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Wavelet Transforms: Continuous and Discrete

The wavelet transform of a signal s is the family C(a,b), which depends on two indices a and b. The set to which a and b belong is given below in the table. The studies focus on two transforms:

From an intuitive point of view, the wavelet decomposition consists of calculating a "resemblance index" between the signal and the wavelet located at position b and of scale a. If the index is large, the resemblance is strong, otherwise it is slight. The indexes C(a,b) are called coefficients.

We define the coefficients in the following table. We have two types of analysis at our disposal.

Continuous Time Signal
Continuous Analysis
Continuous Time Signal
Discrete Analysis




Next we will illustrate the differences between the two transforms, for the analysis of a fractal signal (see the following figure).

Figure 6-6: Continuous Versus Discrete Transform

Using a redundant representation close to the so-called continuous analysis, instead of a nonredundant discrete time-scale representation, can be useful for analysis purposes. The nonredundant representation is associated with an orthonormal basis, whereas the redundant representation uses much more scale and position values than a basis. For a classical fractal signal, the redundant methods are quite accurate.


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