International Journal of Engineering Trends and Technology (IJETT) - Volume4Issue5- May 2013
Marginal Moment Generating Function Based
Analysis of Channel Capacity for Rayleigh Fading
Tarun Kumar #1, Narayan Kumar Sharma*2, Ashish Soni#3
diversities as well as rate adaptation and transmit power
Abstract: In this paper, we have investigated the marginal
schemes.
In this paper, we have presented a marginal
moment generating function (MMGF) for the Rayleigh
fading channel for the computation of channel capacity for
various adaptive transmission schemes such as: 1) Optimal
simultaneous power and rate adaptation, 2) Optimal rate
adaptation with constant transmit power, 3) Channel
inversion with fixed rate, and 4) Truncated channel
inversion with fixed rate the channel capacity obtained
using this proposed approach for all schemes is compared
moment generating function (MMGF) based channel
capacity analysis over Rayleigh fading channel. The main
contribution of this paper consists of the evaluation of
MMGF function and the derived MMGF function is used to
obtain a closed-form mathematical expression for the
channel capacity with optimal rate and power adaptation
(COPRA), Truncated CIFR (CTCFIR) approach and MGF is
used to derive expression of the channel inversion with
with reported literature.
fixed rate (CCIFR), channel capacity under optimal rate
Key-words: Adaptive transmission techniques, MMGF,
adaptation (CORA) because MGF is a special case of MMGF
MGF, COPRA, CTCIFR.
by using lower limit equal to zero. MMGF is converted into
MGF as shown in next section. The derived expression is
Introduction
expressed in terms of well known Meijer G function and
Recently, the demand of wireless communication is
other special functions, which can easily implemented by
growing explosively; therefore it is very important to
using Maple or Mathematica software. The remainder of
determine the capacity limits of fading channels. In general,
the paper is organized as follows Section 2 describes the
the capacity in fading channel is a complex expression in
MMGF. In Section 3, MMGF is evaluated. The channel
terms of the channel variation in time and/or frequency
capacity evaluation under different policies is performed In
depending upon the transmitter and/or receiver knowledge
the Section 4. Numerical results are discussed In the
of the channel side information. Earlier, the channel
Section 5. Finally, the Section 6 concludes the work.
capacity has been studied by various researchers for several
2.Marginal Moment Generating Function:-
fading
have
In this section, the MMGF of the SNR evaluated and
examined the capacity of Rayleigh fading channels under
further it is used to obtain the channel capacity. The
different adaptive transmission techniques.
MMGF is defined as [24]:
environments.
Goldsmith
and
Alouini
Lee has derived an expression for the channel capacity for
Rayleigh fading channel. Gunther has extended the result of
capacity of Rayleigh fading channels under
different
M (s, a) = ∫
.
( )
(1)
( ) is probability distribution function for Rayleigh
fading.
diversity scheme. Alouini and Goldsmith have derived the
capacity of Rayleigh fading channels under different
ISSN: 2231-5381
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(2)
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International Journal of Engineering Trends and Technology (IJETT) - Volume4Issue5- May 2013
Where ̅ is average SNR of received signal and
is SNR of
channel information, then the channel capacity for the
received signal.
Putting value of
When both the transmitter and receiver have perfect
( ) from equation (2) to (1)
M(s, a) = ∫
COPRA
(
M(s, a) =∫
(
M(s, a)= [
optimal power and rate adaptation (COPRA) is given by [3]
(
)
( )
,
− ∫
[∫
− ∫
[∫
− [∫
−
]
}
]
(3)
=
Moment Generating Function Evaluation :
When putting a=0 it is converted into MGF
{M(s)}
MGF M(s)= ( + )
[∫
{
=
]
( )
dγ
)
)
=
(4)
=
( )
( )
∫
The channel capacity has been regarded as the fundamental
Putting value of A& B then
information theoretic performance measure to predict the
COPRA=
( )
maximum information rate of a communication system. It
[∫
]
)
∫ ( +
)
(
COPRA=
detrimental effects of the multipath fading propagation via
opportunistic and adaptive communication methods. The
)
(
−
spectral efficiency of modern wireless communication
systems and to gain insight into how to counteract the
]
A= ̅
B=-
design of new and more efficient techniques to improve the
)
∫ ( + 3.Marginal MGF based Channel Capacity Analysis:-
is extensively used as the basic tool for the analysis and
(
∫
)
( )
]
)
[∫
− {∫
+
}]
main reason for the analysis of the spectral efficiency over
fading channels is represented by the fact that most
framework described in various literature make use of the
so-called PDF based approach of the received SNR has to
be used, which is a task that might be very cumbersome for
=
=
( )
( )
[∫
(1 −
)ds+
[∫
(1 −
) ds+
I=∫
most system setups and often require to manage expression
including series. It is also well known that a prior
knowledge of channel state information at the transmitter
may be exploited to improve the channel capacity, such that
Let
(
)
∫
+ = After Differentiating we get
= = dt
so
When s=0 then t=1, when s=∞ then t=∞
of a fading channel might be much larger than that of
So from equation (6) I become
without fading.
