Asymptotic Performance of Digital Communication over Fading Channels

Ning Kong and Laurence B. Milstein, ECE. UCSD

Abstract

This paper1 presents a simple way to obtain the asymptotic performance of various digital communications systems operating over fading channels. The resulting expression for performance contains only the inverse average SNR raised to the system’s diversity order, by averaging the conditional error probability (conditioned on the fading) with the dominant term in the fading density function. The proposed approach often renders simple closed-form asymptotic error probabilities (ASEPs), even when the corresponding closed-form actual error probabilities (ACEPs) are too difficult to find, such as those in generalized selection combining, denoted by ),(GSC Lm , where the strongest m branches from L )1( Lm ≤≤ total branches are selected and combined. Using the proposed approach, the paper derives simple closed-form ASEPs for ),(GSC Lm which show its diversity order and SNR gaps among different m, over both iid and non-iid Rayleigh fading channels for both MPSK and QAM. It is found that the SNR gaps of ),(GSC Lm are not a function of modulation type or orders. In addition, the paper also presents simple closed-form ASEPs for a Nakagami-m fading channel with both MPSK and QAM in conjunction with maximal ratio combining (MRC). I. Introduction

The performance of digital communication systems over generalized fading channels has been well studied (see, e.g.,

[1-3], to name only a few). Due to the complexities of both the density functions of the fading channel, such as Nakagami and log-normal, and the conditional error probabilities for different modulation schemes, it often not easy to obtain the closed-

form ACEPs. Further, those that exist in closed form are often very complex, especially with multiple diversity branches, and are not explicit functions of diversity order. An approximation which reduces the ACEP to a single term containing the inverse of the average SNR (per branch) raised to the power equal to the diversity order is very attractive due to the following three advantages: simplicity, explicit function of the diversity order, and the closeness to the corresponding ACEP as the SNR increases.

For ),(GSC Lm where the strongest m branches from L )1( Lm≤≤ total branches are selected and combined another advantage is that the simple closed-form ASEP provides straightforwardly the different SNR gaps (BER curve shifts in SNR to reach the same performance) resulting from different m , relative to the performance of selecting the largest (in SNR) branch. These kinds of ASEPs have been presented and used more often than their ACEP counterparts. For example, [4] reduces the ACEPs of MRC with binary signaling over both iid and non-iid multipath Rayleigh fading channels [4, Eq.14-4-15, Eq. 14-5-28] to their ASEPs [4, Eq. 14-4-18, Eq. 14-5-29], and [17] also presents a way to reduce the ACEPs to the ASEPs. However, obtaining the ASEP from its ACEP is not always possible, such as when there is no closed–form

1 Acknowledgement: This research was partially supported by a grant from the Air Force Office of Scientific Research Grant # FA 9550-06-1-0210

ACEP expression available or the available ones are too complex to reduce the desired APEP. For instance, the ACEP of GSC for binary signaling over a Rayleigh fading channel [5, Eq.9] is too lengthy and complex to reduce to a one-term APEP.

When the ASEP is not easy to find, a Chernoff upper bound sometimes has been used to simplify the Q function to an exponential function. Then, if the fading density is simple, for example, Rayleigh fading, which results in an exponentially distributed SNR, a closed-form error probability upper-bound can be found [16, Eq.5.5]. This upper bound also has the advantage of containing only the term of the inverse SNR raised to its diversity order, which means that the Chernoff bounds so obtained are also simple and show their diversity order clearly. But when the fading density is more complex, simplifying the Q function to an exponential is not sufficient to be able to obtain the Chernoff bound. II. Some mathematical foundations

In this paper, we propose an easy and reliable way to find this one-term ASEP. For example, the error probability of

),(GSC Lm , ),,;( 1 LcceP , where ic is the inverse average SNR for the ith diversity branch, is given by

)1( ),,;,,()}{(),,;( 11110 0

1 1 mLmZZmllL dzdzcczzfzePcceP

m=

∞ ∞

∫ ∫=

where ),,;,,( 111 LmZZ cczzfm

is the joint density function of the m selected branches, and )}{( 1

mllzeP = is the conditional error

probability and is not a function of Lcc ,,1 . The Maclaurin series of ),,;( 1 LcceP in terms of Lcc ,,1 is given by

)0,,0;(!

