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Performance bounds for diversity receptions over a new fading model with arbitrary branch correlation
EURASIP Journal on Wireless Communications and Networking volume 2020, Article number: 97 (2020)
Abstract
The performance of a new (BeaulieuXie) fading model is analyzed using bounds. This recently proposed fading model can be used to describe both lineofsight and nonlineofsight components of a fading channel having different diversity orders. We consider the outage probability and error rate performance of maximal ratio combining, equalgain combining, and selection combining over arbitrarily correlated BeaulieuXie fading channels. Closedform expressions for upper and lower bounds to the outage probability and error rate are obtained, and it is shown that these bounds are asymptotically tight in the high signaltonoise ratio regime. The analytical results are verified via Monte Carlo simulations. It is shown that the BeaulieuXie fading model can be more useful than the Ricean and Nakagamim fading models in characterizing environments with both lineofsight and multiple reflected specular components.
Introduction
Numerous models have been developed to characterize wireless communication systems. The Ricean fading model has been largely employed to characterize wireless systems with the presence of both a lineofsight (LOS) signal and multipath (nonLOS) signals. However, the Ricean fading model has a disadvantage in terms of its inadaptability to fading variations in the environment and its restriction to a diversity order of one [1]. The Nakagamim fading model, which can be derived from the central chidistribution, was proposed to characterize wireless systems over different fading variations. This is enabled by its flexible fading parameter m. It has been shown that the Nakagamim fading model is practical when characterizing systems with multipath signals. However, it is only suitable for systems with multipath (nonLOS) signals [2–4].
A new fading model [1], which we and others refer to as the BeaulieuXie (BX) fading model [6–9], was recently proposed to overcome the limitations of both the Ricean and Nakagamim fading models. This model, derived from the noncentral chidistribution in an identical manner in which the central chidistribution was transformed to the Nakagamim fading model, has a fading parameter m, LOS power λ^{2}, and nonLOS power Ω. More importantly, it has the ability to characterize wireless systems with multiple LOS and nonLOS components. The normalization that leads to the BX fading model is carried out by scaling the random variable (RV), R, which is distributed according to noncentral chi distribution by a factor \(\sqrt {\frac {2m}{\Omega }}\) to have another RV, Z which is distributed according to the BX distribution, such that \(R=Z\sqrt {\frac {2m}{\Omega }}\). In doing this, the noncentrality parameter λ in the noncentral chi distribution is also scaled by the factor \(\sqrt {\frac {2m}{\Omega }}\) so that its relationship with the parameter λ in the BX distribution, which for clarity we refer to as λ^{∗}, becomes \(\lambda =\lambda ^{*}\sqrt {\frac {2m}{\Omega }}\) [1]. This normalization process ensures that the effect of the anomaly of unbounded fading power found in the noncentral chidistribution is eliminated. It is important to comment that the κ−μ (or generalized Ricean) distribution, unlike the BX fading model, does not have a proper normalization. In fact, the BX fading model is a normalized form of the κ−μ (or generalized Ricean) model [1], as will be shown in Section 3. This leads to the κ−μ model having an unbounded fading power just as it is found in the noncentral chidistribution. This major flaw is evident in its second moment (also known as the instantaneous power) which is dependent on the degrees of freedom. Hence, the κ−μ (or generalized Ricean) distribution violates the principle of physical fading channels and should not be used to describe a fading environment.
The flexibility presented by the BX fading model, by way of its three parameters, makes it of interest in representations of practical fading for future femtocell, millimeterwave (mmWave), and terahertz (THz) communication systems, as well as 6G shortrange random access channels, where multiple components are present due to signal reflections [1]. In addition, the BX fading model exhibits a seamless relationship to other fading models, such as the Ricean, Nakagamim, and Rayleigh fading models. The relationship between the BX, Ricean, Nakagamim, and Rayleigh distribution is feasible because these four fading models all have bounded fading powers which is evident in their second moment. In the absence of LOS components, the BX fading model becomes the Nakagamim fading model just as the Ricean fading model becomes the Rayleigh fading model. Also, when the fading parameter m is equal to 1, the BX fading model becomes the Ricean fading model just as the Nakagamim fading model becomes the Rayleigh fading model.
