New Accurate Approximation for Average Error Probability Under $\kappa-\mu$ Shadowed Fading Channel

This paper proposes new accurate approximations for average error probability (AEP) of a communication system employing either $M$-phase-shift keying (PSK) or differential quaternary PSK with Gray coding (GC-DQPSK) modulation schemes over $\kappa-\mu$ shadowed fading channel. Firstly, new accurate approximations of error probability (EP) of both modulation schemes are derived over additive white Gaussian noise (AWGN) channel. Leveraging the trapezoidal integral method, a tight approximate expression of symbol error probability for $M$-PSK modulation is presented, while new upper and lower bounds for Marcum $Q$-function of the first order (MQF), and subsequently those for bit error probability (BER) under DQPSK scheme, are proposed. Next, these bounds are linearly combined to propose a highly refined and accurate BER's approximation. The key idea manifested in the decrease property of modified Bessel function $I_{v}$, strongly related to MQF, with its argument $v$. Finally, theses approximations are used to tackle AEP's approximation under $\kappa-\mu$ shadowed fading. Numerical results show the accuracy of the presented approximations compared to the exact ones.


I. INTRODUCTION
Wireless technologies are becoming part of our daily lives and their utilization increase rapidly due to many advantages such as cost-effectiveness, global coverage and flexibility.

arXiv:2009.13159v1 [cs.IT] 28 Sep 2020
Nevertheless, these technologies are infected by many phenomena including shadowing which is relatively slow and gives rise to long-term signal variations and multipath fading which is due to constructive and destructive interferences as a result of delayed, diffracted, reflected, and scattered signal components [1].A great number of communication channels' models have been proposed in the literature to describe either the fading or the joint shadowing/fading phenomena [2]- [5].Recently, the κ − µ shadowed fading proposed in [6], has attracted a lot of interest due to its versatility and wide applicability in practical scenarios.For instance, it was used for characterizing signal reception in device-to-device communications, body-tobody communications, underwater acoustic, fifth-generation (5G) communications, and satellite communication systems [7]- [11].In addition, it was shown that numerous statistical models can be derived from the κ − µ shadowed one by setting the parameters to some specific real positive values [12].Particularly, when the parameters µ and m are integer, such a model is equivalent to what's referred to as composite fading, namely, mixture Gamma distribution [13].
The average error probability (AEP) is a fundamental performance evaluation tool in digital communications, quantifying the reliability of an instantaneous received signal.Furthermore, dealing with the average EP (AEP) is quite practical in most applications as it states the average performance irrespective of time.Nonetheless, evaluating AEP in closed form remains a big challenge for numerous communication systems because of the complexity of either the end-to-end fading model or the employed modulation technique.Essentially, depending on the employed modulation scheme, EP is provided in either complicated integral form [1] or first-order Marcum Q-function (MQF) and the zeroth-order modified Bessel function (MBF) of the first kind [14] for various M -ary and differential quadrature phase-shift keying (DQPSK) modulation schemes, respectively.That integral form can be reexpressed also in terms Gaussian Q-function (GQF), which is not known in closed form.By its turn, the MQF integral-form involves the MBF with exponential term [1], that can be rewritten appropriately as an upper incomplete upper Fox's H-function (UIFH), or equivalently, an infinite summation of the product of upper incomplete Gamma functions [15].Thus, obtaining AEP requires the averaging of a UIFH over a generalized fading distribution, which is not evident particularly for fading model with probability density function (PDF) involving the product of exponential and Fox's H-functions (e.g.κ − µ shadowed model).Obviously, deriving accurate bounds or approximations for the AEP is strongly depending on the EP's ones.To this end, several EP's bounds and approximations for the EP are proposed in the literature, for instance, in [16]- [19], numerous bounds for the symbol error probability (SEP) in the case of M -ary PSK modulation are derived in terms of GQF and its powers.Such a function is itself mathematically intractable when involved in complicated integrals resulting from generalized fading distributions.To remedy this problem, several various works deal with simple, and accurate approximations or bounds for GQF when applied to inspect the performance of a communication system experiencing to a particular bivariate-Fox's H-fading model [20], [21].In contrast, evaluating the performance of GC-DQPSK modulation requires simple bounds or approximate expressions for MQF due to its complicated closed-form and intractability when involved in the computation of AEP [22]- [25].In [26] and [27], bounds for EP are investigated, while in [28], new lower and upper bounds for EP were proposed, based on which a novel approximation was derived.Despite the good accuracy of the latter's bounds and/or approximation for both MPSK and GC-DQPSK, they remain useless for AEP computation because of their forms' complications.

