A New Asymptotic Analysis Technique for Diversity Receptions Over Correlated Lognormal Fading Channels

# A New Asymptotic Analysis Technique for Diversity Receptions Over Correlated Lognormal Fading Channels

## Abstract

Prior asymptotic performance analyses are based on the series expansion of the moment-generating function (MGF) or the probability density function (PDF) of channel coefficients. However, these techniques fail for lognormal fading channels because the Taylor series of the PDF of a lognormal random variable is zero at the origin and the MGF does not have an explicit form. Although lognormal fading model has been widely applied in wireless communications and free-space optical communications, few analytical tools are available to provide elegant performance expressions for correlated lognormal channels. In this work, we propose a novel framework to analyze the asymptotic outage probabilities of selection combining (SC), equal-gain combining (EGC) and maximum-ratio combining (MRC) over equally correlated lognormal fading channels. Based on these closed-form results, we reveal the followings: i) the outage probability of EGC or MRC becomes an infinitely small quantity compared to that of SC at large signal-to-noise ratio (SNR); ii) channel correlation can result in an infinite performance loss at large SNR. More importantly, the analyses reveal insights into the long-standing problem of performance analyses over correlated lognormal channels at high SNR, and circumvent the time-consuming Monte Carlo simulation and numerical integration.

{IEEEkeywords}

Communication channels, correlation, diversity methods, lognormal distributions.

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## 1 Introduction

Diversity reception systems combine signals suffering different channel fading in order to obtain a more reliable output signal[1]. The simplest diversity reception scheme is selection combining (SC) which selects the channel with the highest signal-to-noise ratio (SNR). Maximum-ratio combining (MRC) is the optimal linear diversity reception technique that combines all of the channels with the optimal weights, but such operation requires phase and fading amplitude information of the channels. Equal-gain combining (EGC) combines the channels with equal weights, and it usually provides performance close to MRC without requiring the amplitude information.

Exact performance analyses of diversity receptions over Rayleigh, Rician and Nakagami- fading channels are relatively straightforward. Closed-form or single-fold-integral outage probability expressions have been derived for independent channels [2, Chaps. 6, 9][1, Chap. 7]. Existing works have also provided closed-form or single-fold-integral expressions for error rates [3, 4, 5, 6] and outage probabilities [7, 6, 8] for diversity receptions over correlated channels. In contrast, performance analysis of diversity receptions over correlated lognormal fading channels is much more challenging, which leads to -fold nested integrals [9] where is the number of links, and these integrals are troublesome to be estimated using numerical methods. The time complexity of numerical integration increases exponentially with the number of channels, thus it is not practical to perform numerical integration when the number of receptions becomes large. Even for the dual-branch cases, the exact outage probability expressions of SC over correlated lognormal fading channels can only be simplified to a single-fold integral [10].

To circumvent the difficulty of the exact performance analyses and numerical estimation, various approximation techniques have been proposed for the diversity systems over lognormal fading channels. In [11, 12, 13, 14, 15], the authors approximated the probability density function (PDF) of sum of lognormal random variables (RVs) using another lognormal RV by matching their moments, and these techniques were widely applied in subsequent studies due to its simplicity [16, 17, 18]. In [19, 20, 21], the authors applied the Gaussian-Hermite integration technique to numerically estimate the outage probability of MRC over lognormal fading channels. In [22, 23, 24], the authors approximated the cumulative distribution function (CDF) of sum of lognormal in a transformed domain. In [25], various ways of approximating the sum of lognormal RVs are compared. However, all of the aforementioned approximation methods cannot provide reliable estimation in high SNR region. Bounds on the CDF of sum of lognormal RVs were studied in [26, 10], but these bounds cannot provide accurate outage probability estimation at large SNR either. Some works rely on more complicated random variables to approximate the sum of lognormal RVs, and determine the associated parameters using numerical methods [27, 28, 29]. However, these approximation techniques suffer larger time complexity and reveal few insights.