1),,;(10

1 ePc

cn

ccePn

L

i ii

nL

∂∂= ∑∑

=

∞

= , (2)

where, for example, if 2=L and 1=n ,2

21

1

2

1 cc

cc

cc

i ii ∂

∂+∂∂=

∂∂

∑=

,

and if 2=m and 2=n , 22

222

21

2

2121

221

22

12)(

cc

cccc

cc

cc

i ii ∂

∂+∂∂

∂+∂∂=

∂∂

∑=

.

Note that the right-hand side of (1) can also be expressed by expanding the density in a Maclaurin series. This yields

mmZZ

nL

i ii

n

mll

mLmZZmll

dzdzzzfc

cn

zeP

dzdzcczzfzeP

m

m

1110

10 0

11110 0

)0,,0;,,(!

1)}{(

),,;,,()}{(

1

1

∂∂= ∑∑∫ ∫

∫ ∫

=

∞

==

∞ ∞

=

∞ ∞

Therefore,

mmZZ

L

i

n

ii

n

mll

L

i

n

ii

n

dzdzzzfc

cn

zePePc

cn

m 1110

10 010

)0,,0;,,()(!

1

)}{()0,,0;()(!

1

1∑∑

∫ ∫∑∑

=

∞

=

=

∞ ∞

=

∞

=

∂∂×

=∂∂

(3)

Since the diversity order can be found by expanding the ACEP in a power series and retaining the term with the inverse of the SNR raised to the lowest power (the diversity order), the ASEP, denoted by ),,;( 1 La cceP , can be found by keeping only the lowest power term in its power series (in terms of

1-4244-0264-6/07/$25.00 ©2007 IEEE 910

inverse SNR). Specifically,

)0,,0;()(!

1),,;(1

1 ePc

cn

ccePL

i

n

ii

LLa

L∑= ∂

∂=

where Ln is the lowest power term in (2) and it is known that LnL = [16]. Since this term on both sides of (3) needs to be

equal, we have =),,;( 1 La cceP

mmZZ

LL

i ii

mll dzdzzzf

cc

LzeP

m 111

10 0

)]0,,0,,,(!

1)[}{(1

∂∂

∑∫ ∫=

=

∞ ∞

Hence, the ASEP can be achieved equivalently by averaging the conditional error probability )}{( 1

mllzeP = over the fading

density, where we retain only the lowest power of its expansion in terms of the inverse SNR. We call this term, which produces the dominant part (ASEP) of the ACEP, the dominant term of the fading density. Obviously, using this term to achieve the ASEP is much simpler than using the fading density itself to obtain ACEP and then simplifying it to the ASEP. It is not hard to see that this term, denoted by

),( Lmfdom (which is, in general, a function of both m and L), can be found in the following manner if the diversity order is known ( L ):

)/(),,;,,(lim),(1

110,,11

1

∏∏=→=

=L

llLmZZcc

L

lldom ccczzfcLmf

mL

(4)

For example, if the fading is flat Rayleigh, then the density function is czceczf −=),( and 1== Lm . Thus,

=)1,1(domf ( ) cccec cz

c=−

→/lim

0, so the APEP for flat Rayleigh

fading is given by

cdc

dzdeccdzzQdzfzePcePz

doma

41sin

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

2/

0

2

0

sin2/

000

2

==

===

∫

∫ ∫∫∫∞ −∞∞

φφπ

φπ

π

φπ

which is the same as presented in [4, Eq. 14-3-13]. Alternatively, if we have frequency-selective Rayleigh fading with L multipaths, the joint density of these L paths is

∏=

−=L

i

zciLLZZ

ii

Leccczzf

111 ),,,,,(

1and Lm = . Then the joint

density function has a dominant term given by

∏∏∏∏===

−

→==

=L

ii

L

ii

L

i

zcicc

L

iidom cceccLLf ii

L 1110,,1 1

lim),( . Hence the APEP for

maximal ratio combining all L paths is given by

∏∫∏∫ ∫ ∫∏

∏∑∫ ∫∏∫ ∫

=−

=

∞∑

−∞

=

==

∞ ∞

==

∞ ∞

==

=

==

=L

iiL

LL

LL

ii

z

L

ii

L

ii

L

ii

L

ii

Llla

cCdcdzdec

dzczQdzczePeP

L

ii

112

2/

0

2

10

sin

0

2/

01

110 011

0 0

41sin11

2)}{()(

21

φφπ

φπ

πφπ

(5)

where, )!1(!)!12(

12 −−=− LL

LC LL and the last equality is because

LLL

L Cd41sin1

12

2/

0

2−=∫ φφ

π

π[13]. Eq. (5) is same as that presented in

[4, Eq.14-4-18]. For the iid case , (5) reduces to LLLL cC 4/12 − .

