Groupwise Generalized Selection Diversity Combining and Its Applications in Wireless Communication

Ning Kong and Laurence B. Milstein , ECE. UCSD

Abstract -The paper introduces a new concept of groupwise generalized selection combining (GGSC), denoted by

),,( LmnGGSC , in which there are n groups of generalized selection combining ),( LmGSC where the largest m (in SNR) elements are selected from L independent elements. The paper proves that

),,( LmnGGSC has the same diversity order (DO) as that of MRC with nL elements, denoted by )(nLMRC and ),( nLnmGSC which selects the same number of branches globally from the same total of nL elements. The paper quantifies the performance differences among ),,( LmnGGSC , ),( nLnmGSC and )(nLMRC . The paper shows the SNR loss of ),,( LmnGGSC relative to ),( nLnmGSC is less than 2.59dB for arbitrary m, n and L. For diversity orders of practical interest (<12), the loss is less than 0.6dB

I. INTRODUCTION Due to the benefits of diversity in wireless communications, for example, diversity order reduces the SER exponentially,

),( LmGSC with special cases of ),( LLGSC namely MRC(L), and ),1( LGSC namely selection, has been well studied in references such as [1-7], to name only a few and implemented in many cellular systems. However, in reality ),( LmGSC is not general enough to cover all diversity selection cases, some of which are of great practical interest. For instance, consider a MIMO system, where L Tx antennas transmit the same signal, as shown in Fig.1. For a frequency non-selective fading channel, each of n Rx antennas receives L independently faded signals. The receiver selects and switches to the n strongest signals from the total nL incoming signals, assuming the composite signal can be separated, for example, by Walsh codes [8]. Since the symbol timing synchronization after switch consumes time and when this time is greater than the channel coherence time, the switch can only be done among the signals having the same timing (so there is no need to resynchronize when choosing the stronger signal). In other words, ),( nLnGSC is not applicable here before grouping the elements with the same symbol timing. Another example shows that grouping is not necessary as the above example, but it reduces the number of switches. Consider a MIMO system, with four Tx antennas for either diversity gain or spatial multiplexing gain, as shown in Fig.2a. A cost-effective design is to have two radio frequency (RF) chains (instead of four, since the components of an RF chain are much more expensive than a sole antenna) connecting to the two antennas which have the strongest signals [8]. Then the RF chains need to switch to the antennas with the strongest signals every channel coherence time. Because the switching has negative effects due to the making and breaking of connections, it is desirable to switch as little as possible. One way to accomplish this is as shown in Fig. 1b, where each RF chain only connects to one (m=1) of the two (L=2) antennas, so that there are fewer antennas for each RF chain to switch to compared to the system of Fig. 1a, and this results in a smaller number of times. The average number of switch is in Fig. 1b is 0.5+0.5=1, which 40% less than that in Fig.2a, where the average switch time is ¾ +2/3=1.4. Fig.2a is an

example of GSC(m,L) where m=2 and L=4. Fig. 1b is an example of ),,( LmnGGSC where n=2, m=1 and L=2. It is obvious that as the

total number of antennas increases, the saving in switch also increases. Note that the two signals that are chosen in Fig.1b are not necessarily the strongest two signals from the total incoming four signals, as in case of Fig. 1a, since the two signals are not selected globally from the four signals, but rather in groups. Note also that more switches are reduced as n and L increase. Since all elements of ),,( LmnGSC are independent and the n groups are independent, the joint density of the nm selected elements in ),,( LmnGGSC is the product of the densities of the selected m elements in ),( LmGSC for all n groups. The average combined SNR of ),,( LmnGGSC is the sum of the combined SNR for ),( LmGSC for all n groups. This paper presents the closed-form combined average SNR expressions for ),,( LmnGGSC , as well as single-integral expressions for the SEPs of both MPSK and M-QAM over Rayleigh fading channels. We prove that the combined SNR/SEP of ),,( LmnGGSC is smaller/greater than that of ),( nLnmGSC . We prove that ),,( LmnGGSC has the same DO (nL) as that of both ),( nLnmGSC and )(nLMRC and we derive a simple closed-form asymptotic SEP (ASEP) for ),,( LmnGGSC . We also derive the SNR gaps among these three combining schemes. It is found that the gap between ),,( LmnGGSC and ),( nLnmGSC decreases with m, as expected since both approach )(nLMRC as m increases, so the maximum gap occurs when 1=m with fixed n and L. Further, it proves that for a fixed DO, the maximal SNR gap is achieved either when L=2 or 3 and the SNR gap between

),( nLnGSC and ),1,( LnGGSC increases as n increases with a fixed L. Therefore, the maximal gap for any n, m and L is found by letting

∞→n in ),1,( LnGGSC and choose the greater values between L=2 and 3. This max is 2.59dB with L=3 as ∞→n . In addition, for a fixed L1 of practical interest (<12), this gap is shown to be less than 0.6dB. The remainder of this paper is organized as follows: Section II presents the closed-form expression for the combined SNR of and a compact integral expression for the SEP of MPSK and QAM signals in a Rayleigh fading channel for ),,( LmnGGSC . It also proves that they are upper and lower bounded by SNR and SER of ),( nLnmGSC . Section III proves that ),,( LmnGGSC has the same diversity order as that of )(nLMRC and presents a simple closed-form expression for the ASEP of ),,( LmnGGSC for both MPSK and M-QAM signals. Section IV derives the SNR gap between

),,( LmnGGSC and ),( nLnmGSC . It is found that this gap is maximized when m=1 with respect to m, and the gap with m=1 monotonically increases with n (when L=2 or 3) and can be at most 2.59dB as L1 goes to infinity. When the diversity order is less than 12, the gap is less than 0.6dB. Section V presents BER numerical results to show the performance difference between ),,( LmnGGSC and ),( nLnmGSC with different m and n for Rayleigh fading channels. They show a good agreement with the theoretical analysis. Finally, Section VI presents the conclusions.

