We study the identification capacity of discrete-time Gaussian channels impaired by correlated noise and inter-symbol interference (ISI). Our analysis is formulated for deterministic encoding functions subject to a peak power constraint and colored noise whose covariance matrix features a polynomially bounded singular value spectrum, i.e., $\sim [n^{-μ} , n^{μ/2}]$ where $n$ is the codeword length and $μ\in [0,1/2)$ is the spectrum rate. A central result establishes that, even when the ISI memory length grows sub-linearly with $n,$ i.e., $\sim n^κ$ where $κ\in [0,1/2)$ and $κ+ μ\in [0,1/2),$ the codebook size continues to exhibit super-exponential growth in $n$, i.e., $\sim 2^{(n \log n)R},$ with $R$ representing the associated coding rate. Moreover, by employing the well-known Mahalanobis-distance decoder induced by colored Gaussian noise statistics, we characterize bounds on the identification capacity, with the resulting bounds parameterized by $κ$ and $μ.$

翻译：我们研究了受相关噪声和码间干扰(ISI)影响的离散时间高斯信道的辨识容量。本分析针对受峰值功率约束的确定性编码函数以及协方差矩阵具有多项式有界奇异值谱（即$\sim [n^{-μ} , n^{μ/2}]$，其中$n$为码字长度，$μ\in [0,1/2)$为谱速率）的有色噪声进行建模。核心结果表明，即使ISI记忆长度以$n$的亚线性增长（即$\sim n^κ$，其中$κ\in [0,1/2)$且$κ+μ\in [0,1/2)$），码本规模仍呈现关于$n$的超指数增长（即$\sim 2^{(n \log n)R}$，$R$为编码速率）。进一步，通过采用基于有色高斯噪声统计特征的马氏距离译码器，我们刻画了辨识容量的边界，所得边界由参数$κ$和$μ$决定。