ON THE THEORY OF CYCLOSTATIONARY SIGNALS

Cyclostationary signals are time-series whose statistical (average) behavior varies periodically with time. Closely related are almost cyclostationary (AC) signals with statistical behavior characterized by almost periodic functions of time, i.e., trigonometric series with possibly non-harmonically related frequencies. In many signal processing problems involving modulated communication signals, the waveforms encountered are appropriately modeled as AC. This dissertation presents a statistical theory of these signals, along lines recently suggested by W. A. Gardner, based on the principles of time-averaging.; The motivation for studying almost cyclostationary signals is discussed. A basic second-order statistical theory for complex valued AC signals--which involves idealized measurements of hidden periodicity in lag product waveforms--is presented. This leads to a description of AC signals in terms of the cyclic autocorrelation and cyclic conjugate correlation functions and their Fourier transforms, the cyclic spectrum and cyclic conjugate spectrum. The cyclic spectrum is shown to admit an interpretation as a spectral correlation density, i.e., a cross-spectrum between two distinct frequency shifted versions of a waveform. Similarly, the cyclic conjugate spectrum can be interpreted as a cross-spectrum between a frequency shifted version of a waveform and its frequency shifted complex conjugate. Together, the cyclic spectrum and cyclic conjugate spectrum are shown to constitute a complete second-order statistical description of an AC waveform. Cyclic spectral analysis concepts are then applied to the optimal linear-conjugate-linear almost periodic filtering problem.; Systems designed to estimate the cyclic spectrum and cyclic conjugate spectrum, i.e., cyclic spectrum analyzers, are found to be usefully characterized as quadratic almost periodically time-varying systems. The performance of such systems is described in terms of properties of the system kernel. Guidelines for system design are developed and several novel architectures are proposed. The AC fraction-of-time density for complex valued signals is defined and the Gaussian AC signal model is introduced. The appropriateness of the Gaussian AC model is discussed. The variance performance of cyclic spectrum analyzers driven by Gaussian AC waveforms is then studied. Finally, the above theory is applied to the problem of detecting the presence of an AC signal obscured by noise using a quadratic almost periodic system.