amount of fading revisited

7/29/2019 Amount of Fading Revisited

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Amount of Fading Revisited

Yan Li and Shalinee Kishore

Abstract

The fading statistics in various diversity systems are uniquely determined by the distribution

of the signal-to-noise ratio (SNR) of combined signals. In this study, the amount of fading

(AF), introduced in [1] as a unified measure of the severity of fading, is revisited and utilized to

parameterize the SNR distribution. Specifically, two approaches are proposed, approximating the

SNR as Gaussian and Gamma distributions. The approximated SNR distributions are then used

to evaluate various conventional performance measures: Shannon capacity, outage probability

and average error rates. The accuracy of these approximations based on the AF implies that

systems with the same AF have roughly the same performance, given the receiver is optimally

designed. Therefore a complicated system can be equivalent to a much simpler diversity system.

This equivalence can be used to simplify certain theoretical analysis. As illustrations, applications

to receive antenna diversity and multiuser diversity are investigated; the results confirm this claim

and establish the broader utility of the AF.

I. Introduction

The random, time-varying channel characteristics between a transmitter and a receiver

pose many challenges to reliable wireless communications. A key challenge lies in variable

fading of the received signal power which can lead to variable performance over the commu-

nication link. Diversity methods attempt to counter this disparity; in particular, they focus

on reducing the fluctuations of received signal power to overcome the unreliability caused by

poor fading conditions. In order to evaluate the performance of various diversity schemes,

different fading measures have been introduced and investigated.

The authors are with the Department of Electrical and Computer Engineering, Lehigh University, Bethlehem, PA18015, USA. (Email: {liyan,skishore}@lehigh.edu). This research was supported by the National Science Foundationunder CAREER Grant CCF-03-46945.


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Average signal-to-noise ratio (SNR) at the diversity combiner output is a useful and

perhaps the simplest performance measure. However, this performance criterion does not

adequately capture diversity benefits as it ignores the variation of the received SNR. To

account for its variance, we need to look at performance measures which take into account

higher moments of the SNR.

In 1979, Charash introduced a measure of the severity of the fading channel [1], referred

to as the amount of fading (AF), which can be computed using only the first and second

central moments of the SNR at the diversity combiner output. This measure was gener-

alized by Simon and Alouini when describing the behavior of diversity combining systems

over correlated log-normal fading channels [2]. Another measure, diversity factor (DF) wasintroduced by Awoniyi, Mehta and Greenstein [3] to study the impact of dispersive channels

on the throughput of code division multiple access (CDMA) schemes. Calculable for any

channel delay profile, it captures the effective number of equal mean gain paths offered

by a delay profile to a RAKE receiver. In [5] and [6], this definition was used to quantify

the uplink and downlink user capacities of a hierarchical cellular CDMA system in various

multipath environments. We note here that the DF is simply the reciprocal of the AF. The

normalized standard deviation, defined by Win and Winters in [4], is another quantity with

a definition similar to that of the AF.

Although the AF is widely used as an alternative performance criterion (e.g. [7] [11]),

there is, to the best of our knowledge, no answer to the following questions: Does the AF,

which depends only on the first two moments of the combined output SNR, accurately

reflect the behavior of the diversity combining techniques and capture the statistics of the

fading channels? If the answer is yes, then what is the relationship between the AF andother conventional performance measures, such as Shannon capacity, outage probability,

etc.? We try to give some answers to these questions in the paper. Specifically, we answer

these questions by proposing approximations to the distribution of the output SNR in terms

of AF which are valid in a variety of scenarios. With these approximations at hand, we then


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compute other conventional performance measures as functions of AF.

The rest of the paper is outlined as follows: Section II describes the system model and

provides the definition of AF. Section III contains the theoretical results of approximations

to the distribution of the output SNR. Based on these results, we study the relationships

between AF and other performance criteria in Section IV. As illustrations to our theoretical

results, some applications are investigated and the numerical results are given in Section V.

We summarize the contributions of this paper in Section VI.

II. System Model

In this study, we use the following standard notations: superscripts T for vector transpose,

for conjugate and for transpose conjugate; E[] for expectation and V ar[] for variance.The complement of a set S will be denoted by S.

