Abstract: Compressive sensing (CS) can achieve spectral detection and estimation from far fewer samples at sub-Nyquist sampling rates. However, efforts to design CS reconstruction algorithms for wideband spectrum sensing are very limited. Considering the characteristics of spectrum distribution are partially known in advance, we proposed a new algorithm, called weighted basis pursuit denoising(WBPDN), based on the famous -minimization algorithm in order to add prior information on the support of the sparse domain. The WBPDN incorporates the statistical properties of spectrum usages to reweight the coefficients of the previous -minimization algorithm. This underlying optimization corresponds to encouraging nonzero coefficients to gain performance improvement. There are empirical evidences of the fact that the proposed algorithm not only guarantees accurate spectral estimation from fewer measurements, but also outperforms basis pursuit denoising (BPDN) and OMP in terms of measurements requirements and reconstruction error. Moreover, WBPDN has faster convergence speed and significantly reduces the computation time required by BPDN.