A New Statistical Channel Model for Emerging Wireless Communication Systems

A new composite fading model is introduced. This shadowed Beaulieu-Xie model is developed to characterize wireless communication in an environment with an arbitrary number of line-of-sight and non-line-of-sight signals, in contrast to the existing Rayleigh, Ricean, generalized Ricean (i.e., <inline-formula> <tex-math notation="LaTeX">$\kappa -\mu $ </tex-math></inline-formula>), and Nakagami-<inline-formula> <tex-math notation="LaTeX">$m$ </tex-math></inline-formula> models. The proposed model benefits from four parameters that characterize a wide range of fading conditions, unlike existing composite models such as the shadowed Ricean model, the two-wave with diffuse power model, and the fluctuating two-ray model. The proposed shadowed Beaulieu-Xie model is used here to characterize experimental data obtained from fading measurements in 28 GHz outdoor millimeter-wave channels, and it is found to describe the communication environment accurately. We conclude that the proposed composite fading model is particularly useful for characterizing emerging (millimeter-wave and terahertz) wireless communication systems.


I. INTRODUCTION
N EW FORMS of wireless systems are emerging to meet the growing demand for mobile communications [1]. Of particular interest are the millimeter wave (mmWave) and terahertz (THz) wireless communication systems, which increase capacity by pushing beyond the traditional microwave spectrum. Such systems hold great promise, but their channels can be challenging to characterize and ultimately understand. This is because they demand a model of fading that offers both flexibility, in terms of adaptable parameters, and compatibility, in accommodating signal transmission via line-of-sight (LOS) 1 and non-lineof-sight (NLOS) components. Contemporary models, such as the Rayleigh, Ricean, generalized Ricean (i.e., κ-μ), and Nakagami-m models, are limited in their ability to provide the required flexibility and LOS/NLOS compatibility. The Rayleigh fading model has been used to characterize a wide range of communication environments. However, its 1. Throughout this work, the term LOS will be used to refer to signal transmission via direct propagation and/or specular reflections. fixed (two) degrees of freedom limit its flexibility in describing wireless communication environments. Moreover, it is incapable of characterizing transmission via LOS components, because it is based upon the central chi-distribution. Thus, it cannot support the needed LOS/NLOS compatibility.
The Ricean and generalized Ricean (i.e., κ-μ) distributions [4] are obtained from the non-central chi-distribution and are thus able to characterize transmission via LOS and NLOS components. They have gained notoriety because of this LOS/NLOS compatibility. However, the Ricean fading model only has two degrees of freedom and is therefore limited in its flexibility for characterizing a broad range of signal fades [5]. The generalized Ricean (i.e., κ-μ) model, in contrast, has 2n degrees of freedom, where n is the number of independent Gaussian random variables (RVs) and is thus flexible. However, it is not a meaningful fading model because it does not satisfy the power constraint, i.e., its fading power is unbounded [9]. This is evident from its second moment (or the instantaneous power) and parameter, K, in the generalized Ricean model, or κ in the κ-μ model, for which changes to the severity of fading demand adjustments This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ to these parameters to keep the total power of the scatter component invariant.
The Nakagami-m model is a beneficial model in that it can characterize communication environments with multiple NLOS components by way of its flexible fading parameter, m. However, it is derived from the central chi-distribution via normalization and thus cannot characterize LOS components. Notable attempts have been made to apply the Nakagami-m model to environments with both LOS and NLOS components, for the needed LOS/NLOS compatibility, but they have been unsuccessful [8]- [10].
The two-wave with diffuse power fading model [11] can characterize an environment with two dominant LOS components together with NLOS components. However, the probability density function (PDF) and cumulative distributive function (CDF) of its signal envelope are complicated and have no closed-form expressions. Hence, its application for performance analysis of communication systems is limited.
In light of the abovementioned challenges, various composite models have been proposed to characterize the random fluctuations suffered by the received signal given LOS and NLOS components. For example, the shadowed Ricean fading model [12], which considers only a shadowed LOS component, the fluctuating two-ray (FTR) fading model [13], which is limited to two fluctuating LOS components, the κ-μ shadowed distribution [14], whose parent PDF is not meaningful in characterizing fading environments, and the Loo's model [15], whose PDF is not known in closed-form because the LOS signal has a lognormal distribution. Details of more composite models can be found in [12]. Ultimately, all of these composite models have one or more limitations in characterizing fading environments with both LOS and NLOS components.
