On the Optimality of OFDM for Aperiodic Ranging
Abstract
We study the expected aperiodic autocorrelation sidelobes of random communication symbols transmitted through a unitary modulation matrix. We consider proper sub-Gaussian constellations, namely those with fourth absolute moment μ4<2 and zero pseudo-variance, including the commonly used uniform square quadrature amplitude modulation (QAM) and uniform phase-shift keying (PSK) constellations. For every prescribed lag, we determine the exact minimum expected sidelobe level (ESL) and a unitary modulation attaining it by applying a normalized inverse discrete Fourier transform (IDFT) separately on each chain of sample indices spaced by that lag. The full orthogonal frequency division modulation (OFDM) modulation attains this minimum if and only if the lag divides the block length N. Combining this criterion with the complete characterization of the minimizers at lag one, we show that a unitary modulation minimizes the ESL simultaneously over the first r positive lags, called the prefix {1,…,r} with 1≤ r<N, if and only if every lag in that prefix divides N. In particular, simultaneous optimality over all positive lags is impossible for N≥ 3. We also prove that OFDM globally minimizes the partial expected integrated sidelobe level (EISL), defined as the sum of the ESLs over a selected subset of positive lags, for every such prefix. This also resolves the full EISL conjecture when r=N−1.
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BibTeX
@misc{xiong2026optimality,
title = {{On the Optimality of OFDM for Aperiodic Ranging}},
author = {Yifeng Xiong and Fan Liu and Ya-Feng Liu and Shi Jin and Jianhua Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.02843},
eprint = {2610.02843},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.02843},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - On the Optimality of OFDM for Aperiodic Ranging
AU - Yifeng Xiong
AU - Fan Liu
AU - Ya-Feng Liu
AU - Shi Jin
AU - Jianhua Zhang
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2610.02843
UR - https://arxiv.org/abs/2610.02843
AB - We study the expected aperiodic autocorrelation sidelobes of random communication symbols transmitted through a unitary modulation matrix. We consider proper sub-Gaussian constellations, namely those with fourth absolute moment μ_4<2 and zero pseudo-variance, including the commonly used uniform square quadrature amplitude modulation (QAM) and uniform phase-shift keying (PSK) constellations. For every prescribed lag, we determine the exact minimum expected sidelobe level (ESL) and a unitary modulation attaining it by applying a normalized inverse discrete Fourier transform (IDFT) separately on each chain of sample indices spaced by that lag. The full orthogonal frequency division modulation (OFDM) modulation attains this minimum if and only if the lag divides the block length N. Combining this criterion with the complete characterization of the minimizers at lag one, we show that a unitary modulation minimizes the ESL simultaneously over the first r positive lags, called the prefix {1,…,r} with 1≤ r<N, if and only if every lag in that prefix divides N. In particular, simultaneous optimality over all positive lags is impossible for N≥ 3. We also prove that OFDM globally minimizes the partial expected integrated sidelobe level (EISL), defined as the sum of the ESLs over a selected subset of positive lags, for every such prefix. This also resolves the full EISL conjecture when r=N−1.
ER -