We consider a simplified fading channel model with L-branch diversity

Yl = alX+ Nl, l = 1, 2,...,L, (1)

where Yl is the processed received signal of the l-th sub-channel or diversity branch (out of

L); X is the scalar transmitted signal with power constraint

E[|X|2] = P; (2)

{al}Ll=1 are the mutually independent fading coefficients; and {Nl}Ll=1 are the independent,identically distributed (i.i.d.) complex Gaussian noise (plus interference in multiuser environ-

ments under the standard Gaussian assumption) components with zero-mean and variance

N0/2 per dimension. In addition, we assume the channel state information (CSI) is known

at the receiver thus enabling coherent detection.

For simplicity, we can rewrite (1) in the vector form

Y = AX+N, (3)


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where Y= [Y1, Y2,...,YL]T, A = [a1, a2,...,aL]

T and N= [N1, N2,...,NL]T.

Among different types of diversity combiners, e.g., generalized selection combiner (GSC),

equal-gain combiner (EGC), etc., the optimal one is the maximal ratio combiner (MRC).

The output Z of the MRC can be written as

Z = AY

=Ll=1

al Yl. (4)

In this paper, we focus primarily on this type of an MRC system.

A. Definition of the AF

Let denote the total instantaneous SNR at the combiner output,

=Ll=1

|al|2PN0

=Ll=1

l, (5)

where l =

|al|2P

N0 is the instantaneous SNR on the l-th sub-channel. The AF is defined as

AF V ar[]

E2[]

=

Ll=1 V ar[|al|2]Ll=1 E[|al|2]2 . (6)

Note that AF 1 and that it is the reciprocal of the diversity factor defined in [3].

III. Approximations to the Distribution of the Output SNR

The distribution of the MRC output SNR depends on the fading environment. As an

example, let us consider the simplest case in which all L subchannels experience Rayleigh

fading. The distribution of is given in [14, p.847] and depends on whether the L branches

experience i.i.d. fading or have different means. Let E[] = and let k denote the mean


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SNR of branch k.

1) When {k}Lk=1 are all the same, we can denote c = k = /L. Then, we have theprobability density function (pdf) of as

f() =1

(L)LcL1e/c

G(; L, c), (7)

where () is a complete gamma function defined as

(a)

0

ta1etdt. (8)

The pdf in (7) describes the density of a Gamma random variable with the shape parameter

L and the scale parameter c. It is straightforward to show that E[] = and V ar[] = 2/L,

implying that AF = 1/L for this system.

2) When {k}Lk=1 are distinct, the pdf of is

f() =

Lk=1

kk

e/k , (9)

where

k =L

i=1,i=k

kk i . (10)

Under such fading conditions, it can be shown that AF > 1/L (see Appendix I for proof).

In comparing case 1 and 2 above, we find that in Rayleigh channels, uniform i.i.d. fading

over the L branches leads to lower AF than the non-uniform case. Therefore, for the same ,

uniform fading has lower variation about this mean SNR, implying a smaller probability of

deep fades and a smaller probability of high instantaneous SNRs than non-uniform fading.

In many diversity schemes (e.g., RAKE reception), smaller values of AF are desirable for

a fixed as they yield a fewer instances of deep fades. On the other hand, in multiuser

diversity systems with some M users, larger AF values are desirable for a fixed value of


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at the receiver. The reason is because multiuser diversity capacity grows with max, the

instantaneous value of the maximum received SNR among the M users. Specifically, when the

value of AF is large so is the probability that max is high, leading to a higher overall capacity

for the system. We see therefore that the parameter AF is in fact related to the performance

of a diversity scheme. It is the goal of this paper to give more explicit descriptions of the

relationship between AF and conventional performance metrics (like capacity).

To do so, we begin by developing approximations to the output SNR distributions as

functions of AF. It is difficult to solely use AF to characterize the exact distribution of the

output SNR since it is comprised only of the first and second moments of the SNR. However,

we argue here that the AF is accurate enough to reflect the statistics of the fading channelunder various conditions. For example, when L is large, the central limit theorem (CLT)

applies and we have:

Proposition 1 (Gaussian Approximation (GA)): The distribution of the MRC combiner

output SNR tends to a Gaussian with mean and variance AF 2, s.t.

f() 1

2 AF exp

( )2

2AF 2

N(; , AF 2) (11)

given the Lindeberg condition [15, p.256] is satisfied. The condition requires that the indi-

vidual variances {V ar[k]}Lk=1 are small as compared to their sum AF 2, i.e., that for any > 0

V ar[k]

AF 2 < , k = 1,...,L, (12)

for L sufficiently large.

