# BEP/SEP and Outage Performance Analysis ofL-Branch Maximal-Ratio Combiner for κ‐μ Fading

Authors: Mirza Milišić; Mirza Hamza; Mesud Hadžialić
Journal: International Journal of Digital Multimedia Broadcasting
 (2009)

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## Abstract

Maximal-ratio combiner (MRC) performances in fading channels have been of
interest for a long time, which can be seen by a number of papers concerning this topic.
In this paper we treat bit error probability (BEP), symbol error probability (SEP) and outage
probability of MRC in presence ofκ‐μ fading.
We will present κ‐μ fading model, probability density function (PDF), and cumulative distribution function (CDF). We will also present PDF, CDF, and
outage probability of the L-branch MRC output. BEP/SEP will be evaluated for broad class of modulation types and for coherent and noncoherent types of detection. BEP/SEP and outage performances of the MRC will be evaluated for different number of branches via Monte Carlo simulations and theoretical expressions.


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## Body

## 1. Introduction

MRC
performances in fading channels have been of
interest for a long time, which can be seen by a numerous published papers
concerning this topic. Most of these papers are concerned by Rayleigh,
Nakagami-m, Hoyt (Nakagami-q), Rice (Nakagami-n), and Weibull fading [1–5]. Beside MRC, performances
of selection combining, equal-gain combining, hybrid combining, and switched
combining in fading channels have also been studied. Most of the papers
treating diversity combining have examined only dual-branch combining because
of the inability to obtain closed-form expressions for evaluated parameters of
diversity system. Scenarios of correlated fading in combiner branches have
also been examined in numerous papers. Nevertheless, depending on system used
and combiner implementation, one must take care of resources available at the
receiver, such as: space, frequency, and complexity. Moreover, fading
statistic does not necessary have to be the same in each branch, for example, PDF can be
the same, but with different parameters (Nakagami-m fading in ith and jth branches, with mi≠mj), or PDFs in different branches are
different (Nakagami-m fading in ith branch, and Rice fading in jth branch). This paper treats MRC
outage performances in presence of κ‐μ fading [6, 7]. This type of fading has
been chosen because it includes, as special cases, Nakagami-m and Nakagami-n (Rice) fading, and their entire special cases as well (e.g., Rayleigh and one-sided Gaussian
fading). It will be shown that the sum of κ‐μ squares is κ‐μ square as well (but with different
parameters), which is an ideal choice for MRC analysis. Concerning this, in
this paper, we will present model for κ‐μ distribution  and closed form expressions for
outage probability, BEP and SEP at the MRC output will be derived for a broad
class of modulation types. Based upon generic expressions for BEP/SEP for
coherent and noncoherent detection, BEP/SEP will be evaluated in further
analysis. Outage and BEP/SEP performances will be presented for L-branch
combining via Monte Carlo simulations and theoretical expressions. This paper
is organized as follows. In Section 2, we review physical model of the
distribution. In Section 3, we examine κ‐μ MRC, and we show that the sum of κ‐μ squares is κ‐μ square. Throughout Section 4 we analyze
BEP/SEP for κ‐μ MRC based on generic expressions for BEP/SEP
for coherent and noncoherent detection types for various modulation
techniques. Discussion and simulation results are presented in Section 5, where
some conclusions have been drawn.


