This paper presents a neuromorphic processing system integrating a compressed sensing spiking neural network (CSSNN) designed for sparse signal classification. The proposed CSSNN combines data coding, data compression, and SNN classification, enabling end-to-end optimization of network performance and model compression. Evaluated on the MNIST, N-MNIST, and DVS Gesture datasets, under uniform compression ratios (CRs) of 0.1, 0.05, 0.025, and 0.01, the proposed CSSNN consistently reduces the total number of network operations (OPs) by at least 80% compared with compressed learning methods using fixed Gaussian random matrix (GRM) sampling matrices, while maintaining minimal accuracy loss. A specialized CSSNN processor is designed based on a spike-driven processing flow. Validated on field-programmable gate arrays (FPGAs) and evaluated in the 40 nm CMOS process for application-specific integrated circuit (ASIC) design, this CSSNN processor achieves 96.12% classification accuracy with 8-bit fixed-point quantization on the MNIST dataset. The energy consumption of the ASIC is estimated to be 2.089 mW under a 1.1-V supply voltage and 100 MHz frequency.