Four characteristic points of pulse pressure waves-the systolic peak, the anacrotic notch, the dicrotic notch, and the diastolic foot-are used to estimate various aspects of cardiovascular function, such as heart rate and augmentation index. We propose a novel approach to extracting these characteristic points using a topological signal processing framework. We characterize the topology of the signals using a collection of persistence intervals, which are encapsulated in a persistence diagram. The characteristic points are identified based on their time of occurrence and their distance from the identity line in the persistence diagram. We validate this approach by collecting radial pulse pressure data from twenty-eight participants using a wearable tonometer, and computing the peripheral augmentation index using a traditional derivative-based method and our novel persistence-based method. The augmentation index values computed using the two methods are statistically indistinguishable, suggesting that this representation merits further exploration as a tool for analyzing pulse pressure waves.