Mirza Miliši , Mirza Hamza, Mesud HadžialiUniversity of Sarajevo, Faculty of Electrical Engineering, Zmaja od Bosne bb 71000 Sarajevo, BiH

e-mail : mmilisic@etf.unsa.ba

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. Most of these papers treat Rayleigh, Nakagami-m, Hoyt, Rice or Weibull fading. This paper treats outage performances of MRC in presence of generalized κ µ− fading. In this paper, we will present κ µ−fading model, probability density function (PDF), cumulative distribution function (CDF), expression for n-th order moment and outage probability. We will also present PDF, CDF and outage probability of the L-branch MRC output. Outage performances of the MRC will be presented for different number of branches via Monte Carlo simulations and theoretical expressions.

Key words – fading channel, multipath fading, generalized κ µ− fading, maximal-ratio combining, outage, simulations.

I INTRODUCTION

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. Most of these papers treat Rayleigh, Nakagami-m, Hoyt (Nakagami-q), Rice (Nakagami-n) or Weibull fading [1], [2], [4]-[6]. Beside MRC, performances of selection combining, equal-gain combining, hybrid combining and switched combining in fading channels have also been examined. 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 combiners branches, have also been examined in numerous papers. Nevertheless, depending on system used and combiners implementation, one must take care of resources available at the receiver, such as : space, frequency, complexity, etc. Moreover, fading statistic doesn't necessary have to be the same in each branch, e.g. PDF can be the same, but with different parameters (Nakagami-m fading in i-th and j-th branches, with i jm m≠ ), 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 generalized κ µ− fading [3] and [7]. This type of fading has been chosen because it includes, as special cases, Nakagami-m and Nakagami-n (Rice) fading, and all of their 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 generalized κ µ−distribution, closed form expressions for probability density function (PDF), cumulative distribution function (CDF), n-th order moment and outage probability. In addition, closed form expressions for PDF, CDF and outage probability at the MRC output will be derived. MRC performances will be presented for different number of combiners branches via Monte Carlo simulations and theoretical expressions.

II PHYSICAL MODEL AND DERIVATION OF THE κ µ−DISTRIBUTION

The fading model for theκ µ− distribution considers a signal composed of clusters of multipath waves, propagating in a nonhomogenous 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. Given the physical model for theκ µ− distribution, the envelope R, can be written in terms of the in-phase and quadrature components of the fading signal as :

( ) ( )2 2 2

1 1

n n

i i i ii i

R X p Y q= =

= + + + (1)

where Xi and Yi are mutually independent Gaussian processes

with 0i iX Y= = and 2 2 2i iX Y σ= = . pi and qi are respectively

the mean values of the in-phase and quadrature components of the multipath waves of cluster i, and n is the number of clusters of multipath. Total power of the i-th cluster is

( ) ( )2 2 2i i i i i iR X p Y qγ = = + + + . Since iγ equals to the sum of

two non-central Gaussian random variables (RVs), its PDF yields in non-central 2χ PDF with two degrees of freedom :

2

02 2 21( ) exp

2 2i

i ii ii

ddf Iγγγγ

σ σ σ+= ⋅ − ⋅ (2)

where 2 2 2i i id p q= + , and 0 ( )I is the modified Bessel function

of the first kind and order zero [10, Eq. 9.6.16]. Laplace transform of the ( )

i ifγ γ is found to be [10, Eq. 29.3.81] :

Outage Performance of L-branch Maximal-Ratio Combiner for Generalized κ µ− Fading

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2

2 21( ) exp

1 2 1 2ii

isdL f

s sγ γσ σ

= ⋅ −+ +

(3)

where s is the complex frequency (the Laplace variable). Knowing that the , 1, 2,...,i i nγ = , are independent RVs, the

Laplace transform of the ( )fγ γ , where 1

n

ii

γ γ=

= , is found to

be :

( )

2

2

21

exp1 2

( ) ( )1 2

i

n

i ni

sds

L f L fs

γ γσ

γ γσ=

−+

= =+

∏ (4)

where 2 21

nii

d d=

= . The inverse of (4) is given by [10, Eq.