A. Motivation
The performance of wireless communication systems, with perfect channel state information (CSI) knowledge at the receiver, is widely examined by the scientific community.However, imperfect estimation of channel coefficients is dealt with various practical scenarios, leading to a significant degradation of the system performance.To overcome this limitation, differential modulation (DM) can be considered as an alternative solution particularly for low-power wireless systems, such as wireless sensor networks and relay networks [29].The main advantage of this scheme is its simplicity of detection due to the unnecessary channel coefficients estimation and tracking, leading to significant reduction in the receiver computational complexity [30], [31].However, this comes at a cost of higher error rate or lower spectral efficiency.As a result, selecting the most suitable modulation scheme depends on the considered application and both coherent and non-coherent detections.To this end, this paper is devoted to analyzing the performance of two modulation schemes, namely M -PSK and DQPSK over κ − µ shadowed fading channel.

B. Contribution
Capitalizing on the above, we aim at this work to propose accurate approximations for AEP under κ − µ shadowed fading and aforementioned modulation schemes.Specifically, utilizing the trapezoidal integral method, the EP integral form for various M -ary modulation schemes is tightly approximated particularly for M -PSK scheme, while for DQPSK technique, we start by deriving simple lower and upper bounds for EP by bounding MQF, to be used jointly in finding AEP for generalized fading models.
Pointedly, our main key contributions can be summarized as follows • We propose a new exponential type approximation for the EP's first form applied toM -PSK modulation by using the trapezoidal technique integral.To the best of the authors' knowledge, such accurate EP's approximation outperforms those presented in the literature, • We derive new upper and lower bounds of EP in the case of DQPSK modulation based on which an accurate approximation of SEP is proposed, • We provide, relying on the two proposed EP's approximations, a tight approximate expression for AEP over κ − µ shadowed fading channel, • We provide the asymptotic analysis for both forms of AEP and we demonstrate that the diversity order over κ − µ shadowed fading channel remains constant.
Motivated by this introduction, the rest of this paper can be structured as follows.Methods and analysis used are described briefly in Section II.In section III, a new approximation for the first EP form (i.e., M -ary modulation) is presented for M -PSK while, new lower and upper bound of EP in the case of DQPSK are derived, based on which an accurate approximation for the EP is deduced.In Section IV, the expression of AEP under κ − µ shadowed fading for both modulation schemes is evaluated.In section V, the respective results are illustrated and verified by comparison with the exact ones using simulation computing.Section VI summarizes the main conclusions.

II. METHODS
The present work deal with the performance analysis of a wireless communication system subject to κ − µ shadowed fading.To this end, various methods are applied to derive new accurate approximate expressions for the error probability.Pointedly, the Trapezoedial integral technique is utilized to propose a simple approximation for the SEP in the case of M -PSK modulation, while bounding technique is used to derive upper and lower bounds for the bit error probability (BEP) in the case of GC-DQPSK modulation.Furthermore, by using Matlab curve fitting application, a refinement is applied on these two new bounds to obtain a novel accurate approximation for the BEP of this latter modulation scheme.Next, the approximations of the EP are used to derive tight approximation for the AEP under the considered fading model.Lastly, the accuracy of the derived AEP's approximation is validated using Monte carlo simulation.

III. BOUNDS ON THE SEP
In this section, we propose new approximate expressions for the two potential different forms of EP, namely (i) complicated integral form, and (ii) MQF form, applied to M -PSK and DQPSK modulations with Gray coding, respectively.

A. EP with integral form
Proposition 1.The SEP for M -PSK modulation can be tightly approximated by while A l and B l are given in Table I.
Proof.The SEP for M -PSK modulation is given as [1,Eq. (8.22)] with and γ denotes the signal-to-noise (SNR) ratio per bit.
The integral I can be approximated using numerical integration rules.The trapezoidal rule for definite integration of an arbitrary function between [x 0 , x 0 + nφ] is given by where g i = f (x 0 + iφ) for i = 0..n, n refers to the number of sub-intervals equally spaced trapeziums, and φ defines the spacing.Note that greater n the higher the accuracy's approximation and the computational complexity as well.
Table II confirms the accuracy of the proposed approximation.Interestingly, one can ascertain that this tightness can be further improved by increasing n, i.e., by increasing the number of terms in the approximation.