The merits of the BX fading model and its improved ability to characterize channels for emerging wireless communication systems can be demonstrated using the experimental data obtained from crosspolarized LOS measurements of 28GHz outdoor mmWave channels published in [5]. It can be shown via the goodnessoffit of the fading models, quantified by the KolmogorovSmirnov (KS) test, that the BX fading model shows an improved fit in comparison to the fit for the Ricean fading model. Despite its theoretical and practical importance, to the best of the authors’ knowledge, there has been no performance analysis of wireless systems on BX channels reported beyond the works in [6–11].
Although the performance of correlated κ−μ (or generalized Ricean) fading channels [12] have been analyzed in [13], it is not meaningful to study arbitrary correlation of such a fading model as it does not satisfy the power constraint. This is evident in its parameter κ (and the Kfactor in the generalized Ricean model) in that changes in the degrees of freedom, according to the severity of fading, must be compensated for by adjusting the nonLOS parameter. This is done to keep the total power of the scattering component invariant [1]. In contrast, adjusting the fading parameter, m, in BX fading does not affect the power in the scatter component [1].
Given the merits of the BX fading model for future femtocell, mmWave, and THz communication systems, as well as 6G shortrange random access channels, it is important to recognize the assumption of independent fading in implementing spatial diversity. Such an assumption requires that there be sufficient spacing between the channels and this may often not be the case [14]. A separation of 30 to 50 wavelengths between channels is typically required to obtain correlation coefficients between 0 and 0.33 [15, 16]. With this in mind, we consider the effects of correlation in this work for BX fading channels with maximal ratio combining (MRC), equalgain combining (EGC), and selective combining (SC). The analysis targets performance bounds to evaluate the performance—in recognition of the fact that intractable integrals/infinite series arise in the pursuit of closedform expressions. A recently developed bounding technique [17] is applied here. The key idea of this bounding technique is to handle correlated fading amplitude involving BX RVs by transforming the original problem into a new problem involving correlated Gaussian RVs, whose joint probability density function (pdf) can lead to tractable upper and lower bounds of the joint pdf near the origin. This technique enables two main contributions. First, asymptotically tight error rate and outage probability (OP) bounds (at high SNR) are derived in closedform for diversity receptions over arbitrarily correlated BX channels. These tight bounds are obtained by bounding the pdf of the associated nonzeromean Gaussian RVs. Second, we show that the diversity receptions over arbitrarily correlated BX fading channels outperform those of Nakagamim fading channels, given that this latter model lacks the ability to characterize LOS components. Ultimately, the analytical results are verified by Monte Carlo simulations and are compared to those in literature.
The remainder of the paper is organized as follows. In Section 2, we summarize the analytical methods of this work. In Section 3, we discuss the physicality (and justification) of the BX fading model. In Section 4, we introduce the system model for linear diversity receptions in BX fading. We discuss the merits of the BX fading model and the employed bound analysis. In Section 5, we show the relationship between the power correlation and Gaussian correlation coefficient, and in Section 6, we derive bounds on the pdf of the channel coefficient. In Sections 8 and 9, we analyze and discuss the asymptotically tight bounds on the OP and error rate for MRC, SC, and EGC schemes. Section 10 presents numerical results, and Section 11 gives some concluding remarks.
Method
This paper analyses performance bounds on the OP and error rate for diversity reception over arbitrarily correlated BX fading. The work considers MRC, EGC, and SC techniques, with expressions for the upper and lowers bound in closed form. The closedform expressions are obtained by transforming the original problem involving correlated fading amplitude RVs to a new problem involving correlated Guassian RVs, whose joint pdf leads to amenable bounds close to the origin. The results obtained analytically are confirmed by Monte Carlo simulations in MATLAB with different parameters and are compared to those in literature. The effect of correlation on system performance is also illustrated.
Physicality and justification of the BX fading model
The performance analyses put forward in this work are motivated by the BX fading model’s ability to characterize wireless systems with multiple LOS and nonLOS components while satisfying the physical constraint of power conservation. It will be shown in this section that the BX fading model’s ability to handle this power constraint comes about from its normalization—and this sets it apart from the existing κ−μ fading model and the equivalent generalized Ricean fading model. We consider here various distributions for the fading models. For the κ−μ fading model, the pdf is defined for a RV of G by
