ODDM: Orthogonal Delay-Doppler Division Multiplexing

• ODDM is a multicarrier modulation scheme that maps data onto a fine delay-Doppler grid, enhancing robustness in fast-varying channels.
• It employs a delay–Doppler orthogonal pulse (DDOP) design that achieves local orthogonality, thereby reducing inter-symbol interference.
• ODDM enables efficient implementation via pulse-shaped or filtered OFDM techniques, offering improved spectral efficiency and reduced out-of-band emission.

Orthogonal Delay-Doppler Division Multiplexing (ODDM) is a multicarrier modulation scheme developed to address the challenges of reliable and efficient wireless communications over linear time-varying (LTV) channels, particularly in high-mobility scenarios. ODDM leverages a delay–Doppler domain orthogonal pulse (DDOP) structure, allowing for direct mapping of information onto a fine resolution grid in the delay-Doppler (DD) plane, thereby enabling robust performance in environments characterized by doubly selective fading. The design circumvents classical limitations imposed by global Weyl–Heisenberg (WH) frame theory through the use of locally orthogonal pulse trains, and can be efficiently implemented through pulse-shaped or filtered OFDM-like techniques tailored for fine DD grid resolutions.

1. Design Principles and DDOP Foundations

ODDM is fundamentally built to match the fine delay and Doppler resolutions of the channel. Its core design advancements are:

• Fine Resolution Grid: ODDM modulates symbols on a DD grid with spacing $\Delta T = T_0/M$ (delay) and $\Delta F = 1/(N T_0)$ (Doppler), with $T_0$ the base time interval, $M$ the number of multicarrier symbols, and $N$ the number of Doppler bins.
• Delay–Doppler Orthogonal Pulse (DDOP): The transmit pulse is constructed as a sum of $N$ time-shifted sub-pulses:

$u(t) = \sum_{n=0}^{N-1} a(t - nT)$

where $a(t)$ is a square-root Nyquist (or root-raised cosine) pulse with duration $2Q (T_0/M)$ and unit energy.

• Local Orthogonality: The DDOP achieves local (finite-dimensional) orthogonality, formalized by the ambiguity function condition

$$\mathcal{A}_{u,u}(mT_0/M, n/(N T_0)) = \delta(m)\delta(n)$$

for $m \in [0, M-1]$, $n \in [0, N-1]$, ensuring perfect orthogonality with respect to the actual finite DD grid, even when the global (infinite) WH orthogonality condition is unattainable due to the fine grid area ($1/(MN) \ll 1$).

The result is a pulse train that is neither narrowly time- nor frequency-localized, but forms a pseudo two-dimensional impulse sharply confined to the support needed for symbol separation in the DD domain (Lin et al., 2022, Lin et al., 2022, Lin et al., 2023).

2. Mathematical Model and Input–Output Relations

In ODDM, symbols $X[m, n]$ are mapped onto the fine DD grid, leading to the analog transmit signal: $$x(t) = \sum_{m=0}^{M-1} \sum_{n=0}^{N-1} X[m, n] \cdot u(t - mT_0/M) \cdot \exp\left\{j2\pi \frac{n}{N T_0}(t - mT_0/M)\right\}$$ This structure couples the DD domain grid, the underlying channel, and the pulse $u(t)$ for efficient transmission.

After passing through a doubly-selective channel with delay–Doppler response $$h(\tau, \nu) = \sum_{p=1}^P h_p\delta(\tau - \tau_p)\delta(\nu - \nu_p)$$, and subsequent matched filtering with $u(t)$, the DD domain input–output relation is: $$Y[m, n] = \sum_{p=1}^P h_p \cdot \tilde{X}(m - l_p, [n - k_p]_N) \, e^{j2\pi \frac{k_p(m - l_p)}{MN}} + z[m, n]$$ where $z[m, n]$ denotes noise and $\tilde{X}$ accounts for cyclic effects. In vectorized form: $\mathbf{y} = \mathbf{H} \mathbf{x} + \mathbf{z}$ with $\mathbf{H}$ exhibiting a block-circulant-like and sparse structure due to the DD domain's geometrical properties—facilitating efficient, low-complexity receiver algorithms (Lin et al., 2022, Lin et al., 2022).

3. Pulse Shaping, Orthogonality, and Spectral Properties

The DDOP is engineered as a pulse train for simultaneous time and frequency localization while remaining orthogonal on the DD grid. The main features are:

• Pulse-Train Construction: By summing shifted sub-pulses, ODDM creates a waveform that matches the fine delay and Doppler resolutions.
• Frequency Domain Structure: The Fourier transform of $u(t)$ reveals a multi-sinc pattern, reflecting the local orthogonality and the distributed energy in both time and frequency:

$$U(f) = \frac{e^{-j2\pi f\tilde{T}}}{T} A(f) \sum_{k=-\infty}^{\infty} e^{j2\pi(N-1)k/2} \operatorname{sinc}(f N T - kN)$$

where $A(f)$ is the subpulse spectrum, and $N$ is the number of subcarriers (Lin et al., 2023).