II. THE COMBININED SNR AND COMPACT SEP OF

),,( LmnGGSC FOR RAYLEIGH FADING CHANNELS The average combined SNR of ),,( LmnGGSC over any fading channel, denoted by ),,( LmnGGSCΓ , can be found by averaging the sum of mn× instantaneous SNRs of the selected diversity branches

from the n groups, denoted by ∑∑= =

n

j

m

ljlz

1 1, where jlz is the

instantaneous SNR of the jth group and lth branch and it is simply the sum of the average combined SNR in each group, Hence

),(),,( )(

11 11 1

LmzEzELmn jGSC

n

j

n

j

m

ljl

n

j

m

ljlGGSC Γ=

=

=Γ ∑∑ ∑∑∑

== == =

where ),()( LmjGSCΓ is the average combined SNR of ),( LmGSC for the

thj group. For the iid case, where ),(),(),( )()( LmLmLm GSCi

GSCj

GSC Γ=Γ=Γ for all ij and , so that ),(),,( LmnLmn GSCGGSC Γ=Γ , For Rayleigh fading channels, from [6,9], ),( LmGSCΓ has the simple closed-form expression

∑−

= ++=Γ

mL

iGSC imc

mLm

1

Rayleigh )1

1(),( . (1)

Where c is the inverse average SNR per diversity branch. Therefore,

∑∑∑∑−

=

−

=== ++=

++=Γ=Γ

mL

i

mL

i

n

j

n

j imcm

nimc

mLmLmn

1111

RayleighGSC

RayleighGGSC )

11()

11(),(),,(

From (1), the average SNR of ),( nLnmGSC over Rayleigh fading, denoted by ),(Rayleigh nLnmGSCΓ , is obtained by replacing m with nm and L with nL in (1) as

∑−

= ++=Γ

)(

1

Rayleigh )1

1(),(mLn

iGSC inmc

mnnLnm

Obviously, when 1=n , )1,( LmGSC = )1,,1( LmGGSC , and when 1Ln = or 1=L , GGSC and GSC are both MRC with 1L branches. However, when 1≠n and 1Ln ≠ , we prove that ),,( LmnGGSCΓ is upper-bounded by

),(Rayleigh nLnmGSCΓ as follows:

(2) ),,()1

1()1(

)1

1())1(

11(

))1(

11()

11(),(

RayleighGGSC

11

1 11 1

1 1

)(

1

RayleighGSC

Lmnkmc

mn

knnmn

cm

n

knnmcm

nnnknmc

mn

jnknmcm

ninmc

mnnLnm

mL

k

mL

k

mL

k

nj

j

mL

k

nj

j

mL

k

nj

j

mLn

i

Γ=+

+=+

+

=+

+=+−+

+≥

+−++=

++=Γ

∑∑

∑ ∑∑ ∑

∑ ∑∑

−

=

−

=

−

=

=

=

−

=

=

=

−

=

=

=

−

=

Note that the change of the index of nkji )1( −+= on the second equation in (2) transforms the sum over i to sums of k and j, equivalently from one group to n groups and in each group. Then use only one value, n, instead of nj ≤≤1 to represent the restricted selectivity. Its restriction made greater sign in (2). Or the average SNR of GCS is always greater than or equal to that of GGSC due to GSC has a better selectivity. Equality holds when 1=n or L1, where GGSC is the same as either GSC or MRC. The conditional error probability of MPSK and M-QAM when selecting and coherently combining the m elements with the largest instantaneous SNR in n groups, denoted by njzz jmj ,,1,,, 1 LL = , can be generalized to the following form

[10, Table I, 11] (which is the ideal form for a MGF-based analysis [1]):

φπ

β α φ dezeP i

m

ljl

n

j

zg

i

N

i

mll ∫

∑∑∑ ==

−

== =

0

sin

11

1121

)}{( (3)

where g, N, iα and iβ are constants depending on modulation types. g N

1α 2α 1β 2β

MQAM )]1(2/[3 −M 2 2/π 4/π MA 11−= 2A

MPSK )/(sin 2 Mπ 1 πMM )1( − 0 1 0

Table I: Parameters of M-QAM and MPSK in (5) We consider the decorrelation transformation [6]

,1+−= jljljl zzx 1,,2,1 −= ml L and jmjm zx = (4)

Note that this transformation is more general than the Sukhatme transformation [1], where the ranking and decorrelation are on all L branches, even though only m branches are selected. In (4) we decorrelate only among the selected branches. When Lm = , (4) reduces to the Sukhatme transformation. Using (4), the conditional error probability (3) becomes

φπ

β α φ dexePzeP i

m

ljl

n

j

lxg

i

N

imlnjjlmlnjjl ∫

∑∑∑ ==

−

===== ==

0

sin

1,,1,,1,,1,,1

1121

)}{()}{(LLLL

(5)

The density function of njzz jmj ,,1,,, 1 LL = for ),,( LmnGGSC is

simply the product of n density functions of ),( LmGSC . For Rayleigh fading case,

{ }

nj

zzeec

zzzzf

jmjzcL

ml

n

j

L

ii

m

k

zcji

nmnmZZZZ

jmlji

m

jkkji

k

nmnm

,,1

, )},1({

),,,,,,,(

1

11

,,1

, 1

1111,,,,,,,

1

)()1()1()11(

L

L

LLL

L

L

LLL

=

≥≥≥∞−

=−

+== =

−∏∏ ∑ ∏

(6)

where{ }

)1(1

,,1

, 11

jmlji

m

jkkji

k

zcL

ml

L

ii

m

k

zcji eec

−

+==

− −∏∑

∏

L

L

is the joint density of the

elements in the j th group [6], jkkji

k

zcji ec − is the density for the thj

group and the thki branch, { }Lik L,2,1 ∈ , )1( jmlji zc

e−

− is the

cumulative distribution function for the thj group and thli branch,

{ }Lmml L,2,1 ++∈ , and { }∑ Lii m

,,1,,1

L

Ldenotes the sum of all permutations

of mii ,,1 L taking values in the set { }LL,2,1 . When hl

kiji cc = for

all hl iikj and , , , using the same transformation (4) on the density function of (6), we have

nj

mlxxfeec

mLL

xxxxf

jln

j

m

ljlX

n

j

mLcxclxm

l

nmnmXXXX

jl

jmjl

nmnm

,,1

1 ,0,)( )1(

)!(!

