WAVELET-BASED REPRESENTATIONS FOR A CLASS OF SELF-SIMILAR SIGNALS WITH APPLICATION TO FRACTAL MODULATION

A potentially important family of self-similar signals is introduced based upon a deterministic scale-invariance characterization. These signals, which are referred to as "dy-homogeneous" signals because they generalize the well-known homogeneous functions, have highly convenient representations in terms of orthonormal wavelet bases. In particular, wavelet representations can be exploited to construct orthonormal "self-similar" bases for these signals. The spectral and fractal characteristics of dy-homogeneous signals make them appealing candidates for use in a number of applications. As one potential example, we consider their use in a communications-based context. Specifically, we develop a strategy for embedding information into a dy-homogeneous waveform on multiple time-scales. This multirate modulation strategy, which we term "fractal modulation," is potentially well-suited for use with noisy channels of simultaneously unknown duration and bandwidth. Computationally efficient modulators and demodulators are suggested for the scheme, and the results of a preliminary performance evaluation are presented. Although not yet a fully developed protocol, fractal modulation represents a potentially viable paradigm for communication.