To model the multipath fading impacts of wireless channels, different distributions have been used in the open technical literature such as Rayleigh, Nakagami-, and Nakagami- . However, in -, Yacoub has explained that the results of the aforementioned conventional distributions don’t match well with the field measurements. Consequently, several generalised distributions have been suggested to model various scenarios of wireless communication channels with better fitting to the practical data as well as unify most of the well-known classic fading conditions. For instance, the and the fading channels are employed to represent the line-of-sight (LoS) and the non-LoS (NLoS) fading conditions, respectively . In addition, the fading channel is used for the non-linear environment of communication systems . In , the fading channel is presented as a unified framework for and as well as the fading is proposed as a combined representation for both and distributions.
Recently, the so-called fading channel has been proposed by Yacoub  also as a unified model for both and distributions. Accordingly, this fading model includes nearly all the LoS, NLoS, and non-linear fading scenarios. Based on these observations, the fading has been used in  and  to analyse the channel capacity under different transmission policies and the physical layer security, respectively. In , the outage and error probabilities as well as the channel capacity are given.
Motivated by there are still some applications of wireless communication systems, such as the effective capacity (ER) that have not been yet studied over
fading channels, this paper is devoted to derive the aforementioned metric. Furthermore, the behaviour of the energy detection (ED) based spectrum sensing (SS) is also analysed via deriving the average for both the detection probability (DP) and area under the receiver operating characteristics curve (AUC). To this end and to the best of the authors’ knowledge, novel mathematically tractable exact expressions of the ER, ADP, and average AUC are obtained in terms of the Fox’s-function (FHF). Additionally, the asymptotic expressions are provided to gain more insights into the impacts of the fading parameters on the system performance. It is worth mentioning that the derived results are expressed in terms of an infinite series. Accordingly, we have employed a highly accurate method to compute the number of the truncated terms that makes the infinite summation of the probability density function (PDF) of the instantaneous signal-to-noise (SNR) converges steadily and quickly for a certain figure of accuracy.
Ii The Fading Channel Model
The PDF of the instantaneous SNR over fading channels is expressed as [5, eq. (29)]
where , , and stand for the non-linearity parameter, and real extension of the multipath clusters, respectively. In addition, is recursively calculated by the mathematical expression that is given in [5, eq. (15)]. It can be noted that the computation of depends on , , and which respectively denote the ratio between the total powers of the in-phase and quadrature components of the scattered waves of the multipath clusters, the ratio of the total powers of the dominant components and scattered waves, the ratio of the number of the multipath clusters, and the power of the dominant components to the power of the scattered waves of the in-phase and quadrature components. Moreover, is the incomplete Gamma function [9, eq. (8.310.1)] and is the Pochhammer symbol defined in [10, eq. (1.42)].
Invoking the identity [10, eq. (1.39)], (1) can be equivalently rewritten as
where is the FHF defined in [10, eq. (1.2)].
For the asymptotic analysis, the following approximation can be utilised [10, eq. (1.94)]
where is given in [10, eq. (1.4)] and is defined in [10, eq. (1.96)] as follows
With the help of (3), the FHF of (2) when approximately equals to , i.e. . Consequently, the asymptotic expression of (2) is given as
Iii Performance of Fading Channels
Iii-a Effective Rate
The ER has been proposed to measure the performance of the wireless communication systems under the quality of service (QoS) constraints, such as system delays, that have not been taken into consideration by Shannon .
The ER can be calculated by [11, eq. (4)]
where with , , and are the delay exponent, block duration, and bandwidth of the system, respectively.
Substituting (2) into (6) and using the property [10, eq. (1.126)] to express in terms of the FHF, the following integral is yielded
Making use of [10, pp. 60] to evaluate the integral of (7) and inserting the result in (6) along with the remaining terms of (2), the ER is derived in exact expression as given
For the asymptotic behaviour of the ER, , we substitute (5) into (6). Hence, the following integral is the result
where follows [9, eq. (3.194.3)] and is the incomplete Beta function [9, eq. (8.384.1)].
Inserting the right side of (9) and the terms of (5) that have not been used in (8), in (6), we have as
Iii-B Energy Detection Based Spectrum Sensing
Iii-B1 Average Detection Probability
The detection and false alarm probabilities are considered as the main performance metrics that are used in the analysis of an ED. In this paper, we have employed the system model that was presented in .
