Communications in Nonlinear Science and Numerical Simulation
Chaos-based communication systems in non-ideal channels
Highlights
► Chaos-based communication systems present unsatisfactory bit error rate in non-ideal channels. ► Strategies for circumventing band-limitation, noise and delay usually present in communication channels are described. ► Results are auspicious, although the presented ideas can be taken as works in progress.
Introduction
Chaotic signals are characterized by a deterministic, limited and aperiodic behavior, as well as sensitive dependence on initial conditions [1]. Their application has been considered in a variety of areas, see e.g. [2]. Signal processing and telecommunications are no exception specially after the seminal works by Pecora and Carroll [3] and Ott et al. [4]. Applications of chaos ranging from digital and analog modulation to cryptography, pseudorandom sequences generation and watermarking have been proposed [5], [6], [7], [8]. Chaos has also been shown in connection to devices used in signal processing as nonlinear adaptive filters and phase-locked loop networks [9], [10], [11].
In this work, we classify a digital communication system as chaos-based when each symbol is transmitted using a fragment of a chaotic signal; their counterparts that do not use chaotic signals are referred to here as conventional.
Due to their properties, chaotic signals can occupy a wide bandwidth and have impulsive autocorrelation [12]. Furthermore, the cross-correlations between signals generated by different initial conditions present low values [5], [7]. These characteristics have been behind the rationale for using chaotic signals as candidates in spectral spreading information signals.
By employing chaotic signals to modulate independent narrowband sources lead us to obtain systems that transmit signals with increased bandwidths and lower levels of power spectral density in a fashion similar to that of conventional spread spectrum systems [7]. This fact endows them with similar features, namely (i) they are difficult to intercept for an unauthorized user; (ii) they are easily hidden, i.e. for an unauthorized receiver, it is difficult to even detect their presence in many cases; (iii) they are resistant to jamming; and (iv) they provide a measure of immunity to distortion due to multipath propagation [13]. Furthermore, chaos can supposedly provide a low-cost increase in security [5]. A practical 120 km fiber optic link using chaotic signals was recently reported [14].
Many of the proposed chaos-based communication systems are based on chaos synchronization, the property that coupled chaotic systems starting from different initial conditions can synchronize, under certain constrains, despite the sensibility to initial conditions [3], [15].
Although the proposed schemes work well in almost ideal environments, the presence of the usual amount of additive noise, distortion or delay of almost any practical channel brings unsatisfactory results in terms of Bit Error Rate (BER) when compared to conventional communication systems [5], [16].
In the last years, many researches have been conducted with the objective of approximating the performance of chaos-based communication systems to that of conventional ones in realistic environments. In this paper we review some of these new techniques that can allow chaotic signals to be used in practical applications in the near future. The results presented here were exposed in the two mini-symposia Communication with Chaos at Dynamics Days South America 2010 in São José dos Campos, Brazil.
In Section 2.1, we report a way of transmitting chaotic signals in bandlimited channels. The idea is simply to insert identical Finite Impulse Response (FIR) digital filters [17] in the transmitter and receiver feedback loops so that the synchronization process is unaffected but the chaotic signals spectra is fitted to the channel.
Next, in Section 2.2, we succinctly describe two approaches to minimize the effect of the additive noise in chaos synchronization. Firstly, we show numerical evidences that using lattices instead of single maps in discrete-time chaos synchronization problems increase the robustness to noise. After this, in SubSection 2.2.2 we address the use of blind signal separation techniques to separate deterministic chaos and stochastic noise and thus improving Signal-to-Noise Ratio (SNR).
Following, we briefly discuss bidirectional chaos-based systems that need to synchronize transmitter and receiver in scenarios that involve signal delays, as in satellite formation flying [18].
Finally, we draft our conclusions and perspectives in Section 3.
Section snippets
Combating channel impairments
In this section, we provide an overview of some recent and innovative works whose objective is to provide a better chaos-based communication performance in realistic environments. In the sequence, we succinctly describe strategies that allow us to properly handle effects of band limiting properties of the communication channel (Section 2.1), additive noise (Section 2.2) and delay (Section 2.3).
Conclusions and perspectives
In this review paper, we succinctly described innovative and up-to-date techniques that can be applied to allows for using chaos-based communication systems in more realistic channel conditions.
We provide a glimpse of new techniques for deal with bandwidth limitation, lack of robustness of chaos-synchronization in respect to additive noise and delay in bidirectional communications.
All these ideas can be taken as works in progress and, certainly, there is a long path before chaos-based
Acknowledgements
M.E. thanks CNPq, J.M.V.G. thanks FAPESP, grant nr. 2008/11.684-0, D.C.S. thanks CAPES, A.M.B. thanks CAPES and Fundação Araucária, L.H.A.M. thanks CNPq and E.E.N.M. thanks CNPq and FAPESP for financial support.
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