Ratio of Products of Mixture Gamma Variates with Applications to Wireless Communications

07/17/2020 ∙ by Hussien Al-Hmood, et al. ∙ 0

In this paper, the statistical properties of the product of independent and non-identically distributed mixture Gamma (MG) random variables (RVs) are provided first. Specifically, simple exact closed-form expressions for the probability density function (PDF), cumulative distribution function (CDF), and moment generating function (MGF) are derived in terms of univariate Meijer's G-function. The statistical characterisations of the distribution of the ratio of products of MG variates are then derived. These statistics are used to analyse the outage probability (OP), the average error probability for different modulation schemes, the effective rate (ER) of communications systems and the average area under the receiver operating characteristics (AUC) curve of energy detection over cascaded fading channels. Additionally, the lower bound of secure outage probability (SOP^L) and probability of non-zero secrecy capacity (PNSC) of the physical layer and the OP of the multihop communications systems with decode-and-forward (DF) relaying protocol and co-channel interference (CCI) are studied by utilising the statistics of the ratio of the products. The derived performance metrics are applied for the Beaulieu-Xie and α-λ-η-μ shadowed fading channels that have not been yet investigated in the literature. Accordingly, the equivalent parameters of a MG distribution for the aforementioned channels are given. A comparison between the numerical results and the Monte Carlo simulations is presented to verify the validation of our analysis.

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I Introduction

The performance of several wireless communications systems can be analysed via using the statistics of the distribution of products of the random variables (RVs) and ratio of products of the variates. For instance, the statistical properties of products of RVs can be employed for cascaded fading channels, multihop transmission with non-regenerative relays [1], multi-antenna systems operating in the presence of keyholes [2], and vehicle-to-vehicle (V2V) communications [3]. In addition, the performance of wireless communications systems in the presence of co-channel interference (CCI) [4] and physical layer security (PLS) [5] can be investigated by the ratio of products of RVs.

Based on above practical usefulness, many efforts have been devoted in the open technical literature to derive the distribution of products of RVs using various fading models. For instance, in [6], the statistical properties, namely, the probability density function (PDF), the cumulative distribution function (CDF), and the moment generating function (MGF), of product of independent and non-identically distributed (i.n.i.d.) Nakagami- variates were derived. In [7], the PDF of the product of independent Rayleigh variates was accurately approximated to be expressed in terms of elementary functions such as power and exponential. Simple approximate expressions for the PDF and CDF of the cascaded independent Rayleigh, generalised Gamma, Nakagami-, Gaussian, and Weibull fading channels were given in [8]. The statistical characterisations of the product of i.n.i.d. squared generalised- () variates were presented in [9] with applications to the orthogonal space-time block codes (STBC) over multiple-input multiple-output (MIMO) keyhole fading channels.

On the other hand, different distributions of ratio of products of RVs have been analysed by the previous works. For example, the performances of the maximal ratio combining (MRC) diversity reception and multi-carrier code-division multiple access (MC-CDMA) system affected by an interference were studied in [10] by using the statistics of independent and identically distributed (i.i.d.) exponential variates. In [11], the ratios of products of Gamma and Weibull RVs were analysed. The distribution of some mathematical operations, such as, products, powers, and ratios, of arbitrary -function variates that can be used as a unified representation for a large number of fading channels, was derived in [12].

Recently, the products of generalised , , , [13]-[18], and Fisher-Snedecor [5], [19] RVs, have been given a special attention by many studies in the literature. This is because these distributions give better fitting to the practical data and provide various composite multipath/shadowing fading conditions. Moreover, these distributions unify most of the well-known classic fading conditions where they include Rayleigh, Nakagami-

