978-1-4577-2019-2/11/$26.00 ©2011 IEEE 455

The Second Order Statistics of a Two-Branch MRC with Power Unbalanced Nakagami Distributed Branches

Predrag Ivanis1, Vesna Blagojevic1, Dusan Drajic1 and Branka Vucetic2

Abstract - In this paper, the exact expressions for the second order statistics of the maximum ratio combiner (MRC) output for Nakagami fading channel are derived. The joint characteristic function (CF) for the MRC output and its time derivative is derived, and applied to obtain the level crossing rate (LCR) expression for the case of independent but unbalanced diversity branches, with unequal fading parameters. For the special case of two-branch diversity, the LCR expressions are derived in closed form. The exact closed-form expression for the autocorrelation function (ACF) of the MRC output is derived for an arbitrary number of diversity branches. The analytical results are validated by simulations.

Keywords - Autocorrelation function, level crossing rate, maximum ratio combining, Nakagami fading, unbalanced branches.

I. INTRODUCTION

The performances of wireless communication systems depend on the dynamics of fading channels and cannot be completely captured by the long-term averaging. The level crossing rate (LCR) and the autocorrelation function (ACF) are performance measures that are usually used to provide the better insight in the time-varying nature of the fading channel. When the fading can be described with Nakagami distribution, the LCR of the received signal envelope is derived in [1].

On the other hand, it is well-known that a maximum ratio combiner (MRC) gives the maximum output signal-to-noise ratio and minimizes the bit error rate compared to other diversity techniques [2]. For the Nakagami fading channel with the MRC diversity receiver, the LCR is calculated only for independent and identically distributed (i.i.d.) case [3].

The second order statistics of the MRC output signal envelope for the case of unbalanced diversity branches were analyzed in [4], where a novel approach for calculating the LCR, based on characteristic functions (CF) was proposed. This principle is applied to obtain the closed form expressions for the LCR and ACF, for an arbitrary number of independent MRC branches with unequal powers [5]. The closed-form results are extended for the case of spatially correlated space-time coded MIMO systems [6]. However, these results are limited to the case of a Rayleigh fading channel.

In the paper [7], the authors stated that the closed-form expression was derived for the LCR, when the fading in the

1Predrag Ivanis, Vesna Blagojevic and Dusan Drajic are with Faculty of Electrical Engineering, University of Belgrade, Serbia, E-mail: predrag.ivanis@etf.rs

2Branka Vucetic is with School of Electrical and Information Engineering, University of Sydney, NSW 2006 Australia

branches undergoes a Nakagami distribution and the branches are unbalanced (with the same fading parameter m). However, due to an idealistic assumption [7, Eq. 14] that the output envelope and its time derivative are statistically independent random variables, the corresponding expression can only be considered as an approximation.

To the authors' best knowledge, the exact LCR and ACF expressions are not obtained for the MRC diversity system with unbalanced and unequally Nakagami distributed branches. In this paper, we derive these expressions in a closed form and verify it using an independent simulation model.

II. SYSTEM AND CHANNEL MODEL

The signal at the output of an N-branch diversity MRC combiner, for the case of independent non-identically distributed fading channels, is given by [2, Eq. 5-2-1]

1

N

nn

, (1)

where 2n nr , and rn denotes fading envelope of the n-th

branch of the MRC receiver. In a Nakagami fading case, the PDF of rn is represented by [1, Eq. 1]

2 1 22( )

( ) exp( )( )

n n

n

m mn n n n

n mnn n

m r m rp r

m

, (2)

where 2[ ]n nE r and mn=2 2var( )n nr / , respectively, denote

the signal power and fading factor in the n-th branch, and (.) denotes the Gamma function [8].

A mathematical model, suitable for description of the second order statistics in a Nakagami fading channel is proposed in [1]. If 2mn is an integer, the fading envelope of the n-th branch can be represented as the sum of squares of Gaussian zero-mean random variables (RVs), according to the expression [1, Eq. 7]

2 2, ,12

0.52 2 2,0 , ,1

( ), 2 even

( ), 2 odd

n

n

m

n i n i nin m

n n i n i ni

c s mr

c c s m

. (3)

In the previous expression, each pair ,n ic and ,n is represents

a scattered wave, consisting of an in-phase and quadrature

component of a complex RV , ,n i n ic js ( 1j ), with a

Rayleigh distributed magnitude.