ISSN: 2231-5381
)
in the low SNR regime, the maximum achievable data rate
(
3.1Optimal Simultaneous Power and Rate Adaptation
(
I=∫
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)
dt
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International Journal of Engineering Trends and Technology (IJETT) - Volume4Issue5- May 2013
(
)
(
.
then
( )
( )
[∫
(1 −
)ds+
.
COPRA=
( )
.
− ( ) [∫
)
ds
+
=1
=1
......(7)
obtained numerically by standard software like Maple and
Mathematica.
( )]
3.2 Channel Inversion With Fixed Rate in terms of
COPRA=
( )
∫
For evaluation of ‘a’ , in equation (7) which can be
(1 −
− () exponential integral function
Where
COPRA=
.
ds = when s=0, t=1 & when s=∞ t=∞
)
I= ∫
I=
put
dt
I= ∫
[(1 −
MGF.
)∫
ds+(1 −
The channel capacity for channel inversion with fixed rate
[(1 −
)∫
ds + ( )]
(CCIFR) requires that the transmitter exploits the channel
state information such that the constant SNR is maintained
)∫
ds is non integral function
at receiver. In this method, fixed transmission rate is used
So,
since the channel after fading inversion appears. The
COPRA= ( )
[ 1−
(1) + ( )]
.
channel capacity with fixed channel inversion rate can be
expressed as:
...................(6)
Equation (6) represents optimal power & rate adaption
channel capacity for Rayleigh distribution.
To obtain the optimal cut-off SNR
= log (1 + = log (1 + ‘ ’ in equation (6) .
Here, we are presenting MMGF based approach for
∫
(
)
= log (1 + optimization of cut-off SNR ‘ ′.
)
( )
∫
)
1
1
1
̅ ∫ ( + ̅ )
)
We know that
( , )
( , )
−∫
= 1
Where M(s, a) is MMGF .
The CIFR suffers from a large capacity penalty relative to
For Rayleigh distribution estimation of ‘a’(cut-off SNR).
−
1
̅
3.3 Truncated Channel Inversion in terms of MMGF:
1
+
̅
=1
other techniques. The truncated CIFR is a better approach
than that of CIFR, where channel fading is inverted above a
cut-off SNR (‘a’). The channel capacity for a truncated
CIFR is defined as :
−
Let
∫
=1
+ = ISSN: 2231-5381
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1 +
∫
1
( , )
{ (0, )}
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International Journal of Engineering Trends and Technology (IJETT) - Volume4Issue5- May 2013
CTCIFR = log
1 +
{
∫
= log
Where
1 + }
{
}
() exponential integral function
4 .Result:In this section, we have presented some numerical results
for the channel capacity related to Rayleigh Fading . Fig. 1
shows Outage probability versus Average. As the value of
cut off SNR increases Outage Probability will also
increases. Fig. 2 shows the channel capacity for optimal
power and rate adaptation (COPRA) versus Average SNR for
various cut off (threshold) SNR. As the Average SNR
increases COPRA Will also increases but the increases fixed
point depends on the cut-off SNR. Fig. 3 shows the channel
capacity with truncated channel inversion with fixed rate
(CTCFIR) versus cut-off SNR for various values of the
Fig (1) plot of outage probability vs Average SNR
Average SNR. From Fig. 3, it is seen that as the SNR
Whereγ(avgSNR) &γ (cut − offSNR)
increase, the cut-off rate also increases. Figures 4 and 5
show the comparison of channel capacity under various
adaptive transmission with the reported literature [16] for
correlated Rayleigh fading channel . The result of the
proposed method is similar with that of [16]. The results of
the proposed method are comparable with that of the [16].
In
Fig.(2 ) plot of C
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vsAverageSNR
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International Journal of Engineering Trends and Technology (IJETT) - Volume4Issue5- May 2013
Where ̅ &
−
Fig. 5 shows the comparison of the characteristics of
channel capacity for truncated channel inversion with fixed
rate with average SNR of the proposed method with [16]
by considering the Rayleigh fading channel . The results of
the proposed method are comparable with that of the [16].
