On the spectral properties of polynomial-phase signals

Polynomial-phase signals (PPS's), i.e., signals parameterized as s(t) = A exp(j2πΣ_m=0^M a_mt^m), have been extensively studied and several algorithms have been proposed to estimate their parameters. From both the application and the theoretical points of view, it is particularly important to know the spectrum of this class of signals. Unfortunately, the spectrum of PPS's of generic order is not known in closed form, except for first- and second-order PPS's. The aim of this letter is to provide an approximate behavior of the spectrum of PPS's of any order. More specifically, we prove that: i) the spectrum follows a power law behavior f^-γ, with γ = (M - 2)/(M - 1); ii) the spectrum is symmetric for M even and is strongly asymmetric for M odd; and iii) the maximum of the spectrum has an upper bound proportional to T^(M-I)/M and a lower bound proportional to T^1/2. These results are useful to predict the performance of the so-called high order ambiguity function (HAF) and the Product-HAH (PHAF), specifically introduced to estimate the parameters of PPS's, when applied to multicomponent PPS's.