Asymptotic analysis is a kind of approximations that can provide accurate performance estimation in large SNR region. Over Rayleigh, Rician, Nakagami- and most other fading channels, closed-form asymptotic error rate and outage probability expressions of MRC, EGC and SC have been obtained with arbitrary correlation structure [30, 31, 32, 33]. Asymptotic outage probability expressions have also been derived for free-space optical (FSO) communications following Gamma-Gamma fading channels [34, 35]. Unfortunately, the classical asymptotic analysis techniques in [30, 31, 32, 33, 34, 35] fail to provide meaningful result due to the followings:

• Moment-generating function (MGF) of lognormal PDF does not have a unified explicit expression[10]. Therefore, all methods based on MGF fail to work with the lognormal channels.

• The Taylor series of the PDF of a lognormal RV is zero at the origin, and this results in an infinite diversity order [36]. This implies that the asymptotic outage probability is zero, which is a meaningless result because it cannot quantify the performance gap between two systems with different branch number and correlation status.

• CDF of a sum of lognormal RVs does not have a closed-form expression and it is challenging to be accurately approximated at the origin.

Due to the above difficulties, few works studied the asymptotic outage probabilities of diversity systems over lognormal fading channels. For EGC and MRC, the problem is equivalent to the CDF left tail approximation of sum of lognormal RVs, and for SC the problem can be reduced to the asymptotic approximation of multi-variate lognormal CDF. In [37], the authors used the CDF tail of another lognormal RV to approximate the CDF tail of sum of independent lognormal RVs, but it was subsequently proved in [38] that “any lognormal, reciprocal Gamma or log shifted Gamma cannot be used to fit the left tail, under the independence hypothesis”. The authors in [38] derived the approximation of the left tail of the PDF of sum of two correlated lognormal RVs, but asymptotic CDF expression was not derived. In [39], the approximate CDF of the sum of lognormal RVs was transformed into a quadratic optimization problem, which relies on recursive algorithms. For lognormal fading channels, asymptotic performance expression is both theoretically and practically important. This is because the diversity order of lognormal fading channels is infinite [36], which results in a dramatic decrease in outage probability as SNR increases, thus it can be unacceptably time-consuming to estimate the performance of diversity systems using Monte Carlo simulation in large SNR region because it requires many channel coefficient samples to reliably estimate the ultra-low outage probability ().

Since it is challenging to perform exact analyses, accurate approximations and asymptotic analyses for correlated lognormal channels, much fewer insights have been revealed for FSO multiple-input multiple-output (MIMO) links suffering weak turbulence-induced fading [40] and wireless MIMO links suffering slow fading [10, 9], which hampers the system design.

In this work, a new theorem is developed to simplify the asymptotic analyses of lognormal fading channels. Based on this theorem, we derive closed-form asymptotic outage probabilities of SC, EGC and MRC over equally correlated lognormal fading channels. For SC, the derived asymptotic outage probability is expressed using elementary functions. For EGC and MRC, the derived asymptotic outage probabilities are expressed using Marcum- functions. Two properties of lognormal fading channels are revealed: i) the outage probability of EGC or MRC is an infinitely small quantity compared to the outage probability of SC as SNR approaches infinity where the channel correlation coefficients are fixed. ii) Channel correlation will induce infinite SNR loss at high SNR. Both properties are in sharp contrast with the other fading channels (Rayleigh, Rician, Nakagami-, Gamma-Gamma, etc.). Compared to the methods in prior works [38, 39], the derivation in this work is much simpler and has an elegant geometrical interpretation. Numerical results show that the proposed asymptotic expressions are highly accurate in medium to high SNR region. More importantly, new insights into the long-standing problem are revealed, and one can efficiently evaluate the performance of a diversity system over lognormal fading channels without resorting to the expensive Monte Carlo simulation or numerical integration.

