Wavelet prototype

An intuitive solution to fliis problem would be to allow for sine waves with finite duration to appear as building blocks in the transformed data. This is the basis of wavelet analysis. The wavelet transform is based on such building blocks or elementary functions, which are obtained by dilatations, contractions, and shifts of a unique function called the wavelet prototype (or mother wavelet)... 

The wavelet prototype l/(t) is hence shifted in time by increasing n or in frequency by increasing m. An increased m will reduce the time duration and thereby increase the center frequency and frequency bandwidth of j/(t). 

This vast spectral bandwidth illustrates the necessity of a reliable scale and time resolved decomposition of available observations to separate and describe single processes as individual parts of the whole system. Often, the comlex interplay between climate subsystems plays an essential role and the understanding of coupling mechanisms is of crucial importance for the study and prediction of at first sight independent phenomena. Continuous wavelet transformation (CWT) is the prototypic instrument to address these tasks As an important application, it transforms time series to the time/scale domain for estimating the linear non-stationary spectral properties of the underlying process. 

The orthogonal basis functions denoted by are obtained by scaling and shifting a prototype wavelet function if t) (also sometimes called a mother wavelet) by scale a and time b, respectively, as shown below ... 

The Fourier transform of one of the prototype wavelets (called a Morlet wavelet) is given by... 

The model was subjected to a total of eight earthquakes during two flights. The earthquakes were pseudo-harmonic wavelets except for the last earthquake fired during the first flight that was a sine sweep. This signal is actually a sine wave of decreasing acceleration amplitude and frequency. The main characteristics of the input motions are tabulated in Table 22.3 both in model and prototype scale (bracketed values), while the time histories are depicted in Fig. 22.5. 

The wavelet transform [16]-[18] performs the decomposition of a signal into the family of functions generated from a prototype function mother wavelet) /(x) by dilation and translation operations... 

---