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符号间干扰（ISI）影响的离散时间高斯信道的识别容量。我们的分析针对确定性编码函数进行，该函数受限于峰值功率约束，且有色噪声的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。核心结果表明，即使 ISI 记忆长度随 $n$ 亚线性增长（即 $\sim n^κ$，其中 $κ\in [0,1/2)$ 且 $κ+ μ\in [0,1/2)$），码本规模仍呈超指数增长，即 $\sim 2^{(n \log n)R}$，其中 $R$ 表示关联的编码速率。此外，通过采用有色高斯噪声统计特性所诱导的经典马氏距离译码器，我们刻画了识别容量的界，所得界由参数 $κ$ 和 $μ$ 决定。