Orthogonal Chirp Delay-Doppler Multiplexing (CDDM)

• Orthogonal Chirp Delay-Doppler Division Multiplexing (CDDM) is a multi-carrier modulation framework that uses orthogonal chirp waveforms mapped onto a delay-Doppler grid for robust performance in doubly selective channels.
• The framework combines advanced precoding transforms, such as IDFnT and CZT, to achieve full diversity gain and superior BER improvements under high mobility.
• CDDM balances tradeoffs between PAPR, spectral efficiency, and computational complexity, making it ideal for applications in high-speed communications, radar, and integrated sensing.

Orthogonal Chirp Delay-Doppler Division Multiplexing (CDDM) is a multi-carrier modulation framework designed for efficient, resilient communication in doubly selective (delay and Doppler dispersive) channels. CDDM achieves robust performance under high mobility by combining the orthogonality of linear frequency-modulated (chirp) waveforms with delay–Doppler domain multiplexing and targeted precoding transforms. Several concrete instantiations—including Affine Frequency Division Multiplexing (AFDM), Vector Orthogonal Chirp Division Multiplexing (VOCDM), and Chirp-Zak Transform (CZT)-based CDDM—demonstrate modular signal design, full diversity gain, and tractable detection complexity. The approach has demonstrated superior BER, diversity order, and OOBE in recent simulation studies under realistic channel conditions (Bai et al., 22 Nov 2025, Lu et al., 10 Aug 2025, Yin et al., 7 Feb 2025, Li et al., 22 May 2025).

1. Mathematical Foundations and Modulation Principles

CDDM relies on the synthesis of orthogonal chirp waveforms and their mapping onto a delay–Doppler (DD) grid. In the discrete-time case, the orthogonal chirps are constructed as quadratic-phase signals:

$$\varphi_i(w) = \exp(j\pi/4) \exp\left(-j \frac{\pi}{N} (w - i)^2\right)\,,\quad w=0,\ldots,N-1$$

with orthogonality:

$$\sum_{w=0}^{N-1} \varphi_i(w)\, \varphi_{i'}^*(w) = N\,\delta_{i,i'}$$

Modulation in CDDM usually employs an inverse discrete Fresnel transform (IDFnT), affine Fourier transform (IDAFT), or the more general Chirp-Zak Transform (CZT). For example, VOCDM modulation applies $M$ parallel IDFnTs of size $N$ to partitioned data blocks (Lu et al., 10 Aug 2025):

$$\mathbf{u} = (\Phi_N^{\mathcal{H}} \otimes I_M) \mathbf{s}$$

where $\Phi_N$ is the DFnT matrix and $\mathbf{s}$ contains $K=MN$ symbols. In CZT-based CDDM, the transmit signal is synthesized by superimposing time-domain chirps, mapped via block-partitioning and the Zak transform to the DD-domain (Bai et al., 22 Nov 2025).

2. Orthogonality, Delay-Doppler Spreading, and Diversity

The essential property of CDDM waveforms is orthogonality in the delay–Doppler domain under proper grid design. Assuming delay step $\Delta\tau=T/L$ and Doppler step $\Delta\nu=1/T$, the DD shifted chirps are constructed as

$$s_{\ell,k}(t) = s(t - \ell\Delta\tau)\exp(j2\pi k\Delta\nu t)$$

and satisfy

$$\langle s_{\ell,k}, s_{\ell',k'} \rangle = \delta_{\ell,\ell'} \delta_{k,k'}$$

provided $\Delta\tau \Delta\nu = 1$ (Li et al., 22 May 2025). In a channel with maximum delay index $L$ and Doppler index $Q$, full diversity is achieved when the number of parallel transforms $M \ge L+1$ and transform size $N \ge 2Q+1$ (Lu et al., 10 Aug 2025):

$|\mathbb{O}(L,Q,M,N)| = \rho = (L+1)(2Q+1)$

where $\mathbb{O}(L,Q,M,N)$ collects sub-diagonal positions of the effective channel matrix after chirp modulation. Failure to meet these parameters leads to diversity loss due to coherent superposition in the DD domain.

3. Channel Modeling and Input–Output Relationships

CDDM is formulated for doubly selective channels, leveraging representations such as the complex-exponential basis expansion model (CE-BEM) (Lu et al., 10 Aug 2025) and canonical DD channel models (Bai et al., 22 Nov 2025, Yin et al., 7 Feb 2025). Given sampled symbols and received blocks after cyclic prefix removal, the input–output relation for VOCDM is

$\mathbf{r} = H \mathbf{u} + \mathbf{v}$

$$\mathbf{y} = (\Phi_N \otimes I_M) \mathbf{r} = H_{\rm eff}\, \mathbf{s} + \bar{\mathbf{v}}$$

where $H$ models combined delay–Doppler convolution and $H_{\rm eff}$ is the effective channel in the transformed domain. In CZT-based CDDM, after receive filtering and time-domain sampling, the DD-domain observation is

$$Y_{DD}[m,n] = \sum_{p=1}^{P} h_p X_{DD}[\widehat m, \widehat n] e^{j 2\pi \frac{k_p (m - l_p)}{M_D N_D}} + Z[m,n]$$

with $X_{DD}$ the DD-mapped symbol block. Detection leverages the knowledge of nonzero grid locations, sparse path support, and chirp-phase rotations, enabling efficient data recovery via correlators or low-complexity linear equalization.

