Compressive sensing is the newly emerging method in information technology that could impact array beamforming and the associated engineering applications. However, practical measurements are inevitably polluted by noise from external interference and internal acquisition process. Then, compressive sensing based beamforming was studied in this work for those noisy measurements with a signal-to-noise ratio. In this article, we firstly introduced the fundamentals of compressive sensing theory. After that, we implemented two algorithms (CSB-I and CSB-II). Both algorithms are proposed for those presumably spatially sparse and incoherent signals. The two algorithms were examined using a simple simulation case and a practical aeroacoustic test case. The simulation case clearly shows that the CSB-I algorithm is quite sensitive to the sensing noise. The CSB-II algorithm, on the other hand, is more robust to noisy measurements. The results by CSB-II at $\mathrm{SNR}=-10\,$dB are still reasonable with good resolution and sidelobe rejection. Therefore, compressive sensing beamforming can be considered as a promising array signal beamforming method for those measurements with inevitably noisy interference.