Yi-Wen Liu's Model

Liu has recently considered the continuous time linear frequency chirp case. He has presented a result showing that the second derivative of the Fourier transform of such a signal with respect to frequency yields a convenient expression in terms of $\alpha$. Specifically, at the continuous frequency corresponding to DC ($\omega = 0$), Liu finds that

$$\frac{d^2Y_W^{\mathrm{Hann}}(\omega)}{d\omega^2} \Big\vert _{\omega=0} = \frac{-j}{2\alpha}(Y_W^{\mathrm{Hann}}(0)).$$ (132)

We now extend Liu's result to discrete time. Equation  136 thus becomes:

$$\left(\frac{K}{2\pi}\right)^2\frac{\Delta^2 Y^{\mathrm{Hann}}(k)}{\Delta k^2} \Big\vert _{k=0} = \frac{-j}{2\alpha}(Y^{\mathrm{Hann}}(0)).$$ (133)

where we note that second order differencing operation with respect to frequency bin $k$ must be normalized by twice multiplying by $\frac{K}{2\pi}$. This expression may be solved for alpha to obtain:

$$\fbox{$\displaystyle \alpha \approx \frac{-jY^{\mathrm{Hann}}(0)}... ...frac{\Delta^2 Y^{\mathrm{Hann}}(k)}{\Delta k^2} \Big\vert _{k=0}\right)^{-1}.$}$$ (134)