Delay-Doppler QAM: A DD-Domain Modulation Framework

• Delay-Doppler QAM is a modulation framework that maps QAM symbols onto a two-dimensional delay-Doppler lattice using specialized basis functions and transforms.
• It enables joint communication and radar sensing by leveraging techniques such as OTFS, Zak-transform constructions, and DD-orthogonal pulse shaping.
• The approach balances exact lattice orthogonality with improved localization, enhancing channel estimation and mitigating inter-symbol interference in high mobility scenarios.

Delay-Doppler Quadrature Amplitude Modulation (DD-QAM) denotes a class of delay-Doppler-domain signaling architectures in which ordinary complex constellation symbols are associated with a two-dimensional lattice indexed by delay and Doppler, then synthesized into a physical waveform through DD-domain basis functions, Zak-transform constructions, symplectic Fourier mappings, or DD-orthogonal pulse shaping. In the recent literature, the explicit label appears most directly in a sensing-centric chirp waveform for joint communication and radar, where data are modulated across delay, Doppler, and complex amplitude dimensions (Li et al., 16 Dec 2025). In the broader DD-modulation literature, however, closely related systems are usually described as OTFS, general delay-Doppler communications, DD Nyquist communications, DDMC, or ODDM, and DD-QAM is best understood as QAM signaling instantiated within those DD-domain frameworks rather than as a universally standardized standalone family (Li et al., 2024, Lin et al., 2023, Zhang et al., 2023).

1. Terminology and conceptual scope

The literature does not use a single canonical label. General DD-communication papers explicitly state that they do not define a standalone scheme named “DD-QAM,” but they formulate a modulation framework in which complex-valued data symbols occupy an $M\times N$ DD lattice and are transmitted through DD-domain basis functions or OTFS/ODDM transforms. In that sense, if the symbol alphabet is QPSK, 16-QAM, 64-QAM, or another QAM family, the resulting system is naturally interpretable as DD-QAM (Li et al., 2024, Lin et al., 2023).

This terminological dispersion is substantial because different subliteratures emphasize different structural layers. OTFS emphasizes DD-to-TF spreading through ISFFT/SFFT and Heisenberg/Wigner operators; ODDM and DDMC emphasize DD-orthogonal pulse design aligned with the equivalent sampled DD channel; Zak-transform-based DD communications emphasize TF consistency, quasi-periodicity, and DD basis construction; the recent chirp DD-QAM work emphasizes a sensing-centric chirp/FMCW waveform whose information is embedded directly in delay, Doppler, and amplitude (Zhang et al., 2023, Li et al., 2024, Li et al., 16 Dec 2025).

Literature label	Core object	Relation to DD-QAM
OTFS	QAM-like symbols on a DD grid, spread to TF by ISFFT	Structured DD-QAM realization
DD communications / DD Nyquist communications	DD basis functions and Zak-consistent pulse shaping	General DD-QAM framework
DDMC / ODDM	DD multicarrier signaling matched to ESDD resolution	Direct theoretical basis for DD-QAM-like modulation
Chirp DD-QAM	Chirp waveform with data in delay, Doppler, amplitude	Explicit use of the term DD-QAM

A recurrent misconception is that DD-QAM is merely OFDM with different indexing. The DD literature argues otherwise: in conventional TF-domain QAM such as OFDM, each symbol occupies one TF resource element, whereas DD-domain modulation spreads or synthesizes symbols relative to the channel’s physically meaningful coordinates, yielding DD-domain convolutional or shift-structured input-output laws rather than ideal per-subcarrier scalar multiplication (Yuan et al., 2023).

2. Signal models and transform-domain realizations

A general DD linear modulation model is written as

$$s_{\rm DD}(\tau,\nu)=\sum_{l=0}^{M-1}\sum_{k=0}^{N-1}X_{\rm DD}[l,k]\, \Phi_{\rm DD}^{\tau_l,\nu_k}(\tau,\nu),$$

with DD-lattice locations

$\tau_l=l\frac{T}{M},\qquad \nu_k=k\frac{1}{NT}.$

This is the DD analogue of ordinary linear modulation: the information-bearing symbols $X_{\rm DD}[l,k]$ are ordinary complex digital symbols, while $\Phi_{\rm DD}^{\tau_l,\nu_k}$ are DD-domain basis functions (Li et al., 2024).

