The high-order ambiguity function (HAF) was introduced for the estimation o
f polynomial-phase signals (PPS) embedded in noise. Since the HAF is a nonl
inear operator, it suffers from noise-masking effects and from the appearan
ce of undesired cross terms and, possibly, spurious harmonics in the presen
ce of multicomponent (mc) signals. The product HAF (PHAF) was then proposed
as a way to improve the performance of the HAF in the presence of noise an
d to solve the ambiguity problem. In this correspondence we derive a statis
tical analysis of the PHAF in the presence of additive white Gaussian noise
(AWGN) valid for high signal-to-noise ratio (SNR) and a finite number of d
ata samples. The analysis is carried out in detail for single-component PPS
but the multicomponent case is also discussed. Error propagation phenomena
implicit in the recursive structure of the PHAF-based estimator are explic
itly taken into account. The analysis is validated by simulation results fo
r both single- and multicomponent PPS's.