Fundamentals of Delay-Doppler Communications: Practical Implementation and Extensions to OTFS

The recently proposed orthogonal time frequency space (OTFS) modulation, which is a typical Delay-Doppler (DD) communication scheme, has attracted significant attention thanks to its appealing performance over doubly-selective channels. In this paper, we present the fundamentals of general DD communications from the viewpoint of the Zak transform. We start our study by constructing DD domain basis functions aligning with the time-frequency (TF)-consistency condition, which are globally quasi-periodic and locally twisted-shifted. We unveil that these features are translated to unique signal structures in both time and frequency, which are beneficial for communication purposes. Then, we focus on the practical implementations of DD Nyquist communications, where we show that rectangular windows achieve perfect DD orthogonality, while truncated periodic signals can obtain sufficient DD orthogonality. Particularly, smoothed rectangular window with excess bandwidth can result in a slightly worse orthogonality but better pulse localization in the DD domain. Furthermore, we present a practical pulse shaping framework for general DD communications and derive the corresponding input-output relation under various shaping pulses. Our numerical results agree with our derivations and also demonstrate advantages of DD communications over conventional orthogonal frequency-division multiplexing (OFDM).

Low Complexity Turbo SIC-MMSE Detection for Orthogonal Time Frequency Space Modulation

Recently, orthogonal time frequency space (OTFS) modulation has garnered considerable attention due to its robustness against doubly-selective wireless channels. In this paper, we propose a low-complexity iterative successive interference cancellation based minimum mean squared error (SIC-MMSE) detection algorithm for zero-padded OTFS (ZP-OTFS) modulation. In the proposed algorithm, signals are detected based on layers processed by multiple SIC-MMSE linear filters for each sub-channel, with interference on the targeted signal layer being successively canceled either by hard or soft information. To reduce the complexity of computing individual layer filter coefficients, we also propose a novel filter coefficients recycling approach in place of generating the exact form of MMSE filter weights. Moreover, we design a joint detection and decoding algorithm for ZP-OTFS to enhance error performance. Compared to the conventional SIC-MMSE detection, our proposed algorithms outperform other linear detectors, e.g., maximal ratio combining (MRC), for ZP-OTFS with up to 3 dB gain while maintaining comparable computation complexity.

From OTFS to DD-ISAC: Integrating Sensing and Communications in the Delay Doppler Domain

Next-generation vehicular networks are expected to provide the capability of robust environmental sensing in addition to reliable communications to meet intelligence requirements. A promising solution is the integrated sensing and communication (ISAC) technology, which performs both functionalities using the same spectrum and hardware resources. Most existing works on ISAC consider the Orthogonal Frequency Division Multiplexing (OFDM) waveform. Nevertheless, vehicle motion introduces Doppler shift, which breaks the subcarrier orthogonality and leads to performance degradation. The recently proposed Orthogonal Time Frequency Space (OTFS) modulation, which exploits various advantages of Delay Doppler (DD) channels, has been shown to support reliable communication in high-mobility scenarios. Moreover, the DD waveform can directly interact with radar sensing parameters, which are actually delay and Doppler shifts. This paper investigates the advantages of applying the DD communication waveform to ISAC. Specifically, we first provide a comprehensive overview of implementing DD communications, based on which several advantages of DD-ISAC over OFDM-based ISAC are revealed, including transceiver designs and the ambiguity function. Furthermore, a detailed performance comparison are presented, where the target detection probability and the mean squared error (MSE) performance are also studied. Finally, some challenges and opportunities of DD-ISAC are also provided.

Extended Target Parameter Estimation and Tracking with an HDA Setup for ISAC Applications

We investigate radar parameter estimation and beam tracking with a hybrid digital-analog (HDA) architecture in a multi-block measurement framework using an extended target model. In the considered setup, the backscattered data signal is utilized to predict the user position in the next time slots. Specifically, a simplified maximum likelihood framework is adopted for parameter estimation, based on which a simple tracking scheme is also developed. Furthermore, the proposed framework supports adaptive transmitter beamwidth selection, whose effects on the communication performance are also studied. Finally, we verify the effectiveness of the proposed framework via numerical simulations over complex motion patterns that emulate a realistic integrated sensing and communication (ISAC) scenario.

