In [1], the impulse response of the first arrival position (FAP) channel of 2D and 3D spaces in molecular communication (MC) is derived, but its Shannon capacity remains open. The main difficulty of depicting the FAP channel capacity comes from the fact that the FAP density becomes a multi-dimensional Cauchy distribution when the drift velocity approaches zero. As a result, the commonly used techniques in maximizing the mutual information no longer work because the first and second moments of Cauchy distributions do not exist. Our main contribution in this paper is a complete characterization of the zero-drift FAP channel capacity for the 2D and 3D spaces. The capacity formula for FAP channel turns out to have a similar form compared to the Gaussian channel case (under second-moment power constraint). It is also worth mentioning that the capacity value of 3D FAP channel is twice as large as 2D FAP channel. This is an evidence that the FAP channel has larger capacity as the spatial dimension grows. Finally, our technical contributions are the application of a modified logarithmic constraint as a replacement of the usual power constraint, and the choice of output signal constraint as a substitution to input signal constraint in order to keep the resulting formula concise.
翻译:在[1]中,2D和3D分子通信空间(MC)第一个抵达位置(FAP)频道的脉冲反应来源于2D和3D空间的2D和3D空间,但其香农能力仍然开放。描述FAP频道容量的主要困难在于,当漂移速度接近零时,FAP密度会变成多维孔状分布。因此,在最大程度共享信息方面常用的技巧不再起作用,因为Cauch发行的第一和第二时刻不存在。我们在本文件中的主要贡献是完整地描述2D和3D空间的零驱动FAP频道容量。FAP频道的容量公式与Gausian频道的情况(在第二动动力限制下)具有相似的形式。还值得一提的是,3DFAP频道的容量值是2D FAP频道的两倍。这证明FAP频道随着空间尺寸的增长而拥有更大的容量。最后,我们的技术贡献是对2D和3D空间空间空间空间空间的零驱动力限制应用经修改的对调调控带能力。FAP频道的能力公式的公式与Gaissian频道(在控制下)的容量限制是替代常规电力限制,并且将输出信号作为输入的制约。