In remainder of this paper, we use the proposed approach to obtain some ASEPs whose closed-form ACEPs are not available. Specifically, we derive closed-form ASEPs for GSC with both MPSK and QAM modulation schemes. We present these ASEPs as a scaling factor multiplying the APEP for case where Lm = , thereby determining the SNR gaps for

different m. It is found that the gap for one or two-dimensional linearly modulated signals does not depend on the modulation schemes or modulation orders. We also derive the ASEPs of GSC with QAM modulation with MRC combining over a general Nakagami-m fading channel, where its ACEP is presented in [12] in terms of the hypergeometric function. III. ASEPs of GSC for MPSK and M-QAM signals and

the SNR gaps over iid & non-iid Rayleigh fading channels

The closed-form ACEPs for GSC of MPSK signals are presented in [6], but they are lengthy and complex [Eq. 38-42]. For BPSK they are simplified as shown in [5, Eqs. 4 and 9]. In this section, we derive the ASEPs for GSC with MPSK and M-QAM signaling. We start by finding the dominant part of ),(GSC Lm . Recall that the joint density function of the selected m largest (in SNR) branches (i.e., ordered), denoted by )()1( ,, mZZ , is [1,5]

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

1

,,1

,11,,

11)()1( mm

L

mlzmz

L

iizmZZ zzzFzfzfzzf

limi

m

im>>>∞= ∏∑

+=

where )( 11

zfiz is the fading density and )( mZ zF

liis the fading

cumulative distribution function for the thli branch,

{ }Lmil ,2, ∈ . { }∑

L

ii m

,,1

,,1

denotes the sum of all permutations of

mii ,,1 taking values in the set { }L,2,1 . For Rayleigh

fading channels, jji

jji

zcijz eczf −=)( and )1()( mli

li

zcmz ezF

−−= ,

therefore { }

),1(),,(1

,,1

, 11,,

1

)()1(

mli

m

jji

jm

zcL

ml

L

ii

zci

m

jmZZ eeczzf −

+=

−

=∏∑ ∏ −=

where Lici ,,2,1 ,/1 = is the average SNR for the thi branch. After a decorrelation transformation [1,5] obtained by letting

1,,2,1,1 −=−= + mizzx iii and mm zx = , the joint density becomes { }

mjxeecxxf jxcL

ml

L

ii

xc

i

m

jmXX

mli

m

j

j

kki

jm≤≤∞<≤−=

−

+=

−

=∏∑ ∏

∑= 1 ,0 ),1(),,(

1

,,1

, 11,,

1

1

)()1(

Note that the Jacobian of the transformation is unity. Next we use (4) to find ),( Lmfdom of the above density for ),(GSC Lm . It is proved in [18] that

)!(

!),(1

mLm

L

iidom x

mLLcLmf −

= −= ∏ (6)

For an iid channel, Eq (6) reduces to

LmLmdom cx

mLLLmf −

−=

)!(!),( (7)

Note also from (6), letting Lm = , we have

∏=

=L

iidom cLLLf

1!),( (8)

which is the dominant part of the density function that results from MRC when the underlying fading has Rayleigh statistics. Consequently, (6) can be rewritten as

)!(),(

)!(!),(

1 mLxLLfcx

mLLLmf

mLm

dom

L

ii

mLmdom −

=−

=−

=

− ∏

Note that the diversity order is L no matter what m is used, and the term )!/( mLx mL

m −− , resulting from selecting Lmm ≠ , rather than L , does not contribute to the diversity order. It

911

contributes only to the SNR gaps between the performances of Lm = and Lm ≠ . )!/( mLx mL

m −− becomes unity if Lm = . The simplification of (6) to achieve the dominant parts, namely (6) and (7) for iid and non-iid cases, respectively, makes it very easy to find the APEP for both MPSK and QAM modulated signals. Consider first the following conditional error probability (called an ideal form for MGF-based analysis [6]), which can be converted to the cases of both MPSK and QAM:

φπ

β α φ dezeP i

m

llz

g

i

N

i

mll ∫∑

∑=

−

== = 0

sin

11

121)}{( (9)

where g is a modulation constant. In Table I, we list the values of all the parameters in (9) for both MPSK and M-QAM. The values of MPSK are based on the conditional error probability given by [8, Eq. (71), 9, Eq. (3.119)], and the values of M-QAM are based the conditional error probability given by [9,Eqn.(10.32)].

g N 1α 2α 1β 2β

M-QAM )1(2

3−M

2 2π

4π

M11 − 2)11(

M−

MPSK )/(sin 2 Mπ 1 πMM )1( − 0 1 0

Table I: Parameters of M-QAM and MPSK in (9) Performing the same transformation described earlier [1,4],

i.e., ∑=

=i

mjji xz , and ∑∑ ∑∑

== ====

m

ii

m

ij

i

mj

m

ii ixxz

111, we have == )}{( 1

mllxeP

φπ

β α φ dei

m

lllx

g

i

N

i∫∑

∑=

−

=0

sin

1

121 . Then the ASEP, denoted by )/( Lm

aPe , is

obtained by averaging )}{( 1mllxeP = over (6). It is shown in [18]

that

!!)/()/(

mmLPePe mL

LLa

Lma −= (10)

where, in (10), )/( LLaPe is the ASEP when all L diversity

branches are combined and is given by [18]

( ) πααπ

παβ

≤≤−−−−

+=

∑

∏∑

−

=−

==

0 , } )(2

])(2sin[)1(2

1

21{1

2

1

012

2211

)/(

jLjLC

Ccg

Pe

jL

L

j

jL

L

LLL

L

iiL

N

nn

LLa

.

Therefore,

( ) 0 ,!

! } )(2

])(2sin[)1(2

1

21{1

2

1

012

2211

)/(

πααπ

παβ

≤≤−−−−

+=

−

−

=−

==

∑

∏∑

mmL

jLjLC

Ccg

Pe

mLjL

L

j

jL

L

LLL

L

iiL

N

nn

Lma

(11)

For MPSK, substituting the parameter values from Table I into (11), we have the ASEP, denoted by )/(

_Lm

aMPSKPe as

−−−

−+−

= ∑∏ −

=

−−

=

)(]/)1)((2sin[

)1(11

2!!

2

1

022

1)/(_ jL

MMjLCC

MM

gmmcL

Pe jL

L

j

jLLLLL

MPSKmL

Li iLm

aMPSKπ

π(12)

For the iid Rayleigh fading case, ccc ij == for all ij and ,

−−−−+−= ∑

−

=

−− )(

]/)1)((2sin[)1(112!

!2

1

022

)/(_ jL

MMjLCCM

MgmmcLPe j

L

L

j

jLLLLL

MPSKmL

LLmaiid

ππ

(13)

If 1=L , from (12) or (13), the APEP for an MPSK modulated signal for a single path is

−−−= ])1(2sin[

21)1/1(

_ MM

MM

gcPe

MPSKaMPSK

π . (14)

When 2=M , (12) reduces from the symbol ASEP to bit ASEP, 1== BPSKMPSK gg , and

∏=

+−=L

iiLmL

LLLm

aBPSK cmmCLPe

112

2)/(_ 2!

! . (15)

For the iid Rayleigh fading case, ccc ij == for all ij and , and

LLmL

LLLm

aBPSK cmmCLPe 12

2)/(_ 2!

!+−= . (16)

Compared to the BERs of ),(GSC Lm with BPSK for both iid and non-iid Rayleigh fading presented in [5, Eq.4 and Eq. 9], (15) and (16) are much simpler. To elaborate a little, we write [5, Eq. 4], which is the BER for the iid Rayleigh fading case, below:

(17) )})1(1/11()(

)1(

])12(2!