, Rician, exponential distributions as special cases. Additionally, the

fading channel can be used to model the non-linear environment of communication systems whereas the and the fading channels can be employed to represent the line-of-sight (LoS) and the non-LoS (NLoS) scenarios, respectively. Hence, the PDF, the CDF, and the MGF of the product of two i.n.i.d. and variates and their applications to double and composite fading channels were analysed in [13] and [14], respectively. In [15], the statistics of non-identical cascaded fading channels were given in terms of Fox’s -function and applied in the analysis of the lower bound of secure outage probability (SOP) and probability of non-zero secrecy capacity (PNSC) of the PLS. The products of two envelopes that are modelled by , , and distributions were investigated in [16] via providing infinite series expressions and closed-form results in terms of multivariate Fox’s -function (FHF). The statistical characterisations of independent but not necessarily identically distributed cascaded Fisher-Snedecor fading channels that are composite of Nakagami-/inverse Nakagami- distributions, were studied in [17].

For the ratio of products, the PDF and the CDF of i.n.i.d. variates were explained in [18]. Similar to the mathematical results of [16], the statistics of the ratio of two envelopes taken from , , and RVs were given in [19] and employed in the analysis of the PNSC of the PLS over different combined non-identical fading conditions of V2V communications. In the same context, the statistical properties of the distribution of ratio of products of i.n.i.d Fisher-Snedecor fading channels were derived in [5] in terms of single variable Meijer’s -function and applied in studying the performance of the PLS and full-duplex (FD) relaying systems with CCI.

In this paper, we derive the statistical properties of the distribution of ratio of products of i.n.i.d. mixture Gamma (MG) variates that have not been yet provided by the previous work. The MG distribution has been widely utilised as a highly accurate approximation for large numbers of fading models [20]-[25]. For instance, in [20], the outage probability (OP), the average bit error probability (ABEP), the average channel capacity (ACC), and the performance of energy detection (ED) over Nakagami-, , and fading channels were first analysed using a MG distribution. The average detection probability (ADP) and average area under the receiver operating characteristics (AUC) curve of ED over /Gamma, /Gamma, and /Gamma fading channels were studied in [21]. Furthermore, unified analysis of the channel capacity under different transmission protocols and the effective rate (ER) of communications systems over /Gamma fading conditions were investigated in [22] and [23], respectively. The performance of the PLS over MG distribution based model of both main and wire-tap channels were reported in [24] and [25]. The sum and the maximum of i.i.d. and i.n.i.d. /gamma fading channels were derived in [26] and [27], respectively, using a MG distribution with applications to the selection combining (SC) and the MRC diversity receptions.

The main contributions of this work are summarised as follows

  • We derive exact closed-form computationally tractable expressions of the PDF, the CDF, and the MGF of the distribution of products of i.n.i.d. MG variates in terms of a single variable Meijer’s -function. To the best of the authors’ knowledge, these unified statistics have not been yet reported in the open technical literature.

  • Based on the above results, novel statistical properties of ratio of products of i.n.i.d. MG variates are provided in simple exact closed-form expressions.

  • The equivalent parameters of a MG distribution for Beaulie-Xie [28] and shadowed fading channels [29] that are not available in the open literature, are derived. Additionally, the analysis of these fading channels via utilising the exact PDF of a single RV which would lead to mathematically intractable statistical properties due to its including the modified Bessel function of the first kind.

  • Capitalising on the above statistics, the OP, the ABEP, the average symbol error probability (ASEP) for different modulation schemes, and the ER of communications systems and the average AUC of ED over non-identical cascaded fading channels are given in exact closed-form expressions. To the best of the authors knowledge, both the performance metrics ER and the average AUC of ED over cascaded fading channels have not been yet analysed by the previous works.

  • We utilise the CDF of the ratio of products to obtain the OP of multihop communications systems with decode-and-forward (DF) relaying protocol and subject to multiple interferes, the SOP, and PNSC of the PLS. Unlike [30] in which the SOP and PNSC over Beaulieu-Xie fading channels are expressed in terms of infinite series, our derived results are given in simple exact closed-form expressions.

