Orthogonal frequency division multiplexing (OFDM) is a technique that will prevail in the next-generation wireless communication. Channel estimation is one of the key challenges in OFDM, since high-resolution channel estimation can significantly improve the equalization at the receiver and consequently enhance the communication performances. In this paper, we propose a system with an asymmetric digital-to-analog converter/analog-to-digital converter (DAC/ADC) pair and formulate OFDM channel estimation as a compressive sensing problem. By skillfully designing pilots and taking advantages of the sparsity of the channel impulse response, the proposed system realizes high-resolution channel estimation at a low cost. The pilot design, the use of a high-speed DAC and a regular-speed ADC, and the estimation algorithm tailored for channel estimation distinguish the proposed approach from the existing estimation approaches. We theoretically show that in the proposed system, a N-resolution channel can be faithfully obtained with an ADC speed at M=O(S2log(N/S)), where N is also the DAC speed and S is the channel impulse response sparsity. Since S is small and increasing the DAC speed to N >; M is relatively cheap, we obtain a high-resolution channel at a low cost. We also present a novel estimator that is both faster and more accurate than the typical l1 minimization. In the numerical experiments, we simulated various numbers of multipaths and different SNRs and let the transmitter DAC run at 16 times the speed of the receiver ADC for estimating channels at the 16 t resolution. While there is no similar approaches (for asymmetric DAC/ADC pairs) to compare with, we derive the Cramer-Rao lower bound.