Average LCR and AFD of Dual MRC and SC Diversity in Correlated Small-

Scale Fading Channels

Professor:Joy Iong-Zong Chen

Reporter:Kuo_Tung Chen

20.July 2006

Dep. Of Communication Eng. Da Yeh University

Outlines

• AbstractAbstract

• IntroductionIntroduction

• Level Crossing Rate and Average Fade Duration

• LCR and AFD of MRC

• LCR and AFD of SC

• Conclusions

Abstract

• The performance of average level crossing rate (LCR) and average fade duration (AFD) of the output signal of the maximum ratio combining (MRC) and the selection combining (SC) with a dual-branch receiver was analyzed.

• The channel model of the diversity branches are assumed characterized as correlated Nakagami-m statistics.

• The numerical analysis is conducted for verification.

Introduction

• The MRC and SC diversity techniques.

• The correlated Nakagami-m fading Channels.

• The evaluation methods of LCR and AFD.

Level Crossing Rate and Average Fade Duration

0

In general, the LCR is given as [12]

( , )

: denotes the total received signal envelope

: time derivative

( , ) : is the joint probability density function (joint pdf)

RN f r R R RdR

where

R

dR R

dt

f r R R

of and

R R

(1)

(2)0

The average duration of envelope fades (AFD),

1 1( ) ( )

where

( ) ( ) : is the cumulative distribution function (cdf)

of the signal envelope.

R

r

R RR R

T

T p R r p z dzN N

F R p R r

LCR and AFD of MRC

1 2

2 2 2

The dual MRC diversity scheme operates

in the case of non-independent, can be expressed as

where

, 1, 2 : denote the two input intensity of MRC and modeled

as corrl

R R R

R l

1 2

elated Nakagami-m fading distribution with equal

fading severity .m m m

(3)

2 1/ 2 1/ 2( )

2

The root-mean-square (rms) value of the combined

signal envelope is given by

( [ ]) ( )

where

[ ]/ , 1, 2,

: denotes the fading figure of Nakagami-m distribution

[ ]: is the operat

rms

l

R E R

E R m l

m

E

or of expectation

, , 1,2 : denotes the correlation coefficient between

the i-th and the j-th branch.

ij i j

(4)

1 2

2 2 2

12 222

11 2 12 2

212

212 12

The pdf of the output signal envelope

can be written as

2 R exp( )( ) ( ) ( )

( )[ (1 )] 2

where the capitals, U and W, are assigned as , (1 ) (1 )

m

r m m

R R R R

U R Rf R I W R

m W

mmU W

2

2 0.52 2

1

( )

when the average power are assumed unit,

then the pdf expression in last equation becomes as

2 exp[ /(1 )]( )

( )(1 ) (1 )

where 0, 1, 2 are assumed, ,

( )

mr mm

l rms

v

mf R m I

m

R l R R

I

: denotes the modified Bessel function of the first kind of order

( ) : expresses the Gamma function

v

(5)

(6)

12 1 2

2 21 2

12 02 21 2

0

is the correlation coefficient between and , which is defined as

cov( , )(2 )

var( ) var( )

where

( ) : is the zero-order modified Bessel function

: is the separation of the

d

R R

R R dI f

vR R

I

d

diversity branches

: (mobile speed/ carrier wavelength) is the maximum Doppler shift

given in hertz.

Thus the normalized correlation is related to the ratio of /

The time derivati

df v

d

2

1

ve of (3), , is given by

1

where

: denotes the time derivative of the -th branch

l ll

l

R

R R RR

R l

(8)

(7)

2

2

2

2 2

21 22

which is characterized with Gaussian distributed having the pdf given as

1( ) exp[ ]

22

where

: expresses the variance value

The variance can be calculated as

[ ]

1 = (

RRR

R

R

Rf R

E R

R RR

2 21 1 2 1 2) [ ] 2 [ ]E R R R E R R

(9)

(10)

22 '

2

2' 1

By means of the same methods applied in [W.C.Y. Lee],

the result of the expectation of is given as

1[ ] ( [ ]) ( ) { }

2

( ) [1 ( ) ] ( ) ( )

i j

i j i j

ij ij ij

ij

R R

dE R R E R R v

dt

E K I d

d

212

1

( ) ( ) ( )

( )

where

( ) : is the modified Bessel function of first order

ij ijij

ij

K EI A

I x

(11)

(12)

2( ) ( ), , 1,2 , ,

where

2 : denotes the wave number

: is the physical separation between two adjacent branches

: expresses the carrier wavelength

( ) nd ( ) : are corresponding to the

ijA i j d i j i j

d

K z a E z

1

2 2 20

2 21

20

complete elliptic integral

of the first and second kinds, respectively.

