Synchronization methods for communication with chaos over band-limited channels

SUMMARY Methods of chaos synchronization applied to communication with chaotic signals over band-limited channels are studied numerically and experimentally. The "rst group of methods employs a "lter in the receiver which is designed to compensate channel distortions of the transmitted signal. The second approach does not use the compensation but relies on the design of chaotic oscillators which shapes the spectrum of chaotic waveforms sent through the channel. Copyright ( 1999 John Wiley & Sons, Ltd. Recent interest in chaos synchronization is supported, in part, by the hope to build a common framework for the design of self-synchronizing analog, digital, and hybrid (analog}digital) chaotic scramblers which can be used for private communication. Several methods of scrambling of an information signal with analog chaotic signal at the transmitter and extracting the message from chaos at the receiver have been proposed. In one class of these methods, the information signal m(t) is added to the chaotic output x % (t) generated by a chaotic oscillator whose dynamics do not depend on m(t). The mixture x % (t)#m(t) is transmitted to the receiver where it is used as a driving signal for response system with matched parameters. Various implementations of the response systems were proposed, see for example References 1}6 and references therein. The common shortcoming of such methods of communication is that the driving signal which is &distorted’ by the message m(t), does not perfectly "t the response system. As a result, recovered message m 3 (t) will always contain some traces of chaotic waveforms no matter how perfectly the parameters of the decoder (response system) match those of the encoder (driving system). The amplitude of the chaotic contamination is small only if m(t) is small as compared to the chaotic signal x(t). Ad i !erent approach to the problem of chaotic encoding and decoding was suggested in a number of papers.7~13 The main idea of this approach is that information signal m(t) is injected into the feedback loop of the chaotic encoder. As a result the distorted chaotic signal x % (t)#m(t) returns back to the generator and in#uences the dynamics of the encoder. Now when the signal x % (t)#m(t) is applied to the decoder, it generates oscillations of the response system which are identical to the oscillations in the encoding generator. The message m(t) can be recovered using the open feedback loop of the response system. In this case (in the absence of channel distortions), after initial transients, the information can be restored exactly. Since in this