Constraints

The magnitude expression inversion method will ultimately require that our model is valid throughout the frequency range representing frequencies in the signal. In terms of equation 63, we thus require that $b\geq 1$. Solving

$$b < 2 - \ensuremath{\frac{2c}{N}}\sqrt{\ensuremath{\frac{2\pi}{\alpha}}}$$ (107)

for $\alpha$ in this case shows that we require

$$\alpha \geq \ensuremath{\frac{8\pi c^2}{N^2}}$$ (108)

in order for our model to be valid. We may substitute this $\alpha$ value into equation 15 to see that we now require

$$k_{\max} \geq \ensuremath{\frac{2 c^2 K}{N}}.$$ (109)

Inspecting equation 113, we may be tempted to think that choosing very small $c$ will allow our model to be valid for an arbitrarily small range. But we recall that choosing $c<1$ causes large error in the model, and that we can never specify $c < \ensuremath{\frac{\sqrt{2}}{2}}$, because doing so leads to a peak narrower than that prescribed by the Hann window transform for a quasistationary sinusoid. Specifically, if we set $c$ this small, we have

$$k_{\max} = \ensuremath{\frac{K}{N}},$$ (110)

which corresponds to the half height width of a Hann window transform. This is clearly not a valid model for $\alpha$, because this narrow width should only be reached when $\alpha = 0$. We may actually view the peak width requirement as a constraint on $\alpha$: it is only realistic to model $\alpha$ values that cause our model peak to be at least as wide as that of a Hann window main lobe. In fact, backsolving for $\alpha$ while requiring that the peak width be at least that of the main lobe width of a Hann window transform again yields $k_{\max} \geq \ensuremath{\frac{2 c^2 K}{N}}$ as given in equation 113.

In summary, if choosing to model only $\alpha$ values that obey equation 112, we can guarantee accuracy specified by $c$ by applying the magnitude inversion model only when $k_{\max} = \ensuremath{\frac{2 c^2 K}{N}}$ results in a peak at least $\ensuremath{\frac{K}{N}}$ wide. As we will show in the next subsection, it is trivial to verify that the $k_{\max}$ requirement is satisfied.