Dynamic range of the detectable parameters for polynomial phase signals using multiple-lag diversities in high-order ambiguity functions

Two lag diversities in the high-order ambiguity functions for single component polynomial phase signals (PPS) was recently explored by Zhou and Wang. The lag diversity enlarges the dynamic range of the detectable parameters for PPS. In this paper, we first find a connection between the above multiple-lag diversity problem and the multiple undersampling problem in the frequency detection using discrete Fourier transform (DFT). Using the connection and some results on the multiple undersampling problem we recently obtained, we prove that the dynamic range obtained by Zhou and Wang is already the maximal one for the detectable parameters for single-component PPS. Furthermore, the dynamic range for the detectable parameters for multicomponent PPS is given when multiple-lag diversities are used. We show that the maximal dynamic range is reached when the number of the lags in the high-order ambiguity function (HAF) is at least twice of the number of the single components in a multicomponent PPS. More lags than twice the number of single components do not increase the dynamic range.