Correlation-Based DAFT Domain Detector (CDD)

• CDD is a robust detector for AFDM systems that leverages the cyclic-shift property in the DAFT domain to achieve full diversity gain without matrix inversion.
• It employs correlation-based equalization and despreading with impulse-like autocorrelation sequences to suppress severe interference and channel distortions.
• The detector attains linear computational complexity and performance comparable to MMSE detectors, making it ideal for high-mobility, adversarial scenarios.

The Correlation-Based DAFT Domain Detector (CDD) is a linear-complexity detector in affine frequency division multiplexing (AFDM) systems designed for robust signal recovery under severe interference and mobility, leveraging the cyclic-shift structure induced in the discrete affine Fourier transform (DAFT) domain. CDD combines correlation-based equalization—matched to the underlying multipath structure—and despreading with impulse-like autocorrelation sequences, resulting in full diversity gain without requiring any matrix inversion. Its operation and theoretical properties render it especially suited for anti-interference AFDM systems in adversarial and high-mobility wireless communication environments (Yuan et al., 17 Dec 2025).

1. System Model and DAFT-Domain Input-Output Relation

Let $x\in\mathbb{C}^{N}$ denote the DAFT-domain symbol vector and $y \in \mathbb{C}^{N}$ the received DAFT-domain vector. For a doubly-selective channel modeled by $L$ taps with path delays $l_i$, Dopplers $v_i$, and complex gains $h_i$, the DAFT-domain input-output relationship is: $$y = \sum_{i=0}^{L-1} h_i H_i x + w,\qquad w\sim\mathcal{CN}(0, P_n I_N)$$ where each $H_i$ represents the DAFT-domain channel matrix for the $i$th path. The DAFT and its inverse (IDAFT) implement chirp signal transformations with parameters $c_1$ and $c_2$: $$S(m) = \frac{1}{\sqrt N}\, e^{-j2\pi c_2 m^2} \sum_{n=0}^{N-1} s(n)\, e^{-j2\pi (\frac{mn}{N}+c_1 n^2)}$$

$$s(n) = \frac{1}{\sqrt N}\, e^{+j2\pi c_1 n^2} \sum_{m=0}^{N-1} S(m)\, e^{+j2\pi (\frac{mn}{N}+c_2 m^2)}$$

A crucial property is that, up to a phase rotation, each $H_i$ acts as a cyclic shift on $x$: $$[H_i]_{p, q} = e^{j\phi_i(p, q)}\, \delta(\langle p + \text{loc}_i \rangle_N - q)$$ where $\text{loc}_i$ encodes the cyclic shift associated with the $i$th path.

2. Spreading Sequence and Autocorrelation Properties

Signal robustness is achieved by spreading each symbol using a sequence $d[n]$ of length $N_d$ characterized by impulse-like periodic autocorrelation: $$R_d(k) = \sum_{n=0}^{N_d-1} d[n] d^*\bigl(\langle n-k\rangle_{N_d}\bigr), \qquad R_d(0)=N_d, \ |R_d(k)| \ll 1 \text{ for } k\ne0$$ This ensures that after correlation (despreading), the desired signal—matched to the cyclic-shifted path responses—accumulates constructive gain $N_d$, while any non-matching (non-coherent) interference is averaged down to near zero.

3. Detector Construction: Correlation-Based Equalization and Despreading

The CDD operation proceeds via two main steps:

A. Correlation-Based Equalization:

Matrix inversion is avoided. Instead, for each path $i$, the received vector is cyclically shifted back by $\text{loc}_i$, multiplied by the conjugate phase $h_i^*$, and summed over a compensation window covering the Doppler spread: $$\widetilde x_i = \frac{1}{N} h_i^* \sum_{k=-k_\nu}^{+k_\nu} \Pi^{\text{loc}_i + k}\bigl(\pi_{i, k} \odot y\bigr)$$ where $\Pi^k$ denotes a forward cyclic shift by $k$, and $\pi_{i, k}$ is a path- and compensation-index-specific phase rotation. Summing these across all paths yields the equalized DAFT-domain symbol vector: $$\widehat x_{\mathrm d} = \sum_{i=0}^{L-1} \widetilde x_i$$

B. Despreading:

The vector $\widehat x_{\mathrm d}$ is segmented into $N/N_d$ blocks, each of length $N_d$, and correlated with the spreading sequence. When reshaped into a matrix $$\widehat{X}_{\mathrm d} \in \mathbb{C}^{N_d \times (\cdots)}$$, despreading yields: $$\widehat c_{\mathrm d} = \bigl(d_{\mathrm s}^{H} \widehat{X}_{\mathrm d}\bigr)^T$$ Despreading realizes a gain of $N_d$ for the matched signal component; incoherent interference does not accumulate.

4. Computational Complexity and Diversity Order

The full detector complexity comprises cyclic shifting and phase weighting performed for each path and Doppler bin, with total cost $\mathcal{O}(L k_\nu N)$, plus a despreading stage of $\mathcal{O}(N_d N)$. This scales linearly in $N$, compared to the cubic $\mathcal{O}(N^3)$ cost of traditional MMSE inversion. Full diversity gain is achieved: the CDD's effective SNR scales as $\sum_{i=0}^{L-1} |h_i|^2$, and the BER decays as $\mathrm{SNR}^{-L}$, matching the theoretical path-diversity order.

5. Interference Modeling in the DAFT Domain

Closed-form DAFT-domain expressions exist for canonical jammer types, including tone, sweep, broadband, and narrowband jamming: $$J_{\rm t}^A(m) = \sqrt{P_i} e^{j\theta_{m, t}},\qquad J_{\rm bb}^A(m) \sim \mathcal{CN}(0, P_i),\qquad \mathbb{E}\big| J_{\rm nb}^A(m) \big|^2 = P_i$$ Most practical jammers induce stationary noise in the DAFT domain, i.e., identically distributed across indices $m$, except for "sweep-jamming" precisely matched to the AFDM chirp slope, which localizes its energy at a single DAFT index. Stationary interference remains white post-equalization and is suppressed by the despreading autocorrelation. In the non-stationary sweep-jamming scenario, only a fraction $R = N_d/N$ of chips are impacted, leaving the rest error-free.

6. Numerical Performance and Comparative Results

Numerical results (RS(31,17) coding, $L=3$, max Doppler ≈ 22 kHz, SNR = –10 dB) substantiate CDD's anti-interference efficacy:

• Stationary jammers (tone, sweep/non-matching slope, broadband, narrowband): with adaptive spreading factor $N_d$, AFDM maintains packet throughput above 1000 packets/s up to 25 dB ISR, substantially outperforming fixed-parameter AFDM.
• Non-stationary sweep-jammer (perfect slope): throughput decreases negligibly, as only a single DAFT index is affected and code redundancy resolves residual errors.
• Relative to OFDM/OTFS: AFDM using CDD exhibits less throughput degradation under strong broadband jamming.
• BER Performance: CDD matches the MMSE detector's BER at both integer and fractional Dopplers, but with linear, rather than cubic, complexity.

7. Significance and Deployment Implications

The CDD's exploitation of cyclic-shift and phase-structural properties in the DAFT domain underpins a matched-filter architecture robust against aggressive, structured interference. Its linear-complexity scaling and attainment of full path-diversity render it suitable for real-time, high-mobility, and hostile spectrum-use scenarios. The closed-form BER and throughput expressions enable principled parameter optimization and performance prediction under a wide variety of adversarial models (Yuan et al., 17 Dec 2025).