Accurate MGF matching technique for diversity reception in correlated lognormal fading channels

Accurate MGF Matching Technique for Diversity Reception in Correlated Lognormal Fading Channels Cong Lam Sinh, Quoc Tuan Nguyen University of Engineering and Technology, Vietnam Nation

Accurate MGF Matching Technique for Diversity Reception in Correlated Lognormal

Fading Channels

Cong Lam Sinh, Quoc Tuan Nguyen

University of Engineering and Technology, Vietnam

National University, Hanoi - Vietnam

Email: congls@vnu.edu.vn, tuannq@vnu.edu.vn

Dinh-Thong Nguyen Faculty of Engineering and Information Technology, University of Technology, Sydney – Australia Email: dinh-thong-nguyen@uts.edu.au

Abstract –The two-point moment generating function

(MGF) matching technique has been used with some

success to approximate the output from an MRC diversity

combiner operating in correlated lognormal fading

channels The technique, however, is very sensitive to the

choice of the location of the two matching points This paper

proposes to apply the principle of power conservation across

the combiner to control the accuracy of the location of the

MGF matching points The technique is both novel and

effective and this is backed by sophisticated simulation

results To demonstrate the accuracy of the proposed MGF

matching technique, the paper presents a closed-form

expression for the estimation of BER of BPSK using MRC

diversity reception in a correlated lognormal shadowing

environment

I INTRODUCTION

In most realistic scenarios of wireless propagation

between a base station and a receiver, the physics of

radio wave propagation encountering random parallel

multipaths and cascaded obstructions is not well

understood The latter is commonly known as shadow

fading, e.g [1], referring to the random fluctuation in

the received average power as the mobile receiver

moves in and out of the shadow of hills or buildings

which obstruct the line-of-sight transmission The

global path is usually modelled as a lognormal

stochastic process while the local path is modelled as

Rayleigh process In most realistic situations, fading is

the result of a mixture of the two fading mechanisms,

and fading mitigation requires both microdiversity

techniques using multiple antennas or multiple OFDMA

subchannels and macrodiversity techniques using

multiple base stations Maximum ratio combining

(MRC) is most effective for microdiversity while

selective combining (SC) is more suitable for

macrodiversity [2] However in this paper, for

simplicity we assume a microdiversity environment and

that MRC is used

Shadowing has a much higher degree of correlation

than short-term multipath fading, therefore the main

objective of this paper is to formulate the performance

of MRC reception in correlated lognormal fading channels The underlining complexity in using the lognormal model for shadowing in MRC diversity reception is that it results in the well-known ‘sum of lognormal powers’ problem [3], [4] Some authors avoid this complexity by using a gamma pdf to approximate the shadowing as first proposed in [5] In [4] the authors propose to approximate the sum of lognormal random variables by a single lognormal random variable whose normal mean and variance parameters are found by a two-point matching of its MGF to that of the sum However, the result of the two-point matching is very sensitive to the choice of the matching points This problem has been briefly addressed by our group in [6] in the context of MRC

diversity reception in the simple case of independent

lognormal fading channels, by evoking the power conservation principle across the combiner In this paper, we extend the problem of MRC diversity reception in a far more complex environment of correlated lognormal fading channels The power conservation principle is used to determine and to control the accuracy of the location of the two MGF matching points, using a simple search algorithm This technique is both innovative and effective and has not been done before to the best of our knowledge

The rest of the paper is organized as follows Section II defines the signal model and briefly describes the maximum ratio combining (MRC) principle for diversity reception In Section III we present the derivation of the pdf of the correlated multivariate

Gaussian vector Z from a given correlation matrix of

the related multivariate lognormal vector p, i.e p=exp(Z) In Section IV we describe the estimation of

the pdf of the sum of correlated lognormal powers using the current two-point MGF matching technique Section

V is the main contribution of our paper in which we present an innovative technique to control the accuracy

of the two-point MGF matching by invoking the power conservation principle across the MRC combiner Section VI briefly describes the essential steps in Monte Carlo simulation of BER of BPSK signal using MRC