),,,,,,(

1 11 1

1111,,,,,,

)(

)()1()1()11(

L

LLLLLL

=

≤≤∞<≤=

−

−

=

∏∏∏ ∏= ==

−−−

=

(7)

Where

=−−

=−≤≤=

−−−

−

),,(/)1(

,,1,11 ,)(

)( mlmLmBee

njmllcexf

mLcxcmx

clx

jlXjmjm

jl

jl

L (8)

and )1(/)()1(),( +ΓΓ+−Γ≡− LmmLmLmB . Note that (8) reduces to the density function in the Sukahtme Theorem if Lm = in (8). The error probability of ),,( LmnGGSC , denoted by )//( Lmn

GGSCPe , is obtained by a MGF approach as

φφπ

βφ

φπβ αα d

gd

gMPe ii

m

l

nN

i

iLmnN

ii

LmnGGSC ∫ ∏∑∫∑

===−Μ=−=

01

21

2

)//(

01

)//( )]sin

([)sin

(1

where in (9), )sin/( 2)//( φgLmn −Μ is the MGF of ),,( LmnGGSC and

jljlX

lxg

dxxfegjl

jl

)()sin/(2sin

02 φφ

−∞∫≡−Μ is the MGF of the rv jlX . Let

φ2sin/gs ≡ and use the density function of (8), a closed-form solution to )sin/( 2 φg−Μ for iid Rayleigh fading is given by

)10( 11 ,1/

1 )sin/(

002 m-l

csdyeedxlceeg yy

csyclx

jlclxlsx jl

jljl ≤≤+

===−Μ −∞ −=−∞ −

∫∫φ

mcsAmLmcs

ye

jmmLcxmxcs

jmmLcxcmxmsx

dyyymLmB

dxee

mLmBc

dxeceemLmB

csm

jmcx

jmjm

jmjmjm

)1/(10

1)1/(0

)(

0

)1(),(

1)1(

),()1(

),(1

);(

+=−−+

=−−∞ +−

−−−∞ −

=−−

=−

−=−

−=−Μ

∫∫

∫

−

(11) )/1/

1()

/)1(1/1

)(1/

1(

!!

))1/(1

()1)1/(

1](

)1/(1

[)!1(

!)()1(1)1)((

)!1()!(!

)1(),(

1 10

1

mlmLcsmmcscsmm

LLmcsmmcsmcsm

LmLAAA

mLmLmmL

Ldyyy

mLmB

mL

mLA

=++++++

=

++++++−=

−++−−−

−−=−

−

−

−−∫

L

L

L

L

Substituting (11) and (10) into (9),

( ) (12) ])/1/

1()

1/1

([1

!!

]/1/

1/)1(1/

1)1/

1(

!!

[1

10

1

01

)//(

φπ

β

φπ

β

α

α

dmimcscsmm

L

dmLcsmmcscsmm

LPe

mL

i

nmnN

i

n

mLi

nmmL

N

ii

LmnGGSC

i

i

∏∫∑

∫∑

−

==−

−=

++++

=

++++++= L

If n=1, (12) is the SEP for ),( LmGSC [7, Eq(13)]. By partial fractional expansion, (12) can be expressed as an integration of the sum of the product of terms, )/(sinsin 22 b+φφ and )/(sinsin 22 a+φφ

raised to various powers. Since φφφ

φφ

πα d

abI jii )

sinsin

()sinsin

(1

2

2

2

2

0 ++= ∫ ,

for ij ≠ , ba ≠ , and an arbitrary iα , does not have a closed solution [1,p136], (12) does not have a closed-form expression. The SEP of

),( nLnmGSC , denoted by )/( nLnmGSCPe , can be obtained from (11) by

replacing m with nm (since we are selecting nm instead of m) and L with nL (due to the fact that there are a total of nL elements in

),( nLnmGSC ) and then finally letting 1=n (because there is only one group now), which yields

( )

( )∏∫∑

∏∫∑

−

=−

=

→→

−

==−

++++=

++++

=

nmLm

i

nmnmnL

N

ii

nmmLmL

mL

i

mN

imLi

nLnmGSC

dmnimncscsnmnm

nL

dmimcscsmm

LPe

i

i

10

1

10

1

)/(

/1/1

)1/

1(

)()!()!(

[1

)]/1/

1()

1/1

([1

!!

φπ

β

φπ

β

α

α

[ ] (13) /)1(1/

1)1/

1(

)()!(

)!([

1

1 10

1∏∏∫∑−

= =−

= +−++++

mL

i

n

l

nmnmnL

N

ii d

mnlninmcscsnmnm

nLi φ

πβ α

We prove that )/()//( nmnL

GSCmLn

GGSC PePe ≥ by showing that the integrand of )//( mLn

GGSCPe is greater than the integrand of )/( nmnLGSCPe , i.e.,

[ ]

[ ](14) 1

]/)(1/[

]/)1(1/[

)()!/()!(!!

]/)1(1/

1

)()!(

)!([])

/)(1/1

(!!

