Algorithm

Having specified a range over which our model is valid, we may now estimate $\alpha$ from the FFT phase characteristic. If we indeed have a linear frequency chirp, we should see a parabolic phase characteristic in the specified range. If this is the case, the second order difference of the characteristic should be a constant in this range. If this is not the case, the second order difference will be zero or nonconstant. In fact, we may use this fact as a ``check'' on whether or not our linear chirp model is valid for the smeared peak.

According to our phase model, the constant achieved by this second order differencing operation is

$$\frac{\Delta^2\angle(Y(k))}{\Delta k^2} \approx \frac{-2\pi^2}{K^2\alpha}.$$ (112)

Since the marginal error in the inverse midpoint approximations will lead to slight but not major deviations from this value, it is prudent to ensure that the deviations are within that allowed by the model (see error bounds above). If this is so, errors may be smoothed out by averaging the values over the frequency range contained in the signal. Thus, we may estimate $\alpha$ as

$$\fbox{ $ \displaystyle \alpha \approx \frac{-2\pi^2}{K^2} ... ...ne{\frac{\Delta^2\angle(Y(k))}{\Delta k^2}}\right)^{-1}$}.$$ (113)

where the bar indicates an average over the relevant frequency bins. Our estimate will be accurate within bounds noted above.