In this work, we propose the orthogonal delay-Doppler (DD) division multiplexing (ODDM) modulation with frequency modulated continuous wave (FMCW) (ODDM-FMCW) waveform to enable integrated sensing and communication (ISAC) with a low peak-to-average power ratio (PAPR). We first propose a square-root-Nyquist-filtered FMCW (SRN-FMCW) waveform to address limitations of conventional linear FMCW waveforms in ISAC systems. To better integrate with ODDM, we generate SRN-FMCW by embedding symbols in the DD domain, referred to as a DD-SRN-FMCW frame. A DD chirp compression receiver is designed to obtain the channel response efficiently. Next, we construct the proposed ODDM-FMCW waveform for ISAC by superimposing a DD-SRN-FMCW frame onto an ODDM data frame. A comprehensive performance analysis of the ODDM-FMCW waveform is presented, covering peak-to-average power ratio, spectrum, ambiguity function, and Cramér-Rao bound for delay and Doppler estimation. Numerical results show that the proposed ODDM-FMCW waveform delivers excellent ISAC performance in terms of root mean square error for sensing and bit error rate for communications.

Orthogonal delay-Doppler (DD) division multiplexing (ODDM) modulation has recently emerged as a promising paradigm for ensuring reliable communications in doubly-selective channels. This work investigates the spectra and orthogonality characteristics of analog (direct) and approximate digital implementations of ODDM systems. We first determine the time and frequency domain representations of the basis functions for waveform in analog and approximate digital ODDM systems. Thereafter, we derive their power spectral densities and show that while the spectrum of analog ODDM waveforms exhibits a step-wise behavior in its transition regions, the spectrum of approximate digital ODDM waveforms is confined to that of the ODDM sub-pulse. Next, we prove the orthogonality characteristics of approximate digital ODDM waveforms and show that, unlike analog ODDM waveforms, the approximate digital ODDM waveforms satisfy orthogonality without the need of additional time domain resources. Additionally, we examine the similarities and differences that implementations of approximate digital ODDM share with the other variants of DD modulations, focusing on the domain changes the symbols undergo, the type of pulse shaping and windowing used, and the domains and the sequence in which they are performed. Finally, we present numerical results to validate our findings and draw further insights.

Delay-Doppler (DD) domain modulation such as orthogonal DD division multiplexing (ODDM) has been recently explored for communications over doubly selective channels. This paper derives equivalent sampled (on-grid) DD (ESDD) channel models for the effective discrete-time channels in ODDM systems over highly-spread off-grid physical channels with delay and Doppler shifts that can exceed the subpulse spacing and subtone spacing, respectively, of the DD orthogonal pulse (DDOP). The derived ESDD models account for i) off-grid delay and Doppler shifts present in practical physical channels, ii) sample-wise pulse shaping at the transmitter, and iii) matched filtering and windowing at the receiver. We then investigate the supports of the ESDD models and their implications on the input-output (IO) relation of ODDM adopting more general pulses and windows over highly-spread physical channels. In particular, we show that the digital sequences of ODDMcouple finelywith on-grid discrete-time DD channels, which leads to compact IO relation. We also analyze the folding and aliasing of the ESDD channel and their influence on the fading effect and predictability of effective channels experienced by the DD-domain symbols of ODDM. Based on the results, we identify conditions under which the ESDD channels can be estimated directly from embedded DD-domain pilots in a single ODDM frame. Finally, numerical results are provided to further demonstrate the findings of this paper.

