[1]-Tse, David, and Pramod Viswanath. Fundamentals of wireless communication. Cambridge university press, 2005.

1-综述

misaligned OAC：预编码矩阵（含噪声）+ 没同步好

2-系统模型

$\theta \in {\mathbb{R}}^d$全局模型.
$M$ 个边缘设备.
大小为$B_m$的本地数据集${\mathcal{B}}_m$，训练得到新模型${\tilde{\mathbf{\theta }}}_m\in {\mathbb{R}}^d;$
${\theta }_m^{\prime }=B_m({\tilde{\mathbf{\theta }}}_m-\mathbf{\theta })\in {\mathbb{R}}^d$上传缩放的模型***.
${\theta }_{+}={\sum }_{m=1}^M{\theta }_m^{\prime }$服务器端估计算术和.
${\theta }_{\mathrm{new}}=\theta +\frac{1}{{\sum }_m{B}_m}{\theta }_{+}$是全局新权重.

${\theta }_m^{\prime }=[(s_m^{\mathrm{t}}{)}^{\top },(s_m^{\mathrm{i}}{)}^{\top }{]}^{\top }$将上行的权重转为复数.
$s_m\in {\mathbb{C}}^{d/2}:s_m=$ $s_m^{\mathrm{r}}+j{s}_m^{\mathrm{i}}$
$s_m=[s_m[1]$, $s_m[2],\dots ,s_m[\tilde{L}]{]}^{\top }.$每个时隙传$L$个符号.
$x_m(t)={\alpha }_m{\sum }_{\ell =1}^L{s}_m[\ell ]p(t-\ell T),$ 一个时隙，某台设备传输的信号.

$p(t)=1/2\left[\mathrm{sgn}(t+T)-\mathrm{sgn}(t)\right]$.持续时间为$T$的矩形脉冲.
${\alpha }_m$信道预编码因子.${\alpha }_m=1/{\bar{h}}_m.$

$r(t)={\sum }_{m=1}^M{\tilde{h}}_m{x}_m(t-{\tau }_m)+z(t),$ 预编码矩阵（含噪声）+ 没同步好.

${\tilde{h}}_m$是平坦衰弱（相对于频率选择性衰弱而言，见【1，P44】）
${\tau }_m.$根据延时排序，e.g., ${\tau }_0$没有延时第一个被接收.

$r(t)={\sum }_{m=1}^M{\tilde{h}}_m{\alpha }_m{\sum }_{\ell =1}^L{s}_m[\ell ]p(t-{\tau }_m-\ell T)+z(t)={\sum }_{\ell =1}^L{\sum }_{m=1}^M{h}_m^{\prime }{s}_m[\ell ]p(t-{\tau }_m-\ell T)+z(t),$

$h_m^{\prime }={\tilde{h}}_m/{\bar{h}}_m$不完美的CSI带来的.