Orthogonal Space Research Articles

Data-driven deep learning for OTFS detection

Recently, orthogonal time frequency space (OTFS) was presented to alleviate severe Doppler effects in high mobility scenarios. Most of the current OTFS detection schemes rely on perfect channel state information (CSI). However, in real-life systems, the parameters of channels will constantly change, which are often difficult to capture and describe. In this paper, we summarize the existing research on OTFS detection based on data-driven deep learning (DL) and propose three new network structures. The presented three networks include a residual network (ResNet), a dense network (DenseNet), and a residual dense network (RDN) for OTFS detection. The detection schemes based on data-driven paradigms do not require a model that is easy to handle mathematically. Meanwhile, compared with the existing fully connected-deep neural network (FC-DNN) and standard convolutional neural network (CNN), these three new networks can alleviate the problems of gradient explosion and gradient disappearance. Through simulation, it is proved that RDN has the best performance among the three proposed schemes due to the combination of shallow and deep features. RDN can solve the issue of performance loss caused by the traditional network not fully utilizing all the hierarchical information.

Joint multi-domain channel estimation based on sparse Bayesian learning for OTFS system

Since orthogonal time-frequency space (OTFS) can effectively handle the problems caused by Doppler effect in high-mobility environment, it has gradually become a promising candidate for modulation scheme in the next generation of mobile communication. However, the inter-Doppler interference (IDI) problem caused by fractional Doppler poses great challenges to channel estimation. To avoid this problem, this paper proposes a joint time and delay-Doppler (DD) domain based on sparse Bayesian learning (SBL) channel estimation algorithm. Firstly, we derive the original channel response (OCR) from the time domain channel impulse response (CIR), which can reflect the channel variation during one OTFS symbol. Compare with the traditional channel model, the OCR can avoid the IDI problem. After that, the dimension of OCR is reduced by using the basis expansion model (BEM) and the relationship between the time and DD domain channel model, so that we have turned the underdetermined problem into an overdeter-mined problem. Finally, in terms of sparsity of channel in delay domain, SBL algorithm is used to estimate the basis coefficients in the BEM without any priori information of channel. The simulation results show the effectiveness and superiority of the proposed channel estimation algorithm.

DNN-Based Fractional Doppler Channel Estimation for OTFS Modulation

In this paper, we proposed a deep neural network (DNN) based fractional Doppler channel estimation scheme for orthogonal time frequency space (OTFS) modulation in the air-to-ground communication scenario with high-dynamic Doppler. Based on the zero-padded OTFS structure, the traditional pilot pattern with guard symbols is adopted. The received pilots in the OTFS domain are used as the inputs of the network to estimate the channel parameters which are used in the MRC algorithm to demodulate the signal. In our proposed method, it can achieve the similar performance with 14 dB boost of pilot energy comparing with the ideal channel estimation case, while the conventional method requires 30 dB higher. Both the accuracy and generalization ability of the DNN network are validated.

Low PAPR channel estimation for OTFS with scattered superimposed pilots

Orthogonal time frequency space (OTFS) modulation has been proven to be superior to traditional orthogonal frequency division multiplexing (OFDM) systems in high-speed communication scenarios. However, the existing channel estimation schemes may results in poor peak to average power ratio (PAPR) performance of OTFS system or low spectrum efficiency. Hence, in this paper, we propose a low PAPR channel estimation scheme with high spectrum efficiency. Specifically, we design a multiple scattered pilot pattern, where multiple low power pilot symbols are superimposed with data symbols in delay-Doppler domain. Furthermore, we propose the placement rules for pilot symbols, which can guarantee the low PAPR. Moreover, the data aided iterative channel estimation was invoked, where joint channel estimation is proposed by exploiting multiple independent received signals instead of only one received signal in the existing scheme, which can mitigate the interference imposed by data symbols for channel estimation. Simulation results shows that the proposed multiple scattered pilot aided channel estimation scheme can significantly reduce the PAPR while keeping the high spectrum efficiency.