[

1

1

1 11

≥+++

+−+++

=

+−++++++

∏∏

∏∏∏

−

=

=−

−

−

= =−

−

=−

mL

in

n

lnmnL

n

mL

mL

i

n

lnmnL

mL

i

nn

mL

mimcs

mnlninmcs

nmnmnLmmL

mnlninmcsnmnm

nLmimcsmm

L

Eq (14) is true for the following two reasons, expressed in (15) and (16):

(15) 1)])1([(/)]([)1()1(])1()1([

)1()1(])1()1([

)()!()!(

!!

1 1 1≥+−++=

+−+−

=+−

+−=

∏ ∏∏−

=

−

= =

−

−

−−

mL

l

mL

l

n

i

nnmL

nmL

nmnL

n

mL

inlnmlmnnmnLnLnmLL

nmnLnLnmLL

nmnmnL

mmL

L

L

L

L

The last equality is true due to the fact that niinlnmlmn ,,1for ,)1()( L=+−+≥+ . Interestingly, in (15), both the

numerator and denominator are products of )( mLn − integers. All integers in the denominator are distinct (reflecting the better selectivity of ),( nLnmGSC ), while there are only mL − distinct integers in the numerator, with each of them repeating n times (reflecting the restricted selectivity of ),,( LmnGGSC due to the groups).

For the same reason, we have )1(

mnlninm

nmninm

mim +−+

≤+

=+

Or 1]/)(1/[}/])1([1/{1

≥++++−+++∏=

nn

lmimcsmnlninmcs .

Consequently )/()//( nmnLGSC

mLnGGSC PePe ≥ and the equality holds obviously

for both 1=n and 1Ln = , where GGSC=GSC and GGSC=GSC=MRC, respectively.

III. THE DO AND ASYMPTOTIC PERFORMANCE OF ),,( LmnGGSC

In this section, using the approach in [10], we first derive the dominant part in the joint density of ),,( LmnGGSC , which contains only the inverse SNR to its DO. The DO is the lowest power of the inverse SNR, i.e., the product of all

Lknjc jk ,,1,,,1 , LL == in the PDF of ),,( LmnGGSC . Then we derive

its ASEP for both MPSK and M-QAM signals. The dominant density term, averaging which over the conditional SER producing the ASER, for ),( LmGSC is shown in [10] to be given by

mi

xxc

mLL

xxf imLm

L

kkm

domGSC

LL

1

,0 ,)(

)!(!

),,(1

1)(

=

∞<≤

−= −

=∏ (16)

When Lm = , we have the dominant density of )(LMRC which is

Li

xcLxxf i

L

kkm

domMRC

LL

1

,0 ,!),,(

11

)(

=

∞<≤= ∏

=

Due to the independence of all groups, this dominant term for ),,( LmnGGSC is simply the product of the n dominant densities of

each ),( LmGSC , or

mLjm

n

j

n

j

L

kjk

nnmnm

domGGSC xc

mLL

xxxxf −

== =∏∏∏

−= )()(]

)!(!

[),,,,,,(11 1

1111)(

LLL (17)

From (17), it is seen that the DO for ),,( LmnGGSC is nL, which is the same as that of )(nLMRC .When Lm = , (17) becomes

∏∏= =

=n

j

L

kjk

nnLnm

domGGSC cLzzzzf

1 11111

)( )!(),,,,,,( LLL (18)

Which is the dominant density of )(nLMRC . Similarly, the dominant MGF (DMGF) of GGSC, denoted by )sin/( 2)//( φgM Lmn

dom − , is also a product of the n DMGFs of GSC,

denoted by njccgM jLjLm

dom ,,1),sin/( 1,2)/(

LL =− φ . From (5) with n=1

and (17)

mmL

m

xgm

L

kLk

lxg

m

mLm

Lk k

lxg

jLjLm

dom

dxxemL

dxdxcLe

dxdxmL

xcLecc

gM

m

L

ll

m

ll

−∞ −

=

∞ ∞ −

−=

∞ ∞ −

∫∏∫ ∫

∏∫ ∫

−

∑=

=−

∑=−

=

=

)()!(

1][![

)!(

!),

sin(

0

sin

11

0 0

sin

11

0 0

sin12

)/(

21

2

12

φφ

φ

φ

LL

LLL

!!

)sin/(/ 1,2)/(sin

0 0

2

mmL

ccgMdxdxe mLjLjLL

domLm

lxg L

mll

−

−∞ ∞

−=∑=∫ ∫ LLL φφ . Also,

)sin/()sin/( 11,2)//(

1,2)/(

1nL

LLndomjLj

LLdom

n

jccgMccgM LL φφ −=−∏

= . Therefore,

(19) )!

!)(,,(]

!!

)([

)()(

11,sin

)//(

11,

sin)/(

1,sin

)/(

111,

sin)//(

22

22

nmLnL

gLLndom

n

jmLjLj

gLLdom

jLjgLm

dom

n

jnL

gLmndom

mmL

ccMmm

LccM

ccMccM

−−

=−

−

−

=

−

=

==

∏

∏

LL

LL

φφ

φφ

Then the ASEP of the signals with the conditional SEP given in (4), denoted by )//(

_LmnaGGSCPe is given by

=)//(_LmnaGGSCPe

(20) )!

!()

!

!)(,,(

1 )//(_11,

sin)//(

01

2

nmL

LLnaGWGSC

nmLnL

gLLndom

N

ii

mm

LPed

mm

LccMi

−−−

==∫∑ φ

πβ

φα

L

where )//(_LLnaGGSCPe is the ASEP of ),,( LLnGGSC given by

φπ

β φα ddxecLPen

jl

L

l

glxn

j

L

ljl

nN

ii

LLnaGGSC

jli ][)!(1

1 1 0

sin/0

1 11

)//(_

2

∏∏ ∫∫ ∏∏∑= =

∞−

= ===

sin1

1 10

2

11 1∏∏∫∑∏∏= === =

− ==n

j

L

kjk

nLN

ii

n

j

L

kjk

nL cCdcg i φφπ

β α (21)

In (21), the constant C equals )/(sin0

121 πφφβ α nLLN

i i gdC ∫∑ =≡

( )}

)1(2])1(2sin[

)1(21

21

{ 12

11

0112

111212

1

jLjL

CCg

jL

L

j

jL

LLLLnL

Ni i

−−

−−

+= ∑∑ −

=−

= αππ

αβ (22)

Substituting (22) into (21) and then (21) into (20), we have the ASEPs for either MPSK or M-QAM given by

nmL

n

j

L

kjk

LmnaGGSC mm

LcCPe )

!