Orthogonal delay-Doppler (DD) division multiplexing (ODDM) modulation has recently been proposed as a promising paradigm for communications in doubly-selective channels. In this work, we propose a novel ODDM with frequency modulated continuous wave (FMCW) (ODDM-FMCW) signal to enable integrated sensing and communication (ISAC) with a low peak-to-average power ratio (PAPR). We first propose the DD-domain embedded root-raised-cosine filtered FMCW (DD-RRC-FMCW) signal, where digital chirp compression is introduced for efficient radar signal processing. By superimposing this signal onto an ODDM data frame, we obtain the ODDM-FMCW signal for the proposed ISAC system. Next, we introduce a modified orthogonal matching pursuit algorithm for data-aided sensing. The algorithm is then combined with the soft successive interference cancellation with minimum mean square error detector to perform joint channel estimation and data detection. Our numerical results show that the proposed ODDM-FMCW signal delivers excellent normalized mean square error and bit error rate performance for ISAC.

The orthogonal delay-Doppler (DD) division multiplexing (ODDM) modulation has been proposed as a promising paradigm for ensuring reliable communications in doublyselective channels. Although ODDM functions as a multicarrier modulation on the DD plane, academia and industry are more accustomed to understanding and altering the time and frequency resources of signals and waveforms. Recognizing this, in this work, we investigate the time and spectral occupancies of the transmitted signals in both analog (direct) and digital (approximate) implementations of ODDM systems. We begin by explicitly determining the basis functions for both ODDM systems in time and frequency domains, and highlight their phase term difference in the time domain and their envelope difference in the frequency domain. We then derive the spectral responses of their transmit signals. We reveal that while the spectral response of signals in analog ODDM systems exhibits a step-wise behavior in its transition regions, the spectral response of signals in digital ODDM systems is confined to that of the ODDM sub-pulse. Finally, through numerical results, we verify our findings and show that out-of-band-emission of ODDM systems can be further improved by increasing the duration of the ODDM sub-pulse.

In this work, we investigate computational complexity reduction for the detection of the emerging Orthogonal DelayDoppler Division Multiplexing (ODDM), where existing time domain detection schemes become computationally demanding when the channel delay spread is large. We start by deriving the frequency domain input-output relation for ODDM over general physical channels. Based on this relation, we develop a frequency domain iterative MMSE detection scheme, which effectively reduces detection complexity in channels with large delay spread. To leverage the unique advantages of both time and frequency domain detections, we further propose an adaptive ODDM detection scheme that selects the optimal detection domain based on channel conditions. Simulation results show that our adaptive time/frequency domain detector reduces computational overhead by multiple folds with minimal impact on error performance or power efficiency, compared to conventional non-adaptive detectors under practical system and channel parameters.

This paper examines orthogonal delay-Doppler (DD) division multiplexing (ODDM) systems over highly-spread channels characterized by delay and Doppler shifts which can exceed, respectively, the sub-pulse separation and sub-tone separation of the DD orthogonal pulse employed by ODDM. We first derive an equivalent sampled DD (ESDD) channel model with on-grid delay and Doppler shifts specified by the sampling interval and frame duration of the transmission scheme. Our model accounts for off-grid delay and Doppler present in practical channels, samplewise pulse shaping at the transmitter, and matched filtering and windowing at the receiver. By examining their interaction, we show that the time-domain sequences of ODDM implemented approximately (digitally) using discrete Fourier transform (DFT) and inverse DFT (IDFT) and the discrete-time on-grid DD channel are finely coupled, and this leads to compact input-output (IO) relation for ODDM over on-grid channels. We then leverage the ESDD model to describe the IO relation of ODDM adopting more general pulses and windows over highly-spread off-grid channels, which reveals the potential folding (aliasing) of the ESDD channel and its implication on the signal model. We finally present numerical results. It is observed that pulses with shorter duration and windows with lower sidelobes in their spectrum lead to sparser ESDD channels.

