Radio monitoring receivers need to autonomously recognize signal modulation in the presence of unknown frequency offset and multipath fading. To address the low recognition performance, complex feature parameters, high resource consumption, and insufficient real-time capability of modulation recognition for low-SNR signals over multipath fading channels, this paper proposes a modulation recognition scheme based on a Welch power spectrum complexity measure and high-order cumulant features, and integrates it into a radio monitoring receiver with modulation recognition functionality. The scheme requires only a small number of feature parameters and an adaptive decision tree, enabling the recognition of twelve modulation types. Simulation results show that the average recognition accuracy exceeds 90% over the COST 207 Typical Urban and Hilly Terrain channel models when the SNR is no lower than 0 dB. The scheme has also been implemented on a programmable logic device, achieving low processing latency and satisfying the real-time processing requirements of radio monitoring receivers.