4. Power, Spectral, and Complexity Tradeoffs

Peak-to-average power ratio (PAPR) is controlled in CDDM largely through selection of the transform size $N$. For VOCDM, the instantaneous PAPR is bounded above by $aN$ ($a$ is the maximum constellation modulus), favoring small $N$ for PAPR reduction (Lu et al., 10 Aug 2025). Spectral efficiency for CDDM is computed as

$R_{\rm eff} = \frac{LK \log_2 M}{TB}$

where $L,K$ are grid dimensions, $M$ constellation size, $T$ block duration, and $B$ bandwidth (Li et al., 22 May 2025). Computational complexity for CZT-based CDDM is $\mathcal{O}(2 M_D N + N \log N_D)$ for both modulation and demodulation (Bai et al., 22 Nov 2025), a modest increase over OFDM’s $\mathcal{O}(N\log N)$ but with better resilience and estimation features.

5. Pilot/Aided Channel Estimation and Implementation

Sparse pilot and embedded pilot designs in CDDM exploit the DD orthogonality, leading to channel estimation methods that use thresholding and chirp correlation, yielding normalized mean-square error (NMSE) improvements over ODDM, OTFS, and AFDM approaches (Bai et al., 22 Nov 2025). Embedded pilots create guard blocks in the DD grid, enhancing SNR for estimation at the expense of some spectral efficiency. Implementation requires matching chirp-phase coding, precise delay–Doppler grid alignment, and management of cyclic prefixes for DD isolation. Hardware challenges in radar–communication contexts include synchronization and preservation of waveform orthogonality under mixed-IF sampling (Li et al., 22 May 2025).

6. Performance Evaluation and Comparisons

CDDM achieves full diversity order $\rho=(L+1)(2Q+1)$ under optimal design, with BER improvements (up to 4 dB over OFDM, 2 dB over OTFS, and 1 dB over ODDM) at high mobility (e.g., 500 km/h) and realistic channel models (Bai et al., 22 Nov 2025). Out-of-band emission (OOBE) is minimized by using square-root Nyquist pulses in CZT-based CDDM, matching or exceeding DFT-windowed OCDM (Bai et al., 22 Nov 2025). Comparative analysis shows that, while AFDM can also reach full diversity, its PAPR is higher relative to VOCDM/CDDM due to single large transform size (Yin et al., 7 Feb 2025, Lu et al., 10 Aug 2025). In radar-integrated settings, CDDM (DDM-based) delivers superior joint sensing and communication hit rates (exceeding 90% for delay, Doppler, and angle below relevant resolution) even at low SNR, outperforming TDM-based chirp schemes under equivalent resources (Li et al., 22 May 2025).

7. Applications and Design Guidelines

CDDM is applicable to high-mobility communications (V2X, satellite links, rail transit), mmWave radar, underwater acoustic channels, and integrated sensing-communication systems (Li et al., 22 May 2025, Yin et al., 7 Feb 2025, Bai et al., 22 Nov 2025). Key design guidelines include: (1) selecting transform sizes to match delay/Doppler spread, (2) balancing diversity and PAPR through $M$ and $N$, (3) employing guard bins for practical resilience, and (4) optimizing pilot structure based on use case. Open research directions encompass full-duplex AFDM, coded-chirp multiplexing for optimal diversity, numerology design for next-generation networks, and index modulation via selective chirp activation (Yin et al., 7 Feb 2025). The framework’s modularity, full diversity, and decoding tractability position CDDM as a promising PHY-layer candidate for resilient 6G and joint radar-communication systems.

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Selected Paper References:

• Vector Orthogonal Chirp Division Multiplexing Over Doubly Selective Channels (Lu et al., 10 Aug 2025)
• Orthogonal Chirp Delay-Doppler Division Multiplexing (CDDM) Modulation for High Mobility Communications (Bai et al., 22 Nov 2025)
• Affine Frequency Division Multiplexing: Extending OFDM for Scenario-Flexibility and Resilience (Yin et al., 7 Feb 2025)
• Chirp Delay-Doppler Domain Modulation: A New Paradigm of Integrated Sensing and Communication for Autonomous Vehicles (Li et al., 22 May 2025)