delay-Doppler Channel Research Articles

Progressive Inter-Path Interference Cancellation Algorithm for Channel Estimation Using Orthogonal Time-Frequency Space.

Fractional delay-Doppler (DD) channel estimation in orthogonal time-frequency space (OTFS) systems poses a significant challenge considering the severe effects of inter-path interference (IPI). To this end, several algorithms have been extensively explored in the literature for accurate low-complexity channel estimation in both integer and fractional DD scenarios. In this work, we develop a variant of the state-of-the-art delay-Doppler inter-path interference cancellation (DDIPIC) algorithm that progressively cancels the IPI as estimates are obtained. The key advantage of the proposed approach is that it requires only a final refinement procedure reducing the complexity of the algorithm. Specifically, the time difference in latency between the proposed approach and the DDIPIC algorithm is almost proportional to the square of the number of estimated paths. Numerical results show that the proposed algorithm outperforms the other channel estimation schemes achieving lower normalized mean square error (NMSE) and bit error rate (BER).

Learning in Time-Frequency Domain for Fractional Delay-Doppler Channel Estimation in OTFS

Learning in Time-Frequency Domain for Fractional Delay-Doppler Channel Estimation in OTFS

Low-Complexity Delay-Doppler Channel Estimation in Discrete Zak Transform Based OTFS

Low-Complexity Delay-Doppler Channel Estimation in Discrete Zak Transform Based OTFS

OTFS-Based ISAC: How Delay Doppler Channel Estimation Assists Environment Sensing?

OTFS-Based ISAC: How Delay Doppler Channel Estimation Assists Environment Sensing?

Variational Bayesian Learning Based Delay-Doppler Channel Estimator for Multi-User OTFS Systems

Variational Bayesian Learning Based Delay-Doppler Channel Estimator for Multi-User OTFS Systems

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.

A Structured Sparse Bayesian Channel Estimation Approach for Orthogonal Time-Frequency Space Modulation.

Orthogonal time-frequency space (OTFS) modulation has been advocated as a promising waveform for achieving integrated sensing and communication (ISAC) due to its superiority in high-mobility adaptability and spectral efficiency. In OTFS modulation-based ISAC systems, accurate channel acquisition is critical for both communication reception and sensing parameter estimation. However, the existence of the fractional Doppler frequency shift spreads the effective channels of the OTFS signal significantly, making efficient channel acquisition very challenging. In this paper, we first derive the sparse structure of the channel in the delay Doppler (DD) domain according to the input and output relationship of OTFS signals. On this basis, a new structured Bayesian learning approach is proposed for accurate channel estimation, which includes a novel structured prior model for the delay-Doppler channel and a successive majorization-minimization (SMM) algorithm for efficient posterior channel estimate computation. Simulation results show that the proposed approach significantly outperforms the reference schemes, especially in the low signal-to-noise ratio (SNR) region.

Orthogonal time chirp space modulation based upon fractional Fourier transform

Linear chirps play a significant role in communications because of their ability to spread spectrum and provide secure and robust communication. Orthogonal time frequency space (OTFS) modulation is suitable for high-mobility transmissions over a stationary delay-Doppler channel. The advantages of linear chirps and OTFS modulation are combined in this study. Specifically, orthogonal time chirp space modulation based on fractional Fourier transform (OTCS-FrFT) is proposed, which is obtained by replacing the two-dimensional (2D) sinusoidal carriers of OTFS modulation with 2D linear chirps. As the Fourier transform is the kernel of OTFS modulation, FrFT underlies the proposed OTCS-FrFT system for its complete and orthogonal linear chirp bases and its fast computation algorithms. The information symbols are originally placed on the delay-fractional Doppler plane, which is a rotation of the delay-Doppler plane, where the rotation angle is a free parameter that enables the OTCS-FrFT to adapt to different time-varying Doppler scenarios. It is proven that the OTCS-FrFT is robust to time-varying channels with linear delay/Doppler spreading functions. For other time-varying channels, simulations are performed to validate the superior performance of the OCTS-FrFT over other FrFT-based OTFS modulation methods, where the bit error rate is reduced by approximately 1 dB for both high-speed and high-acceleration terminals.