For the isotropic scattering the variance of ,n ic and ,n is can

be denoted by 2 2 2, ,{ } { }n n i n iE c E s , and the correlation

functions are given by [2, Eq 1.3-1], [2, Eq. 1.3-2]

~ TELSIKS 2011 Serbia, Nis, October 5 - 8,2011

456

2

, , , , , , , , 0

, , , , , , , ,

{ } { } (2 ),

{ } { } 0, ( , )n i t n i t n i t n i t n

n i t n i t n i t n i t

E c c E s s J f

E c s E s c n i

m

(4)

where t and t+ denote two successive time instants, and fm denotes the maximum Doppler shift [2].

If ,n ic , ,n is respectively, denote the time derivative of ,n ic ,

and ,n is , we can write , , , ,{ } { } 0n i n i n i n iE c c E s s and 2 2 2 2 2 2

m{ } { } 2n n n nE c E s f [2, Eqs. 1.3-5]. The joint

PDF of nr and its time derivative (denoted by nr ) for any real

numbers nm and n is given by [1, Eq. 13]

2 1 2 2

2

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

2( ) 2

n n

n

m mn n n n n

n n mn nn n n

m r m r rp r r

m

. (5)

In [1], it was shown that the RVs nr and nr are not

statistically independent. Therefore, it is not easy to obtain the exact expression for the joint PDF of the MRC output and its time derivative (denoted by ), for an arbitrary number of branches. However, as it is shown in the next section, the joint characteristic function of and can be represented in a compact form, even for unequally distributed fading channels.

III. JOINT CHARACTERISTIC FUNCTION

By using Eq. (1) and the property that the sum of random variables has the characteristic function (CF) that is equal to the product of its CFs [9], it can be written

, 1 2 , 1 21

( , ) ( , )n n

N

n

, (6)

and according to the definition [9, Eq. 7-23], we obtain

, 1 2 1 2

0

( , ) ( , ) exp( ( ))n n n n n n n np j d d

(7

) where joint PDF of 2

n nr and its time derivative can be

easily obtained by using a transformation of RVs as 3/ 2 2

2

( )( , ) exp( )exp( )

82 2 ( )

n

n

m mn n n n n

n n mn n nn n n

m mp

m

. (8)

The integral in (7) is calculated over n by using the

identity [10, Eq. 3.323-2] to obtain

1, 1 2

0

1

02 2

2 1

( )( , ) exp( )

( )

exp( )( )

,( ) ( / 2 )

n

n n n

n

n

n n

mmnn n nm

n n

mn n nm

nm m

n n n n n

md

m

t t dtm

m m j

(9)

where the substitution 2 22 1( / 2 )n n n n nt m j is

applied in the last step. Finally, as the parameter mn is always real valued, we can

observe that the integral in the last expression corresponds to the definition of the Gamma function (Euler's integral [8, Eq. 6.1.1]). Therefore, we have derived the joint CF for the output of the MRC combiner for the case of branches with unequal powers n and unequal fading parameters mn

2 2, 1 2 2 1

1

( , ) [1 (2 / ) ( / )] n

Nm

n n n n nn

m j m

.

(10) By using the property 1 , 1( ) ( ,0) [9, Eq. 7-28],

the derived expression reduces to the previously derived expression for the CF of the MRC output [6]

1 11( ) (1 / ) n

N mn nn

j m

. (11)

On the other hand, when parameters mn are real valued, the corresponding branch power can be expressed as 22n n nm

[1], and the joint CF simplified into

2 2 2 4 2, 1 2 2 m 1

1

( , ) (1 8 2 ) n

Nm

n nn

f j

, (12)

and, when mn=1 ( n ), the previous expression reduces to the well-known identity for the joint CF in a Rayleigh fading channel [4, Eq. 13].