## 2. Physical Model of theκ‐μ Distribution

Physical model and derivation
of theκ‐μ distribution is described in [7].
Nevertheless, for the purpose of integrity of this paper and apprehension of
generality of this model (as well as its applications to the MRC), it is
necessary to revise the basics of the κ‐μ distribution physical model. The fading model
for the κ‐μ distribution considers a signal composed of clusters of multipath waves,
propagating in a nonhomogeneous environment. Within single cluster, the phases
of the scattered waves are random and have similar delay times, with delay-time
spreads of different clusters being relatively large. It is assumed that the
clusters of multipath waves have scattered waves with identical powers, and
that each cluster has a dominant component with arbitrary power. This
distribution is well suited for line-of-sight (LoS) applications, since every
cluster of multipath waves has a dominant component (with arbitrary power). In
special case, if we set all dominant components to zero, then this distribution
can very well describe nonline-of-sight (NLoS) scenarios. Given the physical
model for the κ‐μ distribution, envelope R and
instantaneous power γ, can be written
in terms of the inphase and quadrature components of the fading signal as(1)R2=γ=∑i=1n(Xi+pi)2+∑i=1n(Yi+qi)2, where Xi  and Yi are mutually independent Gaussian processes with Xi¯=Yi¯=0 and Xi2¯=Yi2¯=σ2. pi and qi  are, respectively, the mean
values of the inphase and quadrature components of the multipath waves of
cluster i, and n is the number of clusters of multipath.By performing random
variables (RVs) transformation, in accordance to [7, Section 2.2], we obtain
the instantaneous power PDF of the κ‐μ RV: (2)fγ(γ)=12σ2⋅(γd2)(n−1)/2×exp(−γ+d22σ2)⋅In−1(dγσ2),  where d2=∑i=1ndi2.
It can be seen that(3)Ω=R2¯=γ¯=2nσ2+d2,R4¯=γ2¯=4nσ4+4σ2d2+(2nσ2+d2)2.Therefore, (4)Var(R2)=Var(γ)=4nσ4+4σ2d2.Parameterκ is defined as κ=d2/2nσ2 and represents the ratio between the total
power of the dominant components and the total power of the scattered waves. Although n can be expressed in terms of
continuous physical parameters (mean-squared value of the power, the variance
of the power, and κ), it still has
discrete nature. If these parameters are to be obtained by field measurements,
the value of the parameter n would be
a real number (not an integer). Several reasons exist for this. One of them,
and probably the most meaningful, is that although the model proposed here is
general, it is in fact an approximate solution to the so-called random phase
problem (which has been extensively elaborated in [7]), as are all the other
well-known fading models approximate solutions to the random phase problem. The
limitation of the model can be made less stringent by defining μ to be the real extension of n. Noninteger values of the parameter μ may account for: the non-Gaussian nature of
the inphase and quadrature components of each cluster of the fading signal,
nonzero correlation among the clusters of multipath components, nonzero
correlation between inphase and quadrature components within each cluster,
and so forth. Noninteger values of clusters have been found in practice, and are
extensively reported in literature, for example, [8].Now, using the definitions
for parametersκ and μ,
and the considerations given above, the κ‐μ power PDF can be written from (2) as (5)fγ(γ)=μ(1+κ)(μ+1)/2κ(μ−1)/2⋅exp(μκ)⋅Ω(μ+1)/2⋅γ(μ−1)/2×exp[−μ(1+κ)γΩ]⋅Iμ−1[2μκ(1+κ)γΩ]. From (5), κ‐μ power CDF can be written in closed form
as (6)Fγ(γ)=∫0γfγ(x)dx=1−Qμ[2κμ,2(κ+1)μγΩ],  where(7)Qν(a,b)=1aν−1∫b∞xν⋅exp(−x2+a22)⋅Iν−1(ax)⋅dx  is generalized Marcum Q function [9], as stated in [7].


## 3.κ‐μ  Maximal-Ratio Combiner

There are four principal
types of combining techniques [10] that depend essentially on the complexity
restrictions put on the communication system and amount of channel state
information (CSI) available at the receiver. As shown in [10], in the absence of
interference, MRC is the optimal combining scheme, regardless of fading
statistics, but most complex since MRC requires knowledge of all channel fading
parameters (amplitudes, phases, and time delays). Since knowledge of channel
fading amplitudes is needed for MRC, this scheme can be used in conjunction
with unequal energy signals, such as M-QAM or any other amplitude/phase
modulations. In this paper, we will treat L-branch MRC receiver. As shown in [10]
MRC receiver is the optimal multichannel receiver, regardless of fading
statistics in various diversity branches since it results in an ML receiver. For
equally likely transmitted symbols, the total SNR per symbol at the output of
the MRC is given by [11]  γ=∑j=1Lγj,
where γj is instantaneous SNR in ith branch of L-branch MRC receiver. Repeating the same procedure
as in Section, previous relation can be written in terms of
inphase and quadrature components: (8)γ=∑j=1Lγi=∑j=1LRi2=∑j=1L∑i=1nRi,j2, where Ri,j2 represents total power of the ith cluster manifested in jth branch of the MRC receiver. Using (1) one can obtain (9)γ=∑j=1L∑i=1n(Xi,j+pi,j)2+(Yi,j+qi,j)2. Repeating the same procedure
as in [7, Section 2.2] one can obtain Laplace transform of the PDF of the RV γ (SNR): (10)L{fγ(γ)}=∏j=1LL{fγj(γj)}=exp(−sd2/(1+2sσ2))(1+2sσ2)L⋅n, where d2=∑j=1Ldj2.
Inverse Laplace transform of (21) yields to PDF of the RV γ:(11)fγ(γ)=Lμ(1+κ)(Lμ+1)/2κ(Lμ−1)/2⋅exp(Lμκ)⋅(LΩ)(Lμ+1)/2⋅γ(Lμ−1)/2×exp[−μ(1+κ)γΩ]⋅Iμ−1[2μLκ(1+κ)γΩ]. Note, that sum of L squares of the κ‐μ distributions is κ‐μ distribution with different parameters, which
means SNR at the output of the MRC receiver subdue to the κ‐μ distribution with parameters (12)μMRC=L⋅μ,κMRC=κ,ΩMRC=L⋅Ω. Now, it is easy to obtain CDF (13)Fγ(γ)=∫0γfγ(x)dx=1−QLμ[2Lκμ,2(κ+1)μγΩ]. For fixed threshold, γth,
outage probability is given by (14)Pout(γth)=Fγ(γth)=∫0γthfγ(x)dx=1−QLμ[2Lκμ,2(κ+1)μγthΩ].


