Orthogonal Chirp Division Multiplexing (OCDM)

Updated 5 January 2026

OCDM is a multicarrier modulation technique that maps complex symbols to orthogonal chirp signals via the inverse discrete Fresnel transform, ensuring resilience in dispersive channels.
It employs quadratic phase profiles and cyclic prefix insertion to counteract multipath and Doppler effects, enabling simple one-tap equalization in doubly selective environments.
Advanced variants like GOCDM and VOCDM mitigate high peak-to-average power ratios while enhancing diversity gains, making OCDM effective for RF, THz radar, and underwater communication systems.

Orthogonal Chirp Division Multiplexing (OCDM) is a multicarrier modulation paradigm wherein blocks of complex-valued information symbols are mapped to time-domain signals via the inverse discrete Fresnel transform (IDFnT), generating a superposition of orthogonal discrete-time chirps. Each constituent chirp sweeps linearly across the available bandwidth over the symbol period and achieves mutual orthogonality by virtue of its quadratic phase profile. OCDM generalizes the classical OFDM architecture, providing enhanced resilience to doubly selective channels (both frequency- and time-selective) and enabling robust performance in highly dispersive, high-mobility, and integrated sensing-communication scenarios. Despite its robustness, a key practical challenge is its elevated peak-to-average power ratio (PAPR), which can compromise amplifier efficiency. This has motivated extensions such as Generalized OCDM (GOCDM) utilizing blockwise Fresnel transforms to balance diversity gains and PAPR reduction. OCDM is now established across RF, THz radar, powerline, and underwater acoustic domains, with significant theoretical, architectural, and implementation literature.

1. Mathematical Foundations and Signal Model

In standard OCDM, the ith block of N complex symbols $x = [x_0, ..., x_{N-1}]^\top$ is modulated via an N-point IDFnT, producing discrete-time samples $s[n]$ as

$s[n] = \sum_{k=0}^{N-1} x_k \cdot (1/\sqrt{N}) \, e^{-j\pi/4} \, e^{j\pi(k-n)^2/N}, \quad n=0,...,N-1$

where the DFnT kernel $e^{j\pi(k-n)^2/N}$ ensures mutual orthogonality in the Fresnel (chirp) domain (Liu et al., 2024, Ouyang et al., 2016). The time-domain OCDM block is equivalently written in matrix-vector form as $s = \Phi_N^H x$ , where $\Phi_N$ is the unitary DFnT matrix.

The orthogonality holds by the quadratic phase structure and unitary property of $\Phi_N$ : $\sum_{n=0}^{N-1} s^{(p)}[n] (s^{(q)}[n])^* = \delta_{p,q}, \quad \forall\,p, q \in \{0, ..., N-1\}$ A cyclic prefix (CP) of length at least equal to the channel impulse response is appended to ensure preservation of circular convolution structure under multipath (Ouyang et al., 2016).

Extensions such as Multicarrier Chirp Division Multiplexing (MCDM) and Vector OCDM (VOCDM) exploit parallel IDFnTs, Kronecker-product generalizations, and customized chirp parameterizations for PAPR and diversity control (Lu et al., 10 Aug 2025, Huang et al., 2018).

2. Orthogonality, Diversity, and Channel Effects

The defining property of OCDM is the use of orthogonal chirp waveforms whose instantaneous frequency sweeps linearly and whose cross-correlation over the symbol interval vanishes except for identical indices. This can be formalized for the kth chirp as

$s_k(t) = \sqrt{1/T} \exp \big[j (2\pi k \Delta f t + \pi \mu t^2)\big],\quad 0 \leq t \leq T,\ k=0,\dots,K-1$

with chirp rate $\mu = B_c/T$ and subcarrier spacing $\Delta f = 1/T$ selected to maintain orthogonality (Huang et al., 2018, Huang et al., 2020). The discrete Fresnel transform decomposes convolution operations such that Fresnel-domain processing effectively diagonalizes channel effects, supporting simple one-tap equalization under circulant multipath models (Ouyang et al., 2016).

Under doubly selective channels (large delay and Doppler spreads), each path can be modeled as a combination of circulant delay (cyclic shift) and Doppler-phase multiplication, with the effective Fresnel-domain channel matrix $H_\text{eff}$ exhibiting $L$ -sparse structure: $[H_\text{eff}]_{p,p'} = \sum_{\ell=0}^{L-1} \hat{h}_{p,\ell} \delta[\langle p-p' - d_\ell\rangle_{MN}]$ where $d_\ell$ encodes net cyclic shift and Doppler, and $L$ is the number of nonzero terms per row—drastically simplifying inversion and detection (Liu et al., 2024).

3. Transmitter and Receiver Architectures

OCDM transmitters concatenate blocks of input symbols, perform the appropriate IDFnT involving FFTs interleaved with diagonal quadratic-phase multiplications, append cyclic prefixes, and convert to analog for transmission. For generalized forms (GOCDM, VOCDM), the input symbols $x \in \mathbb{C}^{MN}$ are reshaped into an $M \times N$ array, and an N-point IDFnT is applied independently to each row, followed by reflattening and CP extension (Lu et al., 10 Aug 2025, Liu et al., 2024).

