Higher-order spectral analysis of complex signals

Even though higher-order spectral analysis is by now a mature field, complex signals are still not routinely used, as they are in second-order analysis. The reason is the complexity of the complex case: nth order moment functions of a complex signal can be defined in 2 n different ways, depending on the placement of complex conjugate operators. It is demonstrated that only a few of these different moments are required for a complete nth order description. Properties of nth order moments and spectra with different conjugation patterns are investigated. For the special case of analytic signals, it is shown how spectra with different conjugation patterns provide different information about the signal. Both energy and power signals and deterministic and stochastic signals are discussed. A major focus lies on extending results from continuous-time signals to their sampled versions. Such an extension is not straightforward due to a phenomenon called higher-order or dimension-reduction aliasing. It is demonstrated why spectra of sampled nonstationary signals may suffer from dimension-reduction aliasing unless they are sufficiently oversampled.