The remainder of this paper is organized as follows. Section 2 introduces the system model including the channel model, the correlation model and the outage probability of diversity receptions. In Section 3, we propose a key theorem of the integrals of joint Gaussian PDF, and the theorem is applied in Section 4 to derive the asymptotic outage probability expressions for SC. The asymptotic outage probabilities of EGC and MRC are derived in Section 5. In Section 6, we discuss the essential differences between lognormal fading channels and the other fading channels, and compare the outage probabilities of SC, EGC and MRC. Numerical results are presented in Section 7, and Section 8 draws some conclusions.

## 2 System Model

### 2.1 Correlated Lognormal Fading Channels

We assume that the channel coefficients of the links are , where is a correlated Gaussian random vector, whose elements have identical mean and variance , and denotes the transpose operation. The equality between statistics of links is valid in most FSO and wireless MIMO links. is known as the “dB spread” in mobile radio environment and is proportional to the Rytov variance in FSO communications. The received signal vector is

 y=cx+n (1)

where is a real-value signal; n is an Gaussian random vector denoting the additive white Gaussian noise and we assume without loss of generality, where is an identity matrix, and denotes the expectation.

After photoelectric conversion, the average received electrical power of the channels can be calculated as . The correlated Gaussian RVs ’s are generated by independent Gaussian RVs ’s with the following relationship

 ⎧⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪⎩G1=aX1+X2+⋯+XLG2=X1+aX2+⋯+XL⋮GL=X1+X2+⋯+aXL (2)

where , and is a parameter determining the correlation coefficients of ’s. According to (2), the parameters between ’s and ’s must have the relationship and .

According to (2), the correlation coefficient between and () can be calculated as

 ρ=E[(Gn−μG)(Gm−μG)]σ2G=2a+L−2a2+L−1. (3)

When we have which corresponds to the independent channels, and when we have which implies that the channels are identical. There must exist at least one real solution of given a fixed as

 a=1+√1−ρ(ρ(L−1)−L+2)ρ (4)

because the discriminant of (3) is for .

### 2.2 Diversity Receptions Over The Correlated Lognormal Fading Channels

SC selects the channel with the highest fading amplitude, and the instantaneous output SNR can be expressed as

 γSC=maxl{e2Gl}. (5)

Outage occurs when falls below a predetermined threshold . The outage probability of SC can be calculated as

 PSCout(γth)=Pr{maxl{exp(Gl)}≤√γth}=Pr⎧⎨⎩maxl⎧⎨⎩exp⎛⎝aXl+L∑k=1,k≠lXk⎞⎠⎫⎬⎭≤√γth⎫⎬⎭. (6)

EGC combines the branches with equal weights, and the output SNR is

 γEGC=1L(L∑l=1eGl)2 (7)

and the outage probability of EGC is

 PEGCout(γth)=Pr(L∑l=1exp(Gl)≤√Lγth)=Pr⎛⎝L∑l=1exp⎛⎝aXl+L∑k=1,k≠lXk⎞⎠≤√Lγth⎞⎠. (8)

MRC combines the branches with the optimal weights, and the output SNR is

 γMRC=L∑l=1e2Gl (9)

and the outage probability of MRC is

 PMRCout(γth)=Pr(L∑l=1exp(2Gl)≤γth)=Pr⎛⎝L∑l=1exp⎛⎝2⎛⎝aXl+L∑k=1,k≠lXk⎞⎠⎞⎠≤γth⎞⎠. (10)

## 3 A Useful Theorem For The Asymptotic Analysis

Lemma: For the joint PDF of independent Gaussian RVs

 fx,iid(x)=1√(2π)Lσ2Lexp(−12σ2∣∣x−μ∣∣2) (11)

where , and , and is the variance, and denotes the 2-norm of a vector, the following equation holds

 lim|μ|→∞∫Ω1(μ,x0)fx,iid(x)dx∫ω2(x0)fx,iid(x)dx=0 (12)

where and

 Ω1(μ,x0)Δ={x∣∣∣∣x−μ∣∣>∣∣x0−μ∣∣+√Lε+ε} (13)

where and

 ω2(x0)Δ={x∣∣|xl−xl,0|≤ε,∀l=1,⋯,L} (14)

where is an arbitrarily small positive constant.