At the receiver, after CP removal and sampling, a DFnT (or generalized version) is applied, mapping the received block into the Fresnel domain. Sparse message-passing detectors (MP) are employed over the associated factor graph:

Each observation $y_p$ depends on only $L$ symbols,
Gaussian extrinsic messages pass between observation and variable nodes,
Damping and convergence tracking term $\eta$ are used for stable iterative updates,
Hard decisions are formed when symbol marginals reach near-deterministic values (Liu et al., 2024, Liu et al., 2024, Bomfin et al., 2019).

This approach reduces complexity from $O(N^3)$ (full matrix inversion in MMSE detection) to $O(N L M)$ per iteration, with $L \ll N$ .

4. PAPR Analysis and Reduction Techniques

OCDM, inheriting blockwise multicarrier-style synthesis, exhibits high peak-to-average power ratio (PAPR): $\mathrm{PAPR} = \frac{P_\mathrm{max}}{P_\mathrm{avg}}, \quad P_\mathrm{max} = \max_n |s[n]|^2,\quad P_\mathrm{avg} = \frac{1}{N} \sum_n |s[n]|^2$ This is problematic for amplifier linearity and energy efficiency. GOCDM, by partitioning the transform into parallel smaller blocks, sharply reduces PAPR—simulation with $MN=128$ and $4$-QAM demonstrates several dB improvement over OCDM and OFDM at $\Pr(\mathrm{PAPR}>\mathrm{PAPR}_0)=10^{-3}$ (Liu et al., 2024).

Low-complexity spreading paradigms using WHT, DCT, ZC, and interleaved DFT premodulation further flatten envelope fluctuations, achieving up to $2.2$ dB PAPR reduction without side information, outperforming legacy methods such as SLM and PTS (Ali et al., 3 May 2025).

5. Performance in Communications and Sensing Applications

OCDM robustly achieves full multipath and Doppler diversity due to bandwidth-wide chirp spreading. Empirically, OCDM outperforms OFDM by $0.5$–$2.5$ dB at BER/BLOCK-ERROR $10^{-3}$ across overspread underwater acoustic (high $S_tS_f$ ) and high-mobility vehicular channels with both MMSE and MP-based detectors (Liu et al., 2024, Liu et al., 2024, Bomfin et al., 2019). High-order modulations and adaptive pilot allocation maintain advantages under severe fading (Huang et al., 2020, Huang et al., 2018).

Integrated sensing and communication architectures leverage OCDM's Fresnel structure for radar parameter estimation. THz-band automotive radars partition bandwidth into subbands, transmit multi-chirp OCDM symbols, and combine range/velocity estimates across subbands via CRLB-weighted fusion to achieve sub-millimeter accuracy (Bhattacharjee et al., 2023). Joint radar-communications multiuser access and sector-modulated OCDM deliver superior sidelobe level (PSLR/ISLR), PAPR, and robust range estimation compared to OFDM (Oliveira et al., 2022).

Ultra-massive MIMO implementations embed dedicated sensing subcarriers for near-field object localization, using OCDM for both data and FMCW-style sensing, thereby exploiting spatial and waveform diversity (Wan et al., 29 Dec 2025).

OCDM arises within a unified DFT-based mother waveform framework (SC-IFDM), wherein configuration of lattice indices and phase adjustments seamlessly yields OFDM, OTFS, AFDM, or FMCW waveforms. Resource allocation, back-/forward compatibility, and orthogonality are managed within a Kronecker product–structured frequency-domain grid, allowing mutual coexistence without spectral overlap or hardware modifications. OCDM-specific phase design ensures orthogonality and insensitivity to ISI/ICI (Boudjelal et al., 16 Mar 2025).

Vector OCDM (VOCDM) applies $M$ parallel $N$ -size IDFnTs, supporting tunable tradeoffs between diversity and PAPR, subject to channel delay/Doppler constraints. With proper parameter selection ( $M \geq L+1$ , $N \geq 2Q+1$ under CE-BEM), VOCDM extracts the full diversity $\rho = (L+1)(2Q+1)$ of a doubly selective channel and yields PAPR below that of OTFS (Lu et al., 10 Aug 2025).

7. Comparative Metrics and Implementation Complexity

Scheme	PAPR Reduction (dB)	BER Advantage (dB)	Per-Block Complexity
Standard OCDM	0	0–2.5	$O(N \log N)$
GOCDM	2–3	0.5–1	$O(MN \log N)$
Spreading + OCDM	1.3–2.2	0.5	$O(N \log N)$
VOCDM	up to 3	tunable	$O(M N \log N)$
MP Receiver	—	3–4 (vs. MMSE/OFDM)	$O(N L M)$ per iteration

GOCDM and spreading methods are particularly advantageous in power-/energy-constrained scenarios and large-scale sensor networks. All OCDM variants retain the capacity for FFT-like hardware implementation and adaptation to OFDM backends via pre-/post-chirp phase rotations and block-level transforms (Ali et al., 3 May 2025, Ouyang et al., 2016).

OCDM, including its generalized variants, constitutes a robust multicarrier modulation methodology suitable for high-mobility and doubly selective channels, with full-band diversity, efficient implementation, compatibility with legacy architectures, and scalability across radar, communications, and sensing use cases. Research in iterative receiver design, resource allocation on unified mother-waveform frameworks, and PAPR mitigation continues to refine its role in next-generation network and sensing systems.