Comments on "Tight Bounds and Invertible Average Error Probability Expressions over Composite Fading Channels"

I. INTRODUCTION

RECENTLY , there have been growing interests in evaluating the wireless communication performance by modeling the composite fading channels (multipath fading and shadowing) as mixture gamma (MG) distribution. The MG distribution can provide the tractable form for many composite fading channels, which facilitates to analyze the error probability or channel capacity [1], [2]. Thus, Wang et al. [1] analyzed the average symbol error probability (ASEP) of M-ary phase shift keying (M-PSK) by approximating the signal-to-noise (SNR) distribution of composite fading channels with MG distribution. However, due to the lack of the closed-form expression for exact ASEP (i.e., (9) in [1]), they attempted to derive the tight algebraic expressions of upper bounds for ASEP of M-PSK when M > 2 and lower- and upper- bounds when M = 2, particularly, under Nakagami-m fading, Generalized-K (KG) fading, and Nakagami-lognormal (NL) fading. However, we find that the closed-form expression for the exact ASEP is available by applying Appell and Gauss hypergeometric functions. Therefore, motivated by this aspect, the aim of this paper is to ap- propriately derive the closed-form formulas for the exact ASEP along with the corresponding asymptotic expressions in high SNR regime for generalM-PSK in a class of the composite fading channels.

II. EXISTING EXACT BUT INTEGRAL FORM EXPRESSIONS OF ASEP FOR M-PSK [1]

The MG distribution to approximate the probability density function of SNR [TeX:] $$\gamma$$ is given by [1, (3)]

and the exact ASEP for anyM-PSK over the MG distribution is given in integral form (i.e., not in closed-form) by [1, (9)]

where [TeX:] $$\alpha_{i}, \beta_{i}, \zeta_{i}$$ denote the parameters of i-th MG component depending on the different fading models, N is the number of terms, and [TeX:] $$\Gamma(\cdot)$$ denotes the gamma function represented by [TeX:] $$\Gamma(t)=\int_{0}^{\infty} x^{t-1} e^{-x} d x.$$

By employing the MG distribution, the exact ASEP expressions for M-PSK over various fading channels can be represented in integral form [1]. For examples, the exact ASEP expressions under Nakagami-m, [TeX:] $$K_{G}$$, and NL composite fading channels are expressed as follows.

A. Nakagami-m Fading

The parameters for Nakagami-m fading are given by

and the exact ASEP expression for any M-PSK is given by [1, (26)]

B. KG Fading

The parameters for [TeX:] $$K_{G}$$ fading are given by [1, (27)]

and the exact ASEP expression for any M-PSK is given by

where [TeX:] $$\psi\left(\theta_{i}, \beta_{i}, \zeta_{i}\right)=\theta_{i} /\left(\sum_{j=1}^{N} \theta_{j} \Gamma\left(\beta_{j}\right) \zeta_{j}^{-\beta_{j}}\right), t_{i} \text { and } \omega_{i}$$ are the abscissas and weight factors for Gaussian-Laguerre integration [3], and l and m are shaping parameters representing the multipath fading and shadowing effects of the wireless channel.

C. NL Composite Fading

The parameters for NL composite fading are given by

and the exact ASEP expression for any M-PSK is given by [1, (28)]

where [TeX:] $$C=\left(\sum_{j=1}^{N} w_{j}\right) /\left(\sum_{i=1}^{N} w_{i} e^{\sqrt{2} \sigma t_{i}+\mu}\right),$$ m is the fading parameter in Nakagami-m fading, [TeX:] $$\mu \text { and } \sigma$$ are the mean and standard deviation of the lognormal distribution, respectively, and [TeX:] $$\dot{t}_{i} \text { and } \omega_{i}$$ are the abscissas and weight factors for Gaussian- Hermite integration [3].

III. CLOSED-FORM EXPRESSIONS OF EXACT AND ASYMPTOTIC ASEP FOR M-PSK

In this section, we derive the exact and asymptotic ASEPs in closed-form for general M-PSK.

