Algorithm

We now consider inversion of the magnitude expression to estimate $\alpha$. We begin by observing the half-height width $k_{hh}$ of the smeared peak in the FFT magnitude domain.⁹ Since the half height width will not necessarily occur at an integer $k$ value, linear or other interpolation may be used to more accurately estimate $k_{hh}$. If our $\alpha$ value is large enough to guarantee that our approximation is valid at this point (discussed in the previous subsection), the half height width corresponds to $k_{\max}$ exactly. This is because the raised cosine is at half height when its argument is $\pm\ensuremath{\frac{\pi}{2}}$, and solving for $k$ in this situation yields $\vert k\vert=\ensuremath{\frac{\alpha K N}{4\pi}}$, or $k_{\max}$.

As suggested previously, ensuring that our model is valid now becomes trivial: if $k_{hh}\geq \ensuremath{\frac{2 c^2 K}{N}}> \ensuremath{\frac{K}{N}}$ for our desired $c$ value, the model is valid.

We now calculate $\alpha$ by solving the half height expression for $\alpha$. That is,

$$\fbox{ $ \displaystyle \vert\alpha\vert \approx \ensuremath{\frac{4\pi k_{hh}}{KN}}$}.$$ (111)

Having obtained this estimate of the magnitude of $\alpha$, we need only determine its sign. This can be done by using the phase: if the phase is concave up, we know that $\alpha$ is negative, and that if the phase is concave down, $\alpha$ is positive. Since we only require knowledge of this simple fact, high accuracy in the phase is not required. To determine the concavity of the phase, we simply inspect the sign of the second order difference of the phase.

As noted above, the approximation in equation 115 will prove more reliable than the phase inversion model below due to error propagation. This is not to say that the phase inversion algorithm is devoid of merit; the error simply propagates in a less well-modeled way than in the magnitude case.