Joint Spatio-Temporal Precoding for Practical Non-Stationary Wireless Channels

The high mobility, density and multi-path evident in modern wireless systems makes the channel highly non-stationary. This causes temporal variation in the channel distribution that leads to the existence of time-varying joint interference across multiple degrees of freedom (DoF, <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">e.g</i> ., users, antennas, frequency and symbols), which renders conventional precoding sub-optimal in practice. In this work, we derive a High-Order Generalization of Mercer’s Theorem (HOGMT), which decomposes the multi-user non-stationary channel into two (dual) sets of jointly orthogonal subchannels (eigenfunctions), that result in the other set when one set is transmitted through the channel. This duality and joint orthogonality of eigenfuntions ensure transmission over independently flat-fading subchannels. Consequently, transmitting these eigenfunctions with optimally derived coefficients eventually mitigates any interference across its degrees of freedoms and forms the foundation of the proposed joint spatio-temporal precoding. The transferred dual eigenfuntions and coefficients directly reconstruct the data symbols at the receiver upon demodulation, thereby significantly reducing its computational burden, by alleviating the need for any complementary post-coding. Additionally, the eigenfunctions decomposed from the time-frequency delay-Doppler channel kernel are paramount to extracting the second-order channel statistics, and therefore completely characterize the underlying channel. We evaluate this using a realistic non-stationary channel framework built in Matlab and show that our precoding achieves ⩾4 orders of reduction in BER at SNR⩾15dB in OFDM systems for higher-order modulations and less complexity compared to the state-of-the-art precoding.

Orthogonal Time Frequency Space and Predictive Beamforming-Enabled URLLC in Vehicular Networks

To fulfill the requirements of future intelligent transportation system in 6G era, ultra-reliable and low-latency vehicular communications is of great importance. In high-mobility scenarios, the conventional orthogonal frequency division multiplexing (OFDM) modulation may fail to work due to high Doppler spreads. Moreover, the dynamic network topology imposes challenges on aligning the beams in multiple antenna systems. In this context, this article introduces a new orthogonal time frequency space (OTFS) and sensing-assisted predictive beamforming framework for supporting reliable and low-latency vehicular communications. In particular, we will first overview the OTFS modulation scheme, which performs data transmission in the delay-Doppler (DD) domain and discuss its superiority in improving communication reliability in vehicular networks. Then the sensing-assisted predictive beamforming scheme will be introduced, which does not rely on dedicated pilots for beam pairing, leading to very low overhead and latency. Benefiting from the DD channel representation, we briefly discuss the potential of channel prediction. Lastly, the challenges and future research directions are summarized.

BER performance analysis of OTFS systems with power allocation

Orthogonal time frequency space (OTFS), as a novel 2-D modulation technique, has been proposed to achieve better BER performances over delay-Doppler channels. In this paper, we propose two different power allocation (PA) algorithms in OTFS systems with zero forcing (ZF) or minimum mean square error (MMSE) equalization, where general formulas with PA are derived in advance under the condition of minimum BER (MBER) criterion. On one hand, a suboptimal MBER power allocation method is put forward to achieve better BER performances, and then analytical BER expressions are derived with proposed PA strategy. Considering the case of MMSE equalization, a combined subsymbol allocation (SA) and PA strategy is raised, where some subsymbols may be abandoned due to worse channel conditions, and then it is proven effectively to improve BER performances through theoretical and simulation results. Furthermore, BER performances with proposed joint SA and PA strategy are also investigated in delay-Doppler channels, where an improved message passing (MP) receiver based on equivalent channel matrix with PA is given.

Orthogonal Time Frequency Space Modulation—Part III: ISAC and Potential Applications

The first two parts of this tutorial on orthogonal time frequency space (OTFS) modulation have discussed the fundamentals of delay-Doppler (DD) domain communications as well as some advanced technologies for transceiver design. In this letter, we will present an OTFS-based integrated sensing and communications (ISAC) system, which is regarded as an enabling technology in next generation wireless communications. In particular, we illustrate the sensing as well as the communication models for OTFS-ISAC systems. Next, we show that benefiting from time-invariant DD channels, the sensing parameters can be used for inferring the communication channels, leading to an efficient transmission scheme. As both functionalities are realized in the same DD domain, we briefly discuss several promising benefits of OTFS-based ISAC systems, which have not been completely unveiled yet. Finally, a range of potential applications of OTFS for the future wireless networks will be highlighted.