( )(1 )(1 )

(1 ) ( )

(1 )

dxK k

x k x

k xE k dx

x

(14)

(13)

0

First of all, to determine the LCR for the dual MRC working

in the correlated Nakagami-m fading channels, by using of the definition

of LCR shown in (1) becomes as

( ) ( )

where the co

R r RN f R R f R dR

( )

rrelation between the branches are assumed not strongly,

that is, the joint pdf has adopted as

( )2

By using of combining (6) with (16), and the average LCR of dual MRC

diversity normaliz

RR MRC rN f R

22 2 2(2 2 1)

( )

20

ed to the maximum Doppler frequency, ,

can be obtained as

14 exp[ ( ) ]

(1 )( )

! ( ) ( )(1 )

d

k m k m k

R MRCrm k

kd

f

mN

f Rf k m k m

(15)

(16)

(17)

Next, the AFD of MRC diversity operating in correlated Nakagami-m

channel can be determined follows that the definition of AFD shown in (2).

The cdf, ( ), of can be obtained by integrating (6), andrF R R

0

1 2 2212 12

0 12

( )

1

0

expressed as

( ) ( )

(2) (1 ) 2( ),

1 (1 )! ( ) ( )2

where

,

( 1, ) has been applied

( , ) : is the first i

R

r r

m k k m

k

rms

u n x n

F R f d

mm k

k m m k

R R

x e dx n u

( ) ( )

ncomplete gamma function

Once the cdf of dual MRC diversity is obtained, the AFD for dual MRC

diversity can be obtained by combining (2) and (10) with (17), that is,

( ) /R MRC d r R MRCT f F R N

(18)

LCR and AFD of SC

1 1 2

2 2 1

, ,

21 2

2

The time derivative of the envelope R is given as

with the variance can be written as

[ ] [ ]

( )

2

R R RR R R

R

R

E R E R

v

(19)

(20)

1 2

1 2

1 2 1 2

1 11 2 1 2

1 2 2 11 2

In the case of correlation between and ,

assuming , the joint pdf of and

is given by [Joy I. Z. Chen]

4 ( ) ( )( , )

( )(1 )( )

exp

m m

R R m

R R

m m m R R

m R Rf R R

m

1 2

2 22 1 1 2 1 2

12 21 2 1 2

2

1 2 1 20 0

m( ) 2-

(1 ) (1 )

[ ], 1, 2

The pdf of the output envelope R can be easily derived by

( ) ( ) , and the results is expres

m

l l

R R

r R R

R R m R RI

where

E R l

df R f R R dR dR

dR

2 1

2 2 2

sed as [G. Fedele]

4 ( )( ) exp( ) 1 ( 2 (1 ), 2 (1 )

( )

m m

r m

mf R m Q m m

m

(21)

(22)

1 2 21

( , ) : is the Marcum function conventionally defined as

( , ) ( / ) exp 0.5( ) ( ) , n 1

Then combining (1) and (22), the LCR of dual SC diversity

normalized with the Doppler fr

m

nn nb

Q

Q a b t t a t a I at dt

( ) 2 1 2

2 2

equency, , operating in

correlated Nakagami-m fading environments can be evaluated as

2 2 exp( )( )

1 ( 2 (1 ), 2 (1 )

d

mR SC m

Rd

m

f

N mm

f m

Q m m

(23)

(24)

1 2 1 2 1 20 0

2( )24( )

2 40 12 12

1 1

By use of the joint pdf shown in (21), the cdf can be obtained as

( ) ( , )

1 4 = ( ) (1,2 2 )

! ( ) ( )(1 )

( ;

R R

r r r

m km k

m k kk

F R f R R dR dR

mB m k

k m k m

F m k m k

2

2

12

1 1

( ) ( )

1, )1

( ; , ) : is the hypergeometric function

Then the AFD of SC diversity, ( ) / , can be easily

obtained by combining (24) and (25).

R SC d r R SC

m

where

F

T f F R N

(25)

Fig. 1 Normalized average LCR of dual MRC for different values of /

with fading parameter 2 , and 3 .

d

m m

Fig. 2 Normalized average LCR of dual SC for different values of /

with fading parameter 2 , and 3 .

d

m m

Fig. 3 Normalized average AFD of dual MRC for different values of /

with fading parameter 2 , and 3 .

d

m m

/d

2m

3m.

Fig. 4 Normalized average AFD of dual SC for different values of /

with fading parameter 2 , and 3 .

d

m m

5. Conclusions• The average LCR and AFD are evaluated in this paper

for MRC and SC work with dual correlated Nakagami-m fading channels.

• The small value of branch is assumed in the paper, but the existing performance degradation is fixed.

• The results from the numerical analysis indicate that average LCR and AFD are significantly affected by the correlation between branches with MRC and SC schemes, that is, the fact of correlation characteristic of branches is not negligible in designing the wireless radio systems.

• In future, in order to make the results are confident, some of the detail simulations will be held.