Cascaded Double κ - μ Shadowed Fading Channels

In this study, the cascaded double    shadowed fading model is mathematically introduced. A novel expression is obtained for the cumulative distribution function (CDF) of the cascaded double    shadowed fading channels by using Meijer’s G function and Laplace transform. Based on the obtained CDF expression, the exact and asymptotic expressions are derived for the outage probability (OP), the bit error rate (BER) for various modulations of the considered channel model. The OP and the average BER curves with exact simulations obtained by the proposed expressions are presented for different parameters and modulation schemes of the considered channel model. Finally, the accuracy of the derived OP and the average BER expressions is confirmed.


INTRODUCTION
In the last decades, cascaded fading channels have been a subject of interest due to it can be applied for various wireless communication systems. For this reason, the cascaded fading studies have been presented by considering many different fading channel models such as Nakagami,   ,   ,   , Beaulieu-Xie, and fluctuating two-ray [1]- [5]. While the statistical properties of N independent random variables (RVs) which have Nakagami-m distribution was introduced in [1], two different   RVs such as independent and non-identically was considered in [2]. In [3], a comprehensive work related to the product of two independent and non-identically distributed   ,   , and   variates was presented. The studies in [2] and [3] did not report the cascaded double   shadowed fading channels. These works analyzed only   fading without shadow effect. In another study in [4], Beaulieu-Xie which is more recent fading model was investigated for double cascaded fading scenario. For the same purpose, a different fading model, called by fluctuating two-ray fading, was analyzed [5]. It is noteworthy that the generalized fading models including traditional fading patterns such as Nakagami, Rayleigh and Rician are considered, especially when the recent studies are examined.
Moreover, the   shadowed model that contains the Rayleigh, Rician, one-side Gaussian,   , Rician shadow, and Nakagami-m was proposed [6]. The introduced channel model is considered different from the   fading in that it also includes the shadowing effect in the   channel model. It should be mentioned that the   shadowed fading is appropriate for different wireless communication applications [7]- [12]. In [7], coverage evaluation in cellular system for the   shadowed conditions was investigated. The author in [8] presented the extensive study for quadrature amplitude modulation analysis over the   shadowed channels. In [9], a statistical characterization investigation related to the   shadowed and Beckmann fading models were presented. In [10], the energy detection characteristic analysis for the   shadowed fading channels were proposed. In [11], a mixture Gamma shadowed case was considered by using inverse Nakagami-m. Then, the proposed unified distribution model was applied to double   shadowed fading channels. However, this work did not consider cascaded double   shadowed fading. In another study in [12], the secrecy analysis for the multiple-input multiple-output network for the   shadowed conditions was presented. However, none of these studies ([1]- [12]) addressed the cascaded double   shadowed fading conditions. Motivated by this, in this study, a cascaded double   shadowed fading model is proposed. To This paper is organized as follows. Section II gives the properties of the considered fading model.
Section III provides the OP and the average BER analysis for the cascaded double   shadowed environment. In section IV, some numerical findings are discussed. Finally, In section V, the concluding remarks are presented.

FADING MODEL
Let define 12 Z a a  which is the product of the two RVs. The envelopes 11 ra  and 22 ra  are modeled by the   distribution which is proposed in [13]. Otherwise, the   shadowed model is proposed by the author in [6]. For   shadowed model, the probability density function (PDF) of the instantaneous SNR is written as a is Pochhammer function) the PDF expression in (1) can be rewritten as . From (2) and the Gamma distribution, we get ii G  is the PDF of the Gamma distribution and i i jm   .

Lemma:
The CDF expression is derived as It should be noted that the proposed CDF expression includes Meijer's G function that is available in commonly used computational software such as Matlab and Mathematica.

PERFORMANCE ANALYSES
This section presents the OP and the average BER analyses for the cascaded double   fading channels by using the CDF approach.

OP Analysis
The OP which is significant performance metric for wireless communications systems is described as the probability that the total instantaneous SNR is lower than a specified SNR threshold ( th  ).

Average BER Analysis
For various binary modulations, the average BER is calculated as [17]    

NUMERICAL RESULTS
Here, we provide some numerical illustrations for the OP and the average BER of the cascaded double   shadowed fading channels. We set as 1

Proof of the Lemma
We assume that i a , 1, 2 i  are independent RVs and the PDF is the Gamma distribution which is given by The Laplace transform of 1 a is given as   The expression in (17) is obtained by employing [15].
The proof is complete.