The Zak-transform viewpoint supplies the structural constraint that a realizable DD signal is not arbitrary. For a time-domain signal $x(t)$, the Zak transform is

$${\cal Z}_x(\tau,\nu)=\sqrt{T}\sum_{k=-\infty}^{\infty}x(\tau+kT)e^{-j2\pi k\nu T},$$

with quasi-periodicity

$${\cal Z}_x(\tau+T,\nu)=e^{j2\pi T\nu}{\cal Z}_x(\tau,\nu),\qquad {\cal Z}_x\!\left(\tau,\nu+\frac{1}{T}\right)={\cal Z}_x(\tau,\nu).$$

These relations motivate DD basis functions that are globally quasi-periodic and locally twisted-shifted (Li et al., 2024).

Within OTFS, the DD symbols are first mapped to the TF plane through the inverse symplectic finite Fourier transform,

$$X_{\rm TF}[n,m]=\frac{1}{\sqrt{NM}}\sum_{k=0}^{N-1}\sum_{l=0}^{M-1} X_{\rm DD}[k,l]e^{j2\pi\left(\frac{nk}{N}-\frac{ml}{M}\right)},$$

then synthesized into the continuous waveform by the Heisenberg transform,

$$s(t)=\sum_{n=0}^{N-1}\sum_{m=0}^{M-1}X_{\rm TF}[n,m]g_{tx}(t-nT)e^{j2\pi m\Delta f(t-nT)}.$$

At the receiver, the Wigner transform and SFFT map the signal back to the DD plane. In the cited radar-oriented OTFS formulation, the DD symbols are explicitly drawn from a modulation alphabet $$s_{\rm DD}(\tau,\nu)=\sum_{l=0}^{M-1}\sum_{k=0}^{N-1}X_{\rm DD}[l,k]\, \Phi_{\rm DD}^{\tau_l,\nu_k}(\tau,\nu),$$0, “e.g. QAM,” making the DD-QAM interpretation direct (Zhang et al., 2023).

A distinct but closely related realization appears in ODDM/DDMC, where the continuous-time transmit waveform is written directly on a DD-matched multicarrier lattice as

$$s_{\rm DD}(\tau,\nu)=\sum_{l=0}^{M-1}\sum_{k=0}^{N-1}X_{\rm DD}[l,k]\, \Phi_{\rm DD}^{\tau_l,\nu_k}(\tau,\nu),$$1

Here the symbols $$s_{\rm DD}(\tau,\nu)=\sum_{l=0}^{M-1}\sum_{k=0}^{N-1}X_{\rm DD}[l,k]\, \Phi_{\rm DD}^{\tau_l,\nu_k}(\tau,\nu),$$2 are again ordinary constellation points, explicitly described as QAM-capable information-bearing symbols (Lin et al., 2023).

The common mathematical core is therefore not a single transform, but the assignment of ordinary complex symbols to a DD-structured signal space whose waveform synthesis preserves delay and Doppler geometry.

3. Basis functions, pulse shaping, and orthogonality

A central requirement in DD modulation is TF consistency. The DD basis family is constructed so that every shifted basis function satisfies

$$s_{\rm DD}(\tau,\nu)=\sum_{l=0}^{M-1}\sum_{k=0}^{N-1}X_{\rm DD}[l,k]\, \Phi_{\rm DD}^{\tau_l,\nu_k}(\tau,\nu),$$3

The same shift law appears in time and frequency: $$s_{\rm DD}(\tau,\nu)=\sum_{l=0}^{M-1}\sum_{k=0}^{N-1}X_{\rm DD}[l,k]\, \Phi_{\rm DD}^{\tau_l,\nu_k}(\tau,\nu),$$4

$$s_{\rm DD}(\tau,\nu)=\sum_{l=0}^{M-1}\sum_{k=0}^{N-1}X_{\rm DD}[l,k]\, \Phi_{\rm DD}^{\tau_l,\nu_k}(\tau,\nu),$$5

These relations formalize the statement that DD basis functions are globally quasi-periodic and locally twisted-shifted; in time and frequency they appear as pulse trains modulated by tones, a structure described as a “pulsone” (Li et al., 2024, Li et al., 2023).