• Spectral Characteristics: Due to the pulse shaping, ODDM dramatically suppresses out-of-band (OOB) radiation compared to rectangular-pulse-based OTFS. For example, ODDM exhibits up to 25 dB reduction in OOB emission relative to OTFS (Lin et al., 2022, Lin et al., 2022, Bayat et al., 2023).
• Staircase Frequency Response: Linear pulse-shaping in ODDM induces a "staircase" appearance in the spectrum, arising from Doppler-dependent filtering (distinct for each Doppler bin) (Bayat et al., 2023).
• Out-of-Band Control: The insertion of zero-guard (ZG) symbols further reduces OOB emissions without substantial BER degradation (Bayat et al., 2023).

4. Performance Metrics and Comparative Results

ODDM’s performance has been rigorously validated through simulation and analysis:

• Bit Error Rate (BER): Under high-mobility and doubly selective channels, ODDM achieves up to 2 dB SNR improvement (at low BER, e.g., $10^{-6}$) compared to OTFS and conventional OFDM (Lin et al., 2022, Lin et al., 2022).
• Spectral Efficiency: All available DD subchannels are utilized, avoiding the wastage of subcarrier edges typical in many TF-based multicarrier schemes.
• Spectral Containment: ODDM, with well-designed pulses and possible ZG insertion, features distinctly lower OOB leakage—critical for spectral mask compliance and minimizing adjacent channel interference (Bayat et al., 2023).
• Diversity and Robustness: The orthogonality in the DD plane ensures minimal inter-symbol/carrier interference even in channels with strong Doppler spread, and the block-circulant structure in the DD domain input–output relation supports robust and efficient equalization.
• Detection Complexity: Exploiting the channel matrix structure with advanced detectors (e.g., message passing, OAMP) further improves performance at low computational cost (Lin et al., 2022, Lin et al., 2022).

5. Implementation Strategies

ODDM can be efficiently realized by two primary methods:

• Pulse-Shaped OFDM (PS-OFDM) Implementation: ODDM symbols are constructed as pulse-shaped (by $u(t)$) upsampled OFDM symbols. Standard IDFT/DFT blocks are employed, with additional filtering and staggered scheduling on a fine symbol grid (Lin, 15 Apr 2025).
• Wideband Filtered OFDM Approximation: The DDOP’s unique pulse-train property allows ODDM to be approximated by OFDM sequences filtered by a wideband Nyquist filter. With proper parameters (e.g., pulse roll-off, duration controls), implementation errors are minimized and can be quantified using normalized mean squared error (NMSE) analysis (Lin, 15 Apr 2025).
• Practical Hardware: Advanced optical implementations of ODDM with optically enabled equalization have also been proposed for chip-to-chip communication, achieving high throughput and low power via comb-based ODDM in conjunction with photonic integration (Zazzi et al., 2023).

6. Application Domains and System Implications

ODDM offers significant advantages in high-mobility, doubly selective, and spectrally constrained environments:

• Wireless Communications: Suited for vehicular, rail, or mmWave/THz communication where the channels are highly time and frequency dispersive.
• Satellite and NTN: In LEO satellite scenarios, ODDM’s robustness to Doppler and delay spread enables reliable links (Liu et al., 24 Jun 2025).
• Integrated Sensing and Communications (ISAC): The underlying pulse design allows ODDM waveforms to simultaneously support fine-range and velocity estimation while maintaining communications performance (Lin et al., 2023).
• Optical Short-Reach Links: ODDM’s orthogonality and spectral efficiency translate to low-complexity, high-capacity data links suitable for next-generation datacenter applications (Zazzi et al., 2023).

Table: Comparison of ODDM with Other Multi-Carrier Schemes

Aspect	OFDM/OTFS	ODDM
Physical Domain	TF grid/coarse	DD grid/fine
Orthogonality	Global (WH frame)	Local (M×N grid)
Pulse Design	Rectangular/MSK	Pulse-train (DDOP)
OOB Radiation	Significant	Strongly suppressed
Channel Utilization	Partial (edges unused)	Full (edges used)
Suitable Channels	LTI/low-mobility	LTV/high-mobility
Implementation	IDFT/DFT + windowing	Pulse-shaped/filtered OFDM

7. Future Directions

Research trends and open problems for ODDM include:

• Advanced Detection Schemes: Exploit the block-circulant DD channel structure for low-complexity message passing, OAMP, and machine-learning-based equalizers.
• Pulse Design Optimization: Explore trade-offs between roll-off, symbol rate, and OOB control, including the use of ZG symbols for spectral containment.
• Channel Estimation and Off-Grid Correction: Develop sparse Bayesian and grid refinement methods to deal with mismatches between real-world and assumed DD grids (Shan et al., 9 Jul 2024).
• Integration with MIMO/Hybrid Beamforming: Extend ODDM to massive MIMO and ISAC architectures, particularly in THz/optical wireless applications (Li et al., 25 Feb 2025).
• Joint Communication and Sensing: Exploit DDOP ambiguity function properties for dual-functional radar–communication systems.
• Standardization and System-Level Evaluation: Assess ODDM in standardized frameworks (e.g., 6G physical layer) and real-world deployments.

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ODDM—pioneered through the formalization of the DDOP and fine-grid local orthogonality—represents a distinctive direction in the evolution of multicarrier waveforms. By directly matching modulation to the doubly selective channel structure and practical transmission/reception dimensions, ODDM attains high robustness, spectral efficiency, and implementation flexibility in challenging next-generation wireless and optical environments.