B. EP with MQF form
The bit error probability (BEP) for DQPSK modulation with Gray coding is given by [14] with and Q(.) denotes the Gaussian Q-function [1, Eq. (4.1)], and δ = b a .
Proof.As I v is a decreasing function with respect to the index v [33], yields with Now applying [15, Eq. (8.431.4)] for v = 1 2 , one can ascertain where sinh(.)accounts for the hyperbolic sine function.
By plugging (13) into (12) along with the following identity one can obtain Finally, as t ≥ b √ γ, yields which concludes the proof.
3) Approximate BEP for DQPSK: In this part, a tight approximate expression for the BEP under DQPSK scheme is derived based on the two bounds presented above.In a similar manner to the approach followed in [28], the new proposed approximation is a linear combination of the two aforementioned bounds for H 2 (γ), namely Proposition 4. The function χ (γ) can be chosen as where C i and D i are the best-fit parameters, depending on the SNR interval, summarized in Table III.Proof.First, note that the following function satisfies the identity That is, it is sufficient to look for a tight approximation for (23) so as to approximate H 2 (γ) .
By plotting χ (γ) as shown in Fig. 1, one can clearly notice its exponential behavior.It follows that its approximate expression can be written in the form (22). Furthermore, the optimized coefficients C i and D i outlined in Table III, for various SNR intervals can be straight forward obtained using a curve fitting tool (e.g., Matlab Curve Fitting app); this ends the proof.
Remark 1.It is worthwhile that the first-order MQF can be approximated, relying on ( 6) and (21), by Both exact and approximate functions χ (γ) and χ (γ) are plotted in Fig. 1.One can observe that there exists a strong matching between the two curves over the entire range of γ.Table IV summarizes the accuracy of the proposed approximation compared with the best ones proposed in the literature, namely H i (γ) i=3..5 labeled {BER i+2 } i=3..5 in [28], respectively.
Besides, the relative error corresponding to the aforementioned approximations, namely

IV. AEP ANALYSIS
As mentioned above, the proposed approximate EP is used to derive an approximate AEP when communicating over κ − µ shadowed fading.
The PDF of instantaneous SNR γ under the κ − µ shadowed fading model can be written as [12,Eq. 4] with where γ is the average SNR, κ indicates the power ratio between the dominant waves, µ refers to the scattered components, while m accounts for the shape parameter.Further, 1 F 1 (.; .; .)and Γ (.) denote the Kummer confluent hypergeometric and Euler Gamma functions, respectively.
The AEP approximation for both M -PSK and DQPSK schemes can be straightforwardly evaluated as by setting i = 1 and i = 2, respectively.
Remark 3. P e accounts for the average symbol error probability (ASEP) for M -PSK modulation, while it refers to the ABEP for DQPSK modulation with Gray coding.

A. Asymptotic Analysis
In order to gain further insights into system parameters at high SNR regime, an asymptotic analysis for the SNR is carried out.Firstly, note that for large values of γ, one can see that ω goes to 0 and thus the term k = 0 dominates the others, v also goes to 0 (i.e., ξ i D i ).It follows that the AEP can be asymptotically approximated as and for M -PSK and DQPSK schemes, respectively, with It is worth mentioning from (44) and (45) alongside with ( 27) that the diversity order equals µ.