!(

1 1

)//(_ −

= =∏∏= (23)

From (23), we can also see that the diversity order of ),,( LmnGGSC is nL regardless of the value m . For the iid case, (23)

can be rewritten as

])!

![()

!

!()( /1)//(

_/1)//(

_ cmm

LPe

mm

LcCcPe L

mLLLnaGGSC

nLLmL

nLLmnaGGSC −−

== (24)

Thus, the SNR gap of GGSC for different m , compared with Lm = (MRC), and denoted by ),,( LmnSNRgap

MRCGGSC− , is given by

LmL

gapMRCGGSC

mm

LLmnSNR /1)

!

!(),,(

−− = , or

(dB) !

!log

10 ),,( 10 mm

LL

LmnSNR mLgap

MRCGGSC −− = (25)

Eq.(25) is exactly the SNR gap between ),( LmGSC and )(LMRC [9]. When L and m is fixed, (25) remains the same, regardless of n. This is because (25) is an average over n groups. But if the DO is fixed, L in (25) varies with n. When n increases, or L decreases, (25) decreases. This is intuitively true since, for a fixed m, as L decreases and nm increases (or more branches are combined), so that ),,( LmnGGSC becomes closer to )(nLMRC . This property can be used to improve the performance of ),,( LmnGGSC by varying n for a fixed m. For example, if we select the largest signal from each of the Rx antennas as shown in Fig. 1, the performance loss by using ),1,( LnGGSC compared to that of MRC is given by

dB /L!log01 ) ,1 ,( 10 LLnSNR gapMRCGGSC =− (26)

This is obtained by letting m=1 in (25). In order to reduce (26), one can increase n or decrease L. We plot, in Fig. 3, the performance comparison (only SNR gaps since diversity order is the same) for all possible n with diversity order nL= 12, 6 and 4. Here, n also represents the number of the Rx antennas and L the number of Tx antennas. In Fig.2, the top curve is the SNR gap between

[ ])/12 ,1 ,( nnGGSC and )12(MRC , and we can see that the gap monotonically decreases as n increases because more branches are combined. For the other two curves, corresponding to diversity orders 6 and 4, the same properties are displayed, For all three curves, when n =L1, GGSC is equivalent to MRC, and requires L1+1 antennas. When 1=n and L=L1, GGSC has the worst performance, even though it also requires L1+1 antennas. In particular, when 41=L , the pair (n,L) can take the values (4,1) (2,2) and (1,4). The performance is the best with (4,1), which is MRC(4) and requires 5 antennas. Note that the encircled point on Fig.2 (n=2,L=2) with (gap, n)=(1.5dB ,2) illustrates that the performances of GGSC(2,1,2) and MRC(4) are 1.5dB apart, which can also be seen using (26) i.e., gap= dB 1.5 !2log5.0 10 = . Similarly, the encircled point (n=3,L=2) with (gap, n)=(1.5dB, 3) shows that the performances of GGSC(3,1,2) and MRC(6) are 1.5dB apart.

IV. PERFORMANCE COMPARISON BETWEEN

),,( LmnGGSC AND ),( nLnmGSC

Since the DO of ),,( LmnGGSC is found the same as that of )(nLMRC in the previous section and it is also known that the DO of )(nLMRC is the same as that ),( nLnmGSC , then ),( nLnmGSC and

),,( LmnGGSC have the same DO. The system degradation for high SNR is the SNR gap, and in this section, we quantify this gap. If we replace m with nm and L with nL , (24) becomes the ASEP of

),( nLnmGSC as

)!()(

)!(

)!()(

)!( )/(_

)//(_

)/(_

nmnm

nLPe

nmnm

nLPePe

nmnLnLnL

aGGSCnmnLLLnaGGSC

nLnmaGGSC −−

== (27)

where, in (27), aMRCnLnLaGSC

LLnaGGSC PePePe _

)/(_

)//(_ == , and )/(

_nLnLaGSCPe is the

ASEP of ),( nLnLGSC . We notice that it is identical to )//(_LLnaGGSCPe in

(24) because both are equivalent to )(nLMRC . From (27), the SNR gap between ),( nLnmGSC and )(nLMRC is

),( nLnmSNRgapMRCGSC−

( )( ) ( )!

!log

11

10nmnm

nLL nmnL−= . This gap can also be

obtained by letting L=L1 and m=nm in (25). Then the SNR gap between GGSC and GSC is simply

( )( ) ( )!

!log11

!!

log1

),(),,(),,(

1010nmnm

nLLmm

LL

nLnmSNRLmnSNRLmnSNR

nmnLmL

gapMRCGSC

gapMRCGGSC

gapGSCGGSC

−−

−−−

−

=

−= (28)

Since we have proved that 1)()!()!(

)!!

( ≥−− nmnLn

mL nmnmnL

mmL in (15),

(28) is lower-bounded by zero with equality when 1=n and 1Ln = , which corresponds to single group selection and MRC, i.e.,

),( LmGSC = )1,,1( LmGGSC and )1,1( LLGSC = )1,1,1(LGGSC , respectively. We prove in Appendix I that the gap in (28) is monotonically decreasing as m increases. When 1=m , the gap is maximal for a fixed n and L and is given by

( ) ( )( ) !

