AFDM Chirp-Permutation-Index Modulation with Quantum-Accelerated Codebook Design

We describe a novel index modulation (1M) scheme exploiting a unique feature of the recently proposed affine frequency division multiplexing (AFDM) in doubly-dispersive (DD) channels. Dubbed AFDM chirp-permutation-index modulation (CPIM), the proposed method encodes additional information via the permutation of the discrete affine Fourier transform (DAFT) chirp sequence, without any sacrifice of the various beneficial properties of the AFDM waveform in DD channels. The effectiveness of the proposed method is validated via simulation results leveraging a novel reduced-complexity minimum mean- squared-error (MMSE)-based maximum likelihood (ML) detec-tor, highlighting the gains over the classical AFDM. As part of the work two interesting problems related to optimizing AFDM- CPIM are identified: the optimal codebook design problem, over a discrete solution space of dimension (<tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$N! K$</tex>), where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$N$</tex> is the number of sub carriers and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$K$</tex> is the number of codewords; and the ML detection problem whose solution space is of dimension <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$KM^{N}$</tex>, where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$M$</tex> is the constellation size. In order to alleviate the computational complexity of these problems and enable large-scale variations of AFDM-CPIM, the two problems are reformulated as a higher-order binary optimization problem and mapped to the well-known quantum Grover adaptive search (GAS) algorithm for their solution.