## 4. SEP forκ‐μ
Maximal-Ratio Combiner

When
we analyze SEP, we must focus upon single modulation format because different
modulations result in different SEPs. We must also consider type of detection
(coherent or noncoherent). Although coherent detection results in smaller SEP
than corresponding noncoherent detection for the same SNR, sometimes it is
suitable to perform noncoherent detection depending on receiver structure
complexity.


### 4.1. Noncoherent Detection

To obtain average SEP at MRC
output forκ‐μ fading for noncoherent detection, we will use
generic expression for instantaneous SEP: SEP=a⋅exp(−b⋅γ),
where γ represents instantaneous SNR at MRC output for κ‐μ fading, and nonnegative parameters a and b depend on used modulation format (see Table 1).Table 1
Values ofa and  b  for some noncoherent
modulations.
ab0.510.5BFSKDBPSK1——(M−1)/2MFSK—Average SEP can be obtained
from(15)SEP¯=∫0+∞a⋅exp(−b⋅γ)⋅fγ(γ)⋅dγ=∫0+∞a⋅Lμ(1+κ)(Lμ+1)/2κ(Lμ−1)/2⋅exp(Lμκ)⋅(LΩ)(Lμ+1)/2⋅γ(Lμ−1)/2×exp[−γ(b+μ(1+κ)Ω)]⋅ILμ−1[2μLκ(1+κ)γΩ]⋅dγ. Using [9,equation  (5), page 318] we
obtain closed-form expression for average SEP for noncoherent detection: (16)SEP¯=a⋅[μ(1+κ)bΩ+μ(1+κ)⋅exp(−bκΩbΩ+μ(1+κ))]Lμ.


### 4.2. Coherent Detection

To obtain average SEP at MRC
output forκ‐μ fading for coherent detection, we will use
generic expression for instantaneous SEP: SEP=a⋅Q(b⋅γ),
where γ represents instantaneous SNR at MRC output for κ‐μ fading, Q(⋅) function is defined as (17)Q(x)=12π∫x+∞exp(−t22)⋅dt and nonnegative parameters a and b depend on used modulation format (see Table 2).Table 2
Values ofa and b for some coherent modulations.
ab122sin2(π/M)3/(M−1)1BFSKBPSK——2QPSKDBPSKMPSK—4((M−1)/M)———Rect.
QAMAverage SEP can be obtained
from(18)SEP¯=∫0+∞a⋅Q(b⋅γ)⋅fγ(γ)⋅dγ. Nevertheless, it is
impossible to find closed-form solution for (18). Because of that we have to
find adequate approximation of the Q function. Knowing the continued fraction representation of the Q function [12, equation (06.27.10.0001.01)],
and adopting the first-order approximation: (19)Q(x)≈1x2π⋅exp(−x22), equation (18) now becomes(20)SEP¯≈∫0+∞a2πb⋅γ⋅exp(−b⋅γ2)⋅fγ(γ)⋅dγ=∫0+∞a2πb⋅Lμ(1+κ)(Lμ+1)/2κ(Lμ−1)/2⋅exp(Lμκ)⋅(LΩ)(Lμ+1)/2⋅γ(Lμ−2)/2×exp[−γ(b2+μ(1+κ)Ω)]⋅ILμ−1[2μLκ(1+κ)γΩ]⋅dγ. Using [9,equation (5), page 318] we
obtain closed-form expression for average SEP for coherent detection: (21)SEP¯≈a⋅Γ(Lμ−0.5)2πb⋅Γ(Lμ)⋅(μ(1+κ)exp(κ)Ω)Lμ×(b2+μ(1+κ)Ω)0.5−Lμ×F11(Lμ−0.5;Lμ;μ2Lκ(1+κ)bΩ/2+μ(1+κ)),
where F11(⋅;⋅;⋅) is the Kummer confluent hypergeometric
function defined in [12, equation (07.20.02.0001.01)].


