Abstract: The theory of the wavelet transform using the complex Morlet basis in the context of discrete signal sampling is thoroughly explored. A sequence of steps is developed relating the Fourier transform in the continuum to the wavelet transform on a lattice, including remarks on the admissibility criteria. The wavelet domain is extended to in- clude all coefficients touched by convolution with the data. Nearly perfect reconstruc- tion is obtained for bandwidth limited test signals with the inclusion of a correction factor on the norm of the basis wavelets. The Gibbs effect is observed in the residual near the edges of a finite duration of data, and the conditions required for perfect re- construction are evaluated. An analysis of a coastal sea level time series is performed to demonstrate the utility of the extended wavelet transform.
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