978-1-4799-2903-0/14/$31.00 ©2014 IEEE 140

reception in correlated lognormal fading channels, and

finally a conclusion is presented in Section VII

II SIGNAL AND MRC DIVERSITY RECEPTION

MODEL

The effect of maximum ratio combining is to add up the powers, hence the signal-to-noise ratios, of the

received signals to be combined Expressing it in matrix

form, the diversity receiving system is described as,

y = h*x + n (1)

where

• y = [y 1 , y 2 ,… y N]T is the received symbol vector from all the diversity branches,

• h = [h 1 , h 2 ,… h N]T is the channel gain vector on all the diversity branches,

• x is the transmitted BPSK symbol vector

(complex signals) and

• n = [n 1 , n 2 ,… n N]T is the AGWN noise vector

on all the diversity branches

The equalized received signal from an MRC combiner, used for detection/demodulation, is

h h

y h

H

t)=

and it is well-known that the SNR in this equalized

signal is equal to the sum of the SNRs in all diversity

branches at the input to the MRC combiner [7]

The received SNR is then

0

2

N

E

=

γ (3)

where the transmit signal energy is E s =E[s2(t)]

In view of (3) where the transmit SNR, Es/N o, may

be assumed fixed, in this paper we use the term channel

power gain, p = h2 , and signal-to-noise ratio, γ,

interchangeably where it is appropriate

III CORRELATED LOGNORMAL FADING

CHANNELS

The cascade (product) model of shadowing implies that the path loss is exponentially proportional to the distance to a power α between 3 to 7 and that the standard deviation of shadow fading loss is independent

of the distance and is in the range of 5 to 12dB [8] The power gain of a shadow fading channel is usually modeled as a lognormally distributed random variable, i.e p≡ h2 =e Z ~ LN(µZ,CZ), with Z being normally distributed, i.e Z~ N(µZ,CZ)

Since p is a correlated multivariate lognormal vector, let p=(p1, p2,…,p N) = (eZ1, eZ2,…., eZN) in which

each component pi is a lognormal RV and Z=(Z1, Z2,…,

Z N) is a correlated multivariate normal vector and its pdf

is

2

) ( ) ( exp(

) 2 (

1 )

2 /

Z -1 Z Z Z

μ -z C μ -z C

f

π

where the mean, variance and covariance matrix of Z

are, respectively:

) , , ,

( )

μ Z (5a)

) , , ,

( ) ( Z21 Z22 ZN2

2

) , ( ) ,

Z i j Cov Z i Z j

C = =σ (5c)

Here we use the simple decreasing correlation model by Gudmundson in [8] for the shadow

fading p, then its covariance matrix is, assuming

all pi variates have the same variance σ2p ,

1

1

1

1

3 -2 1

3 2

2

-1 2

2





























=

−

−

−

N N N

-N N N

p p

C

ρ ρ ρ

ρ ρ

ρ

ρ ρ

ρ

ρ ρ

ρ

In which ρ is the correlation coefficient between any two successive pi and p i+1

In general, the mean, variance and covariance matrix

of p are, respectively:

( ) ( , , , )

2

p

p

( ) ( 2 , 2 , , 2 )

2

p

2 p

p

ij

p j p i p Cov j

) , ( ) , (

P

141

The relationship between the two parameter sets in (5)

and (7) can be summarized as follows:

) 1 ( )]

( [ ) 1 (

,

2 2

2 /

2

−

=

−

=

=

+

+

2 Z 2

Z Z

2

Z

Z

σ σ

σ

μ

p

σ

μ

p

σ

μ

e E

e e

e

(8a,b)

Or

ln ,

2 2

2

















+

=

pi pi

pi

Z i

σ μ

μ

μ (9a)

2 ( 2 2 )

/ 1

ln pi pi

σ = + (9b)

C Z(i,j)=Cov(Z i,Z j)=ln(1+σ /pij2 μpiμpj) (9c)

For simplicity, in this paper we assume all random

variates of Z have the same μ Z and σ Z, and all random

variates of p have the same μ p and σ p

IV ESTIMATING PDF OF SUM OF CORRELATED

LOGNORMAL POWERS USING MGF MATCHING

TECHNIQUE

Consider N correlated lognormal RVs, {pi}, with

joint distribution f(p1, p2,… , p N), input to the MRC

The MGF of the MRC output power, Y=∑ipi , is

p

p d

f e

dp dp dp p p p f e s

N

i

sp

N N

N i

sp Y

i

i

 ∏

∏

 

