Despite the broad application of the analytic wavelet transform (AWT), a systematic statistical characterization of its magnitude and phase as inhomogeneous random fields on the time-frequency domain when the input is a random process remains underexplored. In this work, we study the magnitude and phase of the AWT as random fields on the time-frequency domain when the observed signal is a deterministic function plus additive stationary Gaussian noise. We derive their marginal and joint distributions, establish concentration inequalities that depend on the signal-to-noise ratio (SNR), and analyze their covariance structures. Based on these results, we derive an upper bound on the probability of incorrectly identifying the time-scale ridge of the clean signal, explore the regularity of scalogram contours, and study the relationship between AWT magnitude and phase. Our findings lay the groundwork for developing rigorous AWT-based algorithms in noisy environments.

In-home Smart Grid (SG), the integration of Renewable Power Systems (RPSs) with Conventional Power Systems (CPSs), calls for cost-effective management for the electricity usages of end users' household appliances. In this paper, by taking the charging process of RPSs and multiple types of household appliances in to consideration, we have developed analytical models to characterize the electricity cost in the in-home smart grid. Based on these models, we formulate the electricity cost minimization problem as a finite-horizon continuous-time Markov decision process (CTMDP), from which we obtain a threshold policy to minimize the cost. Numerical results show that the threshold policy can manage the electricity usage very effectively.