In conventional sinusoidal modeling parameter estimation [5,4], peaks are treated as representative of quasistationary sinusoids. This is so even when such peaks are wider, which often indicates that the sinusoid1 is changing in frequency. Herein, we seek to determine if the sinusoid is in fact linearly (or approximately linearly) changing in frequency, and if so, at what approximate rate. We then parameterize the sinusoid in terms of its amplitude, center frequency, and rate of linear frequency change.
In order to achieve our objective, we must first have an invertible model of a linear frequency chirp. To this end, we consider a linear chirp signal centered about DC and investigate its STFT phase characteristic. In order to perform this analysis, we see that an understanding of Fresnel integrals is required. Thus, a review is included in section 2. Equipped with this understanding, we proceed in section 3 to show that we may obtain invertible closed form approximations of the magnitude and phase of the STFT of a linear chirp signal. Doing so requires two nontrivial assumptions which are discussed as they are made. Next, in section 4, we invert the expressions formed in section 3 to obtain a method for extracting the frequency change parameter from an STFT peak. Finally, we demonstrate the performance of our algorithm in section 5.
We make an important note that the approximations used herein require significant constraints on the rate of frequency change. Thus, the method outlined in the body of this paper is not in general suitable for use on typical speech or music signals, whose frequency varies more slowly than signals used here. Modifications to the approach used here do, however, allow for such application. Alternative models allowing for this application are summarized in appendix A, and will be addressed in detail in an upcoming paper.