A fundamental challenge in sampled-data control arises when a continuous-time plant is subject to disturbances that possess significant frequency components beyond the Nyquist frequency of the feedback sensor. Such intrinsic difficulties create formidable barriers for fast high-performance controls in modern and emerging technologies such as additive manufacturing and vision servo, where the update speed of sensors is low compared with the dynamics of the plant. This paper analyzes spectral properties of closed-loop signals under such scenarios, with a focus on mechatronic systems. We propose a spectral analysis method that provides new understanding of the time- and frequency-domain sampled-data performance. Along the course of uncovering spectral details in such beyond-Nyquist controls, we also report a fundamental understanding on the infeasibility of single-rate high-gain feedback to reject disturbances not only beyond but also below the Nyquist frequency. New metrics and tools are then proposed to systematically quantify the limit of performance. Validation and practical implications of the limitations are provided with experimental case studies performed on a precision mirror galvanometer platform for laser scanning.