Recently, there is an increasing yet challenging demand on broadband mobile communications for high-speed trains. In this article, cognitive Doppler spread compensation algorithms are proposed for high-speed rail broadband mobile communications, which make use of the dedicated radio environment map (REM) for railway to compensate the time-varying Doppler spread. The dedicated REM for high-speed rail can be viewed as a spatial-temporal database consisting of the radio channel parameters along a given railway. The performance of the proposed Doppler spread compensation algorithms are evaluated with a typical OFDM-based broadband mobile system. Simulation results show that the link-level performance of high-speed rail broadband mobile communications can be improved significantly due to the REM-enabled radio channel condition awareness and the cognitive Doppler spread compensation algorithms. The REM-based cognitive radio approach presents a new paradigm for systems design of high-speed rail broadband mobile communications.