## Abstract

In J. Opt. Soc. Am. A **27**, 797 (2010), Yura and Hanson derived what they claim to be “a general expression for the mean level crossing rate for an arbitrary” random process following any desired probability distribution function. On the other hand, the authors themselves assert that in some cases their results “differ somewhat from the result reported in the literature.” This discrepancy “remains unexplained” by the authors “and is laid open for future discussion.” In this note, we explain the reason for such discrepancy and show that the Yura-Hanson formula is indeed a special-case solution. A more general solution is then given that is applicable to arbitrary random processes and that is fully consistent with the Rice mean level crossing rate formula.

© 2012 Optical Society of America

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### Equations (10)

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(1)
$${\nu}_{Y}(y)=\frac{{\sigma}_{\dot{y}}}{\pi \sqrt{\gamma}}\text{\hspace{0.17em}}\mathrm{exp}(-\frac{{h}^{2}(y)}{2}),$$
(2)
$${\sigma}_{\dot{y}}^{2}=-{\ddot{R}}_{Y}(\tau ){|}_{\tau =0}=\frac{{\int}_{0}^{\infty}{\omega}^{2}{S}_{Y}(\omega )\mathrm{d}\omega}{{\int}_{0}^{\infty}{S}_{Y}(\omega )\mathrm{d}\omega},$$
(3)
$$h(y)\triangleq \sqrt{2}\text{\hspace{0.17em}}{\mathrm{erf}}^{-1}(2{F}_{Y}(y)-1),$$
(4)
$$\gamma \triangleq {\int}_{\mathcal{Y}}\frac{{f}_{Y}(y)}{{\dot{h}}^{2}(y)}\mathrm{d}y,$$
(5)
$${\nu}_{Y}(y)={\int}_{-\infty}^{+\infty}|\dot{y}|{f}_{Y,\dot{Y}}(y,\dot{y})\mathrm{d}\dot{y}.$$
(6)
$$X(t)={F}_{X}^{-1}({F}_{Y}(Y(t)))\triangleq h(Y(t)),$$
(7)
$${f}_{Y,\dot{Y}}(y,\dot{y})={[\dot{h}(y)]}^{2}{f}_{X,\dot{X}}(h(y),\dot{h}(y)\dot{y}).$$
(8)
$$Y(t)={F}_{Y}^{-1}({F}_{X}(X(t))),$$
(9)
$${\nu}_{Y}(y)={\nu}_{X}(h(y)),$$
(10)
$$h(y)\triangleq {F}_{X}^{-1}({F}_{Y}(y)).$$