Multi-Channel Implantable Cubic Rectenna MIMO System With CP Diversity in Orthogonal Space for Enhanced Wireless Power Transfer in Biotelemetry

For wireless power transfer (WPT), an implantable cubic rectenna multi-input-multi-output (MIMO) system (CRMS), operating in dual industrial, scientific, and medical (ISM) frequency bands of 2.45 and 5.8 GHz, is presented as a receiver ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$R_{x}$ </tex-math></inline-formula> ). CRMS is evolved to the proposed biocompatible full-package cubic rectenna (FPCR) by including the power and data management circuit modules. The dual frequency bands differentiated by orthogonal circular polarization (CP) in a spatial quadrature would serve as propagation channels for power, data, and control signals. Four dual-branch rectifiers are designed using both distributed and lumped elements on the backside of individual antenna elements. To demonstrate the feasibility of the proposed work as an integrated system for WPT, the structures are fabricated individually, where the transmitter ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$T_{x}$ </tex-math></inline-formula> ), which is an external antenna, and rectifiers are tested in free space and FPCR in a custom-made canonical phantom. After validation of individual measurements, power delivery from the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$T_{x}$ </tex-math></inline-formula> to the FPCR is conducted experimentally to measure the integrated system’s radiofrequency (RF)-dc total conversion efficiency (TCE). The FPCR is designed to receive low RF power ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$P_{\mathrm {RF}}$ </tex-math></inline-formula> ) of 0 dBm, where a single cubic rectenna element (CRE) provides dc power ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$P_{\mathrm {DC}}$ </tex-math></inline-formula> ) of 0.26 and 0.33 mW at 2.45 and 5.8 GHz, respectively. Thereafter, the interconnection of CREs in a series/parallel configuration improved <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$P_{\mathrm {DC}}$ </tex-math></inline-formula> to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$0.5\mathbf {/}1.39$ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$0.51\mathbf {/}1.64$ </tex-math></inline-formula> mW, respectively. Additionally, human safety is considered by evaluating a maximum permissible exposure (MPE) limit and specific absorption rate (SAR) distribution in a canonical tissue model when both <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$T_{x}$ </tex-math></inline-formula> and FPCR are excited with 1 W power.

Generalized integral type mappings on orthogonal metric spaces

This study is devoted to investigate the problem whether the existence and uniqueness of integral type contraction mappings on orthogonal metric spaces. At the end, we give an example to illustrative our main result.

Orthogonality Spaces Associated with Posets

An orthogonality space is a set equipped with a symmetric, irreflexive relation called orthogonality. Every orthogonality space has an associated complete ortholattice, called the logic of the orthogonality space. To every poset, we associate an orthogonality space consisting of proper quotients (that means, nonsingleton closed intervals), equipped with a certain orthogonality relation. We prove that a finite bounded poset is a lattice if and only if the logic of its orthogonality space is an orthomodular lattice. We prove that that a bounded poset is a chain if and only if the logic of the associated orthogonality space is a Boolean algebra.

A Comprehensive Experimental Emulation for OTFS Waveform RF-Impairments.

The orthogonal time-frequency space (OTFS) waveform exceeds the challenges that face orthogonal frequency division multiplexing (OFDM) in a high-mobility environment with high time-frequency dispersive channels. Since radio frequency (RF) impairments have a direct impact on waveform behavior, this paper investigates the experimental implementation of RF-impairments that affect OTFS waveform performance and compares them to the OFDM waveform as a benchmark. Firstly, the doubly-dispersive channel effect is analyzed, and then an experimental framework is established for investigating the impact of RF-impairments, including non-linearity, carrier frequency offset (CFO), I/Q imbalances, DC-offset, and phase noise are considered. The experiments were conducted in a real indoor wireless environment using software-defined radio (SDR) at carrier frequencies of 2.4 GHz and 5 GHz based on the Keysight EXG X-Series devices. The comparison of the performances of OFDM and OTFS in the presence of RF-impairments reveals that OTFS significantly outperforms OFDM.

