#### 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 of fading.
We will present 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.

#### 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 [1โ5]. 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 *i*th and *j*th branches, with ), or PDFs in different branches are
different (Nakagami-*m* fading in *i*th branch, and Rice fading in *j*th branch). This paper treats MRC
outage performances in presence of 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 squares is 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 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 MRC, and we show that the sum of squares is square. Throughout Section 4 we analyze
BEP/SEP for 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 the Distribution

Physical model and derivation
of the 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 distribution physical model. The fading model
for the 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 distribution, envelope *R* and
instantaneous power , can be written
in terms of the inphase and quadrature components of the fading signal as where and are mutually independent Gaussian processes with and . and 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 RV: where . It can be seen thatTherefore,

Parameter is defined as 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 ), 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 to be the real extension of *n*. Noninteger values of the parameter 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 parameters and ,
and the considerations given above, the power PDF can be written from (2) as From (5), power CDF can be written in closed form
as where is generalized Marcum *Q* function [9], as stated in [7].

#### 3. 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] ,
where is instantaneous SNR in *i*th 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: where represents total power of the *i*th cluster manifested in *j*th branch of the MRC receiver. Using (1) one can obtain Repeating the same procedure
as in [7, Section 2.2] one can obtain Laplace transform of the PDF of the RV (SNR): where .
Inverse Laplace transform of (21) yields to PDF of the RV : Note, that sum of *L* squares of the distributions is distribution with different parameters, which
means SNR at the output of the MRC receiver subdue to the distribution with parameters Now, it is easy to obtain CDF For fixed threshold, ,
outage probability is given by

#### 4. SEP for 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 for fading for noncoherent detection, we will use generic expression for instantaneous SEP: , where represents instantaneous SNR at MRC output for fading, and nonnegative parameters and depend on used modulation format (see Table 1).

Average SEP can be obtained from Using [9,equation (5), page 318] we obtain closed-form expression for average SEP for noncoherent detection:

##### 4.2. Coherent Detection

To obtain average SEP at MRC
output for fading for coherent detection, we will use
generic expression for instantaneous SEP: ,
where represents instantaneous SNR at MRC output for fading, function is defined as and nonnegative parameters *a* and *b* depend on used modulation format (see Table 2).

Average SEP can be obtained
from 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: equation (18) now becomes Using [9,equation (5), page 318] we
obtain closed-form expression for average SEP for coherent detection:
where 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 . ranges from โ10โdB to 10โdB. Figures 1โ8
clearly show that theoretical expressions used are correct because theoretical
results concur with simulations results extremely well. Figures 1โ6 show outage
probability for , and .
For fixed values of and outage probabilities have been compared for
specified numbers of combiners branches, *L*.

From
Figures 1โ6 it can be easily concluded that for fixed values of and 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 and (situation with no combining and dual-branch
combining). Distribution parameters also have a significant impact on outage
probability. When is increasing, is decreasing. Namely, these results were expected because represents ratio between total power of
dominant components and total power of scattered components. Parameter represents fading severity parameter. As decreases, fading severity increases and so
does outage probability. From Figures 1โ6, for fixed ,
as increases so does the slope of the outage
curve. For dual-branch combining (), behavior of ,
for different values of parameters and ,
can be observed in Figures 7 and 8. In Figure 7 parameter is fixed, and parameter changes, and in Figure 8 we have inverse
situation ( is fixed, and changes). We perceive existence of the single
intersection point (point where all curves intersect), and it is determined
with only one parameter ( or ) and fixed number of branches *L*. In that point, outage probability ,
and threshold level ,
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 and ,
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 9โ12 case of
dual-branch combining has been shown because the highest gain is obtained
between outage curves for and (situation with no combining and dual-branch
combining). Figures 9โ12 show theoretical and simulated average BEP/SEP as functions of average SNR . ranges from 0โdB to 15โdB. Figures 9โ12
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 .
This is a consequence of the approximation used for generic expression for coherent
detection (19). Figures 9 and 10 show BEP/SEP for and , respectively,
for noncoherent detection, and Figures 11 and 12 show
BEP/SEP for and , respectively, for coherent detection. By examining
Figures 9 and 10 we notice that if we use dual-branch MRC we will gain 4โdB for
the same BEP/SEP. The same goes for Figures 11 and 12, but we will gain
approximately 7โdB, which is to be expected because there is approximately 3โdB
gain when we use coherent detection instead of noncoherent.

Figures 13, 14, and 15 show comparison between FSK and PSK for 3-branch combining. For Figures 13โ15 various values of and 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.