A. Exact Closed-form ASEP Expressions for M-PSK

The ASEP in (2) can be reformulated as

where

Since M is the modulation order of M-PSK and must be a positive integer power of 2, e.g., [TeX:] $$M=2,4,8,16, \cdots$$, the relationship of [TeX:] $$\frac{\pi}{2} \leq \varphi\left(=\frac{(M-1) \pi}{M}\right) \leq \pi$$ always holds. Therefore, for [TeX:] $$\pi / 2 \leq \varphi \leq \pi,$$ (10) can be written in closed-form by using (5) in [4] as

The first term in (11) can be re-written by changing the variable [TeX:] $$t=\sin ^{2} \theta$$ and after some arithmetic manipulations with the relationship [6, (3.211)] of

as

where [TeX:] $$F_{1}\left(a ; b_{1}, b_{2} ; c ; x_{1}, x_{2}\right)$$ denotes the Appell hypergeometric function [5, (07.36.02.0001.01)] and B(a, b) denotes the beta function. Similarly, the second term in (11) can be re-written by changing the variable [TeX:] $$t=\cos ^{2} \theta / \cos ^{2} \varphi$$ and after some arithmetic manipulations as

Therefore, (11) can be written by

Consequently, the exact closed-form ASEP for M-PSK in (9) can be expressed as (17).

In the following, we provide the closed-form expressions of ASEP for Nakagami-m, [TeX:] $$K_{G},$$, and NL composite fading channels.

A.1 Nakagami-m Fading

For Nakagami-m fading, the exact ASEP of M-PSK can be expressed in closed-form as (18).

A.2 [TeX:] $$K_{G},$$ Fading

For [TeX:] $$K_{G},$$ fading, the exact ASEP ofM-PSK can be represented in closed-form as (19).

A.3 NL Composite Fading

For NL composite fading, the exact ASEP of M-PSK can be given in closed-form as (20).

B. Asymptotic Closed-form ASEP Expressions for M-PSK

From (3), (5), and (7), we see that [TeX:] $$\zeta_{i}$$ can be re-represented in the form of [TeX:] $$\zeta_{i}=\left(c_{i} \bar{\gamma}\right)^{-1},$$ where [TeX:] $$C_{i}$$ is a constant depending

on the channel fading. Accordingly, the ASEP in (9) can be re-written by

where [TeX:] $$K_{i}=c_{i} \sin ^{2}\left(\frac{\pi}{M}\right) \text { and } \phi=\frac{(M-1) \pi}{M}.$$ If [TeX:] $$\bar{\gamma} \rightarrow \infty,$$ then [TeX:] $$\zeta_{i} \rightarrow 0$$ and [TeX:] $$\vec{F}_{1}\left(a ; b_{1}, b_{2} ; c ; x_{1}, 0\right)$$ is simplified to [TeX:] $${ }_{2} F_{1}\left(a, b_{1} ; c ; x_{1}\right),$$ where [TeX:] $${ }_{2} F_{1}(a, b ; c ; x)$$ is the Gauss hypergeometric function [5, (07.23.02.0001.01)]. Thus, through the high SNR approximation [TeX:] $$\text { (i.e., } \bar{\gamma} \rightarrow \infty), I_{\beta_{i}}\left(K_{i} \bar{\gamma}, \phi\right)$$ in (21) can be simplified as

Therefore, the following compact asymptotic ASEP formula can be obtained as

B.1 Nakagami-m Fading

For Nakagami-m fading, [TeX:] $$c_{i}=1 / m, K_{i}=\frac{1}{m} \sin ^{2}\left(\frac{\pi}{M}\right),,$$ and the asymptotic ASEP for high SNR can be expressed as

B.2 [TeX:] $$K_{G}$$ Fading

For [TeX:] $$K_{G}$$ fading, [TeX:] $$c_{i}=\frac{t_{i}}{l m}, K_{i}=\frac{t_{i}}{l m} \sin ^{2}\left(\frac{\pi}{M}\right),$$ and the asymptotic ASEP for high SNR can be presented as