IV. LEVEL CROSSING RATE

In the considered system, the level crossing rate, denoted by NR , is usually defined as the rate at which the signal at the MRC output defined in (1), crosses level R going in one (positive, or negative) direction [1-6]

, ,0

1( , ) | | ( , )

2RN p R d p R d

, (13)

where , ( , )f denotes the joint probability density

function of and . The LCR can be also expressed in terms of the joint CF of

and , denoted by , 1 2( , ) and the CF of ,

denoted by 1( ) [4]. Following (13), to express the

crossings in one direction we include factor 1/2 in [4, Eq. 2] to obtain [5, Eq. 4]

1, 1 2 11 22 2

2

( , ) ( )1.

4j R

RN e d d

(14)

Generally, the LCR can be calculated by substituting (10) in (14) and performing double numerical integration over the variables 1 and 2 . Although valid for arbitrary powers n

and any real-valued fading parameters mn, this approach has some limitations. This method requires a double integration with infinite bounds, using complex analysis numerical methods, and it is usually computationally expensive. Furthermore, the function in the integral in (14) has a discontinuity for 2 0 , which has to be avoided during the

numerical integration , as that often leads to inaccurate results. If we define variables

2 2 4m8n na f , 22n nd ,

1(1 ) /n n nc j d a , the joint CF can be expressed as

2, 1 2 2

1 1

( , ) ( + )n n

N Nm m

n nn n

a c

. (15)

For integer values of mn, the second product in the expression given above can be transformed from the product into a sum by using the partial fraction method to obtain

457

2,

, 1 2 21 1 2

( , )( )

nmn l

ln l n

K

c

. (16)

The corresponding partial resolving coefficients Kn,l are calculated using an approach described in [6, Eq. (15)]. For a special case of a two-branch MRC diversity we obtain

21,

1 ( 1).

1 ( )

n

n k

m lk n

kn l m m lk n k k n

m m lK

m c c

(17)

In this case, the LCR expression can be transformed into the following form

1

2 2

, 121 11

212 2

2 2

1( )

4

1 1 [ ( ) ] ,

( )

n

t

mm j R

R t n ln lt

l ln n

N a K j e

dd

c c

(18)

that can be further simplified by applying the successive integration over the variables 2 and 1 .

As it is shown in [6, Eq. (A2)], the integral in the square brackets can be represented in a closed form and the LCR expression can be simplified into

1

2 2,

12 2 2 1/ 21 11

2 11.

4 2

n

t

mn lm j R

R t l ln lt n

KllN a e d

l c

(19) By substituting (17) in (19), and observing that k=1 if n=2

and vice-versa, and by using the identity derived in [6, Eq. (A8)], we derive the final expression for LCR

22( ) 1 2( )2

2/(2 ) 1/ 21, 2 2

1m

2

, , 1 12 21

( )

1 1 ( ; ; ),

2 2

k n n k

n n k

n k

n

m m m mR m m n kR

k m mk nn n k

lmn

n k l n k n kl k k

Ne R

f

Rp F m m l m m

(20)

where

2 1

, ,

12 1

1( 1) 22 2 1( )!

n n k

k n

m l m m lk

n k lk nk n

k n

m m ll

mllp

m mm m

m m

. (21)

In a Rayleigh fading case, when 1n km m l , we obtain

, , / 3, ,n k lp n k and (20) reduces to [5, Eq. 16].

V. AUTOCORRELATION FUNCTION

By combining (1) and (3), the ACF of the MRC output, denoted by , ( ) , can be obtained as

2 2 2 2, , , , , , , , ,

1 1 1 1

( ) [( )( )].N m N m

n i t n i t k l t k l tn i k l

E c s c s

(22)

Random variables 2,n ic and 2

,k lc are independent for n k

or i l . By using the ergodicity property of the observed RVs and identities in [2, Eqs. 1.3-5], the ACF expression can be rewritten in the following form

2 2 2 2 2 2, , , , , , , , , , , , ,

1

2 2 4 2 2, , , ,

1 1,

( ) { } { } { }

{ } 4( 1) 4 .

