A data analysis algorithm for OFDR-based distributed acoustic sensing (DAS) is proposed, which achieves high acoustic bandwidths of tens of kilohertz with sharp spatial resolutions in the order of centimeters. The non-idealities of the setup as well as the phase noise affecting the measurement are analyzed and a method to compensate them is experimentally demonstrated. The performance of the sensor is evaluated by extensive experimental tests, showing the viability of the proposed technique to achieve high frequency and high spatial resolution distributed acoustic sensing.