!)!12(11

11[)1({)1(

21

)12(2!!)!12(

1111

21

11

2

0

11

0

0

1

0

)/(_

cmi

iimm

jjj

cc

cim

imL

mL

iii

cc

cmL

Pe

m

mm

j

jkm

j

kk

m

i

i

mL

ii

im

i

LmiidBPSK

++−+

−+

++

++−

−−

−

+

++

++−

=

−−

−−

=

+−

=

−

=

−

=

∑∑

∑∑

Comparing (17) with its BER ASEP in (16), the computational complexity of (16) is negligible. We plot (17) and (16) in Figure 1, where there are two groups of curves, one for

3=L and the other for 5=L . For each group of curves, m varies from 1 to L . In the figure, gsc (abbreviated for the actual performance of GSC) and gsca (abbreviated for the approximate performance of GSC) represent ACEP and ASEP, respectively. It is seen that the curves generated from (16) and (17) are very close for the range of SNR displayed. When Lm = , from (15),

∏=

−=L

ii

LLL

LLaBPSK cCPe

1122

)/(_ 2

1 (18)

which is the ASEP for BPSK and MRC and is the same as (5) and [4, Eq.14-4-18]. Note that L

LLL CC 2122 =− . For the iid case,

LLLL

LmaBPSK cCPe 122

)/(_ 2

1−= (19)

and 4/)1/1(_ cPe aBPSK = which is the same as presented in (3) and

in [4, Eq. 14-3-13]. In (15), if 1=m , then we have the APEP for selection combining of non -iid Rayleigh fading with BPSK modulation:

∏∏=

+=

−

−==L

iiL

L

ii

LLL

LaBPSK cLcCLPe

11

1212

)/1(_ 2

!)!12(2

! (20)

Compared to its ACEP counterpart in [5, Eq.18], which is

{ }

{ }

{ } { }

−= ∑∑∑ ∑∑

+

=

+

=

−

= +

l

mji

lm

mjii

L

l

iL

ii

lL

i

LBPSK jj

l

ccwcPe11

0

/,1,1)/1( /)1(

1

1

121

(21)

where { }

{ } { }∑

+

1

12

/,1 iL

ii l

means the indices { }lii +12 take values from

{ }L,,1 but cannot equal the value of 1i , and )/11(5.0)( xxw −= , obviously (20) is much simpler. Note that

912

we can obtain the APEP directly from (21). For example, if 2=L , from (21) we have

{ }

{ } { }{ })

11

11

111(

21]/)1([

2121

2,1 11

0

/2,1)2/1(

1

1

1

12 ccccccwcPe

i

l

mji

lm

mjii

l

i

ii

lBPSK jj

l +++

+−

+−=

−= ∑ ∑∑∑ ∑

+

=

+

== +

(22) .83)]2(

83)(

211

83

211

83

21[5.0 2121

22

2121

222

211 cccccccccccc =++++−+−+−−≈

If we use (20) instead, we have =)2/1(_ aBPSKPe

2121

1 83

2!)!12( cccL

L

L

iiL =

−

==+ ∏ which is identical to (22), but the

calculation is much simpler. From (10), )/( Lm

aPe is the product of )/( LLaPe and ),( LmG

where !

!),(mm

LLmG mL−= , and )/( LLaPe has only one term

containing the SNR raised to its diversity order (i.e., L). Therefore, from (10), the SNR gap for the same error probability between selecting all L branches and selecting

Lm < branches is simply

(dB) !

!log10),(log10),(mm

LL

LmGLmGap mLL

−== (23)

Note that when Lm = , (dB) 0),( =LLGap , and when m decreases, ),( LmGap increases. These gaps are SNR invariant in the region of SNR where the slopes of the ACEPs are constant (equal to the diversity order), Also from (23), it is seen that ),( LmG is not related to the modulation scheme. The reason is that as long as the conditional error probability can be expressed as in the form of (9), the effect of the modulation type on ),( LmG , as embedded in the constant g , is cancelled, as shown in the derivation of Eq. (10) [18]. Therefore, since the Q-function, the Q-squared function and the conditional error probability of MPSK are all special cases of (9), the SNR gap for one and two dimensional linear modulation signals, whose conditional error probabilities are the Q or squared-Q functions, is not a function of the modulation scheme when the fade statistics are Rayleigh. It is a function only of m and L . For M-QAM signals, substituting the parameter values from Table I into (9) and using (10) and (11), we obtain the APEP of M-QAM as

!/!)/(_

)/(_ mmLPePe mLLL

aQAMLm

aQAM−= (24)

where

( ) })(2

]2

)sin[()1(

21

2[

2{

42

1

01222

2212

21)/(_ jL

jLCCaCa

gc

Pe jL

L

j

jL

L

L

LL

L

LL

LQAM

Li iLL

aQAM −

−−−+−= ∑∏ −

=−++

=

π

π (25)

Substituting (25) in to (24), we have

( )!

