A communications system includes a modulator for generating a chirp signal aimed at spreading the frequency spectrum of an information signal over a specified spectral bandwidth of a communications channel. The chirp signal has initial and final instantaneous frequency. The chirp signal is controlled from an in-phase control signal and a quadrature-phase control signal to have, in a complex plane, constant amplitude and instantaneous phase. The instantaneous frequency is defined by the speed the instantaneous phase is changed in the complex plane by the in-phase and quadrature-phase control signals; the instantaneous frequency is linearly changed between initial and instantaneous frequencies over the whole duration of the chirp signal; the initial and final instantaneous phases of the chirp signal are identical. The communications system also described an adapted demodulator capable of working even in presence of a significant frequency and/or timing offset between the transmitting and receiving clocking systems.