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.

OTFS—A Mathematical Foundation for Communication and Radar Sensing in the Delay-Doppler Domain

Orthogonal time frequency space (OTFS) is a framework for communication and active sensing that processes signals in the delay-Doppler (DD) domain. This paper explores three key features of the OTFS framework, and explains their value to applications. The first feature is a compact and sparse DD domain parameterization of the wireless channel, where the parameters map directly to physical attributes of the reflectors that comprise the scattering environment, and as a consequence these parameters evolve predictably. The second feature is a novel waveform / modulation technique, matched to the DD channel model, that embeds information symbols in the DD domain. The relation between channel inputs and outputs is localized, non-fading and predictable, even in the presence of significant delay and Doppler spread, and as a consequence the channel can be efficiently acquired and equalized. By avoiding fading, the post equalization SNR remains constant across all information symbols in a packet, so that bit error performance is superior to contemporary multi-carrier waveforms. Further, the OTFS carrier waveform is a localized pulse in the DD domain, making it possible to separate reflectors along both delay and Doppler simultaneously, and to achieve a high-resolution delay-Doppler radar image of the environment. In other words, the DD parameterization provides a common mathematical framework for communication and radar. This is the third feature of the OTFS framework, and it is ideally suited to intelligent transportation systems involving self-driving cars and unmanned ground/aerial vehicles which are self/network controlled. The OTFS waveform is able to support stable and superior performance over a wide range of user speeds.

Joint Bayesian Channel Estimation and Data Detection for OTFS Systems in LEO Satellite Communications

Lower earth orbit (LEO) satellites play an important role in the integration of space and terrestrial communication networks, which typically encounter high-mobility scenarios. It has been shown that orthogonal time frequency space (OTFS) modulation performs well in such high-mobility scenarios by transforming the time-varying channels into the delay-Doppler domain. In this paper, we develop a joint channel estimation and data detection algorithm for OTFS-based LEO satellite communications. Firstly, we adopt the powerful variational Bayesian inference (VBI) method for estimating the delay-Doppler channel vector, which contains the channel gain, the delay and the Doppler. Secondly, we exploit the unknown data symbols in an OTFS frame as ‘virtual pilots’ for improving the accuracy of channel estimation and detect them simultaneously. Our simulation results demonstrate that the proposed algorithm achieves improved channel estimation mean square error and bit error rate performance than its conventional counterparts.

On the Performance of Nonlinear Corrective Active Constellation Expansion in OTFS Systems

Orthogonal time frequency space (OTFS) modulation has been put forward to achieve better performances over delay-Doppler channels in comparison with orthogonal frequency division multiplexing (OFDM) systems. In this letter, a nonlinear corrective active constellation expansion (NC-ACE) method is firstly proposed in OTFS waveform to achieve precise control of peak-to-average power ratio (PAPR) and BER performances due to its nonlinear compression characteristics. Compared to linear constraint of clipping and filtering process in traditional active constellation expansion (ACE) and bounded distortion ACE (BD-ACE) strategies, nonlinear compressed clipping noise is considered with a flexible curvature control coefficient. According to flexible features of NC-ACE, a bridge between optimal suppression and no reduction of PAPR performances is established in OTFS systems, where tradeoff between PAPR and BER performances can be further obtained.