For other diversity combining techniques (for instance, GSC), various CLTs involving the

asymptotic normality ofL-statistics or trimmed sum of order statistics are needed; they are

usually complicated and beyond the scope of our discussion here.


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As implied by the Lindeberg condition, GA works well for nearly equal-power and diversity-

rich environments. However, in many fading environments the powers on different sub-

channels may vary dramatically. Furthermore, the supported levels of diversity may be

restricted, for instance, due to the affordable receiver complexity. Under such conditions when

the CLT does not hold, the Gamma distribution is found to be a better approximation to the

output SNR for Nakagami-m fading channels than the Gaussian distribution. Nakagami-m

distribution, named and proposed by Nakagami (1943) from his large-scale experiments on

rapid fading in long-distance propagation, has a pdf

f(r) =2mmr2m1

(m)m

e(m/)r2

M(r; m, ), (13)

where r is the received signal amplitude, = E[r2], and m is called the fading figuredefined

as

m =2

E[(r2 2)2] 1

2, always. (14)

To come up with the Gamma approximation, we use the following lemma proposed by

Nakagami [13].

Lemma 1 (Sum of Squares of m-Variables): Let {ri}ni=1 be a sequence of independent Nak-agami m-variables distributed according to M(ri, mi, i) and define

R2 =Li=1

r2i . (15)

Then, we have approximately

f(R) M

R; oM,ni=1

i

, (16)

where

oM =

ni=1

i

2/

ni=1

imi

. (17)


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For more details on Lemma 1, readers are referred to [13, p.20].

Now consider a diversity system where all sub-channels experience Nakagami-m fading so

that k Mk, mk, k. Since =Lk=1 k, we have via Lemma 1,

f(

) M

; oM,

Lk=1

k

= M (

; oM, ) , (18)

where

oM = L

k=1k2

/

Lk=1

kmk . (19)

From (14) we have

mk =k

E[(k k)2]=

kV ar[k]

. (20)

By substituting (20) into (19) we have

oM =

Lk=1

k

2/

Lk=1

V ar[k]

= 2/V ar[]

= 1/AF. (21)

Finally, we obtain the approximate pdf of as

g() = f()2

=1

(AF1)

AF 1/AF

exp

AF

= G(; 1/AF, AF ). (22)


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G(; 1/AF, AF ) is a Gamma distribution with the shape parameter 1/AF and the scaleparameter AF . Recall that the output SNR follows the Gamma distribution with theshape parameter L and the scale parameter /L when all the L sub-channels experience

i.i.d. Rayleigh fading. Subsequently, we have the following proposition:

Proposition 2 (Gamma Approximation (GaA)): A MRC diversity combining system over

Nakagami-m fading channel with the amount of fading AF can be approximated by a

Rayleigh fading channel with DF i.i.d. paths, where DF = 1/AF. In other words, the

distribution of the combiner output SNR can be approximated by a Gamma distribution

with the shape parameter DF and the scale parameter /DF.

IV. Interrelations between the AF and Other Performance Criteria

In this section, we study the relationships between the AF and other performance measures

by applying our propositions.

Given the channel realization A is known at the receiver and the channel input X is

a circularly symmetric complex Gaussian with zero-mean and variance P, the maximum

mutual information between the input X and the combiner output Z can be achieved as

Imax = maxE[|X|2]P

I(X; Z)

= log(1 + ). (23)

Two important information theoretical measures are related to Imax: Shannon capacity and

outage probability.

A. Shannon Capacity

When the channel is ergodic, the Shannon capacity is simply the expectation of Imax.

From Shannons capacity function we have

C = E[log(1 + )] . (24)


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The first order approximation

C1 log(1 + ), (25)

is the simplest approximation. It can, for example, be used as the capacity upper bound,

which directly comes from Jensens inequality

E[log(1 + )] log (1 + ) . (26)

The equality in (26) holds if and only if = with probability of 1 and corresponds to a

non-fading AWGN environment.

Given that the tail probability of is small enough, we can obtain a reliable second order

approximation:

C2 log(1 + ) AF2

1 +

2. (27)

In Appendix II the derivation of (27) is given, which shows the second order approximation

is applicable for any combining scheme and accurate enough in most practical fading cases

in which AF 1. We verify this via numerical results in Section V.Alternatively, if we approximate as a Gaussian, we have

CGA =

R+

log(1 + )N(; , AF 2)d. (28)

If we approximate as a Gamma, we have

CGaA = R+log(1 + )G(; AF1, AF )d. (29)

All the capacity approximations (27)(29) yield functions of AF.

