Fractal Interpolation Functions as PN Waveform Generators for Broadband Communications

This paper is concerned with the novel use of Iterated Function Systems (IFS) as generators of Pseudo-Noise (PN) spreading waveforms for broadband communications applications. Motivation for this proposal indirectly arises from the recent interest in alternative methods of spread spectrum modulation, for example, via the use of syn-chronising chaotic carriers and localized wavelet lters. IFS related signals and systems are already shown themselves signiicantly important in modelling and analysis, from mul-tiresolution wavelet methods to fractal image coding techniques. While these applications tend to emphasis time-frequency localization, here we show IFS capable of creating dispersive PN signatures. Concentrating on Fractal Interpolation Functions (FIF), we show how their construction can be written in terms of a chaotic Markov shift system and use this to qualify a proposed whitening procedure. The role of waveform sampling is emphasised. Finally, we apply them as signatures in the provision of CDMA communications. Motivated by the identiication of chaotic systems capable of remote self-synchronization, there has been much research interest over the last ve years into the possible application of chaos to spread spectrum related communications 1]. One of the properties of low-dimensional chaotic dynamics which makes this proposal attractive is their ability to generate complex, broadband, aperiodic spreading signals from simply speciied systems. Another property, in the case of synchro-nised systems, leads to the opportunity to both synchronize and reconstruct a spreading sequence without the receiver having a priori knowledge of a xed sequence; in this case only the generating system model is required to be known. This can be extended to non-synchronizing systems by interpreting carrier recovery as a generalized state observation or further, estimation, problem 2]. However, there are a number of problems, often not addressed in the literature, with the direct use of chaotic wave-forms in their typically cited application areas, for example, in covert and secure communications, e.g. Low Probability of Interception / Detection (LPI/D). While low-dimensional chaotic signals may, after spectral attening, appear random to simple linear analysis, their deterministic structure will still be detectable to nonlinear techniques based around, say, state space reconstruction. This can be made more dif-cult by increasing either the eeective dimensionality, or the dynamical expansiveness of the generating system with the eeect of making the generated signals appear more random. If, however, synchronization or state observation is being utilized for carrier recovery, then its eeectiveness and robustness to noise will be greatly reduced. Another important area …