Application of Phase Result

We now extend the result in equation 14 to discrete time. This equation thus becomes:

$$\left(\frac{K}{2\pi}\right)^2\frac{\Delta^2 Y^{\mathrm{Hann}}(k)}{\Delta k^2} \Big\vert _{k=0} = \frac{-j}{2\alpha}(Y^{\mathrm{Hann}}(0)).$$ (16)

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:

$$\fbox{$\displaystyle \alpha \approx \frac{-jY^{\mathrm{Hann}}(0)}... ...frac{\Delta^2 Y^{\mathrm{Hann}}(k)}{\Delta k^2} \Big\vert _{k=0}\right)^{-1}.$}$$ (17)

This estimator in general yields excellent results for cases where