Introduction

The sinusoidal model [1,2] has been a fundamentally important signal representation for coding and analysis of audio. In conventional sinusoidal modeling, the parameters used are frequency, amplitude and phase of each frequency component. We presently consider the enhancement of sinusoidal modeling by using an additional parameter for each such component: the rate of linear increase or decrease in instantaneous frequency. Though estimators for this parameter have been investigated for Gaussian-windowed signals [3,4], estimators for Hann windowed signals have proven elusive. In audio applications such as coding, the Hann window is often more desirable than the Gaussian window, especially due to its constant-overlap-and-add (COLA) property. Hence, our estimators provide a valuable tool for such applications. A previous attempt at a chirp estimator for Hann windowed signals [5] relied on a model that did not apply when the chirp parameter became small (as small as often seen in speech and music signals). Though the error manifested itself in a predictable way incorporated into the estimator via a least squares solution, the analysis lacked a rigorous definition of the point at which the basic model required modification. The current models, however, are valid for specific ranges of the chirp parameter. We apply different models for larger and smaller values of the chirp parameter alpha ($\alpha$); both ranges are seen in audio signals [6]. The first model corresponds to a Fresnel integral analysis (for larger $\alpha$) and the second to a Taylor series analysis of the signal's DFT (for smaller $\alpha$). The Fresnel model yields an expression for the DFT magnitude domain peak shape, and both models yield expressions for the curvature of the DFT phase corresponding to the FFT magnitude peak. The Taylor series model has the novel property that it is valid for DFT phase curvature modeling when any zero phase symmetric time windowing function is used. In section 2, we derive expressions for relevant portions of a chirp signal's DFT, based on our models. We use a highly simplified, but completely generalizeable chirp signal, and explicitly state the range of validity of each model in terms of $\alpha$. In section 3, we solve the expressions given in section 2 for the chirp parameter $\alpha$, yielding estimators. We also make important notes about the application of each such solution, and describe a neural net based estimator to cover the $\alpha$ values for which neither model is fully applicable. In section 4, we evaluate the estimators' performance.