Fractal signal models are important in a wide range of signal processing applications. For example, they are often well-suited to analyzing and processing various forms of natural and man-made phenomena. Likewise, the synthesis of such signals plays an important role in a variety of electronic systems for simulating physical environments. In addition, the generation, detection, and manipulation of signals with fractal characteristics has become of increasing interest in communication and remote-sensing. | Wornell . Fractal Signals Digital Signal Processing Handbook Ed. Vijay K. Madisetti and Douglas B. Williams Boca Raton CRC Press LLC 1999 1999 by CRC Press LLC 73 Fractal Signals Gregory W. Wornell Massachusetts Institute of Technology Introduction Fractal Random Processes Models and Representations for 1 f Processes Deterministic Fractal Signals Fractal Point Processes Multiscale Models Extended Markov Models References Introduction Fractal signal models are important in a wide range of signal processing applications. For example they are often well-suited to analyzing and processing various forms of natural and man-made phenomena. Likewise the synthesis of such signals plays an important role in a variety of electronic systems for simulating physical environments. In addition the generation detection and manipulation of signals with fractal characteristics has become of increasing interest in communication and remote-sensing applications. A defining characteristic of a fractal signal is its invariance to time- or space-dilation. In general such signals may be one-dimensional . fractal time series or multidimensional . fractal natural terrain models . Moreover they may be continuous-time or discrete-time in nature and may be continuous or discrete in amplitude. Fractal Random Processes Most generally fractal signals are signals having detail or structure on all temporal or spatial scales. The fractal signals of most interest in applications are those in which the structure at different scales is similar. Formally a zero-mean random process x t defined on 1 t 1 is statistically self-similar if its statistics are invariant to dilations and compressions of the waveform in time. More specifically a random process x t is statistically self-similar with parameter H if for any real a 0 it obeys the scaling relation x t a Hx at where denotes equality in a statistical sense. For strict-sense self-similar processes this equality is in
