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4. Homework

There is not much of a homework for this section of the help. You may just play around and synthesize the various signals, testing how the parameters will affect the outputs. Here are however some combinations worth trying:

Weierstrass function

If you have not done so already, try the test mentioned in the description of the Weierstrass sub-menu. This experiment highlights some specific difficulties one meets when dealing with fractal signals. These are due to the fact that such signals are by definition not band-limited. The classical sampling theory specifies that the signals should then be low-pass filtered before processing. This is not always relevant or even possible in our case. As a matter of fact, a whole new sampling theory is needed for the kind of signals of interest in multifractal analysis. Indeed, the basic assumption is that (a part of) the pertinent information in the signal lies in its regularity structure, e.g. its Hölder function. Low-pass filtering the signal will transform it into an infinitely smooth one, thus loosing essential features. It is beyond our scope here to discuss further this important matter, and we will content ourselves with a simple experiment: Generate two deterministic Weierstrass functions with exponent H = 0.2, 256 points, a time support of 1, and a large lambda, e.g. 25, with both the default sum order and a sum order of 20. Visualize and compare the two synthesized signals and notice that the first one (let's call it Wei0) looks very smooth. This is clear, since the highest frequency allowed in Wei0 will be such that only one term will be included in the summation: Wei0 is thus simply a sine function, and as a consequence it has no fractal features. For instance,it would not make sense to try and estimate a fractional dimension of Wei0. On the other hand, the signal generated without caring about the sampling theorem does look irregular. This simple experiment shows that new rules are needed when one processes "fractal" signals.

mBm and generalized Weierstrass function

If your machine is sufficiently powerful, try synthesizing some mBm-s with 512 samples and various Hölder functions. This will help you understand what exactly the Hölder exponent controls. If synthesizing 512 samples traces of mBm takes ages, do not worry: You can perform the same kind of experiments using Generalized Weierstrass functions, either deterministic or stochastic. Try first the default Hölder functions h(t) = t. Observe the smooth evolution of the pointwise regularity along the graph, and get a feeling of what it means to have regularity t at each point t. Enter then your own functions and make additional tests.


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