Time-frequency characterization of harmonizable random processes

Abstract

In this thesis we study how to characterize nonstationary harmonizable random processes simultaneously in time and frequency. Unlike stationary random processes, harmonizable processes can have a frequency content that changes with time. Rather than working directly with the process itself, we analyze the second-order moment functions of the process and characterize the process from these moments. The second-order moments of a harmonizable process can be represented in the dual-time domain, the dual-frequency domain, the ambiguity domain and the time-frequency domain, where all domains are connected through Fourier transforms. The time-frequency domain often offers the most intuitive descriptions of the process, thus it will be the main focus of this thesis. We propose estimators of the time-frequency spectra, and we analyze the statistical properties of the estimators. The proposed estimators enjoy a great freedom in that they have many parameters that can be adjusted, and different choices of these parameters will be discussed. We demonstrate the estimator on both simulated complex-valued data and real-world real-valued data. The ambiguity domain is connected to the time-frequency domain through a 2-D Fourier transform. We can relate the support of the second-order moments in the ambiguity domain, which again is related to the concept of an underspread processes, to the smoothness of the time-frequency spectra. We propose an estimation procedure for the second-order moments in the ambiguity domain based on thresholding of empirical moments, as this will enable us to determine the support in this domain. The estimator is tested on simulated data, and we compare the estimated mean square error of our proposed estimator to a standard estimation approach. In order to provide objective and dimensionless representations of the time-frequency behavior of a harmonizable process, we define spectral coherence measures. The spectral coherences measure the correlation between the time behavior and frequency behavior of the process (time-frequency coherence) or the correlation across frequencies (dual-frequency coherence). We show how previously defined coherences may be obtained through a linear estimation scheme, and we propose alternative spectral coherence measures based on a widely linear estimation scheme. The time-frequency representations are applied to a specific stochastic problem, namely that of stochastic differential equations. By transforming the stochastic differential equation to the time-frequency domain and thus considering the second-order moments of the processes involved, we avoid the problems related to stochastic integration. We consider both random processes in time and random fields in spatial variables. We develop a general theory, and we consider both theoretical and simulated examples that corroborate the theory.