The interference behavior of DD symbols is governed by the ambiguity function of the prototype pulse. For TF-consistent basis functions, the matched-filter correlation satisfies

$$s_{\rm DD}(\tau,\nu)=\sum_{l=0}^{M-1}\sum_{k=0}^{N-1}X_{\rm DD}[l,k]\, \Phi_{\rm DD}^{\tau_l,\nu_k}(\tau,\nu),$$6

Accordingly, DD symbol separation is a pulse-design problem: a sharply localized $$s_{\rm DD}(\tau,\nu)=\sum_{l=0}^{M-1}\sum_{k=0}^{N-1}X_{\rm DD}[l,k]\, \Phi_{\rm DD}^{\tau_l,\nu_k}(\tau,\nu),$$7 suppresses inter-delay and inter-Doppler interference, while broad sidelobes induce structured leakage (Li et al., 2023).

The practical pulse-shaping results are specific. General DD-communication theory shows that rectangular windows achieve perfect DD orthogonality, whereas truncated periodic signals can obtain sufficient DD orthogonality; a smoothed rectangular window with excess bandwidth gives slightly worse orthogonality but better DD localization (Li et al., 2024). Complementarily, periodic or approximately periodic time and frequency windows yield near-Nyquist DD signaling, and different periodic windows can preserve similar DD orthogonality while producing different transmit spectra (Li et al., 2023).

This establishes the principal design tradeoff in DD-QAM implementations: exact lattice orthogonality under rectangular windows versus improved localization, lower sidelobes, and reduced out-of-band emission under smoother periodic or approximately periodic windows.

4. Channel structure, leakage, and receiver design

The canonical DD channel model is sparse in physical coordinates: $$s_{\rm DD}(\tau,\nu)=\sum_{l=0}^{M-1}\sum_{k=0}^{N-1}X_{\rm DD}[l,k]\, \Phi_{\rm DD}^{\tau_l,\nu_k}(\tau,\nu),$$8 Under DD-domain signaling, the channel action is not a diagonal gain law but a DD shift/convolution law. For practical DD communications, the pre-matched-filter DD input-output relation is a twisted convolution, and after matched filtering the symbol-wise relation is determined by path gains, path delays, path Dopplers, and the ambiguity function of the shaped DD pulse (Li et al., 2024).

In OTFS and ODDM formulations with on-grid delay and Doppler, each path acts as a structured DD shift with a deterministic phase rotation. For ODDM, a lattice-sampled DD input-output relation takes the form

$$s_{\rm DD}(\tau,\nu)=\sum_{l=0}^{M-1}\sum_{k=0}^{N-1}X_{\rm DD}[l,k]\, \Phi_{\rm DD}^{\tau_l,\nu_k}(\tau,\nu),$$9

showing explicitly that the received DD symbol is a convolution-like superposition over delay and Doppler shifts rather than an OFDM-style one-tap multiplication (Lin et al., 2023).

The ideal sparse DD picture is degraded by fractional offsets and practical pulses. Both OTFS radar processing and OTFS channel-estimation studies decompose path parameters as integer-grid indices plus fractional parts; the consequence is leakage into neighboring DD bins and loss of strict sparsity (Zhang et al., 2023, Khan et al., 2021). This is not a minor implementation detail but a structural property: realistic DD-QAM receivers must account for off-grid spreading rather than assume perfectly isolated DD taps.

