In this article we present SHARP, an original approach for obtaining human activity recognition (HAR) through the use of commercial IEEE 802.11 (Wi-Fi) devices. SHARP grants the possibility to discern the activities of different persons, across different time-spans and environments. To achieve this, we devise a new technique to clean and process the channel frequency response (CFR) phase of the Wi-Fi channel, obtaining an estimate of the Doppler shift at a radio monitor device. The Doppler shift reveals the presence of moving scatterers in the environment, while not being affected by (environment-specific) static objects. SHARP is trained on data collected as a person performs seven different activities in a single environment. It is then tested on different setups, to assess its performance as the person, the day and/or the environment change with respect to those considered at training time. In the worst-case scenario, it reaches an average accuracy higher than 95%, validating the effectiveness of the extracted Doppler information, used in conjunction with a learning algorithm based on a neural network, in recognizing human activities in a subject and environment independent way. The collected CFR dataset and the code are publicly available for replicability and benchmarking purposes.