1. A method of separating speech from ambient acoustic noise to generate a speech output signal from a speech source, comprising: generating multiple microphone output signals from corresponding multiple microphones located at multiple physical positions; filtering the multiple microphone output signals to split each of the multiple microphone signals into a plurality of frequency band-limited output signals for each of the multiple microphone signals; forming a spatial beam having a primary lobe having a direction adjusted by a beam-former, wherein the beam-former has multiple inputs for receiving the plurality of band-limited output signals for each of the multiple microphone signals; adaptively filtering at least one of the plurality of frequency band-limited output signals to periodically determine a position of the speech source and generate a steering control value; adjusting the direction of the primary lobe of the beam-former toward the determined position of the speech source according to the steering control value; generating an estimate of the ambient acoustic noise by removing speech from the plurality of band-limited output signals; post-filtering an output of the beam-former in conformity with the estimate of the ambient acoustic noise, wherein the post-filtering applies a transfer function to the output of the beam-former that is frequency-dependent on content of the estimate of the ambient acoustic noise; and processing the output of the beam-former in conformity with a result of the post-filtering to suppress residual noise in the output of the beam-former and generate the speech output signal therefrom.