Given the above search for a fading model that offers both flexibility and LOS/NLOS compatibility, a new model was recently proposed by Beaulieu and Xie [6]. The BX fading model, as it will be referred to in this work, unifies the Rayleigh, Ricean, generalized Ricean (i.e., κ-μ), and Nakagami-m distributions. It is derived from the non-central chi-distribution, in the same manner by which the Nakagamim distribution was derived from the central chi-distribution, via normalization. The normalization is carried out by scaling a RV that is described by the non-central chi distribution to create a new distribution in the form of the BX fading model. Ultimately, this normalization ensures that the BX fading model does not violate the physical principle of the scatter component invariance, unlike the generalized Ricean (i.e., κ-μ) distribution. Moreover, the BX fading model exhibits flexibility by way of its parameter m.
The seamless relationships that exist between the Rayleigh, Ricean, Nakagami-m, and BX models are presented in Figure 1. The Nakagami-m model is a particular case of the BX fading model in the absence of the LOS component, just as the Ricean fading model becomes the Rayleigh fading model in the absence of the LOS component. Also, the BX fading model becomes the Ricean fading In this work, we introduce a new composite model, based on the BX fading model, which we refer to as the shadowed BX fading model. We assume that the amplitude of the LOS components is Nakagami-m distributed, 2 which makes its power a Gamma RV. While other composite models that assumed the LOS components to be fluctuating according to the Nakagami-m model did not justify their choice [12], [13], we will justify our assumption of Nakagmi-m distributed LOS component as follows. Our assumption of a varying LOS signal envelope is based upon physical insight. The physical fading environment causes the received signal power to fluctuate in time and space as a result of shadowing and multipath fading [7]. Multipath fading arises as a result of signal components traveling to the receiver via different paths. When the received signal is a combination of multiple LOS plus NLOS components, we have the envelope become BX distributed [6]. However, a partial or complete blockage of the LOS components can induce random variations to the amplitude of the received signal. It should be noted that shadowing is only considered for the LOS components here because these components are concentrated in beams over narrow solid angles, i.e., small cross-sectional areas, making them preferentially sensitive to degradation from impinging obstructions. This is in contrast to the NLOS components, which are far less susceptible to degradation from impinging obstructions simply because they are distributed over wide solid angles, i.e., large cross-sectional areas. Ultimately, the received signal envelope from this combination of random variations of the LOS and NLOS power presents an envelope not describable by 2. The Nakagami-m random variable (RV) is assumed for describing the fluctuating LOS components because its probability density function (PDF) offers analytical tractability. We have investigated the shadowed BX model using the BX PDF, but we found it to be analytically intractable. the abovementioned unimodal and bimodal distributions. The shadowed BX model overcomes this challenge by incorporating distinct parameters to describe the fading and shadowing severities in the LOS and NLOS signals. This comes about because the constituent Nakagami-m and BX models exhibit flexibility by way of their parameters, m Y and m, respectively. Moreover, the shadowed BX model remains valid when there are more than two LOS components. This is a key feature that other composite models do not incorporate. In this work, we obtain the chief probability functions of this new model in terms of relatively simple (hypergeometric) functions. To justify our claim that the shadowed BX model can be more effective than other fading models, especially in emerging systems, we fit its distribution to field measurements from [3].
The remainder of the paper is organized as follows. In Section I, we introduce the system model of the shadowed BX model. The physical model of the shadowed BX model is presented in Section II. We derive its first-order statistics in Section III, including the PDF of the envelope and instantaneous power, the CDF and the characteristic function (CHF) of the envelope, the moments of the PDF, and the moment generating function (MGF) of the instantaneous power.
In Section IV, we present the fit of the shadowed BX model to measurements. We derive closed-form expressions of the error rates and outage probability (OP) in Section V while considering noncoherent and coherent detections. Section VI presents and discusses the numerical results. Concluding remarks are given in Section VII.

