Temporal cleaning of high-power infrared (IR) pulses generated by a Ti:Sapphire system is demonstrated by the use of the Nonlinear Fourier Filtering (NFF) method. In a proof-of-principle experiment suppression of up to 1000 is achieved for the temporal pedestal prior to the main pulse, with a moderate (20-25%) overall throughput. This includes the same suppression ratio for the picosecond coherent pedestal in the direct vicinity of the main pulse. Based on the instantaneous, intensity-dependent and high-order switching characteristics of NFF, excellent pulse cleaning performance is observed. The efficient, high-contrast removal of the coherent pedestal from the foot of the main pulse even if its duration is shorter than 100 fs is compared with the capability of the plasma mirror technique. Calculations are also performed, supporting the experimentally observed sharp intensity dependence of the switching process, pointing out the dominant role of the ionization-based refractive index change.