The development of advanced autonomous driving applications is hindered by the complex temporal structure of sensory data, as well as by the limited computational and energy resources of their on-board systems. Currently, neuromorphic engineering is a rapidly growing field that aims to design information processing systems similar to the human brain by leveraging novel algorithms based on spiking neural networks (SNNs). These systems are well-suited to recognize temporal patterns in data while maintaining a low energy consumption and offering highly parallel architectures for fast computation. However, the lack of effective algorithms for SNNs impedes their wide usage in mobile robot applications. This paper addresses the problem of radar signal processing by introducing a novel SNN that substitutes the discrete Fourier transform and constant false-alarm rate algorithm for raw radar data, where the weights and architecture of the SNN are derived from the original algorithms. We demonstrate that our proposed SNN can achieve competitive results compared to that of the original algorithms in simulated driving scenarios while retaining its spike-based nature.