Channel estimation in this setting has therefore shifted toward parametric DD-domain methods. For OTFS with fractional delay and Doppler, Modified Maximum Likelihood (M-MLE) and Two-Step Estimation (TSE) exploit the fine DD resolution to decouple multi-path estimation into per-path estimation, and further decouple the per-path search into separate delay and Doppler searches. Simulations reported for these methods show better channel-estimation accuracy at lower complexity than other known accurate OTFS estimators, and SER close to perfect CSI when used with DD-domain message passing detection (Khan et al., 2021).

Equalization complexity remains a major obstacle for DD-domain QAM signaling. In ODDM, time-domain Tomlinson-Harashima precoding has been proposed to make a DD-domain single-tap equalizer feasible. That design explicitly assumes DD-domain $\tau_l=l\frac{T}{M},\qquad \nu_k=k\frac{1}{NT}.$0-QAM symbol mapping, introduces an adaptive-modulus modulo operation because the DD-to-time transform changes the symbol statistics, derives BER lower bounds for 4-QAM and 16-QAM, and states total precoding-plus-equalization complexity $\tau_l=l\frac{T}{M},\qquad \nu_k=k\frac{1}{NT}.$1 (Ma et al., 2024).

Receiver front-end choice is also consequential. For OTFS with rectangular pulses in very high mobility, direct time-domain to DD-domain conversion via Zak sampling yields a different effective DD channel from the conventional two-step TD$\tau_l=l\frac{T}{M},\qquad \nu_k=k\frac{1}{NT}.$2TF$\tau_l=l\frac{T}{M},\qquad \nu_k=k\frac{1}{NT}.$3DD receiver, with direct-conversion complexity $\tau_l=l\frac{T}{M},\qquad \nu_k=k\frac{1}{NT}.$4 and Doppler-robust spectral-efficiency behavior in the analyzed case (Mohammed, 2020). This suggests that DD-native front ends may be preferable when a DD-QAM waveform operates in regimes where Doppler is a significant fraction of bandwidth.

5. Radar sensing, ISAC, and explicit chirp DD-QAM

The relation between DD modulation and radar processing is unusually tight. In radar sensing via OTFS, the range-Doppler matrix computation is shown to be exactly OTFS demodulation under rectangular pulse shaping. When the sampled receive signal is rearranged into a fast-time/slow-time matrix and a DFT is applied along slow time, the resulting matrix is exactly the OTFS DD-domain receive matrix, identified as a discrete Zak transform (Zhang et al., 2023).

Because the OTFS/DD receive matrix is dense when the transmitted communication symbols are random, sensing requires an additional matched-filtering stage. The cited radar work therefore applies a 2D correlation between the received DD matrix and the known transmitted DD symbol matrix, yielding a pulse-compressed surface from which integer delay and Doppler bins are detected and fractional offsets are inferred from neighboring-peak ratios. This establishes a DD-domain radar-processing chain in which delay and Doppler estimation, leakage analysis, and target localization are performed directly on the same DD grid used for symbol placement (Zhang et al., 2023).

The broader DD-ISAC literature extends this viewpoint beyond a single OTFS construction. Delay-Doppler communications are argued to be well matched to ISAC because the waveform directly interacts with the sensing variables, the DD channel is relatively static during a frame compared with its TF counterpart, and the derivation of the range-Doppler matrix is identical to the demodulation process in Zak-based DD communications. The same literature emphasizes the role of the DD ambiguity function, and reports better target-detection probability than OFDM-based ISAC in its simulations, while also identifying unresolved problems in pulse design and deterministic–random waveform tradeoffs (Yuan et al., 2023).

The most explicit DD-QAM formulation appears in the 2025 chirp-based JCR work, which states: “We term this modulation scheme DD-QAM, since data is modulated on delay, Doppler and complex amplitude of the transmitted signal.” In that model, a chirp/FMCW frame carries a QAM symbol $\tau_l=l\frac{T}{M},\qquad \nu_k=k\frac{1}{NT}.$5, a delay-domain offset $\tau_l=l\frac{T}{M},\qquad \nu_k=k\frac{1}{NT}.$6, and a Doppler-domain offset $\tau_l=l\frac{T}{M},\qquad \nu_k=k\frac{1}{NT}.$7, so that communication is embedded into radar-relevant coordinates rather than superimposed on a separate communication-centric waveform (Li et al., 16 Dec 2025).