Proof: see Appendix 9.

The lemma in (12) essentially states that if the multi-variate independent Gaussian PDF is the integrand, the integral in an arbitrarily small hypercube is a high order infinitely large quantity compared to the integral outside the hyperspherical region centered at with radius . Figure 1 illustrates and on a two-dimensional plane.

Theorem: For -variate independent Gaussian PDF in (11), the following equation holds

 lim|μ|→∞∫ω1(μ,x0)fx,iid(x)dx∫Ω2(x0)fx,iid(x)dx=0 (15)

where is any region with non-zero volume1 contained by

 ¯Θ(x0,μ)Δ={x∣∣∣∣x−μ∣∣>∣∣x0−μ∣∣} (16)

and is any region with non-zero volume contained by

 Θ(x0,μ)Δ={x∣∣∣∣x−μ∣∣<∣∣x0−μ∣∣}. (17)

Proof: see Appendix 10.

The theorem in (15) says that the integral over any region within the hypersphere is always an infinitely large quantity compared to the integral over any region outside when . Figure 2 illustrates the relationship between the involved regions in (15) on a two-dimensional plane. As a result, to obtain the asymptotic value of an integral with Gaussian integrand, it is valid to approximate the original integral region with its arbitrary subset as long as the approximating region keeps the dominant term. As an intuitive example, Fig. 3 presents in a region on a two-dimensional plane with various mean vectors, where . It can be observed that the region containing dominant PDF values becomes smaller and smaller as grows.

## 4 Asymptotic Outage Probability of SC Over Correlated Channels

Based on (6) and (11), by letting and , we obtain

 PSCout(γth)=∫maxl{exp(axl+∑k≠lxk)}<√γthfx,iid(x)dx=∫axl+∑k≠lxk

and the integral region can be denoted as

 ΦSC(x)Δ=maxl⎧⎨⎩exp⎛⎝axl+∑k≠lxk⎞⎠⎫⎬⎭−√γth≤0. (19)

Unfortunately, it is challenging to simplify (18) further due to the nested integral region. However, it can be implied by the theorem in (15) that it is valid to approximate the integral region in (18) with its subset as long as the subset contains the dominant term. A necessary condition for the approximating subset to contain the dominant term is that it must contain a continuous set that contains or touches the nearest point to in , which can be proved by the theorem in (15) using the method of contradiction. Therefore, we hope to find a subset of the integral region in (18) that satisfies the followings: i) contains the nearest point to ; ii) can be arbitrarily small so that we can use Taylor series to simplify the Gaussian integrand; iii) results in closed-form expression of the approximate integral.

According to the theorem in (15), eq. (18) has an asymptotic expression as

 PSCout(γth)=∫axl+∑k≠lxk

as long as the dominant term has a nonempty integral region where denotes an infinitely small quantities compared to the other summed terms when . It is proved in Appendix 11 that the nearest point to inside the integral region in (18) is and . We let the vector in (20) where , which ensures that the integral region of the dominant term in (20) contains and has nonzero volume, and we comment that the volume can be set arbitrarily small by adjusting , which is shown in Fig. 4. It is proved in Appendix 12 that the integral region in (20) is contained by another region, i.e.

 ⎧⎨⎩x∣∣ ∣∣axl+L∑k=1,k≠lxk

when . Therefore, by replacing the integral region of the dominant term in (20) with the larger region in (21), we obtain

 PSCout(γth)=∫ln√γth−δ

where can be arbitrarily small when is sufficiently small. Note that the integral regions in (20) and (22) are both subsets of the integral region in (18). Figure 4 illustrates the relationship of the two regions in (21) on a two-dimensional plane.

