Wireless Sensor Network, 2009, 1, 1-60
Published Online April 2009 in SciRes (http://www.SciRP.org/journal/wsn/).
Copyright © 2009 SciRes. Wireless Sensor Network, 2009, 1, 1-60
The Influence of LOS Components on the Statistical
Properties of the Capacity of Amplify-and-Forward Channels
Gulzaib RAFIQ, Matthias PÄTZOLD
Faculty of Engineering and Science, University of Agder, Grimstad, Norway
Received February 26, 2009; revised March 5, 2009; accepted March 7, 2009
Abstract
Amplify-and-forward channels in cooperative networks provide a promising improvement in the network
coverage and system throughput. Under line-of-sight (LOS) propagation conditions in such cooperative
networks, the overall fading channel can be modeled by a double Rice process. In this article, we have stud-
ied the statistical properties of the capacity of double Rice fading channels. We have derived the analytical
expressions for the probability density function (PDF), cumulative distribution function (CDF), level-
crossing rate (LCR), and average duration of fades (ADF) of the channel capacity. The obtained results are
studied for different values of the amplitudes of the LOS components in the two links of double Rice fading
channels. It has been observed that the statistics of the capacity of double Rice fading channels are quite dif-
ferent from those of double Rayleigh and classical Rice fading channels. Moreover, the presence of an LOS
component in any of the two links increases the mean channel capacity and the LCR of the channel capacity.
The validity of the theoretical results is confirmed by simulations. The results presented in this article can be
very useful for communication system designers to optimize the performance of cooperative networks in
wireless communication systems.
Keywords: Amplify-and-Forward Channels, Channel Capacity, Cooperative Networks, Line-of-Sight Component,
Double Rice Process, Double Rayleigh Process, Level-Crossing Rate, Average Duration of Fades
1. Introduction
Increased network coverage, improved link quality, and
provision of new applications with increased mobility
support are the basic demands imposed on future wireless
communication systems. One promising solution to fullfil
these requirements is the use of cooperative diversity
techniques [1-3]. Single-antenna mobile stations in co-
operative networks assist each other to relay the trans-
mitted signal from the source mobile station (SMS) to
the destination mobile station (DMS). The cooperation of
single-antenna mobile stations in such networks to share
their antennas for transmission of the signal makes it
possible to form the so-called virtual multiple-input mul-
tiple-output (MIMO) system [4], thus, achieving the di-
versity gain. Moreover, such cooperation between mo-
bile stations results in increased network coverage with
enhanced mobility support.
For the development and analysis of wireless commu-
nication systems that exploit cooperative diversity, a
solid knowledge of the multipath fading channel charac-
teristics is required. Recent studies illustrate that mo-
bile-to-mobile (M2M) fading channels associated with
relay-based cooperative networks under non-line-of-sight
(NLOS) propagation conditions in different propagation
scenarios can be modeled either as a double Rayleigh
process [5-8] or an NLOS second-order scattering
(NLSS) process [9]. On the other hand, different scenar-
ios under LOS propagation conditions lead to modeling
the overall M2M fading channel either by a double Rice
process [10], a single-LOS double-scattering (SLDS)
process [11], a single-LOS second-order scattering
(SLSS) process [9], or a multiple-LOS second-order
scattering (MLSS) process [12,13]. These studies pro-
vide results for the statistical characterization of M2M
fading channels in cooperative networks under different
propagation conditions. The impact of double Rayleigh
fading on the performance of a communication system is
investigated in [14]. Even with all this research going on,
February 26, 2009. The material in this paper is based on “On the Sta-
tistical Properties of the Capacity of Amplify-and-Forward Channels
Under LOS Conditions”, by Gulzaib Rafiq and Matthias Pätzold which
appeared in the proceedings of 11th IEEE International Conference on
Communications Systems, ICCS 2008, Guangzhou, China, November
2008.
©
2007 IEEE.

8 G. RAFIQ ET AL.
Copyright © 2009 SciRes. Wireless Sensor Network, 2009, 1, 1-60
the important question regarding the maximum possible
information transfer rate in such fading channels is still
unanswered. Thus, the purpose of this paper is to fill in
this gap in information regarding the capacity of am-
plify-and -forward channels in cooperative networks.
Studies pertaining to the analysis of the outage capac-
ity of double Rayleigh channels can be found in [15,16].