The orthogonal delay-Doppler (DD) division multiplexing (ODDM) modulation has been recently proposed as a promising modulation scheme for next-generation communication systems with high mobility. Despite its benefits, ODDM modulation and other DD domain modulation schemes face the challenge of excessive equalization complexity. To address this challenge, we propose time domain Tomlinson-Harashima precoding (THP) for the ODDM transmitter, to make the DD domain single-tap equalizer feasible, thereby reducing the equalization complexity. In our design, we first pre-cancel the inter-symbol-interference (ISI) using the linear time-varying (LTV) channel information. Second, different from classical THP designs, we introduce a modified modulo operation with an adaptive modulus, by which the joint DD domain data multiplexing and time-domain ISI pre-cancellation can be realized without excessively increasing the bit errors. We then analytically study the losses encountered in this design, namely the power loss, the modulo noise loss, and the modulo signal loss. Based on this analysis, BER lower bounds of the ODDM system with time domain THP are derived when 4-QAM or 16-QAM modulations are adopted for symbol mapping in the DD domain. Finally, through numerical results, we validate our analysis and then demonstrate that the ODDM system with time domain THP is a promising solution to realize better BER performance over LTV channels compared to orthogonal frequency division multiplexing systems with single-tap equalizer and ODDM systems with maximum ratio combining.

In this work, we study the time-frequency (TF) localization characteristics of the prototype pulse of orthogonal delay-Doppler (DD) division multiplexing modulation, namely, the DD plane orthogonal pulse (DDOP). The TF localization characteristics examine how concentrated or spread out the energy of a pulse is in the joint TF domain, the time domain (TD), and the frequency domain (FD). We first derive the TF localization metrics of the DDOP, including its TF area, its time and frequency dispersions, and its direction parameter. Based on these results, we demonstrate that the DDOP exhibits a high energy spread in the TD, FD, and the joint TF domain, while adhering to the Heisenberg uncertainty principle. Thereafter, we discuss the potential advantages brought by the energy spread of the DDOP, especially with regard to harnessing both time and frequency diversities and enabling fine-resolution sensing. Subsequently, we examine the relationships between the time and frequency dispersions of the DDOP and those of the envelope functions of DDOP’s TD and FD representations, paving the way for simplified determination of the TF localization metrics for more generalized variants of the DDOP and the pulses used in other DD domain modulation schemes. Finally, using numerical results, we validate our analysis and find further insights.

Orthogonal delay-Doppler (DD) division multiplexing (ODDM) has been recently proposed as a promising multicarrier modulation scheme to tackle Doppler spread in high-mobility environments. Accurate channel estimation is of paramount importance to guarantee reliable communication for the ODDM, especially when the delays and Dopplers of the propagation paths are off-grid. In this paper, we propose a novel grid refinement and adjustment-based sparse Bayesian inference (GRASBI) scheme for DD domain channel estimation. The GRASBI involves first formulating the channel estimation problem as a sparse signal recovery through the introduction of a virtual DD grid. Then, an iterative process is proposed that involves (i) sparse Bayesian learning to estimate the channel parameters and (ii) a novel grid refinement and adjustment process to adjust the virtual grid points. The grid adjustment in GRASBI relies on the maximum likelihood principle to attain the adjustment and utilizes refined grids that have much higher resolution than the virtual grid. Moreover, a low-complexity grid refinement and adjustment-based channel estimation scheme is proposed, that can provides a good tradeoff between the estimation accuracy and the complexity. Finally, numerical results are provided to demonstrate the accuracy, the convergence, and the efficiency of the proposed channel estimation schemes.

Orthogonal delay-Doppler (DD) division multiplexing (ODDM) modulation has recently emerged as a promising candidate for ensuring reliable communications over high mobility channels. One of the key challenges faced by systems based on ODDM modulation and other DD domain modulation schemes (e.g., orthogonal time frequency space modulation), is the excessive receiver complexity. To address this challenge, we propose time domain Tomlinson-Harashima precoding (THP) for the ODDM systems to make the single-tap equalizer feasible, thereby significantly reducing the receiver complexity. Different from previous THP designs, we first propose intersymbol-interference (ISI) pre-cancellation based on the linear time-varying (LTV) channel information. Second, we propose a modified modulo operation with an adaptive modulus to realize DD domain data modulation and time domain ISI precancellation. We analytically investigate the bit error rate (BER) performance of our proposed ODDM system with time domain THP. Particularly, we investigate three types of losses that can degrade the performance, namely the modulo signal loss, the power loss, and the modulo noise loss. Based on these, a lower bound for the BER of our proposed ODDM system with time domain THP under the DD domain single-tap equalizer is derived. Through numerical simulations, our analysis is first validated and finally, we show the significance of our design to attain a low complex receiver and superior BER performance over LTV channels.

