Multisine excitations are widely used for identifying multi-input multi-output systems due to their periodicity, data compression properties, and control over the input spectrum. Despite their popularity, the finite sample statistical properties of frequency-domain estimators under multisine excitation, for both nonparametric and parametric settings, remain insufficiently understood. This paper develops a finite-sample statistical framework for least-squares estimation of the frequency response function (FRF) and its implications for parametric modeling. First, we derive exact distributional and covariance properties of the FRF estimator, explicitly accounting for aliasing effects under slow sampling regimes, and establish conditions for unbiasedness, uncorrelatedness, and consistency across multiple experiments. Second, we show that the FRF estimate is a sufficient statistic for any parametric model under Gaussian noise, leading to an exact equivalence between optimal two stage frequency-domain methods and time-domain prediction error and maximum likelihood estimation. This equivalence is shown to yield finite-sample concentration bounds for parametric maximum likelihood estimators, enabling rigorous uncertainty quantification, and closed-form prediction error method estimators without iterative optimization. The theoretical results are demonstrated in a representative case study.

翻译：多正弦激励因其周期性、数据压缩特性及对输入频谱的可控性，被广泛应用于多输入多输出系统的辨识。尽管其应用广泛，但在多正弦激励下，频域估计器（包括非参数与参数设置）的有限样本统计特性仍未得到充分理解。本文针对频率响应函数的最小二乘估计及其对参数建模的影响，构建了一个有限样本统计框架。首先，我们推导了频率响应函数估计器的精确分布与协方差特性，明确考虑了慢采样机制下的混叠效应，并建立了多实验条件下无偏性、不相关性及一致性的条件。其次，我们证明在高斯噪声下，频率响应函数估计是任意参数模型的充分统计量，从而得出最优两阶段频域方法与时域预测误差及最大似然估计之间的精确等价性。该等价性被证明可导出参数最大似然估计器的有限样本集中界，从而实现严格的不确定性量化，以及无需迭代优化的闭式预测误差方法估计器。理论结果通过一个代表性案例研究进行了验证。