However, to the best of the authors’ knowledge, the sta-
tistical properties of the capacity of double Rice channels
have never been investigated. The analysis of the statis-
tical properties of the channel capacity can be very help-
ful to study the dynamic behavior of the channel capacity.
Here, the statistical properties of interest include the PDF,
CDF, LCR, and ADF of the channel capacity. The PDF
and CDF of the channel capacity provide the information
regarding mean value and variance of the channel capac-
ity. The LCR and ADF of the channel capacity, on the
other hand, give a deep insight into the temporal varia-
tions of the channel capacity [17]. The LCR of the chan-
nel capacity is a measure of the rate at which the channel
capacity crosses a certain threshold level from up to
down or vice versa. However, the ADF of the channel
capacity is defined as the average duration of time over
which the channel capacity is below a certain threshold
level [18,19].
In this paper, we have investigated the statistical prop-
erties of the capacity of amplify-and-forward channels in
cooperative networks. The transmitted signal from the
SMS is received at the DMS via a mobile relay (MR).
The MR amplifies the received signal and forwards it to
the DMS. We have assumed that there is no direct trans-
mission link between the SMS and the DMS. Moreover,
it is also assumed that the LOS components exist in both
of the transmission links, i.e., the SMS-MR and MR-
DMS links. Hence, the overall fading channel is modeled
by a double Rice process [10]. We have derived exact
analytical expressions for the PDF, CDF, LCR, and ADF
of the channel capacity of double Rice channels. The
results are studied for different values of the amplitudes
of the LOS components in the two transmission links of
double Rice channels. It has been observed that the sta-
tistics of the capacity of double Rice channels are quite
different from those of double Rayleigh and classical
Rice/Rayleigh channels. Specifically, for medium and
high signal levels, the presence of LOS components in
the two cascaded transmission links increases the mean
channel capacity and the LCR of the channel capacity.
However, it results in a decrease in the ADF of the
channel capacity.
The rest of the paper is organized as follows. In Sec-
tion 2, we describe briefly the double Rice channel
model and some of its statistical properties. The statisti-
cal properties of the capacity of double Rice channels are
studied in Section 3. Section 4 presents the statistical
properties of the capacity of double Rayleigh channels.
Numerical results are discussed in Section 5. Finally, the
concluding remarks are given in Section 6.
2. The Double Rice Channel Model
In cooperative networks employing amplify-and-forward
relay, the channel between the SMS and the DMS via a
MR can be represented as a concatenation of the SMS-
MR and MR-DMS channels [8,10]. Figure 1 depicts an
example of the transmission link from the SMS to the
DMS via the MR in such amplify-and-forward relay-
based networks. For the case when an LOS component is
present in both of the transmission links, i.e., the SMS-
MR link and the MR-DMS link, the overall fading chan-
nel can be modeled as a product of two non-zero- mean
complex Gaussian processes given by [10].
(2) (1)
MR
()() ()tAt t
ρρ
χμμ = (1)
where MR
A
is a real positive constant representing the
relay gain and )2,1()()()( )()()( =+= itmtt iii μμ
ρ
mod-
els the fading in the ith link. Here, )(
)( t i
μ denotes the
scattered component and )(
)( tm i is the LOS component.
The scattered component )(
)( t i
μ can be modeled in the
complex baseband as a complex Gaussian process with
zero mean and variance 2 2i σ , i.e.,
)()()()()(
1
)(
2tjtt iii
μμμ += where, )(
)(
1t
i
μ and )(
)(
2t
i
μ
are the underlying zero-mean real-valued Gaussian
processes. The LOS component )(
)( tm i having ampli-
tude i ρ , Doppler frequency i f
ρ
, and phase i
ρ
θ
i can
be expressed as )2(
)( )( iitfj
i
ietm ρρ
θπ
ρ
+
=. Let SMS,f
ρ
MR ,f
ρ
and DMS
f
ρ
represent the respective Doppler fre-
quencies of the SMS, MR, and DMS, then it can be eas-
ily observed from Figure 1 that 1SMS MR
f
ff
ρρ ρ
=+ and
2MRDMS
f
ff
ρρ ρ
=
+. The envelope of the process ()t
χ
in
(1) results in a double Rice process given by [10]
(1)( 2 )
MR
MR 12
()| ()||()()|
() ()
ttAtt
Att
ρρ
χμμ
ξξ
Ξ= =
= (2)
where )2,1()(
=
it
i
ξ
represents the classical Rice process.
Figure 1. The propagation scenario describing double Rice
fading channels.

