Introduction

Wavelets are versatile tools of harmonic analysis. Because of their many uses, the word `wavelet' comes with different connotations (and different promises) to users in different fields. Therefore we think it important to set from the beginning the limits of our approach, lest the user be disappointed by unfulfilled expectations.

The wavelet-based tools presented in this brochure are aimed specifically at a more thorough characterization - hopefully leading to better understanding - of experimental time-series. This process comprises three steps:

1. the signal is mapped with some redundancy into the transformed (wavelet) space. An analogy from engineering drafting is to replace a blueprint of some complicated machinery with an exploded view of the same, with each component shown from many different angles or in holographic view. No new information is generated, although the volume of information is greatly increased.
2. features of interest are enhanced in the transformed space. To pursue the analogy, one may wish to select from the data some contact surfaces with specific finish tolerances.
3. the amount of information is then reduced by selection and/or by statistical means. For example, how many different machining/coating operations are required for these surfaces only.

It is important to formulate these three steps as an unambiguous algorithm, so that the end result can be interpreted accordingly. Indeed, the user has great flexibility at each step. For someone trained in the well-worn grooves of Fourier analysis, so much freedom frequently appears arbitrary; the interplay of the user's options can lead to confusion -- or to creativity. The point is that, with each of many possible algorithms being formulated as a question, the interrogation of the data can be much more thorough. Each quantitative answer can be the basis for comparisons between data sets, or for glimpses of the underlying dynamics.

With this goal in mind, one of wavelets' claims to fame - data compaction - is not a dominant concern. Similarly, the computing time is important (particularly in the context of real-time processing), but not at the expense of physical relevance or richness of interpretation. Thus we find that much of the rapidly growing literature on wavelets has little impact on the present work, as `better wavelets' are either too specialized or have limitations from our viewpoint. We recommend that the non-mathematician reader interested in some rigorous background start with Meyer's little book (Meyer, 1993) or the first three chapters of Daubechies' (Daubechies, 1992).