Higher Order Nested Wigner Distributions: Properties and Applications

The concept of Wigner distribution (WD) is extended to higher-order nested Wigner distributions. The nested ambiguity function (NAF) is also derived naturally by employing the known interrelation between the WD and the ambiguity function (AF). These distributions are obtained by nesting a function within itself a number of times. The definitions of these new multidimensional distributions are given in analogy to the standard first-order 2-D distributions in both temporal/spatial and spectral forms and follow the definitions of quartic distributions presented in and . Some important properties are derived and discussed, including interrelationships between the various representations, marginals, correlations, and uncertainties, as well as their geometrical characteristics. The second-order nested Wigner distribution (NWD) is used to showcase the power of such distributions in potential application areas, such as signal estimation, signal separation, and radar signal analysis. For such signal processing tasks, it is demonstrated that the NWDs can be successful in situations where the standard 2-D distributions have proved to be inadequate