AI 中文总结
本文研究了随机通信符号经酉调制后的期望非周期自相关旁瓣，证明了OFDM在特定时延整除块长度时达到最小期望旁瓣电平，并解决了部分期望积分旁瓣电平的猜想。
AI 中文摘要
我们研究了通过酉调制矩阵传输的随机通信符号的期望非周期自相关旁瓣。我们考虑适当的亚高斯星座，即具有第四绝对矩$\mu_4<2$和零伪方差的星座，包括常用的均匀方形正交幅度调制（QAM）和均匀相移键控（PSK）星座。对于每个指定的时延，我们确定了精确的最小期望旁瓣电平（ESL），并通过分别对由该时延间隔的每个样本索引链应用归一化逆离散傅里叶变换（IDFT）来获得达到该最小值的酉调制。完整的正交频分调制（OFDM）调制当且仅当时延整除块长度$N$时才达到该最小值。将此准则与时延一处的极小化器的完整刻画相结合，我们证明了一个酉调制同时在前$r$个正时延（称为前缀$\{1,\ldots,r\}$，其中$1\le r<N$）上最小化ESL，当且仅当该前缀中的每个时延都整除$N$。特别地，对于$N\ge 3$，在所有正时延上同时最优是不可能的。我们还证明了OFDM全局最小化部分期望积分旁瓣电平（EISL），该电平定义为所选正时延子集上ESL之和，对于每个这样的前缀。这也解决了当$r=N-1$时的完整EISL猜想。
英文摘要
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 $r=N-1$.
Comments36 pages, 3 figures
