This dissertation studies the effects of radiation-induced single-event upsets (SEUs) on digital signal processing (DSP) systems designed for field-programmable gate arrays (FPGAs). It presents a novel method for evaluating the effects of radiation on DSP and digital communication systems. By using an application-specific measurement of performance in the presence of SEUs, this dissertation demonstrates that only 5-15% of SEUs affecting a communications receiver (i.e. 5-15% of sensitive SEUs) cause critical performance loss. It also reports that the most critical SEUs are those that affect the clock, global reset, and most significant bits (MSBs) of computation. This dissertation also demonstrates reduced-precision redundancy (RPR) as an effective and efficient alternative to the popular triple modular redundancy (TMR) for FPGA-based communications systems. Fault injection experiments show that RPR can improve the failure rate of a communications system by over 20 times over the unmitigated system at a cost less than half that of TMR by focusing on the critical SEUs. This dissertation contrasts the cost and performance of three different variations of RPR, one of which is a novel variation developed here, and concludes that the variation referred to as "Threshold RPR" is superior to the others for FPGA systems. Finally, this dissertation presents several methods for applying Threshold RPR to a system with the goal of reducing mitigation cost and increasing the system performance in the presence of SEUs. Additional fault injection experiments show that optimizing the application of RPR can result in a decrease in critical SEUs by as much 65% at no additional hardware cost.