29.3.81] :

( )1

2212 2 2 2

1 exp .2 2

n

nddf I

dγγγ γγ

σ σ σ

−

−+= ⋅ ⋅ − ⋅ (5)

It can be seen that 2 2 22R n dγ σΩ = = = + , and

( )24 2 4 2 2 2 24 4 2R n d n dγ σ σ σ= = + + + .

Therefore, ( ) ( )2 4 2 24 4Var R Var n dγ σ σ= = + . We define 2

22dn

κσ

= as the ratio between the total power of the dominant

components and the total power of the scattered waves. Then ,

( )( )

( )( )( )

222 2

2

11 2

Rn

VarVar R

γ κγ κ

+= =

+. (6)

From Equation (6), note that n may be totally expressed in terms of physical parameters, such as mean-squared value of the power, the variance of the power, and the ratio of the total power of the dominant components and the total power of the scattered waves. Note also, that whereas these physical parameters are of a continuous nature, n is of a discrete nature. It is plausible to presume that if these parameters are to be obtained by field measurements, their ratios, as defined in (6), will certainly lead to figures that may depart from the exact n.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 :

( )( )( )

( )( )

( )( )

222

2 22

1 2 1 2

1 1

R

VarVar R

γκ κµ

γκ κ

+ += ⋅ = ⋅

+ + (7)

with µ being the real extension of n. Non-integer values of the parameter µ may account for : the non-Gaussian nature of the

in-phase and quadrature components of each cluster of the fading signal, non-zero correlation among the clusters of multipath components, non-zero correlation between in-phase and quadrature components within each cluster, etc. Non-integer values of clusters have been found in practice, and are extensively reported in the literature, e.g. [11]. Using the definitions for parameters κ and µ , and the considerations as given above, the κ µ− power PDF can be written from (5) as :

( )

112

21 1

2 2

1

(1 )( )exp( )

1 (1 )exp 2

f

I

µµ

γ µ µ

µ

µ κγ γκ µκµ κ γ κ κ γµ

+−

− +

−

+= ⋅⋅ ⋅ Ω

+ +× − ⋅Ω Ω

. (8)

Performing RV transformation, PDF of the κ µ− envelope, as a square-root of power, can be found from (8) as :

( )

12

1 12 2

2

1

2 (1 )( )exp( )

1 (1 )exp 2

Rf R

RR I R

µ

µ µ

µµ

µ κ

κ µκµ κ κ κµ

+

− +

−

+=⋅ ⋅Ω

+ +× ⋅ − ⋅Ω Ω

. (9)

From (8), κ µ− power CDF can be written in closed form as :

0

2( 1)( ) ( ) 1 2 ,F f x dx Qγ

γ γ µκ µγγ κµ += = −

Ω (10)

where 2 2

111( , ) exp ( )

2b

x aQ a b x I ax dxa

νν νν

∞

−−+= ⋅ − ⋅ ⋅ is

generalized Marcum Q function [13]. Using [8, Eq. 03.02.26.0008.01, 07.34.21.0011.01 i

07.34.03.0272.01] respectively, envelope n-th order moment is found in closed form as :

( )/ 2

1 1( / 2) / 2; ;

( )exp( ) (1 )

nn nR F nµ µ µ κµ

µ κµ κ µΓ + Ω= ⋅ ⋅ +

Γ + (11)

where ( )Γ is gamma function [14, Eq 6.1.1], and ( )1 1 ; ;F is the confluent hypergeometric function [10, Eq. 13.1.2].

