Customer Reviews

Signal Processing with Fractals: A Wavelet Based Approach

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This is a very good book that deals fractal self similar stocastic processes,linear scale invariant systems and wavelet theory. This book covers synthesizing,estimation methods and modeling of self similar processes and their broad field of applications like communication and signal processing. These areas are extensively covered and the book is organized in a text book manner.Overall I found this book very useful and this is one of the best books I have read.

Observations of signals, time series in telecommunication, or pictures in medical imaging reveal selfsimilarity, i.e., the signal/picture looks the same as the scale varies. Hence, the name "fractal" ! Pictures in the Intro make this scale-similarity visually apparent. But it is made precise in mathematical statistics, and the book further makes the connection to the engineering of signal/image processing. That's a main point of the book!

Selfsimilar processes are stochastic processes that are invariant in distribution under suitable scaling of time and/or space. Fractional Brownian motion or Brownian sheets are the best known of these. They were found by Kolmogorov long ago, but made popular by Mandelbrot and Ness in 1968. We now speak of 1/f processes. More recent use of wavelet bases in telecommunication and in stochastic integration has revived interest. Other even more recent applications include finance.

While the underlying idea behind all of this is quite simple, and can be traced back to Kolmogorov in the 1930ties, it is only recently, with the advent of wavelet methods, that the *computational* power has been better appreciated. The idea is analogous to that of random Fourier series: Instead of treating the Fourier coefficients as random variables, it is now wavelet coefficients that are analyzed statistically. Since wavelets have computational advantages, it is not surprising that the engineering applications abound. This little book is well written, and should be attractive both to members of the math community and to engineers.

Mathematicians will be pleased to note that wavelet analysis now brings *Hilbert space theoretic features* of the subject back to the fore. Amusingly, this was in fact a dominant feature which motivated both A. N. Kolmogorov and Norbert Wiener in the early days; e.g., curves in Hilbert space.

The author is an authority in the field, and his book brings out beautifully the highpoints of the subject. I further expect that the book will go over well in the classroom; nice summaries at the end of each chapter! (Exercises would have helped though!) The book will help bridge mathematical analysis, probability, and engineering. Engineers may like that proofs are relegated to later in the book. Mathematicians will be pleased with having the proofs, and with the clarity of their presentation. Reviewed by Palle Jorgensen, July, 2005.