In any small neighbourhood of , the integrand in (22) can be well approximated as

 Missing or unrecognized delimiter for \left (23)

when . Substituting (23) into (22), we obtain

 PSCout(γth)=1√(2π)Lσ2LXexp(−L2σ2X(xSCnst−μX)2)×∫ln√γth−δ≤axl+∑k≠lxk≤ln√γth,∀l=1,⋯,Lexp(−xSCnst−μXσ2XL∑l=1(xl−xSCnst)+o(⋯))dx (24)

when . Then we change the integrating variables ’s to ’s in (24) following the mapping rule

 g=Ax (25)

where , and

 A=⎡⎢ ⎢ ⎢ ⎢ ⎢⎣a1111a1111⋱⋮11⋯a⎤⎥ ⎥ ⎥ ⎥ ⎥⎦. (26)

Based on (25), the integral in (24) can be simplified as

 Unknown environment '%' (27)

when , where is the Jacobian determinant. Noting that the integrand and integral region are symmetrical for all ’s, the integral in (27) can be further simplified to

 ∫ln√γth−δ≤gl≤ln√γth,∀l≤Lexp(−xSCnst−μXσ2X(a+L−1)L∑l=1gl)dg=⎡⎢ ⎢⎣ln√γth∫ln√γth−δexp(−xSCnst−μXσ2X(a+L−1)g1)dg1⎤⎥ ⎥⎦L=(−σ2X(a+L−1)xSCnst−μX)L×[exp(−xSCnst−μX2σ2X(a+L−1)lnγth)−exp(−xSCnst−μXσ2X(a+L−1)(lnγth2−δ))]L≈(−σ2X(a+L−1)xSCnst−μXexp(−xSCnst−μX2σ2X(a+L−1)lnγth))L (28)

when , where the last approximation is obtained by discarding the higher order term involving . Substituting (28) into (27), we can simplify (24) and obtain

 PSCout(γth)≈1|A|√(2π)Lσ2LX⎛⎜ ⎜⎝σ2X(a+L−1)μX−lnγth2(a+L−1)⎞⎟ ⎟⎠Lexp⎛⎝−L2σ2X(lnγth2(a+L−1)−μX)2⎞⎠ (29)

when . Noting that and , we can also express (29) using the standard deviation and the transmit power as2

 PSCout(γth)≈1|A|√(2π)L(σ2Ga2+L−1)L2⎛⎜ ⎜⎝(a+L−1)2ln√¯Er/γth−σ2G⎞⎟ ⎟⎠L×exp⎛⎜ ⎜ ⎜⎝−L(a2+L−1)2σ2G⎛⎜ ⎜⎝ln√¯Er/γth−σ2Ga+L−1⎞⎟ ⎟⎠2⎞⎟ ⎟ ⎟⎠ (30)

when .

When , the correlation coefficient according to (3), and the lognormal channels become independent. In such case, eq. (30) specializes to

 PSC,iout(γth)≈σLG(ln√¯Er/γth−σ2G)L√(2π)Lexp(−L2σ2G(ln√¯Er/γth−σ2G)2). (31)

We can also use another simpler approach to obtain (31) as a verification to the proposed analysis in the case of independent lognormal channels. The exact CDF of a lognormal RV is known as

 Fc(x)=1−Q(lnx−μcσc)=Q(μc−lnxσc) (32)

where and are the mean and variance of the associated Gaussian RV. Based on (6) and (32), the exact outage probability of SC over the independent lognormal channels is

 PSC,iout(γth)=QL(μG−ln√γthσG)=QL⎛⎜ ⎜⎝ln√¯Er/γth−σ2GσG⎞⎟ ⎟⎠. (33)

Based on the well-known asymptotic approximation of Gaussian -function[41, eq. (4)]

 Q(x)≈1√2πxexp(−x22) (34)

when , eq. (33) can be approximated as

 PSC,iout(γth)≈σLG(ln√¯Er/γth−σ2G)L√(2π)Lexp(−L2σ2G(ln√¯Er/γth−σ2G)2) (35)

which agrees with (31).

