Time-frequency estimation for cyclostationary signals

This thesis provides detailed analysis and design techniques for Wigner-Ville spectrum (WVS) estimators for use with cyclostationary signals. The resulting class of estimators represent a newly defined subset of Cohen's class characterized by a mixed discrete-time/continuous-frequency smoothing kernel. Although both time-variant and shift invariant versions of the estimator are developed, emphasis is placed on the shift-invariant version which is designed to estimate the WVS over an entire period from a single observation. Bias and variance expressions are derived for the new estimator, and these are compared with the general estimator. For this development, we also derive mean and covariance expressions for the general quasi-stationary based estimators, both for the autocorrelation estimator and for the WVS estimator. The concept of quasi-stationarity is extended to cyclostationary models, and we develop a novel measure of kernel smoothing and variance reduction termed the time-bandwidth area. This is a generalization of time-bandwidth product to describe arbitrary kernel functions, even those which are not governed by the uncertainty principle (such as the newly proposed estimators). The properties of the estimator are examined in terms of constraints on the smoothing kernel. In sharp contrast to the conventional estimators based on the quasi-stationary assumption, the low bias and low variance constraints for the new class of estimators do not contradict one another. The relationship between time dependent spectral estimation for nonstationary processes and classical Blackman-Tukey type spectral estimation for stationary processes is developed next.; Using examples the utility of the new estimator kernels are shown. It is seen that in random or noisy environments it may be difficult to achieve a reasonable trade-off between variance reduction and bias using conventional estimators. In the examples any assumption of quasi-stationarity sufficient to produce a low variance estimate would destroy many or all of the nonstationary features of the signal. However, since the signals are cyclostationary we can employ the new class of estimators to achieve an excellent balance between bias and variance reduction.