∞

+

∞

−

=

−

=

−

∞

+

∞

−

∞

+

∞

−

=

=

) ( ) (

) , , , ( ) (

)

(

1

2 1 2

1 1

ψ

(10)

where s is the variable in the Laplace transform domain

Since f LN (p)dp = f(z)dz, the MGF of the combined

SNR in (10) becomes



∞

∞

−













=

z -z C -z C

1 -Z Z

d z

s

L

i

i

M

2

) ( ) ( exp(

)]

exp(

exp[

1

2

/

1

2

/

)

2

(

1

)

Z

μ

ξ

To decorrelate the above expression so that it can have

a suitable form for Gauss-Hermite expansion for the

integration, we make the variable transformation

C z-1/2(z-μ Z)=√2u, i.e z=√2C1/2u + μ Z, or

i N

j

j Zij i

N

u C z

d d

μ +

=

=



=1

2 / 1

2

) 2

z

where C Zij is the (i,j) element of C Z 1/2 Then (11) becomes

u u

i j u z C s

N i

s M

T N

j

N Y

2 exp(

exp[

1

1 ) (

1

2 /

−

=

=

∞

∞

− ∏ ξ = μξ

π

(12)

The integral in (12) has a suitable form for Gauss-Hermite expansion approximation for the MGF of the

sum of N correlated lognormal SNRs, which is first taken with respect only to variable z1 as

N 2 1

1 2

1 2

2 2

/

z

z )]

2 2

exp(

exp[

) exp(

1

) (

1

1 1

d d a

C z

C s

w z s

M

N l

l n Zl N

j j Zlj

N n n N

i i N

Y

p

∞

∞

−

∞

∞

+ +

−

−

≈

ξ

μ ξ

ξ π

(13)

By proceeding in a similar way for the integrals with

respect to other variables z2,…z N, we obtain

μ Z , C Z

] 2

exp(

exp[

) ,

(

1

+

−

≈

N l

l N

j n Zlj

N

n

N

n n Y

j

p

N

p

N

a C s

w w s

M

ξ

μ ξ

π (14)

in which w n and a n are, respectively, the weights and the abscissas of the Gauss-Hermite polynomial The approximation becomes more and more accurate with

increasing approximation order N p

Finally, the sum of N correlated lognormal RVs can then be approximated by a single lognormal RV [4],

X LN

Y ˆ = 100 1ˆwhereXˆ ~N(μˆX,σˆ2X) By matching the MGF of the approximation YˆLN with the MGF of the

lognormal sum in (14) at two different positive real

values s1 and s2, we obtain a system of two 142

simultaneous equations which can be used to solve for

X

μˆ and σˆ2X using function fsolve in Matlab The two

simultaneous equations, with RHS being completely

known from (14), are:

1,2

, ) , ,

(

}]

/ ) ˆ 2 ˆ exp{(

exp[

1

=

=

+

−



=

i i

s

M

a i

s

w

LN

Y

X X

n p

N

Z

Z C μ

π

ξ μ σ

(15)

For a discussion on the choice of matching points (s1,

s2), see [4][6] From Figure 5 of [4] in which for N=4,

μ=0dB and σ=8dB, i.e average SNR=7.36dB from (8a),

it is recommended that (s1,s2)=(1.0, 0.2) for various

different values of correlation coefficient ρ

V ACCURATE MGF MATCHING USING POWER

CONSERVATION PRINCIPLE

The problem encountered in using the 2-point MGF

matching technique proposed in [4] is that it is highly

sensitive to the location of the two matching points and

also to the initial starting values for μˆ and X σˆ2Xchosen

for the Matlab function fsolve In [4], the values of the

matching points are chosen in an ad-hoc manner to

visually judge the accuracy of the match Furthermore,

as clearly seen in Table 1, the technique does not

guarantee conservation of signal power across the MRC

combiner The power loss is as much as 25% The

accuracy of the 2-point MGF matching in the preceding

section can be greatly improved and controlled to a

specified degree by reinforcing this ‘lossless’ principle

This implies equal system average power on both sides

of the combiner

Since the average input power gain to the combiner,

assuming a micro-diversity environment, is

2

Z Z p

While the estimated average output power gain is

ˆ exp( ˆ ˆ /2)

2

X X out

The percentage power estimation error is defined as

in

in out

P

P P

=100 ˆ

% (17)