Quadrature spatial modulation based orthogonal time frequency spatial system

In this paper, we take the spatial and delay-Doppler dimensions into account and propose a novel transmission scheme combining quadrature spatial modulation (QSM) and orthogonal time frequency space (OTFS) to improve the spectral efficiency, achieve the diversity gains, and resist strong Doppler effect in high-mobility scenes. The system model and the specific signal transmission process of the OTFS with QSM (QSM-OTFS) scheme are presented first. Then, the theoretical average bit error rate (ABER) analysis for QSM-OTFS system is provided. Additionally, a linear enhanced minimum mean square error (EMMSE) detection is investigated to reduce the detection complexity in the QSM-OTFS system. Numerical results illustrate that the QSM-OTFS system is able to obtain an improved ABER performance compared with the typical spatial modulation (SM) based OTFS (SM-OTFS) and that the proposed EMMSE detection outperforms the other linear detection algorithms under the same spectral efficiency.

Nonlinear companding transform for PAPR reduction of OTFS signals

Abstract Peak-to-Average Power Ratio (PAPR) is one of the major limitations of an Orthogonal time frequency space (OTFS) signals. To reduce the PAPR, a simple non-linear companding transform is proposed in this paper. The proposed companding transform mainly compresses the large amplitudes and slightly enhances the small amplitudes such that the peak power reduces but the average power remains unaltered. To validate the proposed companding transform, we compare the our proposed companding with the orthogonal frequency division multiplexing (OFDM), normalized μ− law, and clipping methods. Simulations demonstrate that, the proposed companded OTFS transform provides better PAPR than the proposed companded OFDM, μ− law companded OTFS and clipping methods.

Solution to Integral Equation in an O-Complete Branciari b-Metric Spaces

In this paper, we prove fixed point theorem via orthogonal Geraghty type α-admissible contraction map in an orthogonal complete Branciari b-metric spaces context. An example is presented to strengthen our main result. We provided an application to find the existence and uniqueness of a solution to the Volterra integral equation. We have compared the approximate solution and exact solution numerically.

Deep Learning-Based Signal Detection for Underwater Acoustic OTFS Communication

Orthogonal time frequency space (OTFS) is a novel two-dimensional (2D) modulation technique that provides reliable communications over time- and frequency-selective channels. In underwater acoustic (UWA) channel, the multi-path delay and Doppler shift are several magnitudes larger than wireless radio communication, which will cause severe time- and frequency-selective fading. The receiver has to recover the distorted OTFS signal with inter-symbol interference (ISI) and inter-carrier interference (ICI). The conventional UWA OTFS receivers perform channel estimation explicitly and equalization to detect transmitted symbols, which requires prior knowledge of the system. This paper proposes a deep learning-based signal detection method for UWA OTFS communication, in which the deep neural network can recover the received symbols after sufficient training. In particular, it cascades a convolutional neural network (CNN) with skip connections (SC) and a bidirectional long short-term memory (BiLSTM) network to perform signal recovery. The proposed method extracts feature information from received OTFS signal sequences and trains the neural network for signal detection. The numerical results demonstrate that the SC-CNN-BiLSTM-based OTFS detection method performs with a lower bit error rate (BER) than the 2D-CNN, FC-DNN, and conventional signal detection methods.

Bayesian Neural Network Detector for an Orthogonal Time Frequency Space Modulation

The orthogonal time frequency space (OTFS) modulation is proposed for beyond 5G wireless systems to deal with high mobility communications. The existing low complexity OTFS detectors’ performance is suboptimal in rich scattering environments where there are a large number of moving reflectors that reflect the transmitted signal towards the receiver. In this letter, we propose an OTFS detector, referred to as the BPICNet OTFS detector that integrates NN, Bayesian inference, and parallel interference cancellation concepts. Simulation results show that the proposed detector outperforms the state-of-the-art.