B. Bound on the truncation error
The above approximate ABEP for DQPSK is expressed in terms of infinite series.Truncating such summation and estimating the truncated error is though of paramount importance for numerical evaluation purposes.In what follows, a closed-form bound for such truncation error is provided.
Using (31) , the truncation up to L − 1 terms of the first summation results to the following error By changing the summation index to j = k−L in (49), then using [35, Eq. (06.10.02.0001.01)], and performing some manipulations, the bound can be expressed as In a similar manner, the truncated error of the summation (32) can be upper bounded by (52) Consequently, and having in mind that (j) i are positives, as can be seen from 50 and 52, the absolute value of the total truncated error can be upper bounded by i (−a) + V. RESULTS AND DISCUSSION In this section, the proposed approximation for the AEP versus SNR (in dB) for both M -PSK and DQPSK modulation schemes over κ−µ shadowed fading channel is evaluated and compared with the exact one for various fading severity parameters.
• Figs. 3 and 7 illustrate, for a fixed value of m, the effect of parameter µ on the AEP for M -PSK and DQPSK modulation techniques, respectively under a weak line of sight (LOS) condition.One can notice that the greater the µ is, the better the system's performance.
• Figs. 4 and 8 depict the AEP for both considered modulation schemes under strong LOS (κ = 10) for a fixed value of µ.It is observed that countering the effect of shadowing requires the increase of m.
• To show the versatility of the κ − µ shadowed fading, Figs. 5, 6, and 9 present the AEP for some classical fading models.Noteworthy, the results in all figures are provided for either integer or non-integer values of µ and m.Further, the simulation curves match perfectly with the proposed approximation.
• Fig. 10 depicts the absolute value of the truncated error versus the number of limited terms L for DQPSK modulation.It can be shown that the greater L is, the smaller such an error.
Interestingly, the truncated error decreases with the increase of SNR.
• Lastly, Fig. 11 presents the achievable diversity order for both modulation schemes versus the average SNR, computed by evaluating − log Ps log γ .It is clearly noticed that such a metric goes to µ as γ tends to infinity.

VI. CONCLUSION
New approximate expressions for the EP of a communication system employing either M -PSK or DQPSK modulation have been derived.The proposed approximations ensures optimal accuracy-analytical tractability trade-off that enables its versatility to contribute to the AEP  with such fading and modulation scheme with such a simple approximation.As a future aspect, the authors aim to extend the same approach on more general fading model such as α − κ − µ shadowed [36] and fluctuating Beckmann fading models [37].
Average SNR (dB) • Fig. 1 Comparison between χ (γ) and χ (γ).To show the accuracy of the fitting method, Fig. 1 depicts the curves of both exact and approximated fitting coefficients.
• Fig. 2 Comparison of the relative errors.To demonstrate the tightness of the proposed EP's approximate expressions for GC-DQPSK modulation, Fig. 2 depicts the absolute relative error of the proposed approximation with solid line as well as the best ones proposed in the literature.
• Fig. 3 ASEP for M -PSK under weak LOS scenario (κ = 1) with different values of µ and m = 1.3.The approximated ASEP is presented with solid line, the simulated one with marker and dashed line presents the asymptotic ASEP.
• Fig. 4 ASEP for M-PSK under strong LOS scenario (κ = 10) with different values of m and µ = 2.The approximated ASEP is presented with solid line, the simulated one with marker and dashed line presents the asymptotic ASEP.
• Fig. 5 ASEP for M-PSK under non-LOS scenario (κ = 0) with µ = m.The approximated ASEP is presented with solid line, the simulated one with marker and dashed line presents the asymptotic ASEP.
• Fig. 6 ASEP for M-PSK over various practical fading models with (κ = 5).The approximated ASEP is presented with solid line, the simulated one with marker and dashed line presents the asymptotic ASEP.
• Fig. 7 ABEP for DQPSK under weak LOS scenario (κ = 1) with different values of µ and m = 1.3.The approximated ABEP is presented with solid line, the simulated one with marker and dashed line presents the asymptotic ABEP.
• Fig. 8 ABEP for DQPSK under strong LOS scenario (κ = 10) with different values of m and µ = 2.3.The approximated ABEP is presented with solid line, the simulated one with marker and dashed line presents the asymptotic ABEP.
• Fig. 9 ABEP for DQPSK over numerous practical fading distributions with (κ = 5).The approximated ABEP is presented with solid line, the simulated one with marker and dashed line presents the asymptotic ABEP.

•
Fig. 10 Upper bound for the truncated error for µ = 2.3 and m = 4.7 and various values of (κ) and L. • Fig. 11 Diversity order for κ = 5, m = 4.7 and various values of µ under M-PSK and

TABLE I THE
COEFFICIENTS A l AND B l .

TABLE III OPTIMUM
VALUES OF FITTING PARAMETERS FOR DIFFERENT SNR RANGES.