It is widely accepted that single carrier frequency division multiple access (SC-FDMA) is an excellent candidate for broadband wireless systems. Channel estimation is one of the key challenges in SC-FDMA, since accurate channel estimation can significantly improve equalization at the receiver and, consequently, enhance the communication performances. In this paper, we study the application of compressive sensing for sparse channel estimation in a SC-FDMA system. By skillfully designing pilots, their patterns, and taking advantages of the sparsity of the channel impulse response, the proposed system realizes channel estimation at a low cost. Simulation results show that it can achieve significantly improved performance in a frequency selective fading sparse channel with fewer pilots.