This chirp DD-QAM design is sensing-centric. It uses beacon-frame-aided DDM for MIMO operation, derives an achievable-rate expression from a finite-alphabet DD-domain model in which the transmitted DD vector contains only one nonzero element, and proposes a dual-compensation receiver at the passive vehicle: predicted physical delay and Doppler are subtracted from current estimates to recover the data indices, and the recovered data offsets are then removed so that sensing and EKF-based tracking remain coherent (Li et al., 16 Dec 2025).

The same work reports that the number of bits per frame is

$\tau_l=l\frac{T}{M},\qquad \nu_k=k\frac{1}{NT}.$8

that adding DD-domain modulation improves achievable rate by 14 bits/symbol in DDM and 9 bits/symbol in TDM relative to conventional QAM alone, and that the SERs from delay, Doppler, and complex amplitude are identical in its simulations. It also couples DD-QAM to EKF-based estimation of orientation and tangential velocity beyond conventional 4D radar outputs (Li et al., 16 Dec 2025).

6. Variants, neighboring frameworks, and open limitations

DD-QAM is best regarded as a family resemblance across several DD-native or DD-centered designs rather than a single universally fixed waveform. OTFS is explicitly treated as a typical DD communication scheme, not the whole class; ODDM/DDMC develops a more direct DD multicarrier viewpoint matched to the equivalent sampled DD channel; chirp DD-QAM reinterprets DD modulation from a radar-native FMCW perspective (Li et al., 2024, Lin et al., 2023, Li et al., 16 Dec 2025).

Several neighboring variants clarify the design space. Orthogonal Chirp Delay-Doppler Division Multiplexing (CDDM) begins with ordinary constellation symbols, such as QPSK, but spreads each symbol across sparse DD chirp basis functions by a chirp-Zak transform, so that the DD coefficients are not raw QAM samples but chirp-spread linear mixtures. This is therefore not direct one-cell-per-symbol DD-QAM, but a DD-domain chirp-spread QAM framework (Bai et al., 22 Nov 2025). Doppler Shift Keying (DSK) over ODDM goes further and replaces dense QAM-like DD-bin loading by one-hot or sequence-shifted Doppler-domain index modulation, using the same ODDM backbone while changing the alphabet and reducing PAPR and detector complexity (Wang et al., 3 Mar 2026).

The principal limitations in the DD-QAM literature are equally consistent. Exact orthogonality statements typically depend on ideal or rectangular pulses, or on periodic/approximately periodic window assumptions (Li et al., 2023, Li et al., 2024). Many clean DD input-output laws assume integer delay and Doppler indices on an equivalent sampled grid, while real channels exhibit fractional offsets and therefore leakage (Khan et al., 2021, Zhang et al., 2023). Practical high-mobility equalization remains difficult enough that transmitter-side precoding, message passing, or structured sparse detectors are still active design topics (Ma et al., 2024, Wang et al., 3 Mar 2026).

A second misconception is that DD-QAM necessarily implies a direct symbol-per-bin mapping. The literature does support that interpretation in general DD signaling and in explicit chirp DD-QAM, but it also contains spread or precoded DD realizations such as OTFS and CDDM, where the native data symbols are ordinary QAM points and the DD grid carries transformed or superposed representations (Zhang et al., 2023, Bai et al., 22 Nov 2025). A plausible implication is that “DD-QAM” is most usefully treated as a modulation philosophy—QAM carried by DD-structured waveforms and receivers—rather than a single transmitter equation.

Within that philosophy, the enduring technical themes are stable: DD-domain symbol placement; TF-consistent basis construction; pulse shaping through ambiguity-function control; sparse or nearly sparse DD-channel modeling; explicit treatment of fractional leakage; and receiver architectures that preserve the DD geometry rather than forcing it back into a purely TF interpretation.