II. PHYSICAL MODEL
The shadowed BX model introduced in this work is based upon the BX fading model. The BX fading model was introduced in a bid to characterize an environment with multiple LOS (direct and specular) components and NLOS (diffuse) components in a more meaningful manner than the generalized Ricean (i.e., κ-μ) distribution [6], while keeping some characteristics of the generalized Ricean (i.e., κ-μ) distribution.
For these models, the noncentral chi RV follows where X i is a Gaussian RV with a non-zero mean μ i and variance σ 2 i . The PDF of the RV Z takes the form of where k is the degree of freedom, λ is the noncentral parameter, which is related to the mean of X i by λ = k i=1 μ 2 i , and I v (·) is the vth-order modified Bessel function of the first kind. The BX fading model is obtained from the noncentral chi distribution in (2) via normalization by a factor 2m while noting the relation of k = 2m, to have a new RV D distributed according to the BX fading model where m is the BX shape parameter, λ 2 is the power of LOS component, and is the power of the NLOS components. It is important to note that the noncentral parameter λ in (2) is also scaled by the factor 2m such that the λ in (3), which for clarity we refer to as λ * , becomes λ = λ * 2m .
For the shadowed BX model introduced in this paper, we assume that λ in (3) is fluctuating because of the dynamic nature of the channel. Hence, we use another RV, Y, to represent the fluctuating LOS amplitude, λ. The PDF of the new composite model can then be expressed as denotes the expectation with respect to Y and X is the average power of the NLOS component due to multipath effects. The fading parameter, m X , characterizes the level to which both the LOS and NLOS signals are subject to fading. This parameter can take on values in the range of m X > 0 in the shadowed BX model, characterizing severe fading at low m X values to little fading at high m X values. This is in contrast to the traditional BX fading model, where the fading parameters can only take on values from 0.5. Note that the argument of the expectation operator in (4) is the PDF of the BX fading model.

III. STATISTICAL CHARACTERIZATION
In this section, we obtain expressions for the chief probability functions of the proposed shadowed BX model in closed-form. The goal here is to create an accurate statistical model of the communication environment-with realistic and physical parameters characterizing the environmental effects. It then becomes possible to understand and overcome impairments in the system during the design, simulation, and testing. For this statistical model, the LOS and NLOS components are assumed to experience different forms and levels of fluctuations, as is often seen in practice. The fluctuations are quantified by separate shadowing and fading severity parameters for both the LOS and NLOS signals.
The PDF of the envelope of the shadowed BX model is where (·) is the gamma function, Y is the average power of the LOS components, and m Y is the shadowing parameter. This latter parameter characterizes the level to which the LOS components are subject to shadowing, and it defines the number of the LOS components as 2m Y . It can take on values in the range of m Y ≥ 0, characterizing LOS shadowing from a limiting case of complete obstruction that could be seen in a dense urban environment to partial/no obstruction that could be seen in a suburban/open environment. Thus, the range of m Y for the shadowed BX model is broader than that of the traditional Nakagami-m model, making it better suited for characterizing severe shadowing. It should be noted that shadowing depends upon the positions of the transmitter and receiver, as well as the positions (and materials) of obstacles in the environment. This means that shadowing, and its impact on the LOS components, would have no preferential direction in a generalized wireless communication environment. Therefore, one can, at best, use a statistical characterization for the net result of shadowing on all LOS components. This is done with the single parameter m Y in the proposed model. At the same time, the physicality of the proposed model can be seen by the fact that it satisfies the physical constraint of power conversation, i.e., it has a bounded fading power. This will be shown by its second-order moment, which is independent of the parameters m X and m Y . Thus, its constituent BX PDF becomes an impulse at a constant value of √ λ 2 + , as the parameter m approaches infinity, and its constituent Nakagami-m PDF becomes an impulse at a constant value of √ Y , as the parameter m Y approaches infinity. It is also worth noting that the Nakagami-m distribution characterizes the fluctuating LOS components with a fading power that is bounded, and it does so with a PDF that allows for straightforward mathematical manipulations/expressions via chief probability functions.
By solving (5) according to [16, p. 706], the PDF of the envelope of the shadowed BX model is obtained as (7), as shown at the bottom of the page, where K = Y X is the K-factor given Y as the LOS power and X as the NLOS power, and 1 F 1 (·; ·; ·) is the confluent hypergeometric function defined as [16, p. 1023] We observe that the PDF of the shadowed BX model in (7) is expressed in terms of 1 F 1 (·; ·; ·), which can be obtained with standard mathematical software.
Inserting (8) into (7), the CDF of the shadowed BX model, After some algebraic manipulations, the CDF is ultimately obtained as where g(v, u) is the incomplete gamma function given by [19] g The moments of the PDF in (7) are where the Gauss hypergeometric function is defined as [21] which converges if c is a positive value for all of |z| < 1.
Note that the Gauss hypergeometric function is also available in standard mathematical software. The characteristic function (CHF) of any distribution is defined to be [22] VOLUME 1, 2020 919 For our model, we apply the Taylor series expansion to e jωr , so that the CHF of the shadowed BX model can be obtained in terms of an infinite series as Equation (15) is a convergent infinite series representation of the CHF, which can also be computed using the NIntegrate function in Mathematica software directly from (14). We go on to define the instantaneous power as S(t) = R 2 (t) [12], whose PDF is obtained by making the substitution The CDF of the instantaneous power plays a vital role in obtaining the OP. We obtain the CDF of the instantaneous power, defined by F S (γ ) = γ 0 f S (t)dt, as The MGF of the instantaneous power, defined as M γ (s) =

