On the Optimality of OFDM for Aperiodic Ranging

2026-10-02

AI Summary

This study addresses the optimization of the expected aperiodic autocorrelation sidelobes when random communication symbols are transmitted via unitary modulation. For sub-Gaussian constellations, the minimum expected sidelobe levels at specific lags and the corresponding optimal unitary modulation schemes are derived. The main contributions are threefold. First, it is proven that OFDM achieves the global optimum of expected sidelobes only when the lag divides the block length, and that it cannot be simultaneously optimal for all positive lags when N≥3. Second, this work provides the first complete characterization of the necessary and sufficient conditions for jointly minimizing the sidelobes over the first r positive lags, thereby fully resolving the total integrated sidelobe level conjecture. Finally, the global minimality of OFDM with respect to arbitrary prefix integrated sidelobe levels is established.

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 $μ_4<2$ 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 $\{1,\ldots,r\}$ with $1\le r<N$, if and only if every lag in that prefix divides $N$. In particular, simultaneous optimality over all positive lags is impossible for $N\ge 3$. 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 $r=N-1$.

Problem

• aperiodic ranging
• OFDM
• expected sidelobe level
• unitary modulation
• integrated sidelobe level

Innovation

• OFDM
• aperiodic autocorrelation
• expected sidelobe level
• unitary modulation matrix
• EISL conjecture