## 4.1. Noncoherent Detection

To obtain average SEP at MRC
output forκ‐μ fading for noncoherent detection, we will use
generic expression for instantaneous SEP: SEP=a⋅exp(−b⋅γ),
where γ represents instantaneous SNR at MRC output for κ‐μ fading, and nonnegative parameters a and b depend on used modulation format (see Table 1).Table 1
Values ofa and  b  for some noncoherent
modulations.
ab0.510.5BFSKDBPSK1——(M−1)/2MFSK—Average SEP can be obtained
from(15)SEP¯=∫0+∞a⋅exp(−b⋅γ)⋅fγ(γ)⋅dγ=∫0+∞a⋅Lμ(1+κ)(Lμ+1)/2κ(Lμ−1)/2⋅exp(Lμκ)⋅(LΩ)(Lμ+1)/2⋅γ(Lμ−1)/2×exp[−γ(b+μ(1+κ)Ω)]⋅ILμ−1[2μLκ(1+κ)γΩ]⋅dγ. Using [9,equation  (5), page 318] we
obtain closed-form expression for average SEP for noncoherent detection: (16)SEP¯=a⋅[μ(1+κ)bΩ+μ(1+κ)⋅exp(−bκΩbΩ+μ(1+κ))]Lμ.


## 4.2. Coherent Detection

To obtain average SEP at MRC
output forκ‐μ fading for coherent detection, we will use
generic expression for instantaneous SEP: SEP=a⋅Q(b⋅γ),
where γ represents instantaneous SNR at MRC output for κ‐μ fading, Q(⋅) function is defined as (17)Q(x)=12π∫x+∞exp(−t22)⋅dt and nonnegative parameters a and b depend on used modulation format (see Table 2).Table 2
Values ofa and b for some coherent modulations.
ab122sin2(π/M)3/(M−1)1BFSKBPSK——2QPSKDBPSKMPSK—4((M−1)/M)———Rect.
QAMAverage SEP can be obtained
from(18)SEP¯=∫0+∞a⋅Q(b⋅γ)⋅fγ(γ)⋅dγ. Nevertheless, it is
impossible to find closed-form solution for (18). Because of that we have to
find adequate approximation of the Q function. Knowing the continued fraction representation of the Q function [12, equation (06.27.10.0001.01)],
and adopting the first-order approximation: (19)Q(x)≈1x2π⋅exp(−x22), equation (18) now becomes(20)SEP¯≈∫0+∞a2πb⋅γ⋅exp(−b⋅γ2)⋅fγ(γ)⋅dγ=∫0+∞a2πb⋅Lμ(1+κ)(Lμ+1)/2κ(Lμ−1)/2⋅exp(Lμκ)⋅(LΩ)(Lμ+1)/2⋅γ(Lμ−2)/2×exp[−γ(b2+μ(1+κ)Ω)]⋅ILμ−1[2μLκ(1+κ)γΩ]⋅dγ. Using [9,equation (5), page 318] we
obtain closed-form expression for average SEP for coherent detection: (21)SEP¯≈a⋅Γ(Lμ−0.5)2πb⋅Γ(Lμ)⋅(μ(1+κ)exp(κ)Ω)Lμ×(b2+μ(1+κ)Ω)0.5−Lμ×F11(Lμ−0.5;Lμ;μ2Lκ(1+κ)bΩ/2+μ(1+κ)),
where F11(⋅;⋅;⋅) is the Kummer confluent hypergeometric
function defined in [12, equation (07.20.02.0001.01)].