!log

11

!log11

),1,( 1010_

nn

nLL

LL

LnSNRnnL

ngapMaxGSCGGSC −− −= (29)

We also prove in Appendix II that (29), for a fixed L, monotonically increases as n or L1 increases. In Appendix III, we prove that, when L1 is fixed, (29) is monotonically increasing with n when 3≥L . When 2=L , increasing n can either increase or decrease (29). This means, for a fixed finite diversity order L1, the maximal gap between GGSC and GSC can happen either at 2=L or 3. Then we prove in Appendix IV that

=< −−gapMaxGSCGGSC

gapMaxGSCGGSC SNRLnSNR __ ),1,( 2.59 dB for any diversity order

when L=3 as n approaches infinity.. When 121=L ,

( ) ( )( )

dB6.0)!

!log110

!log110(

2,61010

121

_ =−≤==

−≤

−Ln

nnLn

L

gapMaxGSCGGSC

nn

nLL

LL

SNR

Therefore, for a diversity order of practice interest , for example 121≤L ,

gapMaxGSCGGSCSNR _− <0.6dB (30)

Next, for the same system shown in Fig 1, we plot in Fig. 4 the performance difference (SNR gaps) between ),1,( LnGGSC which selects the largest one from the L incoming signals in n groups and

),( nLnGSC which selects the n elements globally from nL incoming signals. It is seen that, as proven theoretically, as n increases for any L1, the gap increases. The gap also increases as L1 increases. For example, when L=2, and n increases from 2 to 3 to 4 (diversity order increases from 4 to 6 to 12), the gap monotonically increases from 0.3 to 0.425 to about 0.55dB. When n=L1 or L=1, the gap is

zero because )1 ,1()1 ,1 ,1( LLGSCLGGSC = . Also, as shown in (30), gapMaxGSCGGSCSNR _− is 0.5dB, (i.e., less than 0.6dB) when L1=12.

V. NUMERICAL BER COMPARISON BETWEEN

),,( LmnGGSC AND ),( nLnmGSC In this section, we study and present some numerical results to compare the performances between ),,( LmnGGSC and ),( nLnmGSC . If

2=M , (24) reduces from ASEP to the asymptotic bit error rate (ABER) of ),,( LmnGGSC , as shown in (31), and (27) reduces to ABER of ),( nLnmGSC , as shown in (32), respectively.

(31) 2

)!

!(

122)//( nLnL

nLnLn

mLLmn

BPSK cC

mmL

Pe+−

= & ( )( ) ( )( )

)32(2!

!12

2)/( nLnL

nLnL

mLnnLnm

BPSK cC

nmnm

nLPe

+−=

In Fig.5, the curves are the ABERs with BPSK for both ),( nLnmGSC and ),,( LmnGWGSC in an iid Rayleigh fading channel,

where 121=L . In ),,( LmnGGSC , the 12 branches are divided into three groups; each has 4 elements. We plot the ABERs of selecting 4,,1L=m paths, and compare them with the ABERs by using ),( nLnmGSC , where the nm =3m branches are selected directly from the nL =12 branches. It is seen that the performances of ),,( LmnGGSC and ),( nLnmGSC have the same slope (L1) and the SNR gap between ),( nLnmGSC and ),,( LmnGGSC monotonically decreases as m increases. The gap is maximal when m=1, as proved in Appendix I and it is only about 0.5 dB, as we proved in (30). The gap becomes zero when Lm = where both GSC and GGSC are equivalent to MRC. In Fig.6, we plot the two sets of ABERs for GGSC and GSC with diversity orders, 12 and 6. The first set corresponds to n=6 and L=2 and the second corresponds to n=3 and L=2. It is seen that the gap between ),,( LmnGGSC and

),( nLnmGSC increases as the diversity order increases, but does not exceed 0.6dB when L1=12, as shown in (31). In Fig. 7, we plot ),,( LmnSNR gap

GSCGGSC− for 4,,1L=m . and choose L1=12. The n in )/12,,( nmnGGSC and )12,(mnGSC increases from 1 to 12/m, for 4,,1L=m . Since L=12/n, L decreases from 12 to m. It is seen that when 1=m , n takes the values of 1,2,3,4,6, and 12, and gap

GSCGGSCSNR − monotonically increases as n increases (as theoretically proven) until n=12 or 1=L , when the gap becomes zero because )1 ,1 ,1()1,1( LLGGSCLLGSC = . When 2=m , n takes the values of 1,2,3,4 and 6. It is seen that the gap first increases as n increases from 1 to 3, and then starts to decrease as n increases from 3 to 4. The reason is that when n increases, ),,( LmnSNR gap

GSCGGSC− tends also to increase since when the number of groups increases, i.e., the selection become more restrictive. On the other hand, the increasing of n results in m being closer to L, and this helps reduce the gap due to the performance now being closer to that MRC. This same property as to how the gap varies as a function of n is also shown when m=3 in Fig. 6. Therefore, when n varies, gap

GSCGGSCSNR − can either increase or decrease, depending on the values of both m and L.

VI. CONCLUSIONS

In this paper, we have studied the performance of GGSC and compared it both with the optimum counterpart GSC with the same number of selected/total diversity branches and with MRC, which yields the best performance. It was found that all three have the same diversity order. Also the SNR gaps between GGSC and MRC/GSC were presented in closed-form expressions as a function of L, m and n. It was found that the performance degradation of

GGSC, due to restricted selection compared to GSC, which selects optimally, is less than 2.59dB for the worst case and less than 0.62dB for diversity orders of practical interest. The properties of how the performance of GGSC varies with n, m and L were also studied.

ACKNOWLEDGMENT

The authors want to thank Dr Tom Eng for his idea of selecting the largest diversity branch in groups (or GWSC)

APPENDIX I: PROOF OF THE SNR GAP ),,( LmnSNR gapGSCGGSC− IN

(28) IS MONOTONICALLY DECREASING AS m INCREASES We first simplify (28) by using the Sterling formula

10 ,)/(2! 12/ <<= θπ θ LL eeLLL , as follows (note LnL ×=1 ): ( ) ( ) ( )

( ) ( )( ) ( ) ( )

( ) ( )( ) ( ) ( ) 1010

1

101

1

10

log of func. aot terms1/]!/![log

]!1!

