Improving the Body Area Line-of-Sight Density Model: A Theoretical Study

In radio propagation characterization, propagation models have been developed as a suitable alterative solution to measurement campaigns [1]. Statistical models, as an essential kind of propagation model, have been broadly applied to various environments [2–8]. Recently, statistical propagation models for Body Area Networks (BANs) attract more and more attention: in [9], the authors presented a dynamic on-body channel model based on a time-variant measurement campaign at 2.45GHz in the 3–5GHz band; Fort et al. [10] measured electromagnetic waves near the torso and derived relevant statistics; in [11], Kim and Takada presented the characterization of an on-body propagation channel with a specific activity of a body on the basis of the measurement results of a male subject in a radio anechoic chamber. From all these examples, we learned that statistical models have become an indispensable part in BAN channel characterization. The statistical model, essentially, is a set of probability distributions in the sample space [12], when it is applied to channel characterization for BANs; so the specific applications are large-scale fading models and probability models of the two, the latter containing time domain characterization as well. In this paper, we mainly focus on the improvement of density models for BANs, since the results and idea are of certain reference value for the BAN channel characterization method.


Introduction
In radio propagation characterization, propagation models have been developed as a suitable alterative solution to measurement campaigns [1].Statistical models, as an essential kind of propagation model, have been broadly applied to various environments [2][3][4][5][6][7][8].Recently, statistical propagation models for Body Area Networks (BANs) attract more and more attention: in [9], the authors presented a dynamic on-body channel model based on a time-variant measurement campaign at 2.45 GHz in the 3-5 GHz band; Fort et al. [10] measured electromagnetic waves near the torso and derived relevant statistics; in [11], Kim and Takada presented the characterization of an on-body propagation channel with a specific activity of a body on the basis of the measurement results of a male subject in a radio anechoic chamber.From all these examples, we learned that statistical models have become an indispensable part in BAN channel characterization.The statistical model, essentially, is a set of probability distributions in the sample space [12], when it is applied to channel characterization for BANs; so the specific applications are large-scale fading models and probability models of the two, the latter containing time domain characterization as well.In this paper, we mainly focus on the improvement of density models for BANs, since the results and idea are of certain reference value for the BAN channel characterization method.

Density Estimation with Regression Method
From [13,14], we know that the density estimation () is the solution of the equation where () is a step function, and () is the probability distribution function.Meanwhile, the following conditions will be met: It is worth noting that, in (1), the distribution function () is unknown, but a random i.i.d.sample  1 , . . .,   is given; then the empirical distribution function is given [15]: where () is a step function.We cite a common time domain parameter (root mean square delay spread) for characterizing BAN channels as an example to build the empirical distribution function.The sampling points cover the chest, the arms, and the legs (the data were measured at the Body-Centric Lab, Queen Mary, University of London); the empirical distribution function can be seen as an approximation of actual probability density function.
The method and the train of thought for density estimation are shown in Figure 2.

International Journal of Antennas and Propagation
After establishing the empirical distribution function, we used the -insensitive method [16] to approximate   ().In Figure 3, the cumulative distribution function for the SV regression is shown where the SV is only shown in Figure 1.The -insensitive loss function ||  is described by [17] In our case, the spacing between the dotted lines is 2.
We do care about the coefficient (Figure 4) in the regression process; these coefficients   are just the ones which would be used to do the final probability density function regression.
Figure 5 presents the density estimation results, while reasonable approximation demonstrates the feasibility and potentiality of this method.In addition, Figure 5 exhibits the traditional method as well: Nakagami distribution is very common in characterizing on-body RMS delay [18], and the results show that this density distribution is not necessarily the best choice.

Conclusion
Statistical model is an important mean of characterizing radio propagation in the BAN propagation environment and other propagation environments.In this paper, the authors try to improve the density estimation method for limited samples in an on-body line-of-sight scenario.In traditional BAN probability density models, least square technique is used to do the regression analysis; however, when the sample size is small, the precision of this method is debatable.In this paper, we proposed new mathematical mechanism to approximate probability density, the final result demonstrates the correctness of the proposed method, and this newly proposed model has extensive reference meaning for statistical radio propagation modeling.