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Compressive sensing (CS) is a promising technique that enables sub-Nyquist sampling, while still guaranteeing the reliable signal recovery. However, existing mixed-signal CS front-end implementation schemes often suffer from high power consumption and nonlinearity. This paper presents a digital-assisted asynchronous compressive sensing (DACS) front-end which offers lower power and higher reconstruction performance relative to the conventional CS-based approaches. The front-end architecture leverages a continuous-time ternary encoding scheme which modulates amplitude variation to ternary timing information. Power is optimized by employing digital-assisted modules in the front-end circuit and a part-time operation strategy for high-power modules. An S-member Group-based Total Variation (S -GTV) algorithm is proposed for the sparse reconstruction of piecewise-constant signals. By including both the inter-group and intra-group total variation, the S-GTV scheme outperforms the conventional TV-based methods in terms of faster convergence rate and better sparse reconstruction performance. Analyses and simulations with a typical ECG recording system confirm that the proposed DACS front-end outperforms a conventional CS-based front-end using a random demodulator in terms of lower power consumption, higher recovery performance, and more system flexibility.