G. RAFIQ ET AL. 9
Copyright © 2009 SciRes. Wireless Sensor Network, 2009, 1, 1-60
The PDF of double Rice processes )(tΞis given by [10]
22 22
1MR
22
1MR
(/)
22
22
1MR 0
1MR
00
22
1MR
1
()
,0
zy y
z
pze e
y
zy
II dyz
y
ρρ
σσ
σσ
ρρ
σσ
++
∞−−
Ξ=
⎛⎞⎛ ⎞≥
⎜⎟⎜ ⎟
⎝⎠⎝ ⎠
∫
(3)
where )(
0⋅I is the modified Bessel function of the first
kind of order zero [20], 22
MRMR 2
()A
σ
σ =, and MRMR 2 A
ρ
ρ
=
.
In order to derive the expressions for the statistical
properties of the capacity of double Rice channels, we
need the PDF )(
2zpΞ of the squared process )(
2tΞ
as well as the joint PDF ),(
22 zzp &
&
ΞΞ of )(
2tΞ and its
time derivative )(
2tΞ
&1. The joint ),(
22 zzp&
&
ΞΞ can be
found using the joint PDF ),(zzp&
&
ΞΞ [10] and by using
the concept of transformation of random variables [21]
as
()
22
22 22
1MR MR22(,,,)
12
22 2
1MR MR
11 2
12
442
12 12
/cos
1
22 2
24
0
1MR12
/cos
() (,,,)
28() 2
1
12
1
(,) ,
42
11
2(4 )
,
zy
zy yy
K
zy
yzyz KZY
zy zzy z
z
pzzpz
zz
ee ee
zyz
ee dddyz
θθ
ρρ ρθ
ππ
σσ σ
ππ
ρθ
θθ
σββ ββ
ππσσ ββ
θθ
ΞΞ
ΞΞ
++
∞−− −−
−−
−−−
++
⎛⎞
=⎜⎟
⎝⎠
=+
×
∫∫∫
&
&
&&
&
&
0,| |z≥<∞
&
(4)
where
)()(2 2
max
2
max
2
11 21ff +=
πσβ (5)
23
22 2
2MRmaxmax
2() ()ff βπσ =+
(6)
and
2
11MR2
12
12 42
12
2sin2 sin
(, ,,)=.
fy fz
Kzy
zy z
ρρ
πρθ πρθ
θθ ββ
+
+
(7)
Here 1
max
f, 2
max
f, and 3
max
f denote the maximum
Doppler frequencies of the SMS, MR and DMS, respec-
tively. The expression for the PDF )(
2zpΞ can be ob-
tained by integrating the joint PDF ),(
22 zzp &
&
ΞΞ over
z
&.
Alternatively, in our case the PDF )(
2zpΞ can also be
found from the PDF )(zpΞ in (3) using the concept of
transformation of random variables [21] as
2
22
22
/MR
12
22
2MR
1
22
1MR 0
MR
1
00
22
1MR
1
()=( )
2
11
=2
,0.
y
zy
pzp z
z
ee
y
y
z
II dyz
y
ρ
ρ
σ
σ
σσ
ρ
ρ
σσ
Ξ
Ξ
+
+−
−
∞
⎛⎞ ⎛⎞ ≥
⎜⎟
⎜⎟
⎜⎟ ⎜⎟
⎝⎠
⎝⎠
∫ (8)
In the next section, we will derive the statistical prop-
erties of the capacity of double Rice channels using the
results found in this section.
3. Statistical Properties of the Capacity of
Double Rice Channels
The instantaneous capacity )(tC of double Rice channels
can be expressed using a similar formula found in [22] as
(
)
2
2)(1
log
=)( ttC sΞ+ γ
(
)
2
2
=1()(bits / s / Hz) log st γ +Ξ (9)
where s
γ
denotes the average received signal-to-noise
ratio (SNR) at the DMS. Equation (9) can be considered
as a mapping of a random process )(tΞ to another ran-
dom process )(tC. Hence, the expressions for the statis-
tical properties of the channel capacity )(tC can be
derived by using the results for the statistical properties
of the process )(t
Ξ
obtained in the previous section.
