Compressive spectrum sensing techniques present many advantages over traditional spectrum sensing approaches, e.g., low sampling rate, and reduced energy consumption. However, when the spectral sparsity level is unknown, there are two significant challenges. They are: 1) how to choose an appropriate number of measurements, and 2) when to terminate the greedy recovery algorithm. In this paper, a compressive autonomous sensing (CASe) framework is presented that gradually acquires the wideband signal using sub-Nyquist rate. Further, a sparsity-aware recovery algorithm is proposed to reconstruct the full spectrum while solving the problem of under-fitting or over-fitting. Simulation results show that the proposed system can not only reconstruct the spectrum using the appropriate number of measurements, but also considerably improve the recovery performance when compared with the existing approaches.