where μ is the positivevalued fading parameter, \(\hat {g}\) is the squareroot of the second moment defined as \(\hat {g}=\sqrt {E\left [G^{2} \right ]}\), and κ is the ratio of total power for the LOS and nonLOS components. With such definitions, d sets the total power for the LOS components at d^{2} and σ sets the total power for the nonLOS components at 2nσ^{2}, given n as the number of Gaussian RVs [12]. For the generalized Ricean fading model, the pdf is defined for a RV of F by transforming (1) with substitutions of s^{2} for d^{2}, n for μ, and \(\sqrt {2n\sigma ^{2}+s^{2}}\) for \(\hat {d}\). This gives a pdf of
where s sets the total power for the LOS components at s^{2}. With such definitions, K=s^{2}/2nσ^{2} is the Kfactor and E[F^{2}]=2nσ^{2}+s^{2} is the secondorder moment. Ultimately, the above definitions for κ, K, and E[F^{2}] all depend upon n, and this leads to two shortcomings for the κ−μ and generalized Ricean distributions: (i) the power of the nonLOS components has to be adjusted to keep the total power invariant and (ii) the fading powers of both distributions are unbounded. Nonetheless, these shortcomings can be overcome by normalization—and this is done by the BX fading model. The BX RV, Z, can be obtained by normalizing the generalized Ricean distribution in (2) such that \(F=Z\sigma \sqrt {2m/\Omega }\) and \(s=\lambda \sigma \sqrt {2m/\Omega }\). This produces a pdf for the BX fading model in the form of
where m is the fading parameter. With such definitions, K=λ^{2}/Ω is the Kfactor given λ^{2} as the LOS power and Ω as the nonLOS power. For this pdf of the BX fading model, m controls the shape, Ω controls the spread, and λ impacts the location and height of the mode [1, 6].
The physicality and manifestation of the BX fading model can be seen by the fact that its distribution becomes an impulse at \(\sqrt {\lambda ^{2}+\Omega }\) as the fading parameter, m, approaches infinity just like the Nakagamim distribution tends to an impulse at \(\sqrt {\Omega }\). To see this, we define the generalized Ricean RV as
where V_{i} is a Gaussian distributed RV with mean m_{i} and variance σ^{2} and n is the number of Gaussian RVs. We obtain the BX RV, Z, via normalization such that
while noting that n=2m. When m and thus n approach infinity, we are left with the BX RV’s limit of
This limit can be simplified using the Chebyshev’s law of large numbers [18] to
We note here that s in (2) is defined as \(s=\sqrt {\sum \limits _{i=1}^{n}m_{i}^{2}}\), which allows (7) to be cast as
We also note that the parameter λ in (3) is scaled by \(\sigma \sqrt {\frac {2m}{\Omega }}\), which gives \(\lambda = \frac {s}{\sigma \sqrt {\frac {2m}{\Omega }}}\). Applying this scaling to (8) and inserting the result into (6) gives
We see from this last expression that the limiting RV of \(Z_{\lim }\) is a constant at \(\sqrt {\lambda ^{2}+\Omega }\). Thus, the distribution of Z takes the form of an impulse at \(z=\sqrt {\lambda ^{2}+\Omega }\) as m approaches infinity—which cannot be said of the κ−μ or generalized Ricean distributions. Moreover, the location of the impulse for the BX distribution, like the Nakagamim distribution, corresponds to the squareroot of the secondorder moment (or instantaneous power) of its distribution and this quantity is bounded.
System model
We consider linear diversity receptions with N branches operating over the channels described by the BX fading model. The received signal is
where x is the transmitted signal, n is a random vector denoting additive Gaussian white noise (AWGN), and z is the fading channel vector, i.e., the real fading channel amplitudes. In addition, \(\mathbf {z} = {\left [ {{z_{1}}, \ldots,{z_{N}}} \right ]^{T}} = {\left [ {\sqrt {{{\bar \gamma }_{1}}} {Z_{1}}, \ldots,\sqrt {{{\bar \gamma }_{N}}} {Z_{N}}} \right ]^{T}}\), where [·]^{T} represents the transpose, \({{\bar \gamma }_{n}}\) is the average received SNR of the nth branch, Z_{n} is the fading amplitude of the nth branch, and z is distributed according to (3).
The respective output SNRs for MRC, EGC, and SC diversity receptions are
We assume that the BX RV is associated with the nth branch as
where X_{n,i},∀i=1,…2m are Gaussian RVs with mean μ and variance \(\left ({\frac {1}{{2m}}} \right)\). Here, m is a halfinteger representing the fading parameter that controls the shape of the pdf in the fading model. The nth component of z can be obtained by generating an N×2m matrix of Gaussian RVs, B, whose entries are b_{n,i} and whose mth column is denoted as b_{m}, such that B=[b_{1},…,b_{2m}]. We obtain the vector \(\mathbf {b} = {\left [ {\mathbf {b}_{1}^{T}, \ldots,\mathbf {b}_{2m}^{T}} \right ]^{T}} = {\left [ {{b_{1,1}},{b_{2,1}}, \ldots,{b_{n,2m}}} \right ]^{T}}\). We can, therefore, express the fading amplitude over the nth branch as
such that \(b_{n,i}^{2} = {\bar {\gamma }_{n}}X_{n,i}^{2}\).
The pdf of b is expressed as
where μ_{b} is the 2mN×1 mean vector and R_{b} is the 2mN×2mN covariance matrix of b. The variances of the LOS and nonLOS components are defined as \(\sigma _{L,n}^{2} = {\left  {{\mu _{b,n}}} \right ^{2}}\) and \(\sigma _{NL,n}^{2} = E\left [ {{{\left  {{b_{n}}  {\mu _{b,n}}} \right }^{2}}} \right ]\), respectively, where · denotes magnitude and E[·] denotes expectation. The Kfactor for the BX fading channel is therefore defined as \(K = \frac {{\sigma _{L,n}^{2}}}{{\sigma _{NL,n}^{2}}}\). The determinant of the covariance matrix R_{b} is expressed in terms of the correlation matrix C_{b} by \(\left  {{\mathbf {R}_{\mathbf {b}}}} \right  = \frac {{\left ({\prod \limits _{n = 1}^{N} {\bar {\gamma }_{n}^{2m}}} \right)\left  {{\mathbf {C}_{\mathbf {b}}}} \right }}{{{{\left ({2m} \right)}^{2mN}}}}\) [17]. This new fading model has the diversity order mN which is the same as that of the Nakagamim fading model, for the same value of m, but the former predicts improved performance for channels possessing LOS components [1].