The PDF )(rpC of the channel capacity )(tC can be
found using (8), (9), and by applying the concept of
transformation of random variables as
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛−
Ξs
r
s
r
Cprp
γγ
12(2)ln2
=)( 2
22
22
21/ MR
12
22
2MR
1
22
1MR 0
MR
1
00
2
2
MR
1
2ln(2)1
=2
21 ,0.
y
ry
s
r
s
r
s
ee
y
y
IIdyr
y
ρ
γρ
σ
σ
γσσ
ρ
ρ
σ
γσ
+
−+
−
−
∞
⎛⎞
⎛⎞
−
⎜⎟ ≥
⎜⎟
⎜⎟
⎜⎟
⎝⎠
⎝⎠
∫ (10)
1Throughout this paper, we will represent the time derivative of a proc-
ess by an overdot.

10 G. RAFIQ ET AL.
Copyright © 2009 SciRes. Wireless Sensor Network, 2009, 1, 1-60
The CDF )(rFC of the channel capacity )(tC can
now be obtained by solving the integral given by
0
()=() .
r
CC
F
rpzdz
∫ (11)
By substituting (10) in (11) and doing some mathe-
matical manipulations, the CDF of the channel capacity
can be expressed as follows
22
MR
2
2MR
2
MR 0
MR 1
01
2
1 MR 1
1
()=1
21
,,0
y
C
r
s
Fr ye
y
IQ dyr
y
ρ
σ
σ
ρρ
σ σγσ
+
−
∞
−
⎛⎞
⎛⎞ −
⎜⎟
≥ ⎜⎟
⎜⎟
⎜⎟
⎝⎠
⎝⎠
∫ (12)
where ),(
1
⋅
⋅
Q is the generalized Marcum Q-function
[20]. The LCR )(rNC of the channel capacity )(tC is
defined as [19]
0,),(=)(
0
≥
∫
∞
rzdzrpzrN CC
C&&& & (13)
where ),( zzpCC &
& represents the joint PDF of )(tC and its
time derivative )(tC
&. The joint PDF ),( zzp CC &
& can be
obtained from the joint PDF ),(
22 zzp &
&
ΞΞ by applying
again the concept of transformation of random variables as
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛−
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
ΞΞ s
z
s
z
s
z
CC
z
pzzp
γγγ
(2)ln2
,
12(2)ln2
=),(22
2&
&&
&
()()
()
22
21/ 22
12
cos 21/ cos
MRMR 2
21/2 11
2
222
12
2MR MR1
2 3/2224
0
1MR 12
2ln(2)2 1
=(4)221/
zy
sz
yy y
s
zz
z
ss
eee e
y
γρ ρρθ
γ
ρθ
σππ
σσ σ
ππ
ππγσσββγ
−+ +
−−
−−− −
∞
−−
−
+−
∫∫∫
()()
() ()
()
() ()
1/2
21
ln(2)2 21
ln (2)22121
,, ,
12 12
4
421/,,,
221 82112
12 12 2
12
zz
zz yz z
yz
Ky z
z
zKy
s
y
ys
s
s
eee dddy
θθ
γγθθ
γββ γββ
θθ
−
−⎛⎞
−
−−
⎜⎟
−
−⎜⎟
⎛⎞
⎜⎟
−−
⎜⎟
+−
+− ⎝⎠
⎝⎠
×
&
&
(14)
for 0, <zz≥∞
&, where
(
)
⋅⋅
⋅
⋅
,,,K is defined in (7). After substituting (14) in (13) and carrying out some algebraic cal-
culations, we obtain
2 cos22 21/cos
21/MR 211
12
22
2MR
11
124
5/22 2
0
1MR
21 21
()=
(2 )
r yr y
ys
s
rr
C
s
s
Nre ee
y
ρθ
γρ
θ
γρ π
σ
σσ
π
ββ γ
πγσσ
−
−+
−
−−
∞
−
⎛⎞
−−
+⎜⎟
⎝⎠
∫∫
()
12
1221/ ,,,
12
2
12
21
,, ,
21
1,,,1
22
r
rr
Ky s
s
s
Ky
eKy
πγθθ
π
θθ
γ
πθθ
γ
⎛⎞
−
⎜⎟
⎝⎠
−
⎤
⎧
⎫
⎛⎞
⎛⎞
−
⎥
⎪
⎪ ⎜⎟
⎜⎟
⎜⎟
⎡
⎥
⎛⎞
⎪
⎪
⎜⎟
−
⎪
⎪
⎝⎠
⎢⎜⎟
×+ +Φ
⎥
⎜⎟
⎨
⎬
⎜⎟
⎢
⎥
⎜⎟
⎪
⎪
⎝⎠
⎣
⎥
⎜⎟
⎪
⎪
⎜⎟
⎥
⎪
⎪
⎝⎠
⎩⎭
⎦
∫
()
22
MR 12
221/ ,,,
212
2 MR
12,0
y
r
Ky
s
ee dddyr
ρ
γθθ
σ
θθ
+
−⎛⎞
−−
⎜⎟
⎝⎠
×≥
(15)
where )(⋅Φ denotes the error function [20]. Finally,
from (15) and (12), the ADF )(rTC of the channel ca-
pacity )(tC can be obtained using [19]
()
()= .