E[exp(−γ s)], is obtained as
The expressions for the chief probability functions shown above are relatively simple, allowing for easy mathematical manipulation and verification via simulations with Mathematica and MATLAB. The CDF and MGF of the instantaneous power shown in (17) and (18) also allow for straightforward calculation of the OP and average bit-error rate (BER), respectively. The derived expression for the PDF of the shadowed BX model or the MGF of its SNR can present insight into its diversity order. The diversity order of a statistical model can be obtained by observing the behaviour of its PDF at the origin or the asymptotic behaviour of the MGF of the SNR as the SNR approaches infinity [17]. Equally, for the MGF of the SNR, the slope of the log-log plot of the BER or OP can define the diversity order of a system. The diversity order, manifesting as the slope of a log-log plot for the average bit error rate (BER) versus average signal-to-noise ratio (SNR) or outage probability (OP) versus average SNR, is equal to the fading parameter m X , which is also associated with the degree of freedom of the shadowed BX model. Appendix A shows the derivation for the diversity order of the shadowed BX from the expression of its BER.
As a final point on statistical characterization, we can identify the relationships between the parameters of the shadowed BX model and other fading models. The relationships are summarized in Table 1. Of particular note is the relationship between the parameter sets of the shadowed BX and Ricean model. The physicality of the shadowed BX model can also be seen by establishing its relevant relationship with the parameter set of the Ricean model. The parameter set of the shadowed BX model, (m X , X , m Y , Y ), where m X is the fading parameter, X is the power of the NLOS components, m Y is the shadowing parameter, and Y is the power of the LOS components, can be linked to the parameter set of the Ricean fading model, (1, b 0 , ∞, ), where b 0 sets the power of the NLOS components, and is the power of the LOS components, because the Ricean model is a special case of the shadowed BX model. In fitting the measured data of [3], it is found that the independence of the shadowed BX model's parameters, m X and m Y , can better match the concavity and convexity of the given PDF and can provide a better overall fit, in comparison to the Ricean fading model.