Relationship between the power correlation and Gaussian correlation coefficient
As shown in [16], the correlation coefficient between RVs X_{n,i} and X_{j,k} with mean μ and variance \( {\frac {1}{{2m}}}\) is obtained by employing the Cholesky decomposition. It can be shown that \({X_{n,i}} = {\rho _{\left ({n,i} \right)\left ({j,k} \right)}}{X_{j,k}} + \sqrt {1  \rho _{\left ({n,i} \right)\left ({j,k} \right)}^{2}} W\), where ρ_{(n,i)(j,k)} is the correlation coefficient and W is a Gaussian RV with zero mean and variance \({\frac {1}{{2m}}}\), which is independent of X_{j,k}. This leads to a relationship between the power correlation coefficient, \({\rho _{z_{{n_{1}}}^{2}z_{{n_{2}}}^{2}}}\), and correlation coefficient of the Gaussian RVs, ρ_{(n,i)(j,k)}, according to
where
\(T = \frac {1}{{2m}} + {\mu ^{2}}\), and \(U = \frac {1}{{2m}}\). A special case can be obtained when μ=0 such that
which is the same as the one obtained for the Nakagamim fading model in [17], as expected.
Bounds to the pdf
Here, we show that the pdf of b is bounded in the region b^{T}b≤r^{2}, which is a 2mNdimensional sphere with radius r. A Rayleigh quotient is applied, such that \({R_{y}}\left ({\mathbf {R}_{\mathbf {b}}^{ 1},\mathbf {b}} \right) = \frac {{{\mathbf {b}^{T}}\mathbf {R}_{\mathbf {b}}^{ 1}\mathbf {b}}}{{{\mathbf {b}^{T}}\mathbf {b}}} \), where R_{b} is a positive definite matrix. The numerical range of R_{y} is obtained as \({\lambda _{\min }} \le \frac {{{\mathbf {b}^{T}}\mathbf {R}_{\mathbf {b}}^{ 1}\mathbf {b}}}{{{\mathbf {b}^{T}}\mathbf {b}}} \le {\lambda _{\max }}\), such that λ_{i} are the eigenvalues of \(\mathbf {R}_{\mathbf {b}}^{ 1}\) and λ_{i}≥0, where [19] λ_{min} is the smallest eigenvalue of \(\mathbf {R}_{\mathbf {b}}^{ 1}\), and λ_{max} is the largest eigenvalue of \(\mathbf {R}_{\mathbf {b}}^{ 1}\). Thus, considering the region b^{T}b≤r^{2}, we take λ_{min}=0 to give
We expand the exponential component with the knowledge that \({ {\mathbf {R}_{\mathbf {b}}^{ 1}}^{T}} = {\mathbf {R}_{\mathbf {b}}^{ 1}}\); this gives
We employ the 2norm of the matrix as \(\left \ \mathbf {b} \right \ = \sqrt {{\mathbf {b}^{\mathbf {T}}}\mathbf {b}} = r\), such that (18) becomes
The upper and lower bounds of f_{b}(b) near the origin are then obtained as
Performance bounds to outage probability
The OP and error rate of diversity combining schemes are two important performance metrics that can be used to quantify the effects of correlated fading. We analyze the performance of this new fading model here with arbitrary channel correlation by computing the bounds according to [17]. Here, we set an SNR threshold, γ_{th}, below which we declare occurrences of outage.
Outage probability bounds for MRC
We express the OP for MRC with respect to a signal threshold, γ_{th}, as
Substituting (12) into (11a) and then substituting the result into (21) leads to
It should be noted that the above integral is a 2mNfold integral, but for notational simplicity, it is expressed with just one integral sign.
Asymptotic outage probability approximation for MRC
The asymptotic OP approximation is obtained by substituting f_{b}(b)≈f_{b}(0) into (22), where
and simplifying to give
where \({\gamma _{\text {MRC}}} = \sum \limits _{n = 1}^{N} {\sum \limits _{i = 1}^{2m} {b_{n,i}^{2}}} \). The integral in (23) is the volume of a 2mN dimensional sphere with a radius of \(r = \sqrt {{\gamma _{\text {th}}}}\). Ultimately, Eq. (23) can be expressed as
Lower bound outage probability for MRC
The lower bound to the OP is obtained by replacing f_{b}(0) with the lower bound of the pdf obtained in (20) into (23). Substituting the result into (23) gives
where
Eq. 25 can be further simplified to
Upper bound outage probability for MRC
The upper bound to the OP is obtained by replacing f_{b}(0) with the upper bound of the pdf obtained in (20) and by substituting the result into (23). This gives
where \( {{O}}_{\text {UM}} = {f_{\mathbf {b}}}\left ({\mathbf {0}} \right)\exp \left ({\left \ {{{\boldsymbol {\mu }}_{\mathbf {b}}}^{T}\mathbf {R}_{\mathbf {b}}^{ 1}} \right \\sqrt {{\gamma _{\text {th}}}}} \right){{ }}\). Eq. 27 is further simplified to
We compare the results obtained in (24), (26), and (28), observing that O_{LM}≤f_{b}(b)≤O_{UM}, which is the same as (20). It is seen that the difference between the bounds lies in the exponential component of the equations. An increase to the average SNR per branch leads to the lower and upper bounds of the OP converging to the asymptotic approximation.
Outage probability bounds for EGC
We express the OP for EGC with respect to a signal threshold, γ_{th}, as
Substituting (12) into (11b) and then substituting the result into (29) gives
Asymptotic outage probability approximation for EGC
The asymptotic OP approximation is obtained by substituting f_{b}(b)≈f_{b}(0) in (30) and simplifying to give
