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C. Asymptotic Analysis

For large average SIR, a Taylor series expansion of the pdf of end-to-end instantaneous SIR

about the origin, 0 .

1

(0) (0)eq n

N

n

f f

(25)

where

1

01, 0

1(0) ( 1) exp ( )

!

s

n

n

m kbk

nkk

s

d df s

d k ds

B

1

0 0 1 0

1 1

( )! !

s

n

m kb

nkk s

d d

sd k ds

B

0

11

0 0

1 1( ) ( ) 1

! !

s

n

mb

n n n kk

b b kk

B

1

1

0

11 as 0

!

s

n

mb

n n n kk

b b kk

B (26)

Therefore, the limiting pdf of the end-to-end SIR is given by

1

1

1 0

1(0) 1

!

s

n

eq

mNb

n n n kn k

f b b kk

B (27)

Average BER

The limiting ABER is given by

2 12 0

1

( , ) (0)2 ( ) eqEP f d

1

1

2 1

1 02 0

1 11 ( , )

2 ( ) !

s

n

mNb

n n n kn k

b b k d k

B

1

2

1 0 1

1 11 ( )

2 !

s

nn

mN

n n bb kn k

b kk

B (28)

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where

1 1 , with

nb

n s In n n n n

n I s

E mb b

m

B

Diversity and Coding Gain

d nbG (29a)

1/1

2

0

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

!

ns

n

n

bmb

c n n n n b s I kk

b b b k m mk

G

(29b)

Average Capacity

The average channel capacity is given by

0

ln 1 (0)ln(2) eq

avg

WC f d

1

1

1 0 0

11 ln 1

ln(2) !

s

n

mNb

n n n kn k

Wb b k d

k

B (30)

To solve the integral in (30), we use the following relation:

1,02,21,1

ln 11,0

G x

from [R1]- [R3] (31a)

1,22,21,1

ln 11,0

G x

from [R4], [R5] (31b)

Substituting (30a) in (29), we have

1

1 0

1ln(2) !

ns

bmN

nn navg n k

n k n

EbWC b kk

(32a)

while, using (30b), we have

12 2

1 0

1 (1 ) ( )ln(2) !

ns

bmN

nn navg n n nk

n k n

EbWC b k b b

k

(32b)

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References

[R1] Wolfram Functions. Available: http:/functions.wolfram.com/

[R2] A. P. Prudnikov, Y. A Brychkov, and O.I. Marichev. Integrals and Series: Volume 3: MoreSpecial Functions. Gordon and Breach Science Publishers, 1990.

[R3] N.C. Sagias, G.S. Tombras, G.K. Karagiannidis, New results for Shannon capacity in

generalized fading channels,IEEE Comm Lett,vol. vol.9, no.2, pp.97-99, Feb. 2005

[R4] A. M. Mathai, and R. K. Saxena, The H-Function with Applications in Statistics and Other

Disciplines, New York, John Wiley,1978.

[R5] A. M. Mathai, R. K. Saxena and H. J. Haubold, The H-Transform: Theory and

Applications, New York, Springer, 2010.