Orthogonal Time Frequency Space Detection via Low-Complexity Expectation Propagation

Orthogonal time frequency space (OTFS) can provide better error performance than orthogonal frequency division modulation in the high-speed scenario. However, the two-dimensional convolution between the information-bearing symbols and the channel response in the delay-Doppler domain induces high-complexity detection, which hinders the implementation of the OTFS. In this paper, expectation propagation (EP) is introduced in the OTFS system to improve the reliability compared to the message passing approaches. The low-complexity EP detection with a log-linear order is proposed by observing the sparsity and the block quasi-banded matrix structure of the time-domain matrix. The exploitation of the lower-upper factorization for the banded matrix, the forward or the back substitution algorithm for the lower or the upper triangular matrices reduces the complexity of the large-scale matrix inversion involved in the EP detection from a cubic order to a linear order. The error performance of the proposed scheme is further analyzed by utilizing the state evolution and discussed under the different frame sizes. Moreover, the proposed low-complexity detector is then explored together with the likelihood decoder in an iterative manner to further improve the reliability of the proposed receiver. Finally, the better error performance and the lower computational complexity of the proposed receiver are validated by the numerical results.

OTFS Transceiver Design and Sparse Doubly-Selective CSI Estimation in Analog and Hybrid Beamforming Aided mmWave MIMO Systems

Orthogonal time frequency space (OTFS) waveform based millimeter wave (mmWave) MIMO systems are capable of achieving high data rates in high-mobility scenarios. Hence, transceivers are designed for both analog beamforming (AB) and hybrid beamforming (HB), where we commence by deriving the delay-Doppler (DD)-domain input-output relationship considering a delay-Doppler-angular domain channel model. Subsequently, a novel two-stage procedure is developed for transmit beamformer (TBF)/ precoder (TPC) and receiver combiner (RC) design, and for estimating the DD-domain’s equivalent channel state information (CSI). The key feature of the proposed framework is that the RF TBF/ TPC and RC design maximizes the directional beamforming gains. It is also demonstrated that the low-dimensional baseband CSI of the DD-domain becomes sparse for mmWave-AB MIMO OTFS systems, and block-sparse for mmWave-HB MIMO OTFS systems. Subsequently, Bayesian learning (BL) and block-sparse BL (BS-BL) solutions are developed for improved CSI estimation. We also derive the Bayesian Cramer-Rao lower bounds (BCRLB) for benchmarking the mean-squared-error (MSE) of the CSI estimates. Finally, our simulation results demonstrate the improved efficacy of the proposed transceiver designs and confirm the enhanced CSI estimation performance of the BL-based schemes over other competing sparse signal recovery schemes.

Spectral Efficiency of OTFS Based Orthogonal Multiple Access With Rectangular Pulses

In this paper, we consider Orthogonal Time Frequency Space (OTFS) modulation based multiple-access (MA). We specifically consider Orthogonal MA methods (OMA) where the user terminals (UTs) are allocated non-overlapping physical resource in the delay-Doppler (DD) and/or time-frequency (TF) domain. Well known OMA methods include, i) Guard Band (GB) based MA (GBMA), ii) Interleaved Delay Doppler MA (IDDMA), iii) Interleaved Time Frequency MA (ITFMA). The spectral efficiency (SE) performance of these OMA methods has <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">only</i> been studied for <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ideal</i> transmit and receive pulses. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">With ideal pulses</i> , IDDMA and ITFMA are free from multi-user interference (MUI) and unlike GBMA they <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">do not</i> use guard bands in the DD/TF domain. Since ideal pulses are not realizable, in this paper we study the SE performance of these OMA methods with <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">practical rectangular pulses</i> . For these OMA methods, we derive the expression for the received DD/TF domain symbols at the base station (BS) receiver and the effective DD domain channel matrix when rectangular pulses are used. Study of these expressions reveals the presence of MUI when rectangular pulses are used, although the amount of MUI in IDDMA is observed to be significantly smaller than that in ITFMA. We also derive the expression for the achievable sum SE. Through simulations, for practical values of the received signal-to-noise ratio, we observe that with rectangular pulses the sum SE achieved by the IDDMA method is significantly higher than that achieved by the ITFMA and GBMA methods.

