Theory and application of higher-order cyclostationarity

The theory of the periodic structure of nth-order statistics of cyclostationary time-series is developed. This theory is the generalization of the existing theory of the periodic structure of second-order moments of cyclostationary time-series to orders greater than two and is therefore called the theory of higher-order cyclostationarity. The theory’s central parameters are moments and cumulants of both timeand frequency-domain quantities. The notion of a pure nth-order sine wave is introduced and is shown to be characterized by the nth-order time-domain cumulant function, whereas impure nth-order sine waves are characterized by nth-order time-domain moment functions. A pure nth-order sine wave is that portion of a sine wave in an nth-order lag product that is left after all sine waves that result from products of sine waves present in lower-order lag products are subtracted. The relationships between the time-domain and frequency-domain moments and cumulants are determined, and the properties of the various moments and cumulants are discussed in terms of their applicability to the solution of statistical signal processing problems, such as detection and parameter estimation. The cyclic cumulants and their frequency-domain counterparts, the cyclic polyspectra, are shown to possess a unique signal-selectivity property. The theory is compared to the theories of second-order cyclostationarity and higher-order statistics of stationary time-series because it encompasses both. The theory is developed further by accommodating complex-valued signals, and by deriving the input/output relations for the higher-order moments and cumulants for several signal processing operations, including signal addition and multiplication, periodic sampling and modulation by a sine wave, and linear time-invariant filtering. The higher-order moments ii and cumulants are calculated for complex-valued pulse-amplitude-modulated signals, which model the complex envelopes of amplitudeand phase-shift keyed signals. Techniques for measuring the higher-order moments and cumulants are introduced and compared to methods for computing similar quantities for strict-sense stationary signals. Alternative measurement methods are comparatively analyzed and illustrated with numerical examples obtained by computer simulation, and their bias and variance are analytically evaluated. The application of the theory to the problems of interference-tolerant weak-signal detection and signal-selective time-delay estimation is considered. Suggestions for further research are provided.