The orthogonal delay-Doppler (DD) division multiplexing (ODDM) modulation has recently been proposed as a cutting-edge and promising paradigm to ensure reliable communications in high mobility scenarios. In this work, we investigate the time-frequency (TF) localization characteristics of the DD plane orthogonal pulse (DDOP), which is the prototype pulse of ODDM modulation. The TF localization characteristics reveal the concentration or spread of pulse energy in the joint TF domain. We first derive the TF localization metric, TF area (TFA), for the DDOP. Based on this result, we provide insights into the energy spread of the DDOP in the joint TF domain. Then, we explore the potential advantages conferred by the energy spread of the DDOP, particularly in harnessing time and frequency diversities, as well as enabling high-resolution sensing. Furthermore, we determine the TFA for the recently proposed generalized design of the DDOP. Finally, we validate our analysis through numerical results and show that the energy spread of the generalized design of the DDOP in the joint TF domain exhibits a step-wise increase as the duration of sub-pulses increases.

We investigate the coexistence of orthogonal time-frequency space (OTFS) modulation with current fourth- and fifth-generation (4G/5G) communication systems that primarily use orthogonal frequency-division multiplexing (OFDM) waveforms. We first derive the input-output-relation of OTFS in the considered coexisting system. In this derivation, we consider 1) the inclusion of multiple cyclic prefixes (CPs) with unequal lengths to the OTFS signal and 2) edge carrier unloading (ECU), to account for the impacts of CP length, frame structure, and subcarrier arrangement described in 3GPP standards for 4G/5G systems. Our analysis reveals that the inclusion of multiple CPs to the OTFS signal and ECU lead to the channel response exhibiting spreading effects/leakage along the Doppler and delay dimensions, respectively. Consequently, the effective sampled delay-Doppler (DD) domain channel model for OTFS in coexisting systems may exhibit reduced sparsity. We also show that the effective DD domain channel coefficients for OTFS in coexisting systems are influenced by the unequal lengths of CPs. Subsequently, we propose an interference cancellation-based channel estimation (CE) technique for OTFS in coexisting systems. Through numerical results, we validate our analysis, highlight the importance of not ignoring the unequal lengths of CPs during signal detection, and show the significance of the proposed CE technique.

This study examines the coexistence of orthogonal time-frequency space (OTFS) modulation with current fourth-and fifth-generation (4G/5G) wireless communication systems that primarily use orthogonal frequency-division multiplexing (OFDM) waveforms. We first derive the input-output-relation (IOR) of OTFS when it coexists with an OFDM system while considering the impact of unequal lengths of the cyclic prefixes (CPs) in the OTFS signal. We show analytically that the inclusion of multiple CPs to the OTFS signal results in the effective sampled delay-Doppler (DD) domain channel response to be less sparse. We also show that the effective DD domain channel coefficients for OTFS in coexisting systems are influenced by the unequal lengths of the CPs. Subsequently, we propose an embedded pilotaided channel estimation (CE) technique for OTFS in coexisting systems that leverages the derived IOR for accurate channel characterization. Using numerical results, we show that ignoring the impact of unequal lengths of the CPs during signal detection can degrade the bit error rate performance of OTFS in coexisting systems. We also show that the proposed CE technique for OTFS in coexisting systems outperforms the state-of-the-art threshold-based CE technique.

