Lipschitz exponent

The singularity of the signal is measured by the Lipschitz exponent (LE). Assume the signal f t) has the following property in the vicinity of... 

It follows directly that if a determines continuously differentiable times at q5 the Lipschitz exponent (LE) tjrgives an indication of how regular the function f t) is at tg. The higher the <2 the more regular the function f t) is. 

Figure 4.17 demonstrates an example of modulus maxima for detection of milling tool at different wear states. Figure 4.17a shows a cutting force signal from a fresh tool and severely worn tool with their wavelet modulus maxima. Lipschitz exponent (LE) extracted from the modulus maxima line (Figure 4.17b). Though they do not change much,...

In some cases, the limit as r -> 0 is not taken in Equation (2.23) and the ratio r lx By x))/ ar is termed in this case the Holder [36], coarse Holder [37] or Lipschitz-HClder exponent [34]. It is traditionally denoted by a and it may be evaluated for any measure, defined or not by Equation (2.22). This exponent is useful to characterize singular measures, which have no local densities (i.e. for which the limit in Equation (2.23) does not exist), and it plays an important role in the definition of multifractal measures (Section 2.6). 

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