()
C
C
C
F
r
Tr Nr (16)
The results found in this section will be used in the
following section to derive the statistical properties of
the capacity of double Rayleigh channels.
4. Statistical Properties of the Capacity of
Double Rayleigh Channels
The double Rayleigh channel follows as a special case of

G. RAFIQ ET AL. 11
Copyright © 2009 SciRes. Wireless Sensor Network, 2009, 1, 1-60
the double Rice channel when 0→
i
ρ
1,2)=(i. Hence,
by letting 0→
i
ρ
1,2)=(i in (10), (12), and (15), the
PDF, CDF, and LCR of the channel capacity of double
Rayleigh channel can be expressed as
2
21
2
22 2
2MR
1
022
1MR
0
2ln(2) 1
()=,0
2
ry
ry
s
Cis
preedyr
y
σ
σγ
ρ
γσσ
−−
−
∞
→≥
∫
(17)
221
222
22
MR 1
02
MR 0
1
()=1, 0
r
y
y
s
Ci
Frye edyr
σσγ
ρ
σ
−
−−
∞
→−≥
∫
(18)
and
022
1MR
2
21
2
22 2
2MR
1
124
0
21
()= 2
21 ,0
r
Ci
s
ry
ry
s
s
Nr
eedyr
y
ρ
σ
σγ
πγ σ σ
ββ γ
→
−−
−
∞
−
⎛⎞
−
+≥
⎜⎟
⎝⎠
∫
(19)
respectively. The ADF of the channel capacity )(tC of dou-
ble Rayleigh channel can be found using (16), (18), and (19).
5. Statistical Properties of the Capacity of
Rayleigh and Rice Channels
In this section, we will present the PDF, CDF, and LCR
of the capacity of Rayleigh and Rice channels. We will
study these results along with the statistical properties of
the capacity of double Rice channels in the next section
for comparison purposes. The PDF )( rpC of the ca-
pacity )(tC of Rice channels can be found using the
PDF )(
2rp
ξ
of the squared Rice process )(
2t
ξ and by
employing the expression presented in (9) corresponding
to Rice processes )(t
ξ
as
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛−
s
r
s
r
Cprp
γγ
ξ
12(2)ln2
=)(2
()
0,
12
2
(2)ln2
=4
0
2
0
2
0
2
2
12
2
0
≥
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛−
+−
−
rIe
s
r
s
s
r
s
r
γσ
ρ
σγ
γσ
ργ
(20)
where
ρ
represent the amplitude of the LOS com-
ponent and 2
0
σ denotes the variance of the underlying
Gaussian processes. By substituting (20) in
dxxprF C
r
C)(=)( 0
∫, the CDF )(rFC of the capacity of
)(tC of Rice channels can be written as
()
0,
12
,1=)( 2
00
1≥
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛−
−rQrF
s
r
C
γσσ
ρ
(21)
By solving zdzrprN CC
C&&
&),(=)( 0
∫∞, the LCR )(rNC
of the capacity of )( tC of Rice channels can be repre-
sented by
() ()
2
21
2
20
0
442
00
21 21
()=, 0
2/
r
s
rr
s
C
ss
Nre Ir
γρ
σγ
β
πσγσγρ
−+
−⎛⎞
−−
⎜⎟
≥
⎜⎟
⎝⎠
(22)
where
β
under isotropic scattering conditions is
given by
(
)
2
0max
2= σπβ f. Here, ),( zzp CC &
& represents
the joint PDF of the capacity )(tC and its time deriva-
tive )(tC
& and max
f denotes the maximum Doppler
frequency.
The results for the PDF, CDF, and LCR of the capac-
ity )(tC of Rayleigh channels can be obtained from
(20)-(22), respectively, by letting 0→
ρ
as follows:
0,
2
(2)ln2
=)(
2
0
2
12
2
0
0≥
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛−
−
→rerp s
r
s
r
C
σγ
ρ
σγ
(23)
0,1=)(
2
0
2
12
0≥− ⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛−
−
→rerF s
r
C
σγ
ρ
(24)
()
21
2
20
02
0
21
1
()=, 0.
