Wavelet Transform The wavelettransform was introduced at the beginning of the 1980s by Morlet et al., who used it to evaluate seismic data [l05 ],[106]. Since then, various types of wavelet transforms have been developed, and many other applications ha vebeen found. The continuous-time wavelet transform, also called the integral wavelet transform (IWT), finds most of its applications in data analysis, where it yields an affine invariant time-frequency representation. | Signal Analysis Wavelets Filter Banks Time-Frequency Transforms and Applications. Alfred Mertins Copyright 1999 John Wiley Sons Ltd Print ISBN 0-471-98626-7 Electronic ISBN 0-470-84183-4 Chapter 8 Wavelet Transform The wavelet transform was introduced at the beginning of the 1980s by Morlet et al. who used it to evaluate seismic data 105 106 . Since then various types of wavelet transforms have been developed and many other applications havebeen found. The continuous-time wavelet transform also called the integral wavelet transform IWT finds most of its applications in data analysis where it yields an affine invariant time-frequency representation. The most famous version however is the discrete wavelet transform DWT . This transform has excellent signal compaction properties for many classes of real-world signals while being computationally very efficient. Therefore it has been applied to almost all technical fields including image compression denoising numerical integration and pattern recognition. The Continuous-Time Wavelt Transform The wavelet transform Wx b a of a continuous-time signal x t is defined as Wx b a a -5 i x t ip - - dt. J-oo a Thus the wavelet transform is computed as the inner product of x t and translated and scaled versions of a single function ip f the so-called wavelet. If we consider ip t to be a bandpass impulse response then the wavelet analysis can be understood as a bandpass analysis. By varying the scaling 210 . The Continuous-Time Wavelet Transform 211 parameter a the center frequency and the bandwidth of the bandpass are influenced. The variation of b simply means a translation in time so that for a fixed a the transform can be seen as a convolution of x t with the time-reversed and scaled wavelet Wx t a a _2x t V o i V a i The prefactor a -1 2 is introduced in order to ensure that all scaled functions ữ -1 2 t ữ with a IR have the same energy. Since the analysis function ip t is scaled and not modulated like the .