The reset of the paper is organized as follows. Section II explains a preliminary information about a MG distribution. Section III is divided into two Subsections A and B. In the former, the statistical properties of products of MG variates are derived whereas in the latter, the statistics of ratio of products of MG RVs are given. The equivalent parameters of a MG distribution for Beaulie-Xie and shadowed fading channels are provided in Section IV. Based on the results of Section III.A, some performance metrics of wireless communications systems over cascaded fading channels are derived in Section V whereas Section VI uses the expressions of Section III.B for some applications of wireless communications systems. The numerical results are presented in Section VII. Finally, some conclusions are highlighted in Section VIII.

Ii MG Distribution Based Channel Model

The PDF of the instantaneous signal-to-noise ratio (SNR) at

th path, , using a MG distribution is expressed as [20, eq. (1)]

(1)

where stands for the number of terms of th Gamma component with parameters , , and . According to [20], the minimum number of terms, , that can provide a good matching between the approximate and exact distributions can be computed by using the mean square error (MSE) between the PDF of these distributions.

Iii Ratio of Products of MG Variates

In this section, the PDF, the CDF, and the MGF of the distribution of products i.n.i.d. MG variates are derived first. The results are then extended to obtain the statistics of ratio of products of i.n.i.d. MG RVs.

Iii-a Product of i.n.i.d. MG Variates

Theorem 1

Let , where for are i.n.i.d. MG-distributed RVs.

The PDF, the CDF, and the MGF of Y can be respectively derived as

(2)
(3)
(4)

where , and is the Meijer’s -function defined in [31, eq. (1.112)].

Proof:

The Mellin transform of Y, , can be expressed as

(5)

where is the Mellin transform of the PDF of that can be evaluated by [31, eq. (2.1)]

(6)

Substituting (1) into (4) and using [30, eq. (3.381.4)], this yields

(7)

where is the incomplete Gamma function [32, eq. (8.310.1)].

Now, plugging (7) in (5) to obtain the following multiple summation expression

(8)

Inserting (8) in (5) and recalling [33, eq. (1.20)], we have

(9)

where , and is the suitable contours in the -plane from to with is a constant value.

With the help of [31, eq. (1.112)], (9) can be written in exact closed-form as given in (2) which completes the proof of the PDF.

The CDF of Y that is given in (3) can be deduced after substituting (2) into and invoking [31, eq. (2.53)].

The MGF of can be obtained via inserting (2) in and using [31, eq. (2.29)] which finishes the proof.

Iii-B Ratio of Products of i.n.i.d. MG Variates

Theorem 2

Assume where and where for and for are i.n.i.d. MG-distributed RVs. Accordingly, The PDF, the CDF, and the MGF of X can be respectively deduced as follows

(10)
(11)
(12)
Proof:

Using the same methodology that is given in Proposition 1 to compute the Mellin transform for both Y and Z. Thereafter, plugging the result in to obtain

(13)

Substituting (13) into [33, eq. (1.21)], the PDF of X can be expressed as

(14)

With the aid of the definition of Meijer’s -function [31, eq. (1.112)], (14) can be expressed in exact closed-form as shown in (10) and the proof is accomplished.

Following the similar steps of deriving (3) and (4) that are provided in Proposition 1, the CDF and the MGF of X can be deduced as given in (11) and (12), respectively, which completes the proof.

Iv Modelling of Fading Channels using a MG Distribution

Iv-a Beaulieu-Xie Fading Channels

The Beaulieu-Xie fading model is proposed as a simple representation for multiple specular and diffuse scatter components via introducing a special scale for the non-central chi-distribution [28]. Moreover, this model unifies the non-central chi, , and generalized Rician distributions where the relationship between the latter and Beaulieu-Xie distributions is the same as that between the Nakagami- and Rayleigh models. Additionally, the Beaulieu-Xie distribution is related to the Nakagami- model in a similar relationship that is between the Rician and Rayleigh fading models [28].