!}

)(2

]2

)sin[()1(

21

2[

2{

42

1

01222

2212

21)/(_ mm

LjL

jLCCaCa

gc

Pe mLjL

L

j

jL

L

L

LL

L

LL

LQAM

Li iLm

aQAM −

−

=−++

=

−

−−

−+−= ∑∏

π

π (26)

If 1=L ,using (26), we have

−−= )1

21

(1)1/1(_ π

agac

PeQAM

aQAM (27)

Eq. (27) can also be obtained from its ACEP counterpart [14, Eq(44)] as [18]

(28) )1

21(

)1arctan4()1(2

2

2)1/1(

−−≈

−+

++

+−=

π

π

aag

c

gcg

cgg

acg

gaPe

QAM

QAM

QAM

QAM

QAM

QAM

QAMQAM

Hence, (27) and (28) are the same. Fig. 2 shows the performance of )1/1(

_ aQAMPe and )1/1(QAMPe with M=4,16,64. It is

observed that the difference between the curves increases as M

increases, for the same SNR. The reason is that the Maclaurin series of )1/1(

QAMPe in (28) is a function of QAMgc / , so as M increases, )]1(2/[3 −= MgQAM decreases, or QAMgc / increases. Since the smaller QAMgc / is, the more accurate the APEP is, the difference between the two curves increases as M increases. IV. APEP for MPSK and M-QAM signals with MRC combining over Nakagami-m fading channel Closed-form error probabilities with MPSK and M-QAM with MRC combining over a general Nakagami-m fading channel (for arbitrary M) are not available. However, with the proposed approach, we can still obtain their APEPs. First we simplify the Nakagami-m density function to its dominant part. For the thi channel, the actual density function is )(/),;( 1

imi

mi

mi

xmciiii mcxmemcxf iiiiii Γ= −− (29)

where (.)Γ is the gamma function, im is the Nakagami-m fading parameter of the thi channel, which ranges from 0.5 to infinity, and ic is the inverse average SNR for the thi branch. Eq. (33) is simplified by replacing iii xmce− with unity to yield the lowest power inverse ic as

)(/),;( 1i

mi

mi

miiiii mcxmmcxf iii Γ≈ − (30)

Then the dominant part of the joint Nakagami-m density for all L branches is

)(/),,,;,( 1

1111 i

mi

mi

mi

L

iLLL mcxmmmccxxf iii Γ≈ −

=∏ (31)

Averaging (9) over (31), we have the APEP for MPSK when combining all L branches of the Nakagami-m channel. We denote it by )/(

___LL

aMPSKmNakPe , and it is given by [18] assuming integer fading parameters,

( )]

)(2]/)1)((2sin[)1(

211

2[

1

12

1

0122

2

1

)/(___

1

1

1

1

1

1

1

jLMMjLC

MMC

gcm

Pe

jL

L

j

jL

L

L

LLL

imMPSK

mi

mi

LLaMPSKmNak

i

ii

−−−

−−

+−

=

∑∏−

=−

=

ππ

(32)

where iLi mL ∑ == 11 . If 1=im , Li ,,1= , the Nakagami-m

reduces to Rayleigh fading and (32) reduces to (12) after replacing m with L. When 2=M , then (32) reduces to the BER for a Rayleigh fading channel with BPSK and MRC as

12

12

)/(___

)/(___ 2/ +

=∏== LL

i

iLL

LLaBPSKmRayleigh

LLaBPSKmNak cCPePe which is the

same as (18). If the Nakagami-m channel is iid, i.e., ccc ij == and mmm ij == for all ij and , then

( ) ])(2

]/)1)((2sin[)1(2

112

[ 2

1

0122

2

)/(____

jLmMMjLmC

MMC

gcm

Pe

jLm

Lm

j

jLm

Lm

Lm

LmLm

LmMPSK

LmLm

LLaiidMPSKmNak

−−−−−+−

=

∑−

=−

ππ

(33)

(33) reduces to (13) if it is Rayleigh fading after replacing m with L in (13). In the exactly same way, the APEP for QAM with MRC for the Nakagami-m channel can be easily found to be given by