where \({\gamma _{\text {EGC}}} = \frac {1}{N}{\left ({\sum \limits _{n = 1}^{N} {\sqrt {\sum \limits _{i = 1}^{2m} {b_{n,i}^{2}}} }} \right)^{2}}\). The integral in (31) is obtained as [17]
This result in (32) is substituted into (31) to give the asymptotic OP approximation as
Lower bound outage probability for EGC
We compare the integral region for EGC to that of MRC. These integral regions can be expressed as
and
Note that \(\sum \limits _{n = 1}^{N} {\sum \limits _{i = 1}^{2m} {b_{n,i}^{2}}} \le N{r^{2}}\) can be derived from \(\frac {1}{N}\sum \limits _{n = 1}^{N} {\sum \limits _{i = 1}^{2m} {b_{n,i}^{2}}} \le {r^{2}}\). Thus, the integral region IR_{EGC}(r) lies within \(\mathrm {I{R_{MRC}}}\left ({\sqrt N r} \right)\). We then obtain a bound on the pdf for EGC by substituting r with \(\sqrt {N}r\) in (20) as
where \(r=\sqrt {\gamma _{\text {th}}}\). Substituting the lower bound of the result obtained in (34) into (31), the lower bound OP for EGC is obtained as
where
This result can be expressed as
Upper bound outage probability for EGC
The upper bound to the OP of EGC is obtained by replacing f_{b}(0) in (31) by the upper bound of the pdf in (34), which gives
where \({O_{\text {UE}}} = {f_{\mathbf {b}}}\left ({\mathbf {0}} \right)\exp \left ({\left \ {{{\boldsymbol {\mu }}_{\mathbf {b}}}^{T}\mathbf {R}_{\mathbf {b}}^{ 1}} \right \\sqrt {N{\gamma _{\text {th}}}}} \right)\). This result can be expressed as
We compare the results obtained in (33), (36), and (38), observing that O_{LE}≤f_{b}(b)≤O_{UE} is the same as (34). It is seen again that the difference between the bounds lies in the exponential components of the equations. An increase to the average SNR per branch leads to the lower and upper bounds of the OP converging to the asymptotic approximation, as was found previously.
Outage probability bounds for SC
We express the OP for SC with respect to a signal threshold, γ_{th}, as
Substituting (12) into (11c) gives
Asymptotic outage probability approximation for SC
The asymptotic OP approximation is obtained by substituting f_{b}(b)≈f_{b}(0) into (40) and simplifying to give
where \(d{\mathbf {b}_{n}} = \Pi _{n = 1}^{N}d{b_{n,i}}\). The integral in (41) is the 2mdimensional volume of a ball with a radius of \(r = \sqrt {{\gamma _{\text {th}}}}\). Ultimately, Eq. (41) can be expressed as
Lower bound outage probability for SC
We compare the integral region for SC to that of MRC. These integral regions can be expressed as
and
where r is the radius. We can therefore bound the 2mdimensional ball in (41) by the hypersphere in (23). In addition, we note that \(\sum \limits _{n = 1}^{N} {\sum \limits _{i = 1}^{2m} {b_{n,i}^{2}}} \le Nr\) can be derived from \(\mathop {\max }\limits _{n} \left \{ {\sum \limits _{i = 1}^{2m} {b_{n,i}^{2}}} \right \} \le {r}\); therefore, IR_{SC}(r) lies within \(\mathrm {I{R_{MRC}}}\left ({\sqrt N r} \right)\). Equation 34 is therefore an applicable bound to SC. Substituting the lower bound of the result obtained in (34) into (41) gives
where
Equation 43 can be further simplified to
Upper bound outage probability for SC
The upper bound to the OP is obtained by replacing f_{b}(0) in (40) with the upper bound of the pdf obtained in (34). This gives
where \({O_{\text {US}}} = {f_{\mathbf {b}}}\left ({\mathbf {0}} \right)\exp \left ({\left \ {{{\boldsymbol {\mu }}_{\mathbf {b}}}^{T}\mathbf {R}_{\mathbf {b}}^{ 1}} \right \\sqrt {N{\gamma _{\text {th}}}}} \right)\). Equation 45 is further simplified to
We compare the results obtained in (42), (44), and (46), observing that O_{LS}≤f_{b}(b)≤O_{US} is the same as (34). It is once again seen that the difference between the bounds lies in the exponential components of the equations. An increase to the average SNR per branch leads to the lower and upper bounds of the OP converging to the asymptotic approximation, as seen previously.
Performance bounds to error rate
Error rate bounds for MRC
We can express the error rate for MRC as
where the Gaussian Qfunction is defined as [20, 21]
For the special case of coherent binary phase shift keying (BPSK), we have p=1 and q=2.
Asymptotic error rate approximation for MRC
The asymptotic error rate approximation is obtained by substituting f_{b}(b)≈f_{b}(0) into (47) to give
where the integral in (48) is obtained as [17, 22, 23]
Substituting (49) into (48) gives the asymptotic error rate bound as
Lower error rate bound for MRC
The lower bound to the error rate is obtained by changing the integral region in (48) according to
where \({R_{\text {MRC}}} = \sum \limits _{n = 1}^{N} {\sum \limits _{i = 1}^{2m} {b_{n,i}^{2}}} \). Substituting f_{b}(b) with the lower bound of the pdf in (20) into (51), where r=R_{th}, and applying some simplifications, we have
where
and
where \(\gamma \left ({\alpha,x} \right) = \int \limits _{0}^{x} {{e^{ t}}{t^{\alpha  1}}} dt \) is the incomplete gamma function [17, 24].
Upper error rate bound for MRC
We obtain the upper bound to the error rate by splitting the integral region in (48) according to
We consider the upper bound of the pdf obtained in (20) such that (55) can be expressed as [25]
