The beta-Pochhammer and its application to arbitrary threshold phase error probability of a vector in Gaussian noise. (English) Zbl 1533.33002
Summary: This paper develops a calculus around a new \(\beta \)-Pochhammer symbol of two variables, \((a,b)_{ m,n }\) based on the Beta weighting \(t^{a - 1} \left( 1 - t\right)^{b - 1}\). The approach is a natural rising factorial formulation that offers a new way to express hypergeometric functions of two variables. The results are implemented to solve the problem of finding the phase error probability of a vector perturbed by Gaussian noise with an arbitrary phase threshold in closed form for the first time, in terms of the incomplete confluent hypergeometric function \(_1 \mathcal{F}_1\). This has been an unresolved problem in angle modulation dating back to the 1950s. Closed-form solutions are developed around the lower and upper incomplete Humbert second \(\Phi_2\) confluent hypergeometric function of two variables using the new incomplete \(\beta \)-Pochhammer calculus.
{© 2023 John Wiley & Sons Ltd.}
{© 2023 John Wiley & Sons Ltd.}
MSC:
| 33B20 | Incomplete beta and gamma functions (error functions, probability integral, Fresnel integrals) |
| 33C90 | Applications of hypergeometric functions |
| 33C05 | Classical hypergeometric functions, \({}_2F_1\) |
| 33C15 | Confluent hypergeometric functions, Whittaker functions, \({}_1F_1\) |
| 60H40 | White noise theory |
Keywords:
cumulative distribution function; incomplete beta function; incomplete Humbert functions; incomplete hypergeometric functions; MPSK symbol error probabilitySoftware:
DLMFReferences:
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