A Matched Filter Developed for Chaotic Waveforms

A matched filter developed for use in chaos-based communications systems is presented. A matched filter is the optimum filter for maximizing the signal-to-noise ratio of a received signal in the presence of additive Gaussian white noise (AGWN). Chaos-based communications systems encode information into a chaotic waveform using arbitrary small perturbations to control the trajectory of the chaotic oscillator. Chaotic waveforms are deterministic, are sensitive to initial conditions, have aperiodic long-term behavior, have a spread frequency spectrum, and are theoretically immune to interference. There has been great interest in using chaotic waveforms in communication applications. One reason for this interest is that the spread spectrum of a chaotic waveform gives the appearance of noise when observed over a prolonged period of time. This masks the waveform from anyone without prior knowledge of its presence. Another reason is that to retrieve the information encoded in the chaotic waveform, complete knowledge of the waveform must be known. This makes it difficult for anyone other than the intended recipient to decrypt the information. Normally, implementing a matched filter for a chaotic waveform is difficult because such waveforms do not have a fixed basis function and have irregular timing. However, a unique chaotic oscillator has been developed that can be represented as an exact analytic solution. The waveform's solution can be written as a convolution of a symbol sequence and a fixed basis function. This fixed basis function makes it possible to derive a delay differential equation describing the matched filter. In order to test the matched filter, an amplitude modulated communications system was developed. In the system, the oscillator's output is first amplitude modulated with 2.3GHz carrier before being transmitted. The receiver then demodulates the signal and applies it to the matched filter. Any encoded information can then be extracted from the output of the matched filter. The oscillator and matched filter were designed to operate at 1.8MHz. In both simulation and testing, the matched filter was able to detect the chaotic waveform in the presence of AWGN.