Deep Neural Networks (DNNs) have been successfully implemented across various signal processing fields, resulting in significant enhancements in performance. However, DNNs generally require substantial computational resources, leading to significant economic costs and posing challenges for their deployment on resource-constrained edge devices. In this study, we take advantage of spiking neural networks (SNNs) and quantization technologies to develop an energy-efficient and lightweight neuromorphic signal processing system. Our system is characterized by two principal innovations: a threshold-adaptive encoding (TAE) method and a quantized ternary SNN (QT-SNN). The TAE method can efficiently encode time-varying analog signals into sparse ternary spike trains, thereby reducing energy and memory demands for signal processing. QT-SNN, compatible with ternary spike trains from the TAE method, quantifies both membrane potentials and synaptic weights to reduce memory requirements while maintaining performance. Extensive experiments are conducted on two typical signal-processing tasks: speech and electroencephalogram recognition. The results demonstrate that our neuromorphic signal processing system achieves state-of-the-art (SOTA) performance with a 94% reduced memory requirement. Furthermore, through theoretical energy consumption analysis, our system shows 7.5x energy saving compared to other SNN works. The efficiency and efficacy of the proposed system highlight its potential as a promising avenue for energy-efficient signal processing.