!![log

11

]!1!

!![log

11

=+=

==

−

−

−

−

−−

mnLmnnm

Lm

nnmLLLmm

nmnmLL

SNR

nnm

n

nmLn

nnmL

nmLngap

MRCGSCGGSC

( ) ( ) ( )

( ) ( ) o terms]/21

)/([log11

o terms1/])/2(/2[

12/1210

12/)12/(

3

2

32

mnememn

eenmmL

mnLeemmneenmnm

mnnmnm

mnmn

nmmnmmnmn

+=

+−

θθ

θθ

π

ππ

of func. aot terms])[(log11

][log11 12

1012

10

32

mnemL

emL

nmmn +−=θθ

−−=+−+

∂∂

=

−+

∂∂

=∂

∂ −−

10ln1211

10ln121

)]log6

(loglog6

[log

121

)(log)log(log21(

11

22

103

10102

10

1210

1210

32

nmLmLe

mmne

mnm

mLemnem

mLmSNR mmn

gapMRCGSCGGSC

θθθ

θθ

0)]21

1(6[110ln12

)]1

1(6[110ln12

)]1(6[

110ln121

10ln1211

10ln121

10ln1211

10ln12

32323

223

23

<−−<−−=−−

=+−

−<+

mLm

nn

mLm

nnmn

Lmmn

LmLn

mn

LmLn

θθθ

θθ

The last inequality is because that 1>n , 1 0 3 <<θ , and

035.06 33 <−=×− mm θθ .Therefore, 0/ <∂∂ −− mSNR gapMRCGSCGGSC . Hence,

the gap is monotonically decreasing as m increases.

APPENDIX II: THE PROOF THAT ),1,(_ LnSNR gapMaxGSCGGSC−

MONOTONICALLY INCREASes AS n INCREASES WHEN L IS FIXED

( ) ( )( ) ( )

( ) ( ) ( )( )

0]log122

)log(log1

[10

]log122

log122

log1

[10

]log122

loglog1

log1

log122

log1[

10),1,(

) terms(nolog1

12loglog

1log)1(log

112

[log10

) terms(nolog1

log1

log1

log1

log1

][log1[

10

log110

) terms(no

!

!log

10!log

10

!

!log110

!log110

),1,(

0

1031

0

6/110

2102103

1103

2102

1031

10102101032

10

_

101

101010102

10

12101010

)1(10

121010

121210

10101010_

12

12

>+−=+−=

+−+−

+−==

=+−

−−−++=

+−

−−++

=

+=

+=−=

>>

−

−

−

−

−−

44344214444 34444 21

eLn

en

LnL

eLn

en

LnL

eLn

enL

Ln

en

Le

ne

nLA

dnd

dnLndSNR

AnennLe

nLLL

nnLe

nnn

L

nene

nLn

Ln

nn

ene

nnL

eenL

Lneen

nLn

nLnn

nLL

Lnn

nLL

LL

LnSNR

gapMaxGSCGGSC

nL

nLLnn

n

nL

nLnnLn

n

nnL

nnL

ngapMaxGSCGGSC

θθθθ

θθ

θθ

θθ

θθ

Note in the last inequality, we have used the facts that 2≥L and 2≥n . Therefore, ),1,(_ LnSNR gapMax

GSCGGSC− increases as n or L1 increases. APPENDIX III: THE PROOF of WHEN L1 IS FIXED THAT MAXIMAL OF (29) HAPPENS EITHER AT 2=L OR 3=L

We first prove that ),1,(_ LnSNR gapMaxGSCGGSC− monotonically increases as n

increases by proving 0/),1,(_ >∂∂ − nLnSNR gapMaxGSCGGSC for 2>L . When

2=L , 0/),1,(_ >∂∂ − nLnSNR gapMaxGSCGGSC can be either positive or negative.

Therefore, the maximum of ),1,(_ LnSNR gapMaxGSCGGSC− can be at either L=2

or 3.

Since ( ) ( )( ) ( )!

!log110

!log110

),1,( 1010_

nnnL

LL

LLnSNR nnL

ngapMaxGSCGGSC −− −=

( ) ( ) ( )( ) ( ) ]!12)2([log

110

!1!!

log110 11212

10

1

10

21

Lneen

neeL

LLL

nnLL

nLn

nnL

LnLn−

−

==

θθ

ππ

We need to prove that when L1 is fixed and nLL /1= ,

( ) ( ) 0]/2)/2[(log 112/12/10

21 >∂∂ −nLnnnLL neenneeLLn

θθ ππ

Let us define A as

( ) ( )

( ) ( ) ( ) ( )

( )

nennnLen

neLn

nL

L

nLn

ennLen

neLn

eLLLn

enen

eeL

Lneen

neeL

LA

not

not

LnLn

LnnL

nnLn

nn

L

L

no termslogloglog12

log21

log112

1log1

12log

2loglog1log

122logloglog

12

loglog12log2

logloglog2log

loglog2log]22[log

1010102

1010

21

10

101010102

n of func.

1010101

n of func.

