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Performance of multihop CDMA ad hoc networks with diversity combining techniques over fading environments
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Performance of multihop CDMA ad hoc networks with diversity combining techniques over fading environments Mahmoud A. Khodeir∗, Sari Khatalin, Sahar Al Ahmad Department of Electrical Engineering, Jordan University of Science and Technology, Irbid 22110, Jordan
a r t i c l e
i n f o
Article history: Available online xxx Keywords: MRC SC Expected progress per hop κ -μ fading CDMA Multihop Ad hoc networks
a b s t r a c t This study investigates the impact of the diversity combining techniques on the performance of wireless code division multiple access (CDMA) ad hoc networks operating over κ -μ fading channels, where the performance criterion in this study is in terms of the expected progress per hop. The probability density function (PDF) and the cumulative density function (CDF) of the interference power are obtained in κ -μ fading conditions. Furthermore, expression for the unconditional CDF of the signal-to-noise-and-interference ratio (SNIR) is derived, which is utilized to derive expressions for the expected progress per hop with selecting combining (SC) and maximal ratio combining (MRC) diversity. Corresponding expressions for Nakagami-m and Rician fading are presented in this paper as special cases of κ -μ fading. Numerical results are presented for illustration purposes. © 2016 Elsevier B.V. All rights reserved.
1. Introduction An ad hoc wireless network comprises of a collection of wireless nodes that can dynamically self-configure to form a temporary wireless network without the help of any inherent infrastructure. Such a network is established to provide wireless communication services in areas where communication infrastructure is too costly or impractical to build. [1–3]. Multihop routing strategy encompasses transmitting packets from the source to the destination via multiple nodes (i.e., multiple short paths) rather than utilizing one long path. It is well-known that most of the nodes in ad hoc networks are limited in terms of power. Therefore, multihop routing strategy can be utilized in ad hoc networks in order to maximize reliability and throughput while maintain power consumption to minimum since transmitting packets via multiple short paths may consume less power than that of long paths [4–7]. One of significant performance metrics in multihop ad hoc networks is the expected progress per hop, which considers both the throughput per hop and the number of hops [5,6]. The expected progress per hop increases as the probability of successful packet transmission increases, and decreases as the number of hops increases. CDMA allows several users to transmit data simultaneously over a single channel by utilizing spread spectrum technology. CDMA has several desirable properties including capacity improvement, network security, multipath resistance and interference rejection, which makes it appropriate to be utilized in ad hoc wireless networks [8–11]. Wireless ad hoc networks have several applications including military wireless applications (e.g., communication services can be established during battlefields or at sea between ships), rescue missions applications (e.g., wireless communication networks can be formed before or after natural disasters such as earthquakes, tornadoes or hurricanes to coordinate emergency disaster relief efforts), business meetings and interactive lectures applications [4,6,11].
∗
Corresponding author. Tel.: +962 272010 0 0; fax: +962 (0) 27095123. E-mail addresses: [email protected] (M.A. Khodeir), [email protected] (S. Khatalin).
http://dx.doi.org/10.1016/j.simpat.2015.12.005 S1569-190X(15)00176-8/© 2016 Elsevier B.V. All rights reserved.
Please cite this article as: M.A. Khodeir et al., Performance of multihop CDMA ad hoc networks with diversity combining techniques over fading environments, Simulation Modelling Practice and Theory (2016), http://dx.doi.org/10.1016/j.simpat.2015.12.005
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Fading phenomena, which is caused by either multipath propagation or shadowing, can drastically impact the performance of wireless communication systems. Hence, fading mitigation techniques are needed. Diversity combining has long been known as an effective scheme for combating fading [12–14]. Two of the most prevalent diversity combining schemes are MRC and SC. In MRC, the diversity branches are co-phased, individually weighted and then combined. MRC diversity is the optimal combining technique, but this comes at the expense of system complexity. On the other hand, SC, in which the combiner selected the branch with the highest SNR, is less complicated system to implement than MRC. However, SC yields sub-optimal performance relative to MRC [13]. Several studies have been published on wireless multipath ad hoc networks including [5,6,10,15–21]. In [5] the optimum transmission ranges were obtained in multihop CDMA networks over an additive white Gaussian noise (AWGN) channel in order to maximize the expected progress per hop. The authors in [15] have extended the channel model to study the optimum transmission ranges under the effect of fading and shadowing. Expressions of the expected progress per hop in multipath CDMA ad hoc networks were derived in [6] over Rayleigh fading channels with SC diversity and without diversity. In [16] the authors proposed a CDMA-based power controlled medium access protocol for wireless ad hoc networks, which takes into consideration the multiple access interference (MAI). This proposed protocol addresses the near-far problem, which results in a network throughput improvement. The impact of multipath diversity on the performance of wireless CDMA ad hoc networks was studied in [10]. It was shown that by utilizing the rake receiver approach for the CDMA systems, multipath diversity can mitigate the multipath fading effect resulting in network capacity improvement. The transmission capacity of CDMA ad hoc networks was considered in [17] for three different models. In [18] the authors analyzed the outage probability and transmission capacity of ad hoc networks in fading and non-fading environments, where multiple antenna diversity schemes were employed. The distribution of MAI power was derived in [19] for the CDMA ad hoc wireless networks operating in Nakagami-m fading environment. The authors in [19] proposed a multi-code multi-packet transmission technique for the derived distribution to improve the expected forward progress. Closed formed expressions for the expected density of progress of ad hoc networks were obtained in [20] over Nakagami-m fading. The three next hop relay receiver selection approach was considered in [20] to derive those expressions for the expected density of progress. Finally, authors in [21] have studied the expected progress per hop of multihop CDMA ad hoc wireless networks operating over κ -μ fading channels. The aim of this paper is to extend the study in [21] by studying the impact of diversity combining techniques on the performance (i.e., in terms of the expected progress per hop) of multihop CDMA ad hoc networks operating over κ -μ fading channels. The κ −μ fading distribution has been proposed in [22] as a general fading distribution that describes the small scale variations of a fading signal in the presence of line of sight (LOS) components. As a general model, κ −μ includes the Rayleigh, Nakagami-m, Nakagami-n (Rician), and one-sided Gaussian distributions as special cases. In particular, expressions for the PDF and the CDF of the interference power operating over κ −μ fading channels are derived in this paper by employing the characteristic function approach. Furthermore, the unconditional CDF of the SNIR is obtained for κ −μ fading, which is used to derive expressions for the expected progress per hop with SC and MRC, and without diversity. The remainder of this paper is organized as follows. Section 2 describes the system model. Expressions of the expected progress per hop are derived in Section 3. Numerical results and discussions are offered in Section 4. Finally, the conclusion is given in Section 5. 2. System model The system model considered in this paper is comprised of a multihop direct sequence CDMA ad hoc network operating under heavy traffic conditions. The system uses slotted ALOHA as a channel access method, which is an improvement to the original ALOHA protocol (i.e., pure ALOHA) in terms of throughput. More details on slotted ALOHA can be found in [3]. In this system model each terminal transmits its packets independently of others with transmission probability p and at the same fixed transmission power. Assuming the nodes are randomly distributed according to a two dimensional Poisson point process model, and the traffic matrix is uniform. Thus, the probability that there are n nodes within a region, RA , with area A is presented as [6,10,15]:
e−λA (λA ) n!
