Analysis and Compensation of Receiver IQ Imbalance and Residual CFO Error for AFDM

Affine frequency division multiplexing (AFDM) is a promising waveform for future wireless communication systems. In this paper, we analyze the impact of receiver in-phase and quadrature (IQ) imbalance and residual carrier frequency offset (CFO) error on AFDM signals. Our analysis shows that the receiver IQ imbalance may not preserve the sparsity of the AFDM effective channel matrix because of the complex-conjugate operator of the discrete affine Fourier transform (DAFT). Moreover, the residual CFO error causes energy leakage in the effective channel matrix in the affine domain. To mitigate these effects, we extend the linear minimum mean-square error (LMMSE) detector to handle the improper Gaussian noise arising from the receiver IQ imbalance. Simulation results demonstrate that the proposed LMMSE detector effectively compensates for the receiver hardware impairments.

💡 Research Summary

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This paper investigates the impact of two practical receiver impairments—IQ imbalance and residual carrier‑frequency‑offset (CFO) error—on Affine Frequency Division Multiplexing (AFDM), a waveform recently proposed for high‑mobility wireless communications. AFDM relies on the discrete affine Fourier transform (DAFT) to map data symbols onto chirp‑like sub‑carriers, and its key advantage is the sparsity of the effective channel matrix even under doubly‑dispersive (delay‑Doppler) channels. The authors first model the receiver IQ imbalance with complex scalars µ and ν (derived from amplitude mismatch ψ and phase mismatch φ) and the residual CFO as a Gaussian random variable ε∼N(0,σ²_ε). After applying the DAFT to the received signal, the affine‑domain observation becomes

 y = µ H_eff x + ν H_eff* x* + w′,

where H_eff incorporates the channel taps, the CPP, the residual CFO phase rotation Δ_ε, and the Doppler‑induced rotation Δ_f. Crucially, the noise term w′ is improper: its covariance is (|µ|²+|ν|²)σ²_n I_N, while its pseudo‑covariance is 2µνσ²_n AA^T, with AA^T being the complex‑conjugate operator of the DAFT.

The paper then analytically examines how IQ imbalance destroys the sparsity of H_eff. The operator AA^T = Λ_{c2} F Λ_{c1} F Λ_{c2} (F is the unitary DFT, Λ’s are diagonal phase matrices) does not reduce to a simple permutation matrix unless the AFDM parameters satisfy special integer conditions (e.g., c₁=c₂=0, which collapses AFDM to OFDM). By expanding the matrix elements, the authors show that AA^T_{m,ℓ}=0 when (i) m+ℓ is odd, or (ii) m+ℓ is even but the 2‑adic valuation of 2N c₁ exceeds that of m+ℓ. In all other cases the entries are non‑zero, meaning AA^T is generally dense. Consequently, the term ν H_eff* x* spreads energy across the affine domain, eroding the sparsity that many low‑complexity AFDM detectors (MRC‑based iterative DFE, message‑passing) rely on. The effect is mitigated only when |ν|≪|µ| (weak imbalance).

Residual CFO introduces the diagonal matrix Δ_ε, which rotates each chirp by e^{−j2π ε n}. As σ²_ε grows, the energy that should be confined to the diagonal of H_eff leaks into off‑diagonal positions, as illustrated in Fig. 1. This leakage degrades channel estimation and detection performance.

To jointly address both impairments, the authors extend the linear minimum mean‑square‑error (LMMSE) detector. Traditional LMMSE assumes proper Gaussian noise; here the improper nature of w′ requires a widely‑linear processing approach. The received vector y and its conjugate y* are stacked into a 2N‑dimensional vector z =