The PDF of , , over Beaulieu-Xie fading channel can be derived after employing [28, eq. (4)] and performing simple change of variables. Thus, this yields

(15)

where is the average SNR, is the fading parameter, controls the location and the height of the PDF, and is the modified Bessel function of the first kind and order. When , the Beaulieu-Xie fading model reduces to Rician distribution with factor where controls the spread of the PDF whereas the Rayleigh fading is obtained after plugging and .

With the aid of the identity [32, eq. (8.445)], the modified Bessel function of (15) can written as

(16)

Plugging (16) in (15), we have

(17)

It can be noted that the infinite series of (16) can be approximated to the number of terms, , that satisfies the required accuracy of the MSE. Accordingly, by matching the PDF of (17) with that of (15), the equivalent parameters of a MG for the Beaulieu-Xie fading channel are expressed as

(18)

Iv-B Shadowed Fading Channels

The shadowed fading channel is a composite model of and Nakagami- distributions. The is proposed in [29] as a generalised distribution that unifies the , , and fading models. Hence, this fading model can be employed to represent the non-linear medium and the non-line-of-sight (NLoS) environment of the wireless communications. Furthermore, the shadowed can be used to model a wide range of composite multipath/shadowed fading scenarios such as /Nakagami-, /Nakagami-, and /Nakagami- [21]. However, the PDF of the distribution is included the modified Bessel function which would lead to results that are expressed in terms of a non-analytical mathematically complicated functions or include an infinite series. Hence, a MG distribution is used in this effort to approximate the PDF of the composite / Nakagami-.

The PDF of , , over fading channel is given as [29, eq. (10)]

(19)

where with , , , and . The fading parameters are defined as follows, stands for the non-linearity parameter, denotes the correlation coefficient between the quadrature components and in-phase scattered waves, indicates the ratio between the power of the quadrature and in-phase scattered components, and represents the real extension of the multipath clusters. components

The PDF of Nakagami- distribution is expressed as

(20)

where refers to the shadowing severity index in this work.

According to [13, eq. (4)], the PDF of the product of two RVs can be evaluated by

(21)

Substituting (19) and (20) into (21), this obtains

(22)

Using the substitution into (22), this yields

(23)

where .

With the help of a Gaussian-Laguerre quadrature approximation, the integration in (23), , can be expressed as , where and are respectively the weight factors and abscissas defined in [34]. Consequently, (23) can be rewritten using (1) with the following parameters

(24)

V Applications of Products of MG RVs to Cascaded Fading Channels

V-a Outage Probability

The OP is an important performance metric of the wireless communication systems operating over fading channels and its defined as the probability that the output SNR drops below a predefined threshold value [35]. Accordingly, the OP of the cascaded fading channels that are modelled by a MG distribution, can be calculated by [35, eq. (1.4)]

(25)

where is given in (3).

V-B Average Bit and Symbol Error Probability

In this section, the ABEP and the ASEP for several modulation schemes over cascaded fading channels that are represented by a MG distribution, are derived in exact unified closed-from expressions.

V-B1 Non-Coherent BFSK and DBPSK

The ABEP of non-coherent binary frequency shift keying (NC-BFSK) and differential binary phase shift keying (DBPSK), , can be computed by [35, eq. (9.254)]

(26)

where and for NC-BFSK and DBPSK, respectively, and is provided in (4).

(33)
(42)

V-B2 Coherent BPSK, BFSK, and BFSK with Minimum Correlation

The ABEP of coherent BFSK, BPSK, and BFSK with minimum correlation can be evaluated by [35, eq. (9.11)]

(27)

where , , and for coherent BFSK, BPSK, and BFSK with minimum correlation, respectively.

Proposition 1

The ABEP of coherent BFSK, BPSK, and BFSK with minimum correlation, , over cascaded fading conditions is obtained as

(28)
Proof:

Plugging (4) in (27) and using the change of the variable with some mathematical manipulations, we have

(29)

With the aid of the definition of the Meijer’s -function [31, eq. (1.112)] and the Fubini’s theorem that is applied to interchange the order of the linear and closed integrations, (29) becomes

(30)

It can be noted that the inner integral of (30) can be computed as

(31)

where is the Beta function defined in [32, eq. (8.380.1)] and follows [32, eq. (3.191.3)].