IV. FIT TO PUBLISHED MEASUREMENTS
To validate the proposed shadowed BX model for use in emerging wireless systems, we compare its fit to published data, for a mmWave communication environment with smallscale fading in the 28 GHz band [3], against those of the FTR and BX fading models. Fitting can also be done with the Ricean fading model, as employed in [3], but its parameters can instead be extracted from the parameters of the shadowed BX model. The fittings carried out in this section bring a fair comparison between the BX and shadowed BX models and other existing models that have characterized (the same) experimental data, specifically the Ricean [3] and fluctuating two-ray [13] models. At the same time, such characterizations give insight on the differing functionality of the shadowed BX model and its parent BX model. In doing this, we have found that the combined effects of the parameters m X and m Y can better encompass (the more challenging) characteristics with high curvatures and/or inflections leading to better overall fitting. To carry out the fitting, we compare the analytical CDF of the envelope of our composite model, F R (r), with the empirical CDF of the published data,F R (r). We obtain the most suitable parameter values for minimized error between these CDFs and quantify the deviation of the analytical CDFs from the empirical CDF using the Kolmogorov-Smirnov (KS) test [11] in two ways. Parameter values from existing measurements, characterized according to the Ricean fading model in [3], were obtained by establishing and applying the relevant expressions linking the parameters of the shadowed BX model and those of the Ricean fading model. Parameter values that were not based upon existing measurements were obtained by a grid search that minimizes the KS test value. The KS test is given as where |·| is the absolute value operator. Based upon the definition in (19), the smaller the value of KS, the better the fit of the analytical CDF to the empirical CDF obtained from the measured data. Figure 2 compares the empirical CDF obtained from the vertical-vertical co-polarized LOS measurements in [3] with the analytical CDFs of the FTR, the BX, and the shadowed BX models. The analytical CDFs for these fading models are seen to fit the measured data closely. However, the BX fading model provides a better fit. This is because the measured data was obtained in a setting where there was no obstruction to the signals traveling from the transmitter to the receiver, i.e., there was no shadowing or fluctuations to the LOS component. The absence of shadowing was confirmed in the PDF of the shadowed BX model, which remains unchanged despite making significant changes to the value of m Y (quantifying the shadowing severity). The goodness-of-fit for the three models are calculated using (19), which gives KS FTR LOS = 0.1560, KS BX LOS = 0.0612, and KS SBX LOS = 0.0708, respectively. Ultimately, the BX fading model shows its advantage in fitting an environment that is free of shadowing. In addition, the results in Fig. 2 confirm that the shadowed BX model indeed reduces to the BX fading model when m Y → ∞. Figure 3 provides a comparison between the empirical CDF for vertical-horizontal co-polarized NLOS measurements in [3] with the analytical CDFs of the FTR, BX, and shadowed BX models. The goodness-of-fit is obtained using (19) as KS FTR NLOS = 1.5723 for the FTR model, KS BX NLOS = 0.0595 for the BX model, and KS SBX NLOS = 0.0412 for the shadowed BX model. Here, the shadowed BX model provides a better fit. This is because the LOS components experience fluctuations as a result of varying channel characteristics, and the shadowed BX fading model best characterizes these LOS fluctuations.
It should be noted that the value of each data point in the measured CDF is based upon (i) the individual multipath component in the bin over the local area and (ii) the amplitude of the previous data point considering that the distribution is cumulative. Thus, the performance of the shadowed BX model when characterizing the measured data in the low r regime is based on the above-mentioned point. Also, in Fig. 2, the amplitude fluctuations over the local area that produces the values below −15 dB are very high, indicating significant fading, in comparison to the lower amplitude fluctuations that lead to higher values above −15 dB. In Fig. 3, the CDF of the measured data deviates far less from the analytical CDF because the measured data was obtained for an environment with prevalent NLOS propagation. Thus, the local areas are rich in multipath signal components, leading to higher amplitude values and better overall fits in the figure.

V. PERFORMANCE ANALYSIS
In this section, we present performance results for the average BER and OP. Owing to the fact that the shadowed BX model has simple expressions for its first-order statistics and those related to the signal-to-noise ratio (SNR), the results for the BER and OP are obtained with minimal mathematical manipulations.

A. AVERAGE BIT-ERROR RATE
In this section, we calculate the average BER,P E , which is defined as the conditional error probability, P e (γ ), averaged over the output SNR in an additive white Gaussian noise (AWGN) channel. It is given as where f S (γ ) is the PDF of the SNR as obtained in (16). We continue in the following subsections by deriving the expressions for noncoherent and coherent detections.

