-OTFS system in the ABER performance over the multiple-input and multiple-output (MIMO) based OTFS (MIMO-OTFS) system. Meanwhile, the SIM-OTFS system realizes better performance than the spatial modulation (SM) and IM based orthogonal frequency division multiplexing (SM-OFDM-IM) system in high mobility vehicular communication with reasonable complexity sacrifice. Furthermore, the influence of the resolvable multipaths of the channel in vehicular networks on the ABER performance of the SIM-OTFS system is also illustrated.

New delay Doppler communication paradigm in 6G era: A survey of orthogonal time frequency space (OTFS)

In the 6G era, Space-Air-Ground Integrated Network (SAGIN) are anticipated to deliver global coverage, necessitating support for a diverse array of emerging applications in high-mobility, hostile environments. Under such conditions, conventional orthogonal frequency division multiplexing (OFDM) modulation, widely employed in cellular and Wi-Fi communication systems, experiences performance degradation due to significant Doppler shifts. To overcome this obstacle, a novel two-dimensional (2D) modulation approach, namely orthogonal time frequency space (OTFS), has emerged as a key enabler for future high-mobility use cases. Distinctively, OTFS modulates information within the delay-Doppler (DD) domain, as opposed to the time-frequency (TF) domain utilized by OFDM. This offers advantages such as Doppler and delay resilience, reduced signaling latency, a lower peak-to-average ratio (PAPR), and a reduced-complexity implementation. Recent studies further indicate that the direct interplay between information and the physical world in the DD domain positions OTFS as a promising waveform for achieving integrated sensing and communications (ISAC). In this article, we present an in-depth review of OTFS technology in the context of the 6G era, encompassing fundamentals, recent advancements, and future directions. Our objective is to provide a helpful resource for researchers engaged in the field of OTFS.