#### Technical Report USC-SIPI-311

“Generalized Higher-Order Statistics (GHOS) and Generalized Higher-Order Cyclic Statistics (GHOCS): Theory and Applications”

by Mehmet Izzet Gürelli

August 1997

In this dissertation, novel statistical signal processing tools called generalized higher-order statistics (GHOS) and generalized higher-order cyclic statistics (GHOCS) are proposed together with several applications. This dissertation concentrates around GHOS for the most part, and only initial results are included for GHOCS.

The idea of GHOS was initially motivated by the non-existance of higher-order statistics for alpha-stable processes. Our research results showed that GHOS can actually be a very useful tool for signals that do not necessarily contain alpha-stable probability structures.

The theory of GHOS is built upon the idea of "generalized cumulants" which are proposed in this dissertation as the Taylor series coefficients of the second characteristic function (SCF) at some prespecified point in the domain of the SCF. If this point is chosen to be the origin, then the generalized cumulants reduce to the traditional cumulants. In a similar manner, generalized moments are defined as the Taylor series coefficients of the first characteristic function (FCF) at some prespecified point in the domain of FCF with a normalization factor that equals to the multiplicative inverse of the value of the FCF at the specified point. Other quantities such as "generalized cepstrum" are defined accordingly.

Several algorithms based on GHOS and GHOCS are proposed in this dissertation. In particular, blind identification methods for single-channel and co-channel systems based on output measurements are considered. Furthermore, the problem of data rate estimation under co-channel interference or in the presence of alpha-stable signals is considered. The applications and simulation results introduced in this dissertation clearly indicate that GHOS and GHOCS are very promising statistical signal processing tools.

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