)Appendix D
Impulse response of rectangular pupil with positive defocus
The rectangular pupil can be informative as a limiting case, where extreme time-limitation of the object support is required. It can be solved in a closed form following the formulas provided in Klauder et al. (1960), but only for positive values of the defocus \(W_d\). Let us define a rectangular pupil \(P_r\) of width \(T_a\),
| \[ P_r(\tau ) = \mathop{\mathrm{rect}}\left(\frac{\tau}{T_a}\right) = \left\{ \begin{array}{l} 1\,\,\,\,\,\,\,\,|\tau | < T_a/2\\ \frac{1}{2}\,\,\,\,\,\,\,|\tau | = T_a/2\\ 0\,\,\,\,\,\,\,\,|\tau | > T_a/2 \end{array} \right.\ \] | (D.1) |
where \(T_a\) is equal here and in the Gaussian pupil, because of how we defined it in Eq. §13.19. Using this pupil in Eq. §13.18, we obtain the following integral
| \[ \tilde h_{dr}(\tau-\tilde\tau_0) = \frac{1}{2\pi} \exp\left( \frac{i\omega_c\tau^2}{2Mf_T} \right) \int_{-T_a/2}^{T_a/2} \exp ( iv^2 W_d \tilde T^2) \exp \left[ { - i\tilde T (\tau - \tilde\tau_0 )} \right] d\tilde T \] | (D.2) |
Then, the response would be given according to
| \[ \tilde h_{dr}(\tau-\tilde\tau_0) = \frac{1}{2\pi} \exp\left( \frac{i\omega_c\tau^2}{2Mf_T} \right) \sqrt{\frac{\pi}{2v^2 W_d}}\exp \left[-\frac{i(\tau-\tau_0)^2}{4\pi v^2 W_d} \right]\left[C(g_2) + iS(g_2) - C(g_1)-iS(g_1) \right] \] | (D.3) |
Where \(C\) and \(S\) are the real and imaginary parts of the complex Fresnel integrals, respectively, which are defined as
| \[ S(g) = \int_0 ^g \sin (\psi^2) d\psi \] | (D.4) |
| \[ C(g) = \int_0 ^g \cos (\psi^2) d\psi \] | (D.5) |
The variables \(g_1\) and \(g_2\) are defined as
| \[ g_1(\tau) = \frac{\tau-\tilde\tau_0}{\sqrt{2\pi v^2W_d}} + \sqrt{\frac{v^2W_dT_a^2}{2\pi}} \] | (D.6) |
| \[ g_2(\tau) = \frac{\tau-\tilde\tau_0}{\sqrt{2\pi v^2W_d}} - \sqrt{\frac{v^2W_dT_a^2}{2\pi}} \] | (D.7) |
A numerical solution exists only when \(g\) is positive, which is unfortunately not the case in the auditory system using the values of \(v\), \(T_a\), and \(W_d\) obtained in this work.
References
Klauder, John R, Price, AC, Darlington, Sidney, and Albersheim, Walter J. The theory and design of chirp radars. Bell System Technical Journal, 39 (4): 745–808, 1960.