Delay-Doppler and Angular Domain 4D-Sparse CSI Estimation in OTFS Aided MIMO Systems

A convenient delay, Doppler and angular-(DDA) domain representation of the multiple-input multiple-output (MIMO) wireless channel is conceived for deriving the end to end relationship in the delay-Doppler (DD)-domain for orthogonal time frequency space (OTFS)-based communications. Subsequently, a time-domain pilot based model is developed for estimating the DDA-domain channel state information (CSI) of our MIMO OTFS system. The key differentiating feature of the CSI estimation model derived is its ability to exploit the 4-dimensional (4D)-sparsity arising in the DDA-domain, given the limited number of dominant scatterers. Furthermore, the training overhead of the proposed framework is low, and the pilot placement is quite flexible, necessitating no guard-interval. Finally, an orthogonal matching pursuit (OMP) framework is employed for 4D-sparse CSI acquisition, followed by deriving the Oracle minimum mean squared error (Oracle-MMSE) and its Bayesian Cramer-Rao lower bound (BCRLB). Our simulation results confirm the improved CSI estimation performance attained over the benchmarks.

Gaussian AMP Aided Model-Driven Learning for OTFS System

Orthogonal time frequency space (OTFS) modulation, which converts the fast time-varying channel in the time-frequency domain into a time-invariant channel in the delay-Doppler(DD) domain, can mitigate the fading caused by the dynamic change of the channel. Nevertheless, due to the influence of the Doppler effect, it is challenging for traditional algorithms to acquire the desired detection results. To this end, this letter proposes a Gaussian approximate message passing (GAMP) aided model-driven signal detection algorithm. It first expands the existing GAMP algorithm into a corresponding deep neural network. Then, a unique set of trainable parameters, including one damping factor and two stepping factors, are constructed to improve the algorithm’s convergence. Simulation results demonstrate that the designed model-driven signal detection algorithm outperforms the original GAMP by about 2 dB at the bit error ratio (BER) of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$1\times 10^{-4}$ </tex-math></inline-formula> .

Low-Complexity LMMSE Receiver Design for Practical-Pulse-Shaped MIMO-OTFS Systems

Orthogonal time frequency space modulation (OTFS) scheme establishes reliable communication in a rapidly time-varying wireless channel with a high Doppler spread. We design a low-complexity linear minimum mean squared error (LMMSE) receiver for practical-pulse-shaped multiple-input multiple-output (MIMO)-OTFS systems. The proposed receiver exploits the inherent channel sparsity and the channel-agnostic structure of matrices involved in the LMMSE receiver, and has only a log-linear complexity. It provides exactly the same solution, and hence the same bit error rate (BER), as that of the conventional LMMSE receiver with a cubic order of complexity. We also derive, by using the Taylor series expansion and the results from random matrix theory, a tight closed-form approximation for the post-processing signal-to-noise-plus-interference ratio (SINR) expression of the proposed receiver. This expression is derived by assuming imperfect receive channel state information. We show using extensive numerical investigations that the derived SINR expression, when averaged over multiple channel realizations, accurately characterizes the BER of a MIMO-OTFS system.

Orthogonal Delay-Doppler Division Multiplexing Modulation

Inspired by the orthogonal time frequency space (OTFS) modulation, in this paper, we consider designing a multicarrier (MC) modulation on delay-Doppler (DD) plane, to couple the modulated signal with a doubly-selective channel having DD resolutions. A key challenge for the design of DD plane MC modulation is to investigate whether a realizable pulse orthogonal with respect to the DD plane’s fine resolutions exists or not. To this end, we first indicate that a feasible DD plane MC modulation is essentially a type of staggered multitone modulation. Then, analogous to orthogonal frequency division multiplexing, we propose an orthogonal delay-Doppler division multiplexing (ODDM) modulation, and design the corresponding transmit pulse. Furthermore, we prove that the proposed transmit pulse is orthogonal with respect to the DD plane’s resolutions and therefore a realizable DD plane orthogonal pulse does exist. The orthogonality of this particular pulse significantly eases the derivation of the ODDM’s DD domain channel input-output relation, and yields a channel matrix with an elegant block-circulant-like structure. We demonstrate that the ODDM outperforms the OTFS in terms of out-of-band emission and bit error rate, by achieving perfect coupling between the modulated signal and the DD channel.