Fig.(3)plot of C
versusavgSNR
Fig.(5) plot of C
Where γavgSNR & γ cutoffSNR
vsAverageSNR
Where γavgSNR & γ cut − offSNR
5. Conclusion
In this paper, we have investigated the marginal moment
generating function
for Rayleigh fading channel with
marginal MGF is used to evaluate the channel capacity
under different adaptive transmission. Due to their simple
forms, these results gives a useful analytical tool for the
accurate performance evaluation of the various systems.
6.Reference:[1 ]
Homayoun Hashemi, “The indoor radio propagation channel”,
proceeding of the IEEE, vol.81, no.7, July 1993.
[2] Kavehrad , “M. and Joseph, M.(1986), “Entropy and method of
moment in evaluation of probability of error in digital communication ”,
presented at the princeton conference on Information Science and System,
1986
[3] V.K.Dwivedi and G.Singh,”Marginal Moment Generating Function
F ig(4) ) plot of C
ISSN: 2231-5381
vsavgSNR
Based Analysis Of Channel Capacity Over Corelated Nakagami –m
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International Journal of Engineering Trends and Technology (IJETT) - Volume4Issue5- May 2013
Fading WithMaximal Ratio Combining Diversty,” Electromagnetics
[16]Alouini, M. S. and A. J. Goldsmith, “Capacity of Rayleigh fading
Research B, Vol 41, 333-356, 2012.
channels under different adaptive transmission and diversity- combining
techniques,” IEEE Trans. on Veh. Technol., Vol. 48, No. 4, 1165{1181,
[4] G.K. Karagiannidis, "Moments-based approach to the performance
Jul. 1999
analysis of equal gain diversity in Nakagami-m fading", presented at IEEE
Transactions on Communications, 2004, pp.685-690
[17] Andrea Goldsmith, Wireless communication , 3rd edition , Cambridge
University press 2007.
[5] F. Hansen and F. I. Mano, "Mobile fading-Rayleigh and lognormal
superimposed”, IEEE Trans. Veh. Technol., vol. 26, pp. 332-335, Nov.
[18]. I. S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series and
1977
Products, Academic Press, San Diego, Calif, USA, 2000
[6] M. K. Simon and M.-S. Alouini, Digital Communication over Fading
[19] Papoulis, A. Probability, Random Variables, and Stochastic
Channels, 2nd edition. New York, Nov. 2004.
Processes, 2nd ed. New York: McGraw-Hill, pp. 92-94, 1984.
[7] M. K. Simon and M.-S. Alouini, Digital Communication over Fading
[20]. Mahmoud H. Ismail and Mustafa M. Matalgah,”Performance of Dual
Channels: A Unified Approach to Performance Analysis, JohnWiley &
Maximal Ratio Combining Diversty in Nonidentical Correlated Weibull
Sons, New York, NY, USA, 2000
Fading Channel Using Pade Approximation,”IEEE Trans. On Comm., vol
54, No. 3 March 2006.
[8] B.P.Lathi, Modern Digital and Analog Communication System, 3rd
ed., oxford university press,2003
[9] John G. Proakis , Masoud Salehi, Communication System Engineering,
2nd edition ,Pearson education, 2004
[10] T. S. Rappaport, Wireless communications - Principles and Practice,
2nd ed., Pearson Education India, 2009.
[11] G. K. Karagiannidis, “Moments-based approach to the performance
analysis of equal gain diversity in Nakagami-m fading”, IEEE
Transactions on Communications, vol. 52, no. 5, pp. 685–690, 2004.
[12] V.K.Dwivedi and G.Singh,”A Novel Moment Generating Function
based performance analysis over corelated Nakagammi –m- fading
Channel,” Springer vol No.10:373–381 DOI 10.1007/s10825-011-0372-9.
[13] A. Annamalai ,C. Tellambura, Vijay K. Bhargava,”A general method
for calculating error probability over fading channel” IEEE Trans.
Commun. ,vol 53 no.5.
[14] M.D.
Renzo, F. Graziosi, F.sanucci “Channel capacity over
generalized fading channel : A novel MGF based approach for
performance analysis and design of wireless communication system”
IEEE Trans. , vol 59,pp 127-150, Jan 2010
[15]. G. K. Karagiannidis, A. Zogas, N.C. Sagias, S.A.Kotsopoulos,”Equal
Gain Maximal Ratio Combining Over Nonidentical Weibull Fading
Channel,”IEEE Trans. On Wireless Comm, Vol.4, No.3,May 2005.
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