where the first integral in (56) can be simplified as in (54), f_{b}(b) is substituted with the upper bound in (20) where \({{E}}_{\text {UM}} = {f_{\mathbf {b}}}\left ({\mathbf {0}} \right)\exp \left ({\left \ {{{\boldsymbol {\mu }}_{\mathbf {b}}}^{T}\mathbf {R}_{\mathbf {b}}^{ 1}} \right \{R_{\text {th}}}} \right)\), and f_{b}(b) is substituted with f_{b}(μ_{b}) obtained as
which is the largest value of f_{b}(b) in the second part of (55). The second integral in (56) is obtained as
Error rate bounds for EGC
We can express the error rate for EGC as
Asymptotic error rate approximation of EGC
The asymptotic error rate approximation is obtained by substituting f_{b}(b)≈f_{b}(0) in (58) to give
where we obtain the integral in (59) according to [11 Eq. (61)]. This is simplified to give
where
Lower Error Rate Bound for EGC
The lower bound to the error rate is obtained by changing the integral bound in (59) as
where \({R_{\text {EGC}}} = \sqrt {\frac {q}{N}{{\left ({\sum \limits _{n = 1}^{N} {\sqrt {\sum \limits _{i = 1}^{2m} {b_{n,i}^{2}}} }} \right)}^{2}}}\). Substituting f_{b}(b) with the lower bound of the pdf in (34) into (62), where r=R_{th}, and simplifying gives
where
and
Upper error rate bound for EGC
The upper error rate bound is obtained by splitting the integral region in (58) according to
We substitute f_{b}(b) with the upper bound of the pdf in (20), for the first part, and f_{b}(b) with f_{b}(μ_{b}), which is the largest value of f_{b}(b) for the second part. This gives
where \({E_{\text {UE}}} = {f_{\bf {b}}}\left ({{{\mu }_{\bf {b}}}} \right)\left ({\mathbf {0}} \right)\exp \left ({\left \ {{{\boldsymbol {\mu }}_{\mathbf {b}}}^{T}\mathbf {R}_{\mathbf {b}}^{ 1}} \right \\sqrt N {R_{\text {th}}}} \right)\). The first integral above is simplified as in (54), and the second integral is simplified as
After some mathematical manipulation, Eq. (68) is further simplified to give
and then
Ultimately, Eq. (68) can be expressed as
Substituting (65) and (71) into (67) gives the upper error rate bound for EGC.
Error rate bounds for SC
We can express the error rate for SC as
Asymptotic error rate approximation for SC
The asymptotic error rate approximation is obtained by substituting f_{b}(b)≈f_{b}(0) into (72) to give
which can be simplified to [17, 26]
Lower error rate bound for SC
The lower bound of the error rate is obtained by changing the integral region in (72) according to
where \({R_{\text {SC}}} = \mathop {\max }\limits _{n} \left \{ {\sum \limits _{i = 1}^{2m} {b_{n,i}^{2}}} \right \}\). Substituting f_{b}(b) with the lower bound of the pdf in (34) into (75), where r=R_{th}, and simplifying gives
where
and
Upper error rate bound for SC
The upper error rate bound is obtained by splitting the integral region in (72) according to
We consider the upper bound of the pdf obtained in (34) such that (79) can be expressed as
where the first integral is simplified as in (78) and \({{E}}_{\text {UB}} = {f_{\mathbf {b}}}\left ({\mathbf {0}} \right)\exp \left ({\left \ {{{\boldsymbol {\mu }}_{\mathbf {b}}}^{T}\mathbf {R}_{\mathbf {b}}^{ 1}} \right \\sqrt {N}{R_{\text {th}}}} \right)\). The second integral is obtained as
The upper error rate bound for SC is obtained by substituting (78) and (81) into (80).
Discussion on the tightness of the bounds
Tightness of the outage probability bound
We consider the exponential component of the lower bounds of the OP for MRC, EGC, and SC, in (26), (36), and (44), respectively, and also that of the upper bound shown in (28), (38), and (46), respectively. Comparing the asymptotic approximations, the lower and the upper bounds of each of the diversity combining schemes, it is seen that the difference lies in the exponential components of each of the expressions. We note that an increase to the average SNR per branch by a factor P results in an increase in the covariance matrix R_{b} and eventually a decrease in \({\mathbf {R}_{\mathbf {b}}^{ 1}}\) by the same factor. This leads to a decrease in the value of λ_{max} (which is the largest eigenvalue of \({\mathbf {R}_{\mathbf {b}}^{ 1}}\)) by that same factor, i.e.,\(\frac {{{\lambda _{\text {max}}}}}{P}\). With this in mind, we can express the limit of the ratio of lower bound OP to the upper bound OP as P tends to infinity as
Equation 82 shows that the lower and upper bounds of the OP converge to the asymptotic approximation as the SNR approaches infinity. This property is highly desirable.
Tightness of the error rate bound
Given the definition of the incomplete gamma function
and the Chernoff bound of the Qfunction [20, 27]
we apply these definitions to G(R_{th}),H(R_{th}),W(R_{th}),Z(R_{th}),S(R_{th}), and V(R_{th}) in (54), (49), (65), (71), (78), and (81), respectively, to show the tightness of the biterror rate (BER) bounds, according to
We then consider the exponential component, where we see that an increase in the average SNR per branch by a factor P results in an increase in the covariance matrix R_{b}. This leads to a decrease in \({\mathbf {R}_{\mathbf {b}}^{ 1}}\) and eventually a decrease in λ_{max} by the same factor P. Thus, we find the limit to be
Expressing R_{th} in terms of P, i.e., \({R_{\text {th}}} = \sqrt P \), gives
This property is highly desirable.
Numerical results