We propose a new spectrum allocation strategy, aided by unsupervised learning, for multiuser terahertz communication systems. In this strategy, adaptive sub-band bandwidth is considered such that the spectrum of interest can be divided into sub-bands with unequal bandwidths. This strategy reduces the variation in molecular absorption loss among the users, leading to the improved data rate performance. We first formulate an optimization problem to determine the optimal sub-band bandwidth and transmit power, and then propose the unsupervised learning-based approach to obtaining the near-optimal solution to this problem. In the proposed approach, we first train a deep neural network (DNN) while utilizing a loss function that is inspired by the Lagrangian of the formulated problem. Then using the trained DNN, we approximate the near-optimal solutions. Numerical results demonstrate that comparing to existing approaches, our proposed unsupervised learning-based approach achieves a higher data rate, especially when the molecular absorption coefficient within the spectrum of interest varies in a highly non-linear manner.

We propose a new spectrum allocation strategy for terahertz (THz) band communication (THzCom) systems. Specifically, we design multi-band-based spectrum allocation with adaptive sub-band bandwidth (ASB), by allowing to divide the spectrum of interest into sub-bands with unequal bandwidths. Due to the frequency and distance-dependent nature of the molecular absorption loss, the variation in this loss between the sub-bands would be very high at the THz band when equal sub-band bandwidth (ESB) is considered, as in the literature. The proposed strategy reduces this variation by allowing changes in the sub-band bandwidth, which leads to an overall improvement in the data rate performance. To study the impact of our strategy, we formulate an optimization problem, with the main focus on spectrum allocation, to determine the optimal sub-band bandwidth and transmit power. Thereafter, we propose reasonable approximations and transformations to solve the formulated problem. Aided by numerical results, we show that by enabling and optimizing ASB, a significantly higher data rate can be achieved by our strategy, compared to adopting ESB, and it is more beneficial to adopt ASB when the spectrum with the highest average molecular absorption loss within the THz transmission window is selected during spectrum allocation.

We study spectrum allocation for terahertz (THz) band communication (THzCom) systems, while considering the frequency and distance-dependent nature of THz channels. Different from existing studies, we explore multi-band-based spectrum allocation with adaptive sub-band bandwidth (ASB) by allowing the spectrum of interest to be divided into sub-bands with unequal bandwidths. Also, we investigate the impact of sub-band assignment on multi-connectivity (MC) enabled THzCom systems, where users associate and communicate with multiple access points simultaneously. We formulate resource allocation problems, with the primary focus on spectrum allocation, to determine sub-band assignment, sub-band bandwidth, and optimal transmit power. Thereafter, we propose reasonable approximations and transformations, and develop iterative algorithms based on the successive convex approximation technique to analytically solve the formulated problems. Aided by numerical results, we show that by enabling and optimizing ASB, significantly higher throughput can be achieved as compared to adopting equal sub-band bandwidth, and this throughput gain is most profound when the power budget constraint is more stringent. We also show that our sub-band assignment strategy in MC-enabled THzCom systems outperforms the state-of-the-art sub-band assignment strategies and the performance gain is most profound when the spectrum with the lowest average molecular absorption coefficient is selected during spectrum allocation.

We derive new expressions for the connection probability and the average ergodic capacity to evaluate the performance achieved by multi-connectivity (MC) in an indoor ultra-wideband terahertz (THz) communication system. In this system, the user is affected by both self-blockage and dynamic human blockers. We first build up a three-dimensional propagation channel in this system to characterize the impact of molecular absorption loss and the shrinking usable bandwidth nature of the ultra-wideband THz channel. We then carry out new performance analysis for two MC strategies: 1) Closest line-of-sight (LOS) access point (AP) MC (C-MC), and 2) Reactive MC (R-MC). With numerical results, we validate our analysis and show the considerable improvement achieved by both MC strategies in the connection probability. We further show that the C-MC and R-MC strategies provide significant and marginal capacity gain relative to the single connectivity strategy, respectively, and increasing the number of the user's associated APs imposes completely different affects on the capacity gain achieved by the C-MC and R-MC strategies. Additionally, we clarify that our analysis allows us to determine the optimal density of APs in order to maximize the capacity gain.