B. Outage Probability

When the fading is slow varying and can be considered as fixed for all uses of the channel,

the maximum mutual information is not equal to the channel capacity anymore. In this


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case the Shannon capacity in the strict sense is zero: no matter how small the rate is,

there is a non-zero probability that the channel is incapable of supporting it even for very

long code length (outage event). There is a tradeoff between the supportable data rate R

(bits/channel use) and the outage probability Pout(R). Based on GA, the outage probability

can be approximated as

Pout(R) = Pr(log2(1 + ) R)

= Q

+ 1 2R

AF

, (30)

where Q() is defined as

Q(x)

x

12

et2/2dt. (31)

Based on GaA, the outage probability can be approximated as

Pout(R) =

2R10

G(, AF1, AF , )d

=

1

AF,

2R 1AF

, (32)

where (a, x) is a incomplete Gamma function defined as

(a, x) 1

(a)

x0

ta1etdt. (33)

C. Average Error Probability

Average error probability is another widely used performance measure. Given the channel

realization, the conditional error probability can be typically formulated as

Ps() = MQ

M

, (34)

where M and M depend on the type of the modulation.


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The average error probability is simply

Ps = E[Ps()]

= E

MQ

M

. (35)

By applying an alternate representation of Q function obtained by Craig [16],

Q(x) =1

/20

exp

x22sin2

d, (36)

we have

Ps =M

/20

E

exp M2sin2 d

=M

/20

M

M2sin2

d, (37)

where M(s) E[es] is the moment generating function (MGF) [12] of .

Based on GA, M(s) can be approximated as

M(s) = exps +AF 2s2

2 , (38)and hence we have

Ps =M

/20

exp

M2sin2

+AF 2M2

8sin4

d. (39)

Based on GaA we have

M(s) = 1/ (1 AF s)1/AF , (40)

and

Ps =M

/20

1 1

1 + 2sin2

AFM

1/AFd. (41)

Thus, we can evaluate the average error probability by a single integral with finite limits.


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It is worth discussing here the limiting behavior of our approximations to the tail error

probabilities. First, our approximations asymptotically converge to the true values as the

average SNR increases, since both the approximations and the true values tend to be

zero. However, this asymptotic equality does not hold in the exponent. For example, as a

well-known result, the outage or error probability for Rayleigh fading is asymptotically equal

to 1/L. We can easily show that the approximation based on GaA is asymptotically equal

to 1/DF = 1/1/AF 1/L. For this reason, we cannot expect to have tight approximationfor high SNRs in the logarithm domain.

V. Applications and Numerical Results

A. Spatially Correlated Antenna Diversity

For simplicity, we consider a single-input multiple-output (SIMO) system equipped with

one transmit antenna and L receive antennas. Extensive discussions on the AF in multiple-

input multiple-output (MIMO) systems can be found in [11]. Following conventional nota-

tions, we denote the channel coefficients as H= [h1, h2, ...hL]T. When the receive antennas

are spatially correlated, H can be represented as a product form

H = R1/2Hw, (42)

where Hw is a circularly symmetric complex Gaussian vector with i.i.d. zero-mean and unit

variance entries, andR is the covariance matrix representing the receive antenna correlations.

In particular, we assume that uniform linear arrays (ULA) are employed at the receiver end.

Thus, for a rich scattering environment, the receive covariance matrix R has the following

Toeplitz structure

R =

1 R (2) (L1)

1 (L2)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

L1 L2 L3 1

.


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The AF can be obtained by

AF =V ar [|H|22]E2 [|H|22]

, (43)

where | |2 denotes 2-norm. Given the AF, we can then apply our theoretical results.Fig. 1 shows the simulated Shannon capacity of the SIMO channels as (a) the function

of the number of antennas for various values of SNR per antenna and a fixed correlation

coefficient = 0.5; (b) the function of the correlation coefficient for SNR per antenna = 5

dB and various values ofL. In addition, we plot in both cases the capacity estimated by the

GA, GaA and second order approximation respectively. The curves match the simulation

results very well, except for the gaps between the simulation and the GA approximations

when is large or L is small. On the other hand, the GaA works well even if L is small, as

long as the sub-channel fading characteristics can be described by Nakagami-m distribution.

