In this paper, the foundations of neuromorphic computing, spiking neural networks (SNNs) and memristors, are analyzed and discussed. Neuromorphic computing is then applied to FPGA design for digital signal processing (DSP). Finite impulse response (FIR) and infinite impulse response (IIR) filters are implemented with and without neuromorphic computing in Vivado using Verilog HDL. The results suggest that neuromorphic computing can provide low-latency and synaptic plasticity thereby enabling continuous on-chip learning. Due to their parallel and event-driven nature, neuromorphic computing can reduce power consumption by eliminating von Neumann bottlenecks and improve efficiency, but at the cost of reduced numeric precision.