## 5. Simulations and Discussion of the Results

As
mentioned previously, MRC outage performances will be examined via Monte Carlo
simulations and theoretical expressions (14). Figures 1, 2, 3, 4, 5, 6, 7, and 8 show theoretical and
simulated outage probabilities as functions of threshold level γth. γth ranges from −10 dB to 10 dB. Figures 1–8
clearly show that theoretical expressions used are correct because theoretical
results concur with simulations results extremely well. Figures 1–6 show outage
probability for L=1,2,3,4, κ=0.55,1,2 and μ=1,2.
For fixed values of κ and μ outage probabilities have been compared for
specified numbers of combiners branches, L.Figure 1
Outage probability forκ=0.55 and μ=1.Figure 2
Outage probability forκ=0.55 and μ=2.Figure 3
Outage probability forκ=1 and μ=1.Figure 4
Outage probability forκ=1 and μ=2.Figure 5
Outage probability forκ=2 and μ=1.Figure 6
Outage probability forκ=2 and  μ=2.Figure 7
Outage probability for dual-branch MRC.Figure 8
Outage probability for dual-branch MRC.From
Figures1–6 it can be easily concluded that for fixed values of κ and μ there is not much sense in increasing the
number of branches (in many cases it is not economically or technically
justified). We can also observe that the highest gain is obtained between
curves for L=1 and L=2 (situation with no combining and dual-branch
combining). Distribution parameters also have a significant impact on outage
probability. When κ is increasing, Pout is decreasing. Namely, these results were expected because κ represents ratio between total power of
dominant components and total power of scattered components. Parameter μ represents fading severity parameter. As μ decreases, fading severity increases and so
does outage probability. From Figures 1–6, for fixed κ,
as μ increases so does the slope of the outage
curve. For dual-branch combining (L=2), behavior of Pout,
for different values of parameters κ and μ,
can be observed in Figures 7 and 8. In Figure 7 parameter κ is fixed, and parameter μ changes, and in Figure 8 we have inverse
situation (μ is fixed, and κ changes). We perceive existence of the single
intersection point (point where all curves intersect), and it is determined
with only one parameter (κ or μ) and fixed number of branches L. In that point, outage probability Pout,
and threshold level γth,
are the same for all curves (Figures  7 and 8). This point is also an inflexion
point. If the threshold value is below the threshold value at inflexion point,
channel dynamic is dominant, and if the threshold value is above the threshold
value at inflexion point, receiver sensitivity is dominant. Namely, for smaller κ and μ,
dynamic in channel is larger. If the threshold is set high enough, then it is
logical to have smaller outage probability with larger channel dynamic apart
from the case of smaller channel dynamic. MRC BEP/SEP, for both coherent and
noncoherent detection, will be examined via Monte Carlo simulations and
theoretical expressions (16) and (21) as well. In Figures 9–12 case of
dual-branch combining has been shown because the highest gain is obtained
between outage curves for L=1 and L=2 (situation with no combining and dual-branch
combining). Figures 9–12 show theoretical and simulated average BEP/SEP as  functions of average SNR Ω. Ω ranges from 0 dB to 15 dB. Figures 9–12
clearly show that theoretical expressions used are correct  because theoretical
results concur with simulations results extremely well,  but
certain deviations of  theory from simulation are noticeable in Figures 11 and
12 for a low values of Ω.
This is a consequence of the approximation used for generic expression for coherent
detection (19). Figures 9 and 10 show BEP/SEP for L=1 and L=2, respectively,
for noncoherent detection, and  Figures 11 and 12 show
BEP/SEP for L=1 and L=2, respectively, for coherent detection. By examining
Figures 9 and 10 we notice that if we use dual-branch MRC we will gain 4 dB for
the same BEP/SEP. The same goes for Figures 11 and 12, but we will gain
approximately 7 dB, which is to be expected because there is approximately 3 dB
gain when we use coherent detection instead of noncoherent.Figure 9
BEP/SEP for noncoherent detection, no diversity.Figure 10
BEP/SEP for noncoherent detection, dual-branch diversity.Figure 11
BEP/SEP for coherent detection, no
diversity.Figure 12
BEP/SEP for coherent detection, dual-branch
diversity.Figures13, 14, and 15 show comparison between FSK and PSK for 3-branch combining. For Figures 13–15
various values of κ and μ have been used, for both coherent and
noncoherent detection. As we can observe, theoretical and simulation results
concur very well. We can also observe gain obtained between no combining,
dual-branch combining, and 3-branch combining cases in Figures 16, 17,  and 18. As number
of branches increases, BEP/SEP decreases, as expected.Figure 13
BEP/SEP for noncoherent detection, 3-branch diversity.Figure 14
BEP/SEP for coherent detection, 3-branch diversity.Figure 15
BEP/SEP for noncoherent
detection, 3-branch diversity.Figure 16
BEP/SEP for coherent detection,L = 1,  2, 3.Figure 17
BEP/SEP for coherent detection,L = 1, 2, 3.Figure 18
BEP/SEP for noncoherent detection,L = 1, 2, 3.



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