Statistical analysis of the product high-order ambiguity function

The high-order ambiguity function (HAF) was introduced for the estimation of polynomial-phase signals (PPS) embedded in noise. Since the HAF is a nonlinear operator, it suffers from noise-masking effects and from the appearance of undesired cross terms and, possibly, spurious harmonics in the presence 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 and to solve the ambiguity problem. In this correspondence we derive a statistical 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 data 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 explicitly taken into account. The analysis is validated by simulation results for both single- and multicomponent PPS's.