n
Pr (n nodes in RA ) =
(1)
where λ is an average density of nodes per unit area. Assuming the system model considered here is operating in an asynchronous Direct Sequence-Binary Phase Shift Keying (DS-BPSK) transmission with a rectangular chip pulse, then the SNIR,η,can be expressed as [15]:
η=
1 1
η0 +
Y 3GPR
(2)
where η0 denotes the received SNIR in the absence of MAI, Y represents the total MAI power, which is assumed to be a Gaussian random variable), G is the processing gain, and PR is the received signal power. The unconditioned probability of successful packet transmission, Ps , can presented as [6]:
Ps = 1 − Fη (ηc )
(3)
where Fη (.) signifies the CDF of the random variable η, and ηc signifies an arbitrary threshold value of η. Please cite this article as: M.A. Khodeir et al., Performance of multihop CDMA ad hoc networks with diversity combining techniques over fading environments, Simulation Modelling Practice and Theory (2016), http://dx.doi.org/10.1016/j.simpat.2015.12.005
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The channel attenuation considered in this paper is a combination of large scale path loss with a path loss exponent, γ , and the generalized κ −μ fading distribution. Our analysis assumes the fading is slow enough in which the channel remains roughly constant over one time slot. The received power, PR , from a transmitter located at a distance r from the receiver is given as [6]:
α2
PR = γ PT r
(4)
where PT is the transmission power, α is the channel fading amplitude, which is κ −μ random variable with a PDF given by [22]:
αˆ κ
(μ−1 ) 2
α μ
(μ+1 )
2μ ( 1 + κ )
f α (α ) =
2
exp (μκ )
αˆ
e
( )
−μ(1+κ ) ααˆ
2
Iμ−1 2μ
α κ (1 + κ ) αˆ
(5)
where Iv (. ) is the modified Bessel function of the first kind of the order v, and αˆ is the root mean square (rms) value of the fading signal envelope. The parameter μ > 0 denotes the number of multipath clusters, and the parameter κ > 0 denotes the ratio between the total power of the dominant components and the total power of the scattered waves. It should be noted that when κ → 0, the κ −μ distribution in Eq. (5) reduces to the Nakagami-m distribution after utilizing the relation v−1 for the small argument of Bessel function Iv−1 (z ) ≈ [(z/2 ) ]/(v ) [23], where μ coincides with the Nakagami-m parameter m. The Rayleigh and the one-side Gaussian distributions can be obtained by setting m = 1 and m = 0.5, respectively, in the Nakagami-m distribution. Furthermore, for μ = 1, Eq. (5) becomes the Rician distribution, where κ coincide with the Rician parameter K = n2 . Now, we consider the derivation of the distribution of the total MAI power, Y, at a given node for the case of κ −μ fading. First of all, we consider only the interferers located within a distance a from a node located at the center of a circle with a radius a. Then, the total interference power caused by all the interferers can be presented as [6]:
Ya =
Yi
(6)
i in Ra
where Yi is the interference power of the ith interferer. Since we assume that all nodes transmit power equally. Then, one γ can normalize the transmit power as Yi = αi2 /ri , where α i is the κ −μ distributed amplitude attenuation due to fading between the ith interferer and the given node, and ri is the distance between the ith interferer and the given node. Given that the attenuation for each interferer is assumed to be independent and identically distributed (i.i.d), then Ya is the sum of i.i.d random variables. Furthermore, providing that the node and the interferer are positioned within a circle of radius a, then the probability that the interferer is positioned within another circle of smaller radius, r, (i.e., r < a) equals to the ratio between the area of the smaller circle to that of the larger circle. Accordingly, the CDF and PDF of r are given, respectively, as [6]:
Fr (r ) = Pr [r < a] =
r2 , a2
0
and
f r (r ) =
dFr (r ) 2r = 2, dr a
0
Next, the characteristic function of Ya is defined as [6]:
∅Ya (ω ) = E eiωYa
which can be written, after utilizing the identity E[x] = E[E[x|y]], as [6