Recalling the property [32, eq. (8.384.1)] for (31) and inserting the result in (30) which completes the proof of (28).

V-B3 M-Psk

The ASEP of -PSK can be calculated by [35, eq. (9.15)]

(32)

where with .

Proposition 2

The ASEP of -PSK over cascaded fading channels using a MG distribution is given in (33) shown at the top of the next page. In (33), is the bivariate Meijer’s -function defined in [36, eq. (10)].

Proof:

After performing simple mathematical manipulations, (32) can be rewritten as

(34)

One can see that can be evaluated by following the same methodology of the Proposition 3 as provided in the first term of (33).

For , we substitute (4) in (34) and assume , to obtain

(35)

Using the definition of the Meijer’s -function [31, eq. (1.112)] and the Fubini’s theorem, the following integral is deduced

(36)

The inner integral of (36) can be expressed as

(37)

where is the hypergeometric function defined in [32, eq. (9.14.1)]. Step follows [32, eq. (8.391)] whereas obtains after using the identities [32, eq. (8.331.1)] and [37, eq. (07.23.26.0005.01)] with some mathematical simplifications.

Recalling [31, eq. (1.112)] for the Meijer’s -function of (37) and substituting the result into (36) to yield

(38)

With the aid of [36, eq. (10)], can be written in exact closed-form as shown in the second term of (33) which completes the proof.

V-B4 -Qam

The ASEP of -QAM can be computed by [35, eq. (9.21)]

(39)

where ,

(40)

and

(41)

with where .

It can be noted that can be calculated by using the same steps of deriving in (28), namely, Proposition 1 as shown in the first term of (42) given at the top of this page. In addition, can be derived by following the same procedure that is employed to derive of (34) after inserting . Consequently, is obtained in exact closed-from expression as shown in (42) given at the top of this page.

V-C Effective Rate of Wireless Communications Systems

The ER has been proposed to measure the performance of the wireless communications systems under the quality of service (QoS) constraints, such as system delays, that have not been neglected by Shannon theorem [38]. Hence, this performance metric has been widely analysed over different fading channels in the open technical literature (see [23] and [39] and references therein). However, there is no work has been achieved to study the ER over cascaded fading channels which is one of our contributions in this paper.

The ER, can be calculated by [23, eq. (1)]

(43)

where with , , and are respectively the delay exponent, block duration, and bandwidth of the system and stands for the expectation.

It can be noticed that (43) can be written as [39, eq. (8)]

(44)

Now, the ER over unified cascaded fading channels using a MG distribution can be derived after inserting (2) in (44) and utilising [31, eq. (1.112)] and the Fubini’s theorem. Thus, this yields

(45)

Invoking [32, eq. (3.194.3)] to compute the inner integral of (45) and making use of the identity [32, eq. (8.384.1)] and [31, eq. (1.112)], the following closed-from expression of the ER is deduced

(46)

V-D AUC of Energy Detection-Based Spectrum Sensing

The ED technique has been widely employed to perform the spectrum sensing in both cognitive radio (CR) and ultra-wide band (WUB) systems. This refers to its low implementation intricacy where the unlicensed user doesn’t need any prior information about the licensed user [20], [21], [39]-[40]. To study the behaviour of an ED, the receiver operating characteristics (ROC) curve which plots the ADP versus the false alarm probability is utilised in many efforts. But, in some cases, this performance metric doesn’t provide a clear result on the superiority of one system on the other. This is due to the intersection between the two compared curves in a specific values of ADP and false alarm probability. Based on this observation, the authors in [40] proposed the average AUC curve as another performance metric of an ED via measuring the total area under the ROC.

The average AUC, , can be evaluated by [39, eq. (21)]

(47)

where is the AUC in additive white Gaussian noise (AWGN) environment which is expressed as [39, eq. (20)]