( ) })(2

]/)1)((2sin[)1(2

112

1

)11(2

1)11{(

1

12

1

01222

222

1

)/(___

1

1

1

1

1

11

1

11

−−−−−+−

−−−

=

∑

∏

−

=−

=

jLMMjLC

MMC

MC

MgcmPe

jL

L

j

jL

LLLL

LLL

L

imQAM

mi

miLL

aQAMmNaki

ii

ππ

913

V. Conclusions

In this paper, we have proposed a simple way to calculate the APEPs for fading channels, which only contains the inverse of the SNR raised to the diversity order, and which asymptotically approaches the precise error probabilities. Using the proposed approach, we have found simple closed-form error probabilities for GSC for both MPSK and QAM signals, as well as their SNR gaps over Rayleigh fading channels. It was also found that these gaps are functions of only m, L and the fading density. We have further derived the APEPs with MRC over a generalized Nakagami-m fading channel. 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. J. Winters, J. Salz and R. Gitlin, “The impact of antenna diversity on the capacity 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. J. Proakis, Digital Communication. New York: McGraw-Hill, the third edition 5. 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 6. M-S Alouini and M. K. Simon, “An MGF-based performance analysis of generalized selection combining over Rayleigh fading channels”, IEEE Tran. On Comm. Vol. 48, No. 3, March 2000 7. 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 8. R. F. Pawula, S. O. Rice, and J. H. Roberts, “Distribution of the phase angle between two vectors perturbed by Gaussian noise,” IEEE Trans. Commu. vol. COM-30, pp. 1828-1841, August 1982 9. M. K. Simon, S. M. Hinedi, and W. C. Lindsey, Digital Communication Techniques- Signal Design and Detection. Englewood Cliffs, NJ: PTR Prentice Hall, 1995 10. J. W. Craig, “A new, simple, and exact result for calculating the probability of error for twodimensional signal constellations,” in Proc. IEEE MILCOM’91, pp.571-575 11. C. –J. Kim, Y. –S. Kim, G. –Y. Jeong, and D. –D. Lee, “Matched filter bound of square QAM in multipath Rayleigh fading channels,” IEEE Trans. Veh. Technol., vol. VT-46, pp. 910-922, Nov. 1997 12. M. –S Alouini and M. K. Simon, “Performance analysis of coherent equal gain combining over Nakagami-m fading channels,” IEEE Trans. Veh. Technol., vol. 50, Nov. 2001 13 .I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 5th ed. San Diego, CA:Academic Press, 1994 14. M. G. Shayesteh and A. Aghamohammadi, “On the error probability of linearly modulated signals on frequency-flat Ricean, Rayleigh, and AWGN channels,” IEEE Trans. Commun., vol. COM-43, pp. 1454-1466, Feb./Mar./April 1995 15. F. Adachi, K. Ohno, and M. Ikura, “Postdetection SC reception with correlated, unequal average power Rayleigh fading signals for π/4 shift QDPSK mobile radio,” IEEE Trans. Veh. Technol., vol VT-41, may 1992 16. A. Paulraj, R. Nabar and D. Gore, Introduction to Space-Time wireless Communications, Cambridge University Press

17. H. S. Abdel-Ghaffar and S Pasuthy, “ Asymptotic performance of M-ary and binary signals over multipath/multichannel Rayleigh and Rician fading,” IEEE Trans Commun., vol. COM-43, Nov. 1995, pp2721-27 18. N. Kong and L. B. Milstein, “A simple approach to the performance analysis of digital communication over fading channels”, submitted to IEEE Trans. On Wireless Communications

15 16 17 18 19 20 21 22 23 2410

-13

10-12

10-11

10-10

10-9

10-8

10-7

10-6

10-5

10-4

SNR

BE

R

gsc51gsc52

gsc53

gsc54

gsc55gsca51gsca52

gsca53gsca54

gsca55

gsc31gsc32

gsc33

gsca31

gsca32gsca33

gsc & gsca with L=3 & m=3

gsc & gsca with L=3 & m=2gsc & gsca with L=3 & m=1

gsc & gsca with L=5 & m=1

gsc & gsca with L=5 & m=2

gsc & gsca with L=5 & m=3

gsc & gsca with L=5 & m=5

gsc & gsca with L=5 & m=4

Figure 1: Comparison between the actual (ACEP) and approximate (ASEP) expressions of GSC with L=5 and L=3

Figure 2: Symbol ACEP and ASEP comparison for QAM with

M=4,16 and 64.

914