Estimation of polynomial frequency modulation law for FM signals based on modified Extended Generalized Chirp Transform

In the paper the parameter estimation of the polynomial frequency law via estimation of parameters A = [a₁, a₂, a_p] of a polynomial phase signal (PPS) is addressed. The term “estimation” is used mainly in statistical community, but in this paper the word “estimation” is understood also as a synonym for the word “assessment”. The frequency law is strongly related to the polynomial phase signal, because the instantaneous frequency (IF) is the first derivative of the phase function. There are many parametric and nonparametric estimation methods for computing PPS. The most popular approach is based on the maximum likelihood (ML) estimator, which has the limitation due to a required multi-dimensional search over the parameter space and has numerous local optima making the application of gradient techniques impossible. Instead of the P-dimensional search of the ML estimation, an alternative solution with iterative reduction of the number of PPS coefficients to be estimated is proposed. The reduction of the PPS coefficients can be performed by the Extended Generalized Chirped Transform (EGCT). In the first step of the EGCT algorithm, the first phase coefficient is estimated. The maximum of the EGCT determines the estimated value of this coefficient. Successive de-chirping of the signal with one just estimated phase coefficient allows to formulate the next index function for one-dimensional search of the next coefficient. The proposed one-dimensional maximization process is implemented with initial “coarse” values of parameters obtained from fitting of a polynomial to the IF trajectory acquired from the Spectrogram, the Polynomial Wigner-Ville Distribution (PWVD) and from the Recursive Least Squares filter (RLS). This approach is analyzed mainly on the example of quadratic frequency modulated signals (QFM), but it can be easily extended to higher orders of polynomials.