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A well established method to detect and classify human movements using Millimeter-Wave ( mmWave) devices is the time-frequency analysis of the small-scale Doppler effect (termed micro-Doppler) of the different body parts, which requires a regularly spaced and dense sampling of the Channel Impulse Response ( CIR). This is currently done in the literature either using special-purpose radar sensors, or interrupting communications to transmit dedicated sensing waveforms, entailing high overhead and channel utilization. In this work we present SPARCS, an integrated human sensing and communication solution for mmWave systems. SPARCS is the first method that reconstructs high quality signatures of human movement from irregular and sparse CIR samples, such as the ones obtained during communication traffic patterns. To accomplish this, we formulate the micro-Doppler extraction as a sparse recovery problem, which is critical to enable a smooth integration between communication and sensing. Moreover, if needed, our system can seamlessly inject short CIR estimation fields into the channel whenever communication traffic is absent or insufficient for the micro-Doppler extraction. SPARCS effectively leverages the intrinsic sparsity of the mmWave channel, thus drastically reducing the sensing overhead with respect to available approaches. We implemented SPARCS on an IEEE 802.11ay Software Defined Radio (SDR) platform working in the 60 GHz band, collecting standard-compliant CIR traces matching the traffic patterns of real WiFi access points. Our results show that the micro-Doppler signatures obtained by SPARCS enable a typical downstream application such as human activity recognition with more than 7 times lower overhead with respect to existing methods, while achieving better recognition performance.