This advantage is more clearly demonstrated in Fig. 2, in which we plot the simulated and

approximated outage probability as the function of the SNR per antenna for L = 2 and 8,

respectively. When L = 2, the GA fails, but the GaA still matches the simulation results.

In Fig. 1 we also note that the second order approximation to the Shannon capacity tightly

follows the GaA results.

B. Multiuser Diversity Scheduling Gain

The basic idea of multiuser diversity (MD) scheduling is to assign radio resources (power,

bandwidth, antenna, etc.) to the sole user with the highest Imax, i.e., the user that can best

utilize these resources to achieve the highest individual throughput.

Our baseline reference is a Round-Robin (RR) scheduler. Given K users, the maximum

average data rate that can be achieved is

DRR = E

Kk=1 I

(i)max

K

=

Kk=1 C

(i)

K, (44)


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where the superscript (i) denotes the i-th user.

With an MD scheduler, the maximum average data rate that can be achieved is

DMD = E

max(I(i)

max : i = 1,...,K)

. (45)

Since I(i)max = log(1 + (i)) is a strictly increasing function of (i), we have

max(I(i)max) = log

1 + max((i))

. (46)

Let

max

max({i} : i = 1,...,K). (47)

Thus, we have

DMD = E[log(1 + max)] . (48)

By applying our propositions, the approximations to the pdf fi and the cumulative dis-

tribution function (cdf) Fi of (i) are obtained. Then, the pdf of max can be expressed

as

f(max) =Ki=1

wi(max)fi(max), (49)

where

wi(max) =K

j=1,j=i

Fj(max). (50)

Knowing the pdf ofmax, we can substitute it into (48) to approximate DMD. We then define

the multiuser diversity gain as

GMD =DMDDRR

. (51)

The accuracy of our proposed approximations to the multiuser diversity gain can be seen


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in Fig. 3. To simplify simulation, we have chosen all users have the same multipath Rayleigh

fading channels with the classic exponentially decaying multipath intensity profile (MIP) of

the form

i =eiLl=1 e

l , (52)

where is a small positive constant that determines the steepness of the power decay.

Fig. 3 shows the multiuser diversity gain as a function of the number of total users for (a)

various values of and a fixed number of resolvable paths, L = 10; (b) various values of L

and fixed = 0.2. We again see the approximations based on GaA matches the simulation

curves very well. The approximation curves based on GA diverge when the is large, e.g.,

= 1.2, or L is small, e.g., L = 2. The reason for this divergence is that the Lindeberg

condition is not satisfied for large or small L.

VI. Conclusions

We used the AF to parameterize two types of approximate distributions of the com-

bined signal SNR for the optimal diversity combining receiver. Specifically, the GA works

for nearly equal-power and diversity-rich environments. A typical working scenario is the

asymptotic analysis of the multiple antenna system. The GaA requires that all sub-channels

experience Nakagami-m fading. Since the Nakagami-m distribution is a generalization of

many frequently-used fading models (such as Rayleigh fading, Ricean fading and so on),

the GaA is valid and applicable for a wide-ranging cases. Given these approximated SNR

distributions, we then evaluated various conventional performance measures. For the cal-

culation of Shannon capacity, we proposed an additional approach which computes the

second order approximation of C as a function of AF. Our approximations based on the

AF imply that systems with the same AF have roughly the same performance, assuming

the receiver is optimally designed. Thus, certain theoretical analysis of complicated systems

can be simplified by examining equivalent, simpler diversity systems. Although most of our

results have been derived for MRC receivers, we note that the reliability of the second


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order approximation to capacity implies that receiver optimality assumption can be relaxed

in some circumstances. As illustrations, applications to the receive antenna diversity and

multiuser diversity were investigated. The results further confirm our claim and establish

the broader utility of the AF.

Appendix I

For the Rayleigh fading channel with L-branch diversity, the amount of fading is given as

AF 1/L with equality if and only if the fades on all diversity branches are independent,identically distributed.

Proof: From Schwartzs inequality we have

(T 1)2 ||22 |1|22, (53)

where 1 is a L 1 vector with each element equal to 1 and = [1, 2,..., k]T. The equalityholds if and only if = 1 with a constant. With this observation we can bound AF as

AF =

Ll=1 l

2

Ll=1 l2

=||22

(T 1)2

1|1|22= 1/L. (54)

Thus, AF = 1/L when the L fading paths have equal mean powers and AF > 1/L, otherwise.


