Concentrating the short-time Fourier transform using its higher order derivatives

In this article we present an improved version of the adaptive time-frequency (TF) paradigm for concentrating the short-time Fourier transform (STFT), introduced in [1] and [2]. The original method concentrates the STFT of an analytic signal near the signal's instantaneous frequency (IF) curve in the time-frequency plane. The concentrated surface is obtained by a morphing process whose acurracy depends on two assumptions: (1) translates of the amplitude and (2) translates of the IF of the signal vary negligibly over the effective length of the STFT window. Under these conditions, it was demonstrated in [1] that the IF of an analytic signal at time T is closely approximated by the first time derivative of the argument of its STFT surface at time T. In this article we present a significantly more accurate method for estimating this IF, based on higher order derivatives of the STFT surface evaluated at time T. In fact, the improved method provides an exact theoretical estimate of the IF of a constant amplitude analytic signal with polynomial phase. When applied to a multicomponent analytic signal which satisfies condition (1) only and a mild separability condition, the process may be used to generate a TF surface representation of the signal in which the value of each component is distributed along a curve which closely approximates that component's IF curve. Moreover, by integrating the surface locally in frequency, the isolated individual components may be recovered. We demonstrate by example that the higher order method produces closer approximations to IF curves than the method introduced in [1] and [2]. Consequently, concentrated STFT surfaces generated with the higher order method exhibit sharper resolution of signal components than the original method.