Figure 1 shows a comparison of OP curves for MRC, SC, and EGC for a 2branch diversity system over correlated BX fading channels with coherent BPSK modulation and a fading parameter of m=1.5. Correlation structure of the form C_{b}=[C_{1},0_{2×2},0_{2×2};0_{2×2},C_{2};0_{2×2},0_{2×2},0_{2×2},C_{3}], where \({\mathbf {C}_{1}}= {\mathbf {C}_{2}}= {\mathbf {C}_{3}} = \left [ {1,\sqrt {0.35} ;\sqrt {0.35},1} \right ]\), is considered. The lower and upper bounds converge to the asymptotic approximation and simulated OPs at high SNR. It is observed that the OPs of MRC and EGC are close to each other unlike MRC and SC. This is largely due to the factor contributed by the integral in (23), (31), and (41).
Figures 2, 3, and 4 show a comparison of OP curves of MRC, SC, and EGC, respectively, for a 2branch diversity system over correlated and independent BX fading channels with coherent BPSK modulation and a fading parameter of m=1.5. Correlation structure of the form C_{b}=[C_{1},0_{2×2},0_{2×2};0_{2×2},C_{2},0_{2×2};0_{2×2},0_{2×2},C_{3}], where \({\mathbf {C}_{1}} = {\mathbf {C}_{2}} = {\mathbf {C}_{3}} = \left [ {1,\sqrt {0.35} ;} {\sqrt {0.35},1} \right ]\) and C_{1}=C_{2}=C_{3}=[1,0;0,1], is considered for correlation and independence, respectively. The independent channels outperform the correlated channels, as expected. The effect of correlation is seen on the rate of convergence exhibited by the three combining techniques.
Figure 5 shows a comparison of BER curves for MRC, SC, and EGC for a 2branch diversity system over correlated BX fading channels with coherent BPSK modulation and a fading parameter of m=1.5 with a correlation matrix of the form C_{b}=[C_{1},0_{2×2},0_{2×2};0_{2×2},C_{2},0_{2×2};0_{2×2},0_{2×2},C_{3}], where \({\mathbf {C}_{1}} = {\mathbf {C}_{2}} = {\mathbf {C}_{3}} = \left [ {1,\sqrt {0.4} ;\sqrt {0.4},1} \right ]\). The lower and upper bounds again converge to the asymptotic approximation at high SNR. It is seen that the BERs for MRC and EGC are close to each other, differing by at most 3 dB.
Figure 6 shows a comparison of BER curves for MRC, SC, and EGC for a 3branch diversity system over correlated BX fading channels with coherent BPSK modulation and a fading parameter of m=1.5 with a correlation matrix of the form C_{b}=[C_{1},0_{3×3},0_{3×3};0_{3×3},C_{2},0_{3×3};0_{3×3},0_{3×3},C_{3}], where \({\mathbf {C}_{1}} = {\mathbf {C}_{2}} = {\mathbf {C}_{3}} = \left [ {1,\sqrt {0.4},\sqrt {0.2} ;} \sqrt {0.4},1,\sqrt {0.4} ;\sqrt {0.2}, {\sqrt {0.4},1} \right ]\). The bounds are seen to be tight at high SNR. It is observed that the BERs for MRC and EGC are again close to each other, differing by at most 3 dB in SNR.
Figures 7, 8, and 9 show a comparison of BER curves for MRC, SC, and EGC, for a 2branch diversity system over correlated and independent BX fading channels with coherent BPSK modulation and a fading parameter of m=1.5 with a correlation matrix of the form C_{b}=[C_{1},0_{2×2},0_{2×2};0_{2×2},C_{2},0_{2×2};0_{2×2},0_{2×2},C_{3}], where \([{\mathbf {C}_{1}} = {\mathbf {C}_{2}} = {\mathbf {C}_{3}} = \left [ {1,\sqrt {0.4} ;} {\sqrt {0.4},1} \right ]\) and C_{1}=C_{2}=C_{3}=[1,0;0,1] for correlated and independent cases. The performance improves when the channels are independent. The effect of the correlation matrix is also seen in the rate of convergence of the bounds, i.e., the higher the determinant of C_{b} (signifying greater independence between channels), the faster the rate of convergence.
Our general results can be formulated into the special case of the Nakagamim fading model, with μ=0, the results of which are shown in [17]. The relationship between the power correlation coefficients and associated Gaussian correlation coefficients when μ=0 is obtained in Section 3. We also compare the bound on the pdf of the channel coefficient obtained in (20) when μ=0 to the result obtained in [17], such that (20) becomes
This leads to the following insights.
The asymptotic approximation and the upper bound of the outage probabilities and error rates are the same for the combining schemes under consideration, which agree with the conclusions in [17].
The results here differ from the results in [17] in that this analysis for the new fading channel include multiple LOS and nonLOS components and can be suitable for environments without LOS components when μ=0.
The expressions for the pdf bounds in (83) are the same as the pdf bound obtained in [17].
Our results bring about meaningful comparison between the BX and abovementioned fading models.
Conclusion
In this work, we derived asymptotically tight lower and upper bounds for the OP and BER of correlated BX fading channels. The effects of certain channel characteristics (e.g., presence of LOS components, diversity combining techniques, and channel correlation) were observed on the performance of the communication system. The special case of our results in the absence of LOS components agrees with those obtained for the Nakagamim fading model in [17], which confirms the accuracy and reproducibility of our work. Ultimately, our results show the accuracy of the BX fading model in characterizing wireless communication environments with both LOS and nonLOS components.
Availability of data and materials
The paper is selfcontained as we provided all mathematical derivations and framework in Sections 3–7. We have also described in details numerical results and parameter settings (values) in Section 8 for ease of result reproduction.
Abbreviations
 BER:

Biterror rate
 BPSK:

Binary phase shift keying
 EGC:

Equalgain combining
 LOS:

Lineofsight
 LB:

Lower bound
 LE:

Lower bound of EGC
 LS:

Lower bound of SC
 LM:

Lower bound of MRC
 MRC:

Maximal ratio combining
 OP:

Outage probability
 RV:

Random variable
 SC:

Selection combining
 UB:

Upper bound
 UE:

Upper bound of EGC
 US:

Upper bound of SC
 UM:

Upper bound of MRC
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Acknowledgements
We thank Dr. Norman Beaulieu for his careful proofreading of an earlier version of the paper.
Funding
This work was supported by the Natural Sciences Engineering Research Council of Canada.
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AO derived the equations and performed the system simulations. JC and JFH revised the equations and contributed to the writing of the manuscript. JC proposed the main idea and the authors read and approved the final manuscript.
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Olutayo, A., Cheng, J. & Holzman, J.F. Performance bounds for diversity receptions over a new fading model with arbitrary branch correlation. J Wireless Com Network 2020, 97 (2020). https://doi.org/10.1186/s13638020017055
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Keywords
 6G
 Bounds
 Correlation
 Diversity
 Fading channels
 Outage probability
 Level crossing rate
 mmWave
 Random access channels
 THz