2) COHERENT DETECTION
For coherent detections, P e (γ ) is defined as [19] P e (γ ) = 1 2 erfc √ γ (25) where is defined in (22) and erfc is the complementary error function defined in terms of the upper incomplete gamma function as Inserting (16) and (26) into (20) gives Using [16, p. 657], we obtain the average BER for coherent detection as where 2 F 1 (·, ·; ·; ·) is defined in (13). We observe from our results of the average BER with coherent detection that the relative error drops from 1 percent to 0.5 percent when the number of terms that are applied in the summations is increased from 50 to 100. We show in Appendix B that the result in (28) approaches zero as n approaches infinity. The error in truncating the infinite series in (28) is given by The expression in (37) approaches zero as n approaches infinity.

VI. NUMERICAL RESULTS
In this section, we present detailed results of the BER for noncoherent and coherent detections and the OP for the shadowed BX model. The displayed figures are obtained by using the closed-form expressions given in (24) for the noncoherent BER detection, in (28) for the coherent BER detection, and in (35) for the OP. These closed-form expressions, in comparison with numerical integration, have the advantage of providing deeper insight into the effects of parameters in wireless communication systems. This enables better evaluations of system performance. In a typical wireless communication environment, it is reasonable to assume that the NLOS components have worse fading than the LOS components. We, therefore, allow the fading parameters to take values in the range m X ≤ m Y ≤ 0.5. Figure 4 shows a comparison of the BER curves for the noncoherent FSK and coherent binary FSK detections as a function of the average SNR for different values of m X . The coherent binary FSK detection is seen to have improved performance over the noncoherent detection, with both having improved system performance as the value of m X increases. For example, the coherent detection shows approximately a 4 dB improvement for m X = 4 over m X = 2 at a BER value of 10 −3 with m Y = 2. For noncoherent detection, a 4 dB improvement is seen for m X = 4 over m X = 2. Figure 5 shows a comparison of the OP curves as a function of the average SNR for different values of m X with consideration to noncoherent and coherent FSK detections. Improved performance is observed for coherent detection over noncoherent detection while keeping m X constant. For example, a value of m X = 3 and an OP of 10 −2 yields a 3.5 dB improvement for coherent FSK detection over noncoherent FSK detection.

VII. CONCLUSION
In this paper, we proposed a new composite model, which we referred to as the shadowed BX model. It was based upon a recently proposed BX fading model for generalized LOS (direct and specular) components and NLOS (diffuse) components. The amplitude of the LOS component was assumed to be Nakagami-m distributed, which makes its instantaneous power a Gamma RV. Such choices were made to ensure that the shadowed BX model is meaningful, according to the physical constraints of power conservation. The chief probability functions of the amplitude and power of this model were obtained in simple closed-form expressions. Closed-form expressions of average BER and OP were also obtained. Ultimately, we fit the new model to measurement data to show its flexibility and compatibility in describing various scenarios. While the experimental verification of the shadowed BX model was inconclusive, suggesting that further experimental studies are required, we were able to see the potential of both the BX and shadowed BX models. Ultimately, these models could lay the groundwork for characterizing emerging mmWave and THz wireless communication systems.

APPENDIX A DERIVATION OF THE DIVERSITY ORDER OF THE SHADOWED BX MODEL
In this section, we derive the diversity order of the shadowed BX model by considering the derived expression for the BER for noncoherent detection as where 2 F 1 (·, ·; ·; ·) is defined as [18] 2 F 1 (a, b; c; z) = 1 + ab c z + a(1 + a)b(1 + b) c(1 + c) z 2 + · · · ; (z → 0).
At high SNR, the BER or OP scale according to SNR −N , where N is the diversity order, meaning that a log-log plot of the BER or OP versus SNR will portray the diversity order as a slope of N = − log(BER or OP) log(SNR) Since the X and Y are the power of the LOS and NLOS components, respectively, we can assume X = Y and let B = ( m X m X + X ). Applying (39) to (38) gives We see that as the SNR approaches infinity, and X → ∞, we get B → 0. Since B → 0,P E is dominated by first term of the square brackets in (41). Therefore,