Phase Approximation

Given the simpler approximations for the real and imaginary parts of $Y_a(k)$ in equations 81 and 82, we may obtain a closed form approximation for the phase as follows (here, we have again used $\phi=\frac{\pi^2 k^2}{K^2\alpha}$ for notational simplicity):

$$\angle Y_a(k) \approx \angle \left( \sqrt{\frac{\pi}{2\alpha}} \left(\ensuremath{\frac{... ...( \cos(\phi) + \sin(\phi) +i\left(\cos(\phi) - \sin(\phi)\right)\right) \right)$$ (85)

$$= \angle \left(\frac{e^{i\phi}+e^{-i\phi}}{2} + \frac{e^{i\phi}-e^{... ...i} + i\frac{e^{i\phi}+e^{-i\phi}}{2} - i\frac{e^{i\phi}-e^{-i\phi}}{2i} \right)$$ (86)

$$= \angle \left(\frac{e^{i\phi}+e^{-i\phi}-e^{i\phi}+e^{-i\phi}}{2} + \frac{e^{i\phi}-e^{-i\phi}-e^{i\phi}-e^{-i\phi}}{2i} \right)$$ (87)

$$= \angle \left(e^{-i\phi}-\frac{e^{-i\phi}}{i}\right) = \angle \lef... ...}\right) = \angle\left(\sqrt{2}e^{i\pi/4}e^{-i\phi}\right) = \frac{\pi}{4}-\phi$$ (88)

$$= \frac{\pi}{4}-\frac{\pi^2 k^2}{K^2\alpha}.$$ (89)

This is clearly a downward pointing parabola as we sought. We repeat our conclusion and conditions for emphasis and clarity:
$\parbox{6.5in}{\begin{eqnarray} \hspace{8cm} \alpha > 0 \nonumbe... ...ease)}\quad \\ \vert k\vert \leq b\cdot k_{\max} \nonumber \end{eqnarray}}$

Figure 10 illustrates the the phase of an example $Y(k)$ as well as the estimate obtained using equation 92. Due to the unwrap function used in the plot, we see that the phases do not line up exactly. Nonetheless, we see the great accuracy of the approximation by inspecting the first and second difference plots included below the phase plot. Error bounds for the large limits approximations are again shown (dot-dash). These were obtained by performing a standard error-propagation analysis [7]. A detailed derivation is given in appendix C. The observed irregularity of the error will ultimately lead to our preference of the magnitude approximation, described in the next subsection.

Figure 10: (a) The phase of an example $Y(k)$(solid) (with $\alpha = 0.004280$, $F=8000$, $K=N=201$, and a Hann windowing function) and the approximation obtained using equation 92 (dashed). (b) The first order difference of (a). (c) The second order difference of (a). Error bounds for the large limits approximation (dot-dash) are also shown.