N

n n i t n i t n i t n i t n i t n i tn

N N

n i t n i t n n n k n kn k k n

m E c c E c s E s c

E s s m m m

(23)

The joint moment of the fourth order of two arbitrary correlated zero-mean Gaussian RVs, denoted by 1x and 2x ,

with variances 2 2 21 2{ } { }E x E x is given by [5, Eq. 18]

2 2 41 2 1 2{ } 2( { }) ,E x x E x x 2

(24)

and can be applied to calculate

2 2 2 2 4, , , , , , , , 0

2 2 2 2 4, , , , , , , ,

{ } { } (1 2 (2 )),

{ } { } ( , ).

n i t n i t n i t n i t n

n i t n i t n i t n i t n

E c c E s s J f

E c s E s c n i

m

(25)

Finally, we derive the ACF of the MRC output for the case of the isotropic environment

4 2 2 2, 0

1 1

( ) 4 [ ( (2 )) 2 ]N N

n n n n k n kn k n

m m J f m m

m .

(26) As the mean power of the signal at the output of the MRC

combiner is determined by , (0) , the corresponding rms

value, denoted by 1/ 2,( (0))rms , is derived as

1/ 2

4 2 2 2

1 1

2 ( ( ) 2 ) .N N

rms n n n n k n kn k n

m m m m

(27)

For the case of a Rayleigh fading channel (m=1), two previous expressions reduce to [5, Eq. 20] and [11, Eq. 16], respectively. To the authors' best knowledge, results (20), (26), and (27) have not been derived before.

VI. NUMERICAL RESULTS

A waveform sequence with 710L samples is generated for every diversity branch by using an improved Jakes fading simulator [12], with the sampling interval of smp m0.02 /T f

and fm=50Hz. An exponential power decay profile model [2] is used, i.e. 2 2

1 exp( ( 1))n n , where n=1,2,...,N. The

MRC combiner output signal is generated by applying relations (1), (3) to each sample, and the number of crossings is counted for various levels R. These results are compared with the analytical results, obtained using (20) and presented for the normalized level / rmsR in Fig. 1.

LCR values are presented in Fig 1, for the various fading factor values m1 and m2 and the case of unbalanced diversity branches that corresponds to power imbalance parameters =1 and =2. It is obvious that the power imbalance of diversity branches results in an increase of LCR values.

458

−35 −30 −25 −20 −15 −10 −5 0 5 1010

−4

10−3

10−2

10−1

100

20log10

(R/rms

) [dB]

Nor

mal

ized

leve

l cro

ssin

g ra

te, N

R/f m

=1, analytical

=1, simulation

=2, analytical

=2, simulation

m1=m

2=1

m1=m

2=10

m1=m

2=4

m1=m

2=2

Fig. 1 Level crossing rate, equal fading factor values

This is especially significant for small fading factor values

(m≤2), and the imbalance impact diminishes for larger fading factor values (m≥4).

The corresponding ACF, normalized to , (0) , is

estimated by simulation and compared with analytical results in Fig. 2, for various normalized time shifts m2 f . The ACF

curves are presented for various fading factors m1=m2 and the case of power unbalanced diversity branches. It is obvious that the power imbalance decorrelates the MRC output signal. This effect is especially significant for small values of the fading factor (m≤2). For the case of unbalanced Rayleigh distributed branches the decorrelation effect is more noticeable for high diversity orders (large number of branches) [5, Fig 2].

VII. CONCLUSION

In this paper, analytical expressions for the second order statistics of the MRC diversity receiver output, for Nakagami propagation with independent, but unequally distributed branches have been derived.

The novel LCR expression is derived in closed form for the case of two diversity branches with unequal powers and integer-valued fading parameters. The ACF expression is derived in an exact closed-form for an arbitrary number of diversity branches and fading parameters. The derived expressions are verified by an independent method, based on computer simulation. In all analyzed scenarios, an increase in Nakagami fading parameters correlates the output MRC signal and lowers the LCR values. The increased power imbalance decorrelates the output signal and increases the number of crossings. It has been shown that an increase in the branch imbalance has a more significant impact on the LCR and ACF for small values of Nakagami fading parameters.

0 1 2 3 4 5 6 7 80.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

2fm

,

()/

,(0

)

=1, analytical

=1, simulation

=2, analytical

=2, simulation

m1=m

2=1

m1=m

2=2

m1=m

2=4

m1=m

2=10

=2

=1

=1

=1

=1

=2

=2

=2

Fig. 2 Autocorrelation function, equal fading factor values

ACKNOWLEDGEMENT

This work was supported by the Serbian Ministry of Science under technology development project TR32028 - "Advanced Techniques for Efficient Use of Spectrum in Wireless Systems”.

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