Delay-Doppler domain decision feedback turbo equalization for OTFS modulation

Emerging evidence indicates that orthogonal time–frequency space (OTFS) modulation is a potential candidate modulation scheme for high mobility wireless communications. However, OTFS may experience significant inter-symbol interference (ISI) and inter-Doppler interference (IDI) in the receiver. In this paper, we propose a soft decision feedback turbo equalization for OTFS transmission over delay-Doppler channels to jointly combat both interferences. A novel block decision feedback equalization (BDFE) algorithm is constructed using the band feature of the channels in the delay-Doppler domain. The feedforward and feedback filters are designed by the delay-time channels coefficients. According to the designed filter, an equivalent system model is employed to allow turbo equalization. The posterior probability is established using the soft prior information and feedback filter, and then fed back to the channel decoder as external information. Both theoretical analysis and simulation results demonstrate the effectiveness of the proposed scheme in improving the bit error rate (BER) performance and combat various interference. Numerical simulations are finally provided to justify the validity of the proposed scheme in improving the bit error rate (BER) performance and combating various interference.

A low-complexity orthogonal time frequency space modulation method for underwater acoustic communication

Compared with the orthogonal frequency division multiplexing (OFDM) modulation, the orthogonal time frequency space(OTFS) modulation has a lower peak-to-average power ratio. It can effectively resist the time selective fading caused by the Doppler effect and has significant performance advantages over doubly dispersive channels. However, the conventional OTFS linear minimum mean square error (LMMSE) method has a high complexity and is not easy to process in real time. In order to solve this problem, we propose a low-complexity equalization algorithm with infinite norm constraints based on the optimal coordinate reduction. The equalization algorithm not only obtains the optimal solution through a certain number of iterations and avoids direct matrix inversion but also equalizes infinite norm constraints to improve the symbol detection performance gains. At the same time, the OTFS delay-Doppler channel matrix we utilize is sparse and the two-norm squares of each column vector equally reduces the complexity of optimal coordinate descent. Finally, the simulation in the underwater acoustic communication scenario we designed verify the effectiveness of the proposed equalization algorithm. The simulation results show that the performance of the proposed equalization algorithm is close to that of the LMMSE method, while its low complexity is ensured.

Design and analysis of spatial modulation based orthogonal time frequency space system

In this paper, we design a spatial modulation based orthogonal time frequency space (SMOTFS) system to achieve improved transmission reliability and meet the high transmission rate and high-speed demands of future mobile communications, which fully utilizes the characteristics of spatial modulation (SM) and orthogonal time frequency space (OTFS) transmission. The detailed system design and signal processing of the SM-OTFS system have been presented. The closed-form expressions of the average symbol error rate (ASER) and average bit error rate (ABER) of the SM-OTFS system have been derived over the delay-Doppler channel with the help of the union bounding technique and moment-generating function (MGF). Meanwhile, the system complexity has been evaluated. Numerical results verify the correctness of the theoretical ASER and ABER analysis of the SM-OTFS system in the high signal-to-noise ratio (SNR) regions and also show that the SM-OTFS system outperforms the traditional SM based orthogonal frequency division multiplexing (SM-OFDM) system with limited complexity increase under mobile conditions, especially in high mobility scenarios.

Receiver Design for OTFS with a Fractionally Spaced Sampling Approach

The recent emergence of orthogonal time frequency space (OTFS) modulation as a novel PHY-layer mechanism is more suitable in high-mobility wireless communication scenarios than traditional orthogonal frequency division multiplexing (OFDM). Although multiple studies have analyzed OTFS performance using theoretical and ideal baseband pulseshapes, a challenging and open problem is the development of effective receivers for practical OTFS systems that must rely on non-ideal pulseshapes for transmission. This work focuses on the design of practical receivers for OTFS. We consider a fractionally spaced sampling (FSS) receiver in which the sampling rate is an integer multiple of the symbol rate. For rectangular pulses used in OTFS transmission, we derive a general channel input-output relationship of OTFS in delay-Doppler domain without the common reliance on impractical assumptions such as ideal bi-orthogonal pulses and on-the-grid delay/Doppler shifts. We propose two equalization algorithms: iterative combining message passing (ICMP) and turbo message passing (TMP) for symbol detection by exploiting delay-Doppler channel sparsity and the channel diversity gain via FSS. We analyze the convergence performance of TMP receiver and propose simplified message passing (MP) receivers to further reduce complexity. Our FSS receivers demonstrate stronger performance than traditional receivers and robustness to the imperfect channel state information knowledge.