Compressed sensing (CS) has emerged as a promising technique for reducing transmission data volume. Despite its significance, achieving a balance between the delay, energy consumption and data distortion caused by CS and transmission remains an understudied area in resource-constrained IoT systems. The emergence of multi-access edge computing provides a potential solution to the aforementioned issue by enabling the strategic implementation of CS either at IoT devices or an edge server (ES), depending on both bandwidth resources and computing resources at ES. In this paper, we investigate where to perform CS computation and how to determine the compression ratio and bandwidth allocation to minimize the weighted energy and distortion cost (WEDC) of all devices under latency requirements. We formulate a WEDC minimizing problem by jointly optimizing the task scheduling, compression ratio, and bandwidth allocation. Since the formulated problem is a mixed-integer and nonlinear programming, which is typically NP-hard, we decompose the original problem into two sub-problems and then develop an iterative algorithm to find the suboptimal solution. Extensive numerical results demonstrate the superiority of the proposed algorithm in reducing WEDC of all devices under delay constraints.