Approximate zero-crossing distributions using the type-B Gram-Charlier series

Abstract

Approximations to zero-crossing distributions using the type-B Gram-Charlier series are the concern here. The type-B series consists of a linear function of the Poisson distribution and its derivatives. The probability <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p(n, \tau)</tex> of exactly <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> zeros, occurring in a given interval <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(t, t + \tau)</tex> of a stationary random process, is expanded in the type-B Gram-Charlier series. The expansion is used to derive approximations to the distribution of intervals between zeros. Specific results are presented, in the case of Gaussian noise, for the probability density function <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">P_{0}(\tau)</tex> of successive zero-crossing intervals and for the probability <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p(0, \tau)</tex> that a given interval <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\tau</tex> contains exactly zero zero-crossings. The accuracy of the approximations are compared with Rainal's experimental results and with upper and lower bounds presented by Longuet-Higgins. The comparisons are most satisfactory for the cases where successive zero-crossing intervals are nearly uncorrelated. For narrow-band Gaussian noise the results are unsatisfactory.