Estimation and classification of polynomial-phase signals

Analysis of underwater mammal vocalisations using time–frequency-phase tracker

“…Obviously, taking into consideration the relation (5), the Nth-order HAF of the signal given in (1) peaks at the frequencyx N : This property gives a practical method for polynomial coefficients estimation [9]. Starting with the highest-order coefficient a N , the maximum of the HAF is evaluated at each order.…”

“…The main property, which makes HIM the main approach for polynomial phase modelling, is that the Nth-order HIM of a PPS given in (1) is reduced to a constant amplitude harmonic with amplitude A 2 NÀ2 , frequencyx N and phase/ N [9]:…”

“…A natural idea to take advantage of this property is to compute the Fourier transform of Nth-order HIM, which leads to the HAF definition [9]:…”

“…The central tool of polynomial phase modelling is the high-order ambiguity function (HAF) which provides the estimation of any slowly varying instantaneous phase function as a polynomial [9]. Mathematically, a polynomial phase signal (PPS) of order N is defined as: …”

“…This step consists of local polynomial phase modelling [9,10] in signal's windows of arbitrary size. The time windowing is the first operation of the step 1 (Fig.…”