2
r
r
s
C
s
Nrer
γσ
ρ
β
πγσ
⎛⎞
−
⎜⎟
−⎜⎟
⎜⎟
⎝⎠
→
−≥ (25)
The expressions (23)-(25) have already been pre-
sented in [23, Equations (23-25)]. However, we have
presented these equations here for the sake of complete-
ness.
6. Numerical Results
This section aims at the validation and analysis of the
analytical results presented in the previous section, using
simulations. We have also included the results for double
Rayleigh, classical Rayleigh [19], and classical Rice
channels in our study for comparison purposes. For the
case of classical Rice channels, we denote the amplitude
of the LOS component as
ρ
. The Rice processes
)()(=)( )()()( tmtt iii + μμ
ρ
1,2)=(i were simulated us-
ing the sum-of-sinusoids model [24]. The model pa-
rameters were computed using the generalized method of
exact Doppler spread (GMEDS1) [25]. The number of
sinusoids ()(
1
i
N and )(
2
i
N) for the resulting determinis-
tic processes )(
)(
1t
~
i
μ and )(
)(
2t
~
i
μ in GMEDS1 were
chosen to be 20== )(
2
)(
1
iiNN for 1,2=i, respectively.
The maximum Doppler frequencies max 2
f and max3
f
were taken to be 91 and 125 Hz, respectively. We have
assumed that the Doppler frequency SMS
f
ρ
equals 0.
Unless stated otherwise, the values of the Doppler fre-

12 G. RAFIQ ET AL.
Copyright © 2009 SciRes. Wireless Sensor Network, 2009, 1, 1-60
quencies MR f
ρ
and DMS f
ρ
were set to be equal to
max 2
f and max 3 f, respectively. The SNR s
γ
was set to
20 dB. The parameters MR
A
and i
σ
1,2)=(i were
chosen to be unity. Finally, using (2) and (9) the simulation
results for the statistical properties of the channel capacity
were found.
The PDF )(zpΞ of double Rice processes )(t
Ξ
are
shown in Figure 2 for different values of the amplitudes of
the LOS components i
ρ
1,2)=(i. In Figure 2, the PDF
of the double Rayleigh process is also shown, which
represents a special case of the double Rice process when
0
21 ==
ρ
ρ
. It is observed that the presence of the LOS
components has a dominant effect on the mean value and
spread of the PDF of double Rice processes. It can also be
seen that the PDF of double Rice processes is identical for
the cases 0=
2
ρ
; 2=
1
ρ
and 2=
2
ρ
; 0=
1
ρ
.
Figures 3 and 4 present the PDF and CDF of the ca-
pacity of double Rice channels, respectively. It is ob-
served that the amplitude of the LOS component has a
significant influence on the PDF and CDF of the channel
capacity. Specifically, the presence of an LOS compo-
nent in one or both of the links (i.e., the SMS-MR and
MR-DMS links) increases the mean channel capacity.
Hence, double Rayleigh channels have a lower mean
channel capacity compared to double Rice channels, (e.g.,
when 2==12
ρ
ρ
). It is also observed that the capacity
of classical Rice channels has a lower mean value com-
pared to that of double Rice channels. These facts are
specifically studied in Figure 5, where the mean channel
capacity of classical Rice as well as double Rice chan-
nels is studied for different values of the amplitudes of
the LOS component. Figure 6 shows the influence of the
amplitude of LOS component on the variance of the
classical Rice and double Rice channels. It is observed
that increasing the value of
ρ
decreases the spread of
the channel capacity for medium and large values of
ρ
,
say 1≥
ρ
. Moreover, the variance of the capacity of
double Rice channels is much higher as compared to that
of classical Rice channels.
Figure 2. The PDF pΞ(z) of double Rice processes Ξ(t).
The LCR and ADF of the channel capacity of double
Rice channels are presented in Figures 7 and 8, respec-
tively. It is evident from Figure 7 that the maximum
value of the LCR of the channel capacity increases with
an increase in the value of the amplitude of the LOS
component i
ρ
1,2)=(i. It is also observed that the
LCR of the capacity of classical Rice channels is much
lower compared to that of double Rice channels. The
converse statements with respect to the LCR of the chan-
nel capacity are true for the ADF, as can be seen in Figure 8.
