Augmented Affine Frequency Division Multiplexing for Both Low PAPR Signaling and Diversity Gain Protection

Research results on Affine Frequency Division Multiplexing (AFDM) reveal that it experiences the same Peak-to-Average Power Ratio (PAPR) problem as conventional Orthogonal Frequency-Division Multiplexing (OFDM). On the other side, some references and also our studies demonstrate that AFDM involves an unneeded matrix, which is based on a parameter typically represented by $c_2$, for signalling. Hence, in this paper, an augmented AFDM scheme, referred to as A$^2$FDM, is proposed to mitigate the PAPR problem of AFDM, which is achieved by replacing the $c_2$ matrix in AFDM by a new unitary matrix that performs both sub-block-based Discrete Fourier Transform (DFT) and symbol mapping. Two symbol mapping schemes, namely interleaved mapping and localized mapping, are proposed for implementing A$^2$FDM, yielding the Interleaved A$^2$FDM and Localized A$^2$FDM. The input-output relationships of these schemes are derived and the complexity and the effects of system parameters on the performance of A$^2$FDM along with AFDM systems are analyzed. Furthermore, simulation results are provided to demonstrate and compare comprehensively the performance of the considered schemes in conjunction with different system settings and various operational conditions. Our studies and results demonstrate that, while A$^2$FDM is capable of circumventing the PAPR problem faced by AFDM, it is capable of attaining the achievable diversity gain, when AFDM is operated in its undesirable conditions resulting in the loss of the diversity gain available.

💡 Research Summary

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The paper addresses two fundamental shortcomings of the recently proposed Affine Frequency Division Multiplexing (AFDM) waveform: (i) a peak‑to‑average power ratio (PAPR) that is as high as that of conventional OFDM, and (ii) the presence of an unnecessary matrix that depends on the parameter c₂ in the inverse discrete affine Fourier transform (IDAFT) stage. While the c₁ parameter is essential for achieving the full diversity order offered by AFDM, c₂ has negligible impact on performance yet adds computational burden.

To overcome these issues, the authors introduce an augmented AFDM scheme, denoted A²FDM (Augmented AFDM). The key idea is to replace the c₂‑based matrix with a unitary matrix that simultaneously performs a sub‑block‑based discrete Fourier transform (DFT) and symbol mapping. The N subcarriers are partitioned into µ sub‑blocks of equal size; each sub‑block undergoes an independent DFT, and the resulting block‑diagonal DFT matrix F_µ is multiplied by a unitary matrix U that implements the required phase rotations. The overall transmit mapping can be written compactly as

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