## 5 Asymptotic Outage Probabilities of EGC and MRC Over Correlated Channels

Based on (8), we can express the outage probability of EGC over equally correlated channels as

 PEGCout(γth)=∫L∑l=1exp(axl+L∑k=1,k≠lxk)≤√Lγthfx,iid(x)dx (36)

where we assume . The integral region in (36) can be expressed as

 ΦEGC(x)Δ=L∑l=1exp⎛⎝axl+L∑k=1,k≠lxk⎞⎠−√Lγth≤0. (37)

According to the theorem in (15), we obtain

 ∫ΦEGC(x)≤0fx,iid(x)dx=∫ΦEGC(% x)≤0,∣∣x−μX∣∣<∣∣xEGCnst−ε−μX∣∣fx,iid(x)dx+o(⋯) (38)

for where and is the nearest point to in the region . Applying the Karush-Kuhn-Tucker (KKT) conditions, it can be shown that , and the procedures are similar to those in Appendix 11. The integral region in (37) can be approximated by the following hyperspherical region

 ~ΦEGC(x)Δ=L∑l=1(xl−(1a+L−1ln(√γthL)−(L−1+a)(1−a)2))2−((L−1+a)√L(1−a)2)2≤0 (39)

where the approximation is valid because if we take as a function of , for hypersurface and , the first-order partial derivatives at are identical, and so do the second-order partial derivatives . The proof of the equalities of derivatives is in Appendix 13. This implies that the two hypersurfaces and has arbitrarily small difference in a sufficiently small neighbourhood of . One can also regard the approximation as a generalization of the curve approximation technique using a circle that has the same curvature. For , the reciprocal of the radius of (39), i.e. , is the curvature of the curve at .

Therefore, according to the theorem in (15), eq. (38) can be approximated as

 ∫ΦEGC(x)≤0fx,iid(x)dx≈∫~ΦEGC(x)≤0,∣∣x−μX∣∣<∣∣xEGCnst−ε−μX∣∣fx,iid(x)dx≈∫~ΦEGC(x)≤0fx,iid(x)dx. (40)

when . An intuitive example for the approximation in (40) is shown Fig. 3, where the integral region is for . Since the dominant term of the integrand is condensed into a small region, we only need to accurately approximate the integral boundary near the dominant term. Based on (39), the last integral in (40) can be expressed as

 Missing or unrecognized delimiter for \right (41)

where . follows the th-order noncentral chi-squared distribution whose CDF can be expressed as Marcum- function defined as where is the modified Bessel function of order , thus we can simplify (41) as

 PEGCout(γth)≈1−QL2⎛⎜ ⎜ ⎜ ⎜⎝√L⎛⎜ ⎜ ⎜ ⎜⎝μX−(1a+L−1ln(√γthL)−L−1+a(1−a)2)σX⎞⎟ ⎟ ⎟ ⎟⎠,(L−1+a)√L(1−a)2σX⎞⎟ ⎟ ⎟ ⎟⎠. (42)

Equation (42) can also be expressed as a function of the average received power and the standard deviation as

 PEGCout(γth)≈1−QL2⎛⎜ ⎜ ⎜ ⎜ ⎜⎝√L⎛⎜ ⎜ ⎜ ⎜ ⎜⎝ln√L¯Er/γth−σ2Ga+L−1+L−1+a(1−a)2σG√a2+L−1⎞⎟ ⎟ ⎟ ⎟ ⎟⎠,(L−1+a)√L(1−a)2σG√a2+L−1⎞⎟ ⎟ ⎟ ⎟ ⎟⎠. (43)

When , we obtain according to (3), thus the channels become independent and (43) becomes

 PEGC,iout(γth)≈1−QL2(√LσG(ln√L¯Er/γth−σ2G+1),√LσG) (44)

which agrees with the prior result in [42, eq. (14)].

According to (10), the associated integral region for MRC can be expressed as

 ΦMRC(x)Δ=