Applications of wavelets to nonlinear wave analysis and digital communication

In this dissertation we deal with two applications of wavelets and wavelet transforms: wavelet-based higher-order spectra and an orthogonal code set based on the orthogonal wavelets for CDMA systems.; Wavelet-based higher-order spectra are applied to analyze irregular ocean waves. Because the frequency characteristic of these waves change as time changes, Fourier-based power spectra are not appropriate.; The wavelet-based bicoherence spectra are utilized to detect short-time duration phase coupling associated with short-lived, yet “relatively strong”, nonlinear interactions between first-order and second-order wave components. Such strong, but short-lived, interactions have been previously shown to be associated with large-amplitude extreme-like waves in the ocean. Confirmation of the wavelet-based bicoherence results is provided by the decomposition of the experimentally observed wave elevation into its first- and second-order components. This is accomplished with the aid of a second-order Volterra digital filter. The efficacy of the approach is demonstrated using model basin irregular wave data.; We also introduce wavelet-based trispectra which can be used to detect short-lived but strong phase coupling phenomena between first-order and third-order components in laboratory generated random sea waves. Such short-lived phase coupling is also responsible for the generation of large-amplitude extreme-like waves. We show that in such simulations, the wavelet-based tricoherence can detect short-duration phase coupling, but Fourier-based tricoherence cannot. Also the statistics of Fourier-based and wavelet-based tricoherence spectra are compared. We also utilize a time domain third-order Volterra model to decompose the sea wave time series into its first-, second-, and third-order components. This decomposition in turn allows us to demonstrate that the non-zero wavelet-based trispectrum is indeed due to short-duration phase coupling of first- and third-order components.; Recently CDMA-based communication systems have received a great deal of attention because they use bandwidth very efficiently. In CDMA all users share the same bandwidth and an orthogonal code set is used for separating different users. We suggest a new orthogonal code set generation method, which uses orthogonal wavelets and scale functions. The bandwidth of this new orthogonal modulation code set based on the Hadamard matrix and the orthogonal wavelets and scale function are strongly related to the choice of the orthogonal wavelets and scale function used to generate the code set. In particular, the bandwidths of these new orthogonal modulation code sets are considerably smaller than those of the older orthogonal code set based on the Hadamard matrix and impulse function.