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Appendix II

Derivation of the second order approximation to the Shannon capacity

Let us define

E1 C C1

=

R+

log

1 +

1 +

f()d (55)

and

E2

t

log

1 +

1 +

f()d, (56)

where

t =

R+| 1 + 2 . (57)Then, it is easy to show

| |1 +

1 if t< 1 otherwise.

Thus, we have

E1 E2 t

1 +

12

1 +

2f()d (58)

and

E2 >

t

1 +

12

1 +

2f()d

E3, (59)

where the approximation and the inequality follow the facts that log(1 + x) x 12

x2 for

|x| < 1 and log(1 + x) > x 12x2 for x 1.


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Given that the tail probability of is small such that |E2| C and |E3| C, we have

C = C1 + (E1 E2) + E2

C1 + (E1 E2) + E3

C1 +R+

1 +

12

1 +

2f()d

= log(1 + ) AF2

1 +

2 C2. (60)

Hence, we obtain the second order approximation.

Intuitively, the tail probability Pr(t) is small if the AF is small since the standard

deviation of is

AF. However, even when AF = 1, which corresponds the Rayleigh

flat fading case, we observe that |E2| C and |E3| C.For Rayleigh fading, is exponentially distributed with the pdf

f() =1

exp

. (61)

Thus, we have

C = exp

1

Ei

1

, (62)

E2 = exp

1 + 2

log2 exp

1

Ei

2(1 + )

, (63)

and

E3 =

12

1 + 2

exp

1 + 2

, (64)

where Ei() is the exponential integral function defined as

Ei(x) x

etdt

t. (65)


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Both |E2|/C and |E3|/C are bounded functions of : E2/C = E3/C = 0 when = 0 or = +, max(|E2|/C) 0.085 when is around 2.92 dB and max(|E3|/C) 0.063 when is around 1.34 dB.

Thus, in most practical fading cases in which AF < 1, the condition that |E2| C and|E3| C is satisfied. This condition ensures the accuracy of the second order approximation.In addition, the derivation does not make use of the MRC assumption, which implies that

the second order approximation is applicable for any diversity scheme.

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[14] J. Proakis, Digital Communications, Fourth Edition, McGraw-Hill, Inc., 2000.

[15] W. Feller, An Introduction to Probability Theory and Its Applications, Volume II, John Wiley & Sons, Inc., 1966.

[16] J. Craig, A new, simple, and exact result for calculating the probability of error for two-dimensional signal

constellations, Proc. IEEE Milit. Commun. Conf. MILCOM91, McLean, VA, Oct. 1991.


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1 2 3 4 5 6 7 8 9 101

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

L

Shannoncapacity,nats/channeluse

Simulation

Second Order

Gaussian Approximation

Gamma Approximation

15 dB

10 dB

5 dB

(a)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.92.2

2.4

2.6

2.8

3

3.2

3.4

3.6

3.8

4

annoncapacity,nats

c

anneluse

SimulationSecond OrderGaussian ApproximationGamma Approximation

L=16

L=8

L=4

(b)

Fig. 1. Antenna diversity: (a) Shannon capacity versus L for = 0.5 and various values of SNR; (b) Shannoncapacity versus for SNR per antenna = 5 dB and various values of L.


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10 9 8 7 6 5 4 3 2 1 00

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SNR per antenna, dB

utageprobability

SimulationGaussian ApproximationGamma Approximation

L=8

L=2

Fig. 2. Antenna diversity: Outage probability versus SNR/antenna for = 0.5, R = 1bit/channel use.


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1 2 3 4 5 6 7 8 9 100.9

1

1.1

1.2

1.3

1.4

1.5

1.6

Number of Users

MultiuserDiversityGain

SimulationGaussian Approximation

Gamma Approximation

= 0.2

= 0.7

= 1.2

(a)

1 2 3 4 5 6 7 8 9 100.9

1

1.1

1.2

1.3

1.4

1.5

1.6

Number of Users

MultiuserDiversityGain

Simulation

Gaussian Approximation

Gamma Approximation

L=2

L=4

L=8

(b)

Fig. 3. Multiuser diversity: Multiuser diversity gain versus number of users for SNR = 5 dB, (a) L = 10 and variousvalues of; (b) = 0.2 and various values ofL.

amount of fading revisited

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