In this paper we derive closed-form formulas of feedback capacity and nonfeedback achievable rates, for Additive Gaussian Noise (AGN) channels driven by nonstationary autoregressive moving average (ARMA) noise (with unstable one poles and zeros), based on time-invariant feedback codes and channel input distributions. From the analysis and simulations follows the surprising observations, (i) the use of time-invariant channel input distributions gives rise to multiple regimes of capacity that depend on the parameters of the ARMA noise, which may or may not use feedback, (ii) the more unstable the pole (resp. zero) of the ARMA noise the higher (resp. lower) the feedback capacity, (iii) certain conditions, known as detectability and stabilizability are necessary and sufficient to ensure the feedback capacity formulas and nonfeedback achievable rates {\it are independent of the initial state of the ARMA noise}. Another surprizing observation is that Kim's \cite{kim2010} characterization of feedback capacity which is developed for stable ARMA noise, if applied to the unstable ARMA noise, gives a lower value of feedback capacity compared to our feedback capacity formula.

中文翻译：

---

非平稳自回归移动平均噪声驱动的AGN信道反馈容量公式

在本文中，我们推导出反馈容量和非反馈可实现速率的闭式公式，用于由非平稳自回归移动平均 (ARMA) 噪声（具有不稳定的一极和零点）驱动的加性高斯噪声 (AGN) 通道，基于时不变反馈代码和通道输入分布。根据分析和模拟得出的令人惊讶的观察结果，(i) 使用时不变的通道输入分布会产生多种容量机制，这取决于 ARMA 噪声的参数，这些参数可能使用也可能不使用反馈，(ii) ARMA 噪声的极点（相应的零）越不稳定，反馈容量就越高（相应地越低），(iii) 某些条件，被称为可检测性和稳定性是必要和足够的，以确保反馈容量公式和非反馈可实现率{\它独立于 ARMA 噪声的初始状态}。另一个令人惊讶的观察结果是 Kim 为稳定 ARMA 噪声开发的反馈容量的 \cite{kim2010} 表征，如果应用于不稳定的 ARMA 噪声，与我们的反馈容量公式相比，反馈容量的值较低。