Current Chirp Parameter Estimation System for Hann-Windowed Signals

• Magnitude and phase of the DFT of Hann-windowed chirp signals behave in a predictable and very interesting manner. (See whiteboard example.)
• Hence, for Hann windowed signals, we need to apply different models for larger and smaller values of the chirp parameter alpha ($\alpha$); both ranges are seen in audio signals [1].
• Basic Idea: determine how the STFT phase and magnitude should behave when we have a chirp signal. Then, invert the model(s) to obtain an estimator.
• Large $\alpha$: Based on Fresnel integral analysis of real and imaginary parts of DFT. Obtain yields an expression for the DFT magnitude domain peak shape, and for the curvature of the DFT phase corresponding to the FFT magnitude peak.
• Small $\alpha$: Taylor series analysis of the signal's DFT. Obtain expression for the curvature of the DFT phase corresponding to the FFT magnitude peak. Model has the novel property that it is valid for DFT phase curvature modeling when any zero phase symmetric time windowing function is used.
• Both models are invertible, and thus may be used to estimate $\alpha$ when $\alpha$ is in the appropriate range.