III κ µ− MAXIMAL-RATIO COMBINER

There are four principal types of combining techniques [9] 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 [9], 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 [9] MRC

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receiver is the optimal multichannel receiver, regardless of fading statistics in various diversity branches since it results in a ML receiver. For equally likely transmitted symbols, the total SNR per symbol at the output of the MRC is given by [12] :

1

L

jj

γ γ=

= , where jγ is instantaneous SNR in i-th branch of L-

branch MRC receiver. Repeating the same procedure as in the previous section, previous relation can be written in terms of in-phase and quadrature components :

2 2,

1 1 1 1

L L L n

i i i jj j j i

R Rγ γ= = = =

= = = where 2,i jR represents total

power of the i-th cluster manifested in j-th branch of the MRC receiver. Using Equation (1) one can obtain :

( ) ( )2 2, , , ,

1 1

L n

i j i j i j i jj i

X p Y qγ= =

= + + + . (12)

Repeating the same procedure as in the previous section and using Equations (1), (2), (3) and (4) one can obtain Laplace transform of the PDF of the RV γ (SNR) :

( )

2

2

21

exp1 2

( ) ( )1 2

j

L

j L nj

sds

L f L fs

γ γσ

γ γσ

⋅=

−+

= =+

∏ (13)

where 2 2

1

L

jj

d d=

= . Inverse Laplace transform of Equation (13)

yields to PDF of the RV γ .

( )( )

112

21 1

2 2

1

(1 )( )exp( )

1 (1 )exp 2

LL

L L

LfL L

I L

µµ

γ µ µ

µ

µ κγ γκ µκ

µ κ γ κ κ γµ

+−

− +

−

+= ⋅⋅ ⋅ Ω

+ +× − ⋅Ω Ω

(14)

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 :

, ,MRC MRC MRCL Lµ µ κ κ= ⋅ = Ω = ⋅ Ω .Now, it is easy to obtain CDF

0

2( 1)( ) ( ) 1 2 ,LF f x dx Q Lγ

γ γ µκ µγγ κµ += = −

Ω (15)

For fixed threshold, thγ , outage probability is given by :

0

( ) ( ) ( )

2( 1)1 2 ,

th

out th th

thL

P F f x dx

Q L

γ

γ γ

µ

γ γ

κ µγκµ

= =

+= −Ω

. (16)

IV SIMULATIONS AND DISCUSSION OF THE RESULTS

As mentioned previously, MRC outage performances will be examined via Monte Carlo simulations and theoretical expressions (16). Figures 1-8 show theoretical and simulated outage probability as a functions of threshold level thγ . thγranges 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 1,2,3,4L = , 0.55,1, 2κ = and

1,2µ = . For fixed values of κ and µ outage probabilities have been compared for specified numbers of combiners branches, L. From figures 1-6 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's not economically or technically justified). We can also observe that the highest gain is obtained between curves for 1L = and

2L = (situation with no combining and dual-branch combining). Distribution parameters also have a significant impact on outage

Figure 1. Outage probability for 0.55κ = and 1µ =

Figure 2. Outage probability for 0.55κ = and 2µ =

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Figure 3. Outage probability for 1κ = and 1µ =

Figure 4. Outage probability for 1κ = and 2µ =

Figure 5. Outage probability for 2κ = and 1µ =

probability. When κ is increasing, outP is decreasing. Namely, these results were expected because κ represents the ratio between total power of dominant components and total power

Figure 6. Outage probability for 2κ = and 2µ =

Figure 7. Outage probability for dual-branch MRC

Figure 8. Outage probability for dual-branch MRC

of scattered components. Parameter µ represents fading severity parameter. As µ decreases, fading severity increases and so does the outage probability. From figures 1-6, for fixed

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κ , as µ increases so does the slope of the outage curve. For dual-branch combining ( 2L = ), behavior of outP , 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 µchanges, and in figure 8 we have inversed 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 outP ,and threshold level thγ , are the same for all curves (figs. 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 dynamic in channel apart from the case of smaller dynamic in channel.

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