Continuous-Time Analysis of AFDM: Fundamental Bounds, and Effects of Pulse-Shaping and Hardware Impairments

TL;DR

A CT-analytical framework based on the affine Fourier series (AFS) representation is developed, which allows us to demonstrate that strictly bandlimited pulses and subcarrier suppression strategies are essential to maintain the multicarrier structure of the signal.

Abstract

Affine frequency division multiplexing (AFDM) has recently emerged as a resilient waveform candidate for high-mobility next-generation wireless systems. However, current literature mostly focuses on discrete time (DT) models, often overlooking effects and hardware non-idealities of actual continuous time (CT) signal generation. In this paper, we bridge this gap by developing a CT-analytical framework based on the affine Fourier series (AFS) representation, which allows us to demonstrate that strictly bandlimited pulses and subcarrier suppression strategies are essential to maintain the multicarrier structure of the signal. In addition, we derive the analytical power spectral density of AFDM and evaluate its spectral characteristics in comparison with other multicarrier schemes, considering the impact of realistic truncated pulse-shaping. Furthermore, we analyze the sensitivity of the CT model to phase noise, carrier frequency offset, and sampling jitter, providing a theoretical analysis of communication performance. Finally, we derive closed-form Cramér-Rao bounds for channel parameter estimation, showing that the chirped modulation peculiar of AFDM increases the estimation variance but enables the resolution of Doppler ambiguities. Our findings provide the necessary theoretical and practical foundations for the implementation of AFDM in realistic wireless transceivers.

Figures (8)

• Figure 1: Magnitude (left) and unwrapped phase (right) of the CT AFDM (with $\lambda_1 = \lambda_2=0.007$) and OFDM signals generated from the same $4$-QAM symbols over multiple periods. Vertical dashed lines indicate the signal period $T$. The following parameters have been employed for generating the CT signals: 1) $N=64$; 2) RRC pulse $p(t)$ with roll-off factor $\alpha=0.15$; 3) oversampling factor $N_{\mathrm{c}}=10$; 4) $T_{\mathrm{s}}=1.04$ µ s.
• Figure 2: Unwrapped phase of the DT sequences $\{\bar{x}_k\}$ for both AFDM and OFDM, generated from the same $4$-QAM symbols. Vertical dashed lines indicate multiples of the symbol period $N=64$. AFDM parameters $\lambda_1 = \lambda_2=0.007$ are used for the chirp-periodic sequence. The OFDM curve ($\lambda_1 = \lambda_2 = 0$) is included for comparison.
• Figure 3: Normalized PSD of OFDM, AFDM ($\lambda_1=\lambda_2=0.007$) and OTFS-DCP. All the considered waveforms occupy the same bandwidth $B=1$ MHz and use $N=64$ subcarriers. The size of the OTFS-DCP modulation is $M \times N$, with $M=32$, it employs the same ideal (untruncated) RRC pulse $p(t)$ adopted for OFDM and AFDM, with roll-off factor $\alpha=0.15$ and FD CP and a postfix of sizes $N_{\mathrm{cp}}^{(\mathrm{FD})}=N_{\mathrm{cpo}}^{(\mathrm{FD})}=1$, respectively.
• Figure 5: Magnitude spectrum of the pulses selected for AFDM signal generation. The RRC, Gaussian and rectangular pulses have been considered. The AFDM signal is characterized by a useful bandwidth $B=1$ MHz and $N=64$ subcarriers. Markers indicate the position of the SC, so that only $N_{\mathrm{u}}=N-N_{\mathrm{sc}}$ useful subcarriers are retained.
• Figure 6: Theoretical CRB for the estimation of normalized delay and Doppler frequencies in AFDM modulation, generated with three different pulses (namely, RRC, Gaussian and rectangular) using SC mechanism to limit or avoid SI. An $\text{SNR}\in [-15, 30]$ dB has been considered. The CRB achieved by OFDM (i.e., $\lambda_1 =\lambda_2 = 0$) employing RRC pulse with $\alpha=0.25$ are also shown for comparison.