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As Internet of Things (IoT) devices become both cheaper and more powerful, researchers are increasingly finding solutions to their scientific curiosities both financially and computationally feasible. When operating with restricted power or communications budgets, however, devices can only send highly-compressed data. Such circumstances are common for devices placed away from electric grids that can only communicate via satellite, a situation particularly plausible for environmental sensor networks. These restrictions can be further complicated by potential variability in the communications budget, for example a solar-powered device needing to expend less energy when transmitting data on a cloudy day. We propose a novel, topology-based, lossy compression method well-equipped for these restrictive yet variable circumstances. This technique, Topological Signal Compression, allows sending compressed signals that utilize the entirety of a variable communications budget. To demonstrate our algorithm's capabilities, we perform entropy calculations as well as a classification exercise on increasingly topologically simplified signals from the Free-Spoken Digit Dataset and explore the stability of the resulting performance against common baselines.