B.3 NL Composite Fading

For NL composite fading, [TeX:] $$c_{i}=\frac{C e^{\sqrt{2} \sigma t_{i}+\mu}}{m}, K_{i}= \frac{C e^{\sqrt{2} \sigma t_{i}+\mu}}{m} \sin ^{2}\left(\frac{\pi}{M}\right), C=\frac{\sum_{j=1}^{N} w_{j}}{\sum_{i=1}^{N} w_{i} e^{\sqrt{2} \sigma t_{i}+\mu}},$$ and the asymptotic ASEP in high SNR can be formulated as

IV. NUMERICAL RESULTS

In this section, we evaluate the average symbol error probability ofM-PSK to validate our analytical results in the previous section by using MATLAB software. Specifically, we compare the Monte-Carlo ASEP, the exact but integral form ASEP, the proposed exact closed-form ASEP, and the asymptotic ASEP in high SNR ofM-PSK for three fading channels (e.g., Nakagami- m fading, [TeX:] $$K_{G}$$ fading, and NL composite fading) and examine how modulation order M and channel parameters affect on the ASEPs. We consider the system parameters as m = 2, [TeX:] $$(m, l, N)=(2,5,10), \text { and }(m, \sigma, \mu, N)=(2,1,0.25,9)$$ for Nakagami-m fading, [TeX:] $$K_{G}$$ composite fading, and NL composite fading, respectively [1].

Fig. 1 plots the Monte-Carlo ASEP, the exact ASEP (4), closed-form ASEP (18), and asymptotic ASEP (25) of M-PSK under Nakagami-m fading versus average SNR [TeX:] $$\bar{\gamma}$$ (dB) for various M, wherem = 2. Fig. 2 depicts the Monte-Carlo ASEP, the

exact ASEP (6), closed-form ASEP (19), and asymptotic ASEP (26) of M-PSK under [TeX:] $$K_{G}$$ fading versus average SNR [TeX:] $$\bar{\gamma}$$ (dB) for various M, where (m, l,N) = (2, 5, 10). Fig. 3 illustrates the Monte-Carlo ASEP, the exact ASEP (8), closed-form ASEP (20), and asymptotic ASEP (27) of M-PSK under NL composite fading versus average SNR [TeX:] $$\bar{\gamma}$$ (dB) for various M, where [TeX:] $$(m, \sigma, \mu, N)=(2,1,0.25,9).$$ We can clearly find that all the Figures 1-3 validate that the derived exact closed-form ASEPs for three fading channels completely match well to the exact but integral form ASEPs in all average SNR regions and the derived asymptotic ASEPs also match well to the exact ASEPs in high SNR region. Those figures commonly showed that the ASEP increases as the modulation order M increases and it decreases as average SNR increases.

Fig. 4 compares the ASEP of QPSK under Nakagami-m fading versus average SNR [TeX:] $$\bar{\gamma}(\mathrm{dB})$$ for various m, which shows that the ASEP decreases as m increases. This is because the fading severity of the channel decreases as m increases. Fig. 5 compares the ASEP of QPSK under generalized-[TeX:] $$K\left(K_{G}\right)$$ fading versus average SNR [TeX:] $$\bar{\gamma}(\mathrm{dB})$$ for various m and l, where l and m are shaping parameters representing the multipath fading and shadowing effects of the wireless channel. This figure shows that the ASEP decreases as l and m increase. This is because the multipath fading and shadowing effects of the wireless channel decrease as l and m increase. Fig. 6 compares the ASEP of QPSK under Nakagami-lognormal(NL) fading versus average SNR [TeX:] $$\bar{\gamma}(\mathrm{dB})$$ for various , where is the standard deviation of the lognormal distribution. This figure shows that the ASEP increases as increases. This is because the lognormal shadowing effect increases as increases.

V. CONCLUSION

In this paper, we have newly derived the closed-form formulas for the exact and asymptotic ASEPs of general M-PSK over various composite fading channels approximated by mixture gamma distribution. Specifically, we have represented the exact and asymptotic closed-form ASEP expressions by judiciously adopting Appell and Gauss hypergeometric functions, respectively, which enable us to obtain more explicit insights into the achievable ASEP performance over a variety of fading channels.