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A wireless underwater acoustic communication system based on the combination of chirp modulation and direct-sequence spread-spectrum signaling is presented. The use of chirp signals takes advantage of the low Doppler sensitivity in the matched filter operation whereas the choice of pseudo noise (PN) sequences allows one to reduce narrowband interference arising from other users and self-interference due to multipath propagation. The system here described is made of a transmitter encoding the bits with linear chirps multiplied by PN sequences, and a rake receiver that allows one to positively exploit the energy present in the most significant propagation paths. Moreover a tracking procedure that allows an adaptation to the instantaneous Doppler shift has been devised and tested. Results, obtained on real data over a long-range shallow-water channel with a moving transmitting platform, are presented and discussed.