In additive white Gaussian noise (AWGN) environment, the detection, , [12, eq. (5)] and false alarm ,, [12, eq. (4)] probabilities are respectively expressed as
where and denote the threshold value and the time-bandwidth product, respectively, and represents the generalized Marcum -function defined in [1, eq. (4.60)] whereas is the upper incomplete Gamma function [9, eq. (8.350.2)].
The ADP, , can be evaluated by [13, eq. (19)]
Expressing the of (14) in terms of double Barnes-type closed contours using [13, eq. (20.())] and substituting the result along with (2) into (11) and recalling , we have
where and and are the suitable closed contours in the complex -plane.
Utilising [10, eq. (2.8)] to compute the inner integral of (13) in terms of the FHF and then invoking the definition of the univariate FHF [10, eq. (1.2)] and [10, eq. (A.1)], we obtain
where is the extended generalised bivariate FHF (EGBFHF) defined in [10, eq. (A.1)]. Since the implementation of this function is not yet performed in the popular software package such as MATLAB and MATHEMATICA, we have employed the programming code that is available in  to calculate the EGBFHF.
When and by using [13, eq. (20.())] and (3), the generalised Marcum -function can be approximated as
where that is evaluated by [10, eq. (1.4)], is the corresponding parameter for the second FHF of [13, eq. (20.())] and is computed by (4).
The solution of the integral of (15) is recorded in [10, eq. (2.51)]. Accordingly, this yields
Now, plugging (16) and (5) in (12) and using the fact that , the result is
The integral of (17) represents the incomplete Gamma function. Hence, the expression of the is obtained as
Iii-B2 Average AUC
The AUC is a single figure of merit that is proposed as an alternative performance metric to the receiver operating characteristic (ROC) curve which plots the versus . This is because, in sometimes, the ROC curve does not give a clear insight into the detectability behaviour of the system which depends on both and .
For AWGN, the AUC, , is given as [13, eq. (24)]
The average AUC, can be computed by [15, eq. (19)]
Plugging (2) and (19) in (20) and making utilise of , we have this integral
where arises after invoking [10, eq. (2.30)].
Plugging the result of (21) and the terms of (2) and (19) that have not been inserted in (21), is deduced as
For the asymptotic behaviour of , , and with the aid of (3), (21) becomes
where follows [9, eq. (3.381.4)].
Inserting the right side of (23) along with the terms of (5) and (19) that have not been plugged in the left side of (23), we obtain
Iv Truncating of the PDF of Fading
One can see that the PDF of fading conditions in (2) is included an infinite series. Therefore, a truncating error should be applied to find the number of terms, , that is required to satisfy a specific figure of accuracy, . In this work, we have employed [16, eq. (5)]
where is the truncating PDF in (2) for terms that is expressed as
Substituting (2) and (26) into (25) and utilising , we have
Recalling [10, eq. (2.8)] to evaluate the integral in (27) and performing some mathematical simplifications to yield
V Analytical and Simulation Results
In this section, the numerical results of our derived expressions are compared with their simulation counterparts that are obtained by Monte Carlo with generations. The asymptotic behaviours for the ER, ADP, and average AUC are also presented to gain more insights into the system performance. The minimum number of a truncated terms, , that satisfies and leads to a perfect matching between the numerical and simulation results is . Moreover, all the performance metrics are plotted versus the average SNR, for fixed and at 1 and various values of , , , and .
Fig. 1 plots the ER for whereas Figs. 2 and 3 show the missed ADP () for and average complementary of AUC (CAUC) which is , respectively, for . From all figures, it can be observed that the performance improves when , , , and/or increase. This is because the high value of means the system tends to be linear whereas a large indicates that the total power of the in-phase components of the scattered waves is larger than that of quadrature counterparts. Additionally, the increasing in and makes the total power of the scattered waves is higher than that of the dominant components, and large number of multipath clusters arrive at the receiver, respectively.
In addition, in Fig. 1, one can see that the ER decreases when becomes 1. This refers to the increasing in the delay of the received signal. In Figs. 2 and 3, the impacts of the and are respectively explained. The increasing in the former improves the performance whereas in the latter degrades the detectability of an ED.
In this letter, the ER and the performance of ED over fading channels were analysed using exact and asymptotic expressions. Moreover, a series truncation for the PDF of fading was applied. The numerical and simulation results for different scenarios were showed. The derived expressions of this work can be used to study the ER, ADP, and average AUC over the special cases of fading channels such as and .
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