Figure 3. The PDF pC (r) of the capacity of double Rice
channels.
Figure 4. The CDF FC (r) of the capacity of double Rice
channels.
Figure 5. The mean capacity E{C(t)} of classical Rice and
double Rice channels.

G. RAFIQ ET AL. 13
Copyright © 2009 SciRes. Wireless Sensor Network, 2009, 1, 1-60
Figure 6. The variance Var{C(t)} of the capacity of classical
Rice and double Rice channels.
Figure 7. The LCR NC (r) of the capacity of double Rice
channels.
Figure 8. The ADF TC (r) of the capacity of double Rice
channels.
Figures 9 and 10 illustrate the effect of the Doppler
frequency on the LCR and ADF of the channel
capacity. From Figures 9 and 10, it can clearly be seen
that the LCR and ADF are strongly dependent on the
Doppler frequencies of the MR and the DMS. It is ob-
served that increasing the Doppler frequencies MR
f
ρ
and DMS f
ρ
from 0 to max 2
f and max 3 f, respectively,
results in a significant increase in the LCR. However,
the ADF decreases by increasing the Doppler frequen-
cies of the MR and the DMS.
Figure 9. The LCR NC (r) of the capacity of double Rice
channels.
Figure 10. The ACF TC (r) of the capacity of double Rice
channels.
7. Conclusions
In this paper, we have studied the statistical properties of
the channel capacity of the double Rice channels. We
have derived analytical expressions for the PDF, CDF,
LCR, and ADF of the channel capacity. The findings of
this paper give a deep insight into the influence of the
LOS components, corresponding to the two links of am-
plify-and-forward channels, on the statistical properties
of the channel capacity. It has been observed that for
medium and high signal levels, the presence of the LOS
components in one or both of the links of the double Rice
channel model increases the mean channel capacity and
the LCR of the channel capacity. However, it decreases
the ADF of the channel capacity. Moreover, the Doppler
frequencies of the MR and the DMS have a significant
impact on the LCR and ADF of the channel capacity. The
validity of all the presented analytical results is confirmed
by simulations, whereby a very good fitting between the
analytical and simulation results is found.
8. References
[1] A. Sendonaris, E. Erkip, and B. Aazhang, “User coopera-

14 G. RAFIQ ET AL.
Copyright © 2009 SciRes. Wireless Sensor Network, 2009, 1, 1-60
tion diversity—Part I: System description,” IEEE Trans-
actions on Communications, Vol. 51, No. 11, pp.
1927-1938, November 2003.
[2] A. Sendonaris, E. Erkip, and B. Aazhang, “User coopera-
tion diversity—Part II: Implementation aspects and per-
formance analysis,” IEEE Transactions on Communica-
tions, Vol. 51, No. 11, pp. 1939-1948, November 2003.
[3] J. N. Laneman, D. N. C. Tse, and G. W. Wornell, “Coop-
erative diversity in wireless networks: Efficient protocols
and outage behavior,” IEEE Transactions on Information
Theory, Vol. 50, No. 12, pp. 3062-3080, December 2004.
[4] M. Dohler, “Virtual antenna arrays,” Ph. D. dissertation,
King’s College, London, United Kingdom, 2003.
[5] J. B. Andersen, “Statistical distributions in mobile com-
munications using multiple scattering,” in Proceedings of
27th URSI General Assembly, Maastricht, Netherlands,
August 2002.
[6] I. Z. Kovacs, P. C. F. Eggers, K. Olesen, and L. G. Peter-
sen, “Investigations of outdoor-to-indoor mobile-to-mobile
radio communication channels,” in Proceedings of IEEE
56th Vehicular Technology Conference, VTC’02 Fall,
Vancouver BC, Canada, pp. 430-434, September 2002.
[7] V. Ercerg, S. J. Fortune, J. Ling, A. J. Rustako Jr., and R.
A. Valenzuela, “Comparison of a computer based propa-
gation prediction tool with experimental data collected in
urban microcellular environment,” IEEE Journal on Se-
lected Areas in Communications, Vol. 15, No. 4, pp.
677-684, May 1997.
[8] C. S. Patel, G. L. Stüber, and T. G. Pratt, “Statistical
properties of amplify and forward relay fading channels,”
IEEE Transactions on Vehicular Technology, Vol. 55, No.
1, pp. 1-9, January 2006.
