A New Hypo-Rayleigh Distribution for Short Range Wide-Band Directive Indoor Channel Fading Modeling at 3.3 and 5.5 GHz Bands

Abstract

In this work, propagation loss models for indoor environment are presented. The directive channel propagation loss in indoor environment at frequency bands of 3.3 GHz with a channel bandwidth of 200 MHz and 5.5 GHz band with 320 MHz bandwidth is measured using vertical polarizations where a set of directive panel antennas and a network analyzer are used in the measurement campaign. It is noticed that, the propagation loss can be modelled by two slopes propagation model giving a rise to two propagation zones. The first zone of propagation is almost free space propagation zone while generally; the second zone has a higher deviation from the mean value of propagation loss. A new Hypo-Rayleigh distribution is proposed to model the channel induced fading in the second zone of the propagation zones.

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Correspondence to Bazil Taha Ahmed.

Appendix

Appendix

The Hypo-Rayleigh induced fading is due to a strong direct ray between the transmitting antenna and the receiving one with lower power indirect rays as shown in Fig. 18 which for simplicity shows only two indirect rays. If the indirect rays diminish, the no fading case will be got.

Fig. 18
figure18

Mechanism of Hypo-Rayleigh induced fading generation in the vertical plane

The PDF of a Rayleigh distribution is given by:

$$ PDF = \frac{r}{{\sigma^{2} }} e^{{{{ - r^{2} } \mathord{\left/ {\vphantom {{ - r^{2} } {2\sigma^{2} }}} \right. \kern-0pt} {2\sigma^{2} }}}} \quad {\text{r}} \ge 0 $$
(5)

The CDF of the Rayleigh distribution is given by:

$$ CDF = 1 - e^{{{{ - r^{2} } \mathord{\left/ {\vphantom {{ - r^{2} } {2\sigma^{2} }}} \right. \kern-0pt} {2\sigma^{2} }}}} \quad {\text{r}} \ge 0 $$
(6)

Subsisting r by (x-a), the PDF of the proposed new distribution (Ahmed Distribution) is given by:

$$ PDF = \frac{{\left( {x - a} \right)}}{{\sigma^{2} }}e^{{{{ - \left( {x - a} \right)^{2} } \mathord{\left/ {\vphantom {{ - \left( {x - a} \right)^{2} } {2\sigma^{2} }}} \right. \kern-0pt} {2\sigma^{2} }}}} \quad {\text{x}} - {\text{a}} \ge 0 $$
(7)

The CDF of the proposed new distribution (Ahmed Distribution) is given by:

$$ CDF = 1 - e^{{{{ - \left( {x - a} \right)^{2} } \mathord{\left/ {\vphantom {{ - \left( {x - a} \right)^{2} } {2\sigma^{2} }}} \right. \kern-0pt} {2\sigma^{2} }}}} \quad {\text{x}} - {\text{a}} \ge 0 $$
(8)

where a is the deviation parameter.

CDF will be 0 when (x−a) is set to zero mean while it will be one when (x−a) is set to infinity.

This distribution is reduced to the Rayleigh distribution one when a = 0 and it reduces to Hyper-Rayleigh distribution when a is negative. From 4 it can be noticed that, the CDF value will be 1 when (x−a) goes to infinity.

Figures 19, 20 and 21 present the PDF of the Hypo-Rayleigh distribution for different values of the parameter (a) for σ of 0.4, 0.8 and 1.2 respectively.

Fig. 19
figure19

PDF of the Hypo-Rayleigh distribution for σ = 0.4

Fig. 20
figure20

PDF of the Hypo-Rayleigh distribution for σ = 0.8

Fig. 21
figure21

PDF of the Hypo-Rayleigh distribution for σ = 1.2

Figures 22, 23 and 24 present the CDF of the Hypo-Rayleigh distribution for different values of the parameter (a) for σ of 0.4, 0.8 and 1.2 respectively. It can be seen that maximum value of CDF is 1.

Fig. 22
figure22

CDF of the Hypo-Rayleigh distribution for σ = 0.4

Fig. 23
figure23

CDF of the Hypo-Rayleigh distribution for σ = 0.8

Fig. 24
figure24

CDF of the Hypo-Rayleigh distribution for σ = 1.2

Figure 25 shows the CDF of the Hypo-Rayleigh, Rayleigh and Hypo-Rayleigh distributions with (a = −0.05, 0 and 0.05 respectively) when σ = 0.8.

Fig. 25
figure25

CDF of the Hyper-Rayleigh, Rayleigh and Hypo-Rayleigh distributions with (a = −0.05, 0 and 0.05 respectively) when σ = 0.8

The distribution reduced to the no fade case when with a = 1 when σ = 0.0. Figure 26 shows the CDF for this case.

Fig. 26
figure26

CDF of the distributions with a = 1 when σ = 0.0 representing the no fade case

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Ahmed, B.T., Plaza, M.E. A New Hypo-Rayleigh Distribution for Short Range Wide-Band Directive Indoor Channel Fading Modeling at 3.3 and 5.5 GHz Bands. Wireless Pers Commun 85, 1905–1923 (2015). https://doi.org/10.1007/s11277-015-2879-z

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Keywords

  • Directive channel
  • Propagation loss
  • Gaussian distribution
  • Rayleigh distribution
  • Hypo-Rayleigh distribution