The assumption of a micro-diversity environment above may not be realistic because diversity paths have different distance and topography However in this paper we apply this assumption for the sake of simplicity of computation and simulation

Thus by systematically searching for the two

matching points (s1, s2) until the power estimation error

is smaller than a specified percentage threshold, an accurate 2-point MGF matching can be achieved as evidenced in Figure 1 In this figure, the MGF matching corresponding to SNR=7.36 clearly shows a significant improvement from the result using the two matching points proposed in [4]

Once the estimated Gaussian parameters are found, the pdf of the estimated SNR from the output of the MRC combiner is

) ˆ

2

) ˆ log 10 ( exp(

2 ˆ

1 ) (

ˆ

2

2 10

,

X X X

MRC LN

f

σ

μ

γ π

σ

ξ γ

where the log conversion constant ξ=10/ln(10)

The BER of BPSK in Gaussian channel with bit SNR γ

is

) 2 ( ) (

BER AWGN BPSK = (19)

γ

BPSK AWGN BER BPSK LN

=

By a change of variable







 +

=

⇔

=

X

X

ξσ ξ

μ γ

σ

μ

exp 2

ˆ

ˆ log

10 10

(20) can be reduced to

X BPSK AWGN BPSK

LN

2 )) ( ˆ 1

0

,

where γˆX(u) =exp(μˆX /ξ +uσˆX 2/ξ)is the argument

of BER AWGN,BPSK (.) in (19) The above expression for BER can then be accurately approximated by an N p -order Gauss-Hermite

polynomial expansion as given in (21)

143

ˆ ( ))

1

1

, ,

, AWGN BPSK X n

p N MRC BPSK

n w n

π =

1

1

,

p N MRC

BPSK

n w n

π =

Table1: Estimation result from two-point MGF

matching for N=4 correlated diversity branches with

ρ=0.3

SNR_dB (s1,s2);

(μ ˆX ,σ ˆX)dB

Output power/input power

PEE

5 (7.2567, 5.7083) (0.003, 0.104); 12.6131/ 12.6491 0.28%

7.36 (9.6505, 5.6957) (0.002 , 0.203); 21.8042/ 21.8002 0.019%

From [4]

7.36dB

(0.2, 1.0);

(9.3283, 4.9462)

16.3868/

21.8002 24.83%

10

(0.017, 0.098);

(12.2157,

5.7340)

39.8201/

40.0000 0.450%

15

(0.005, 0.017)

(17.2240,

5.7287)

125.9572/

126.4911 0.42%

VI SIMULATION SET-UP

In the theory part of the paper, we plot BER of the

MRC output versus average SNR per lognormal

channel (γLN ≡μ p in (8a)) for specified value of the

variance σ z =8dB and specified values of correlation

coefficient, say ρ =0.3 Thus μ z can be calculated from

(8a), then σ p can be calculated from (8b) and C p (i,j)

from (7c), and finally C z (i,j) can be calculated from

(9c)

The intermediate correlated normal variates Z=(Z1,

Z2,…,ZN) can now be generated as

= +

=

i

j c ij U j i

i

Z

1

μ for i, j = 1,2, ,N (22)

in which U j ~ N(0,1) are i.i.d unit normal variates and

c ij is the (i,j) element of C z1/2, obtained from matrix

C z =C z1/2(C1/2)T using Cholesky decomposition

Figure 1: Comparision between 2-point matching+power conservation and 2-point matching in [4]

Finally the N correlated lognormal variates are generated as p i=eZi and the channel gain h i =e Zi/2

Figure 2: BER for BPSK in Correlated Lognormal Fading (ρ = 0 3 ; σZ = 8 dB) using N-branch MRC

diversity reception VII CONCLUSION

We have successfully presented an innovative and simple technique for accurate two-point matching of moment generating functions by evoking the principle

of power conservation between the two matched MGFs The merit of the technique has been demonstrated in the accurate estimation of the ‘sum of lognormal powers’ of 144

the output signal from an MRC diversity combiner The accuracy of the proposed MGF matching technique is also backed by Monte Carlo simulation of the BER of BPSK signal in lognormal fading channel using MRC diversity reception

This work was supported by research grants from QG.12.45 Projects of the University of Engineering and Technology, Vietnam National University Hanoi

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