[9] J. Salo, H. M. El Sallabi, and P. Vainikainen, “Statistical
analysis of the multiple scattering radio channel,” IEEE
Transactions on Antennas Propagation, Vol. 54, No. 11,
pp. 3114-3124, November 2006.
[10] B. Talha and M. Pätzold, “On the statistical properties of
double Rice channels,” in Proceedings of 10th Interna-
tional Symposium on Wireless Personal Multimedia
Communications, WPMC 2007, Jaipur, India, pp.
517-522, December 2007.
[11] B. Talha and M. Pätzold, “On the statistical properties
of mobile to mobile fading channels in cooperative net-
works under line of sight conditions,” in Proceedings of
10th International Symposium on Wireless Personal Mul-
timedia Communications, WPMC 2007. Jaipur, India, pp.
388-393, December 2007.
[12] B. Talha and M. Pätzold, “A novel amplify-and-forward
relay channel model for mobile-to-mobile fading chan-
nels under line-of-sight conditions,” in Proceedings of
19th IEEE International Symposium on Personal, Indoor
and Mobile Radio Communications, PIMRC 2008,
Cannes, France, September 2008.
[13] B. Talha and M. Pätzold, “Level-crossing rate and av-
erage duration of fades of the envelope of mobile-
to-mobile fading channels in cooperative networks under
line-of-sight conditions,” in IEEE Conference Proceed-
ings Globecom’08, New Orleans, LA, USA, November
/December 2008.
[14] J. Salo, H. M. El-Sallabi, and P. Vainikainen, “Impact of
double-Rayleigh fading on system performance,” in Pro-
ceedings of 1st IEEE International Symposium on Wire-
less Pervasive Computing, ISWPC 2006. Phuket, Thai-
land, January 2006.
[15] P. Almers, F. Tufvesson, and A. F. Molisch, “Keyhole
effect in MIMO wireless channels: Measurements and
theory,” IEEE Transactions on Wireless Communications,
Vol. 5, No. 12, pp. 3596-3604, 2006.
[16] D. Gesbert, H. Bölcskei, D. A. Gore, and A. J. Paulraj,
“Outdoor MIMO wireless channels: Models and per-
formance prediction,” IEEE Transactions on Wireless
Communications, Vol. 50, No. 12, pp. 1926-1934, De-
cember 2002.
[17] A. Giorgetti, P. J. Smith, M. Shafi, and M. Chiani,
“MIMO capacity, level crossing rates and fades: The im-
pact of spatial/temporal channel correlation,” Journal of
Communications and Networks, Vol. 5, No. 2, pp.
104-115, June 2003.
[18] B. O. Hogstad and M. Pätzold, “Capacity studies of
MIMO models based on the geometrical one-ring scat-
tering model,” in Proceedings of 15th IEEE International
Symposium on Personal, Indoor and Mobile Radio
Communications, PIMRC 2004, Vol. 3. Barcelona, Spain,
pp. 1613-1617, September 2004.
[19] B. O. Hogstad and M. Pätzold, “Exact closed-form ex-
pressions for the distribution, level- crossing rate, and
average duration of fades of the capacity of MIMO
channels,” in Proceedings of 65th Semiannual Vehicular
Technology Conference, IEEE VTC 2007-Spring, Dublin,
Ireland, pp. 455-460, April 2007.
[20] I. S. Gradshteyn and I. M. Ryzhik, “Table of Integrals,
Series, and Products,” 6th edition, Academic Press, 2000.
[21] A. Papoulis and S. U. Pillai, Probability, “Random vari-
ables and stochastic processes,” 4th edition, New York:
McGraw-Hill, 2002.
[22] G. J. Foschini and M. J. Gans, “On limits of wireless
communications in a fading environment when using
multiple antennas,” Wireless Personal Communications,
Vol. 6, pp. 311-335, March 1998.
[23] G. Rafiq and M. Pätzold, “A study of the influence of
shadowing on the statistical properties of the capacity of
mobile radio channels,” Wireless Personal Communica-
tions, DOI 10.1007/s11277-008-9545-7, June 2008.
[24] M. Pätzold, “Mobile fading channels,” Chichester: John
Wiley & Sons, 2002.
[25] M. Pätzold and B. O. Hogstad, “Two new methods for
the generation of multiple uncorrelated Rayleigh fading
waveforms,” in Proceedings of 63rd IEEE Semiannual
Vehicular Technology Conference, IEEE VTC
2006-Spring, Vol. 6, Melbourne, Australia, pp.
2782-2786, May 2006.