Wavelet decomposition of harmonizable random processes

Abstruct-The discrete wavelet decomposition of second-order harmonizable random processes is considered. The deterministic wavelet decomposition of a complex exponential is examined, where its pointwise and bounded convergence to the function is proved. This result is then used for establishing the stochastic wavelet decomposition of harmonizable processes. The similarities and differences between the wavelet decompositions of general harmonizable processes and a subclass of processes having no spectral mass at zero frequency, e.g., those that are wide-sense stationary and have continuous power spectral densities, are also investigated. The relationships between the harmonizability of a process and that of its wavelet decomposition are examined. Finally, certain linear operations such as addition, differentiation, and linear filtering on stochastic wavelet decompositions are considered. It is shown that certain linear operations can be performed term by term with the decomposition.