Reduced-Complexity Cross-Domain Iterative Detection for OTFS Modulation via Delay-Doppler Decoupling

In this paper, a reduced-complexity cross-domain iterative detection for orthogonal time frequency space (OTFS) modulation is proposed, which exploits channel properties in both time and delay-Doppler domains. Specifically, we first show that in the time domain effective channel, the path delay only introduces interference among samples in adjacent time slots, while the Doppler becomes a phase term that does not affect the channel sparsity. This ``band-limited'' matrix structure motivates us to apply a reduced-size linear minimum mean square error (LMMSE) filter to eliminate the effect of delay in the time domain, while exploiting the cross-domain iteration for minimizing the effect of Doppler by noticing that the time and Doppler are a pair of Fourier dual. The state (MSE) evolution was derived and compared with bounds to verify the effectiveness of the proposed scheme. Simulation results demonstrate that the proposed scheme achieves almost the same error performance as the optimal detection, but only requires a reduced complexity.

On the Pulse Shaping for Delay-Doppler Communications

In this paper, we study the pulse shaping for delay-Doppler (DD) communications. We start with constructing a basis function in the DD domain following the properties of the Zak transform. Particularly, we show that the constructed basis functions are globally quasi-periodic while locally twisted-shifted, and their significance in time and frequency domains are then revealed. We further analyze the ambiguity function of the basis function, and show that fully localized ambiguity function can be achieved by constructing the basis function using periodic signals. More importantly, we prove that time and frequency truncating such basis functions naturally leads to delay and Doppler orthogonalities, if the truncating windows are orthogonal or periodic. Motivated by this, we propose a DD Nyquist pulse shaping scheme considering signals with periodicity. Finally, our conclusions are verified by using various orthogonal and periodic pulses.

Deep Learning-empowered Predictive Precoder Design for OTFS Transmission in URLLC

To guarantee excellent reliability performance in ultra-reliable low-latency communications (URLLC), pragmatic precoder design is an effective approach. However, an efficient precoder design highly depends on the accurate instantaneous channel state information at the transmitter (ICSIT), which however, is not always available in practice. To overcome this problem, in this paper, we focus on the orthogonal time frequency space (OTFS)-based URLLC system and adopt a deep learning (DL) approach to directly predict the precoder for the next time frame to minimize the frame error rate (FER) via implicitly exploiting the features from estimated historical channels in the delay-Doppler domain. By doing this, we can guarantee the system reliability even without the knowledge of ICSIT. To this end, a general precoder design problem is formulated where a closed-form theoretical FER expression is specifically derived to characterize the system reliability. Then, a delay-Doppler domain channels-aware convolutional long short-term memory (CLSTM) network (DDCL-Net) is proposed for predictive precoder design. In particular, both the convolutional neural network and LSTM modules are adopted in the proposed neural network to exploit the spatial-temporal features of wireless channels for improving the learning performance. Finally, simulation results demonstrated that the FER performance of the proposed method approaches that of the perfect ICSI-aided scheme.

Radar Sensing via OTFS Signaling: A Delay Doppler Signal Processing Perspective

The recently proposed orthogonal time frequency space (OTFS) modulation multiplexes data symbols in the delay-Doppler (DD) domain. Since the range and velocity, which can be derived from the delay and Doppler shifts, are the parameters of interest for radar sensing, it is natural to consider implementing DD signal processing for radar sensing. In this paper, we investigate the potential connections between the OTFS and DD domain radar signal processing. Our analysis shows that the range-Doppler matrix computing process in radar sensing is exactly the demodulation of OTFS with a rectangular pulse shaping filter. Furthermore, we propose a two-dimensional (2D) correlation-based algorithm to estimate the fractional delay and Doppler parameters for radar sensing. Simulation results show that the proposed algorithm can efficiently obtain the delay and Doppler shifts associated with multiple targets.

OTFS-SCMA: A Downlink NOMA Scheme for Massive Connectivity in High Mobility Channels

This paper studies a downlink system that combines orthogonal-time-frequency-space (OTFS) modulation and sparse code multiple access (SCMA) to support massive connectivity in high-mobility environments. We propose a cross-domain receiver for the considered OTFS-SCMA system which efficiently carries out OTFS symbol estimation and SCMA decoding in a joint manner. This is done by iteratively passing the extrinsic information between the time domain and the delay-Doppler (DD) domain via the corresponding unitary transformation to ensure the principal orthogonality of errors from each domain. We show that the proposed OTFS-SCMA detection algorithm exists at a fixed point in the state evolution when it converges. To further enhance the error performance of the proposed OTFS-SCMA system, we investigate the cooperation between downlink users to exploit the diversity gains and develop a distributed cooperative detection (DCD) algorithm with the aid of belief consensus. Our numerical results demonstrate the effectiveness and convergence of the proposed algorithm and show an increased spectral efficiency compared to the conventional OTFS transmission.

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.