110101010

110

121010

121010210

1121210

2

121

+−++++

+

=−+++++

−+=++++

+

+=

≡

−

−

θθ

πθπ

θ

ππ

πππ

θ

θθθ

43421

43421

Then

en

Len

nnL

ennL

en

neLn

nL

LnLn

dnd

dndA

1010101010

102

1010

21

1010

log11log

12

12log

21

]loglog1

log12

log21

log112

1log1

12log

2[

−×−

=−+

+++

+

=

π

θθπ

eenL

en

en

en

Le

nL

e

LeenL

en

en

en

L

10101022

10101

1010

1010101022

10101

loglog1

log12

log21

log121

2log1

log21

)2(log21

loglog1

log12

log21

log121

2

−+−++−−

=−+−++

θθ

πθθ

0log121

2log

1)

1221(]log)2([log

23

log23

log21

log12

log121

22log

21

log23

log21

log12

log121

2log1

log1

2log21

0

101

0

102

103/1

10

10101022

101

1010

101022

101

101010

>+−+−=

−+−+=−

+−++−=

>>44 344 21444 3444 21

4444 34444 21

DCB

en

Le

nneL

een

en

en

LLe

en

en

en

Le

nL

enL

L

θθπ

θθπ

θθπ

Since 0>B if 3≥L . When 2=L , 0<B , due to the unknown values 1θ and 2θ take in the range of [0,1], we cannot determine the sign of dndA / . Therefore, for the fixed L1 , we can only conclude that the maximal gap happens either at 2=L or 3=L . APPENDIX IV: THE PROOF THAT gapMax

GSCGGSCSNR _− IS LESS THAN

2.59dB

In order to obtain the maximal ),1,(_ LnSNR gapMaxGSCGGSC− with respect to n,

we let L take either 2 or 3, and n approach infinity and chose the larger gap of the two. When L=2,

( )−=−=

→∞−

→∞dB

nnn

nnSNR nn

gapMaxGSCGGSC

n5.1

!!2

log5

lim2log210

)2,1,(lim 1010_ ( ) ( )3,When

5.1)1

1()1

2(2log5

lim5.1)1()12(2

log5

lim 1010

=

=+−−=+−

→∞→∞

L

dBnnn

dBn

nnnn nnn

LL

( )

( ) ( ) dBnnn

dBn

nnnn

dBnnn

nnSNR

nnn

nn

gapMaxGSCGGSC

n

59.2)1

1()1

2(2log5

lim5.2)1()12(2

log5

lim

59.2!!2

log5

lim!3log310

)3,1,(lim

1010

1010_

=+−−=+−

−=−=

∞→∞→

∞→−

∞→

LL

Therefore 59.2_ <−gapMaxGSCGGSCSNR dB for any diversity order.

REFERENCES

1. M. K. Simon and M. –S. Alouini, “Digital communications over generalized fading channels: A unified approach to performance analysis,” New York, NY: John Wiley & Sons 2. T. Eng, N. Kong and L. B. Milstein, “Comparison of diversity combining techniques for Rayleigh fading channels”, IEEE Tran. on Communications, Vol 44, Sept. 1996 of wireless communications system”, IEEE Trans Comm., 1740-1751, Feb. 1994 3. C. Tellambura and A. Annamalai, “Unified performance bounds for

generalized selection diversity combining in fading channels”, in Proc. IEEE WNCN’2003

4. M. Z. Win; N. C. Beaulieu; L. A. Shepp; B. F. Logan, J. H. Winters, “On the SNR penalty of MPSK with hybrid selection/maximal ratio combining over i.i.d. Rayleigh fading channels,” IEEE Tran. On Comm. June 2003 5. M. K. Simon and M.-S. Alouini, “A Compact performance analysis of generalized selection combining (GSC) with independent but nonidentically distributed Rayleigh fading Paths”, IEEE Tran. on Comm. Sept. 2002 6. N. Kong and L B. Milstein, “Performance of generalized selection diversity combining for both iid and non-iid Rayleigh fading channels”, in Proc. IEEE Milcom’2004 7. N. Kong and L. B. Milstein, “Average SNR of a generalized diversity selection combining scheme for iid Rayley fading channel”, IEEE Communication letters, Vol 3, No 3, pp57-59, March 1999. 8. V. Weerackody, “Diversity for the direct-sequence spread spectrum system using multiple transmit antennas”, AT&A Tech. Memo., 1993 9. N. Kong J. Cartelli and C. Wang, “Simple BER Approximations for Generalized Selection Combining (GSC) over Rayleigh Fading Channels and its SNR Gap Properties”, in Proc. IEEE Milcom’2006 10. N. Kong and L. B. Milstein, “Asymptotic performance of digital communication in fading channels”, Proc. Of IEEE VTC’2007 11. N. Kong and L. B. Milstein, “A Simple Approach to the Asymptotic Performance of Digital Communication over Fading Channels”, submitted to IEEE Tran. On Comm.

Tx ant. 1

Tx ant. 2

Tx ant. L

Rx ant. 1

Rx ant. n

L

L

L

L

L signals coming into Ant 1

Or group 1

L signals coming into Ant n

Or group n

Rrgroup to L

group w the same

time L

Switching withoutChanging timing

Group1 w

sametime

L groups or switchesw n signals inside each

Group L wsametime

Fig. 1 An example of GGSC(n,1,L) in MIMO diversity application

Fig. 2 An example of GGSC(2,1,2) in RF antennas switching

Fig.3 Performance of MIMO GGSC with diversity order L1, n Rx & [L1/n]

Tx antennas

1 2 3 4 5 6 7 8 9 10 11 120

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

n ( # of Rx antennas), Total # of Tx (L)+ Rx antennas = L + n

Gap in dB

The gap between GG(n,1,6/n) and GSC(n,6)

The gap between GG(n,1,4/n) and GSC(n,4)

The gap between GG(n,1,12/n) and GSC(n,12)

Fig.4 Performance comparison between MIMO GGSC and GSC shown in Fig. 1 with diversity order L1 being 12, 6 and 4

Fig. 5: BER comparisons between )12,3( mGSC and )4,,3( mGGSC when m

increases from 1 to L (L=4) 4,,1L=m

Fig. 6: BER comparisons between GWGSC and GSC with diversity order

12 and 6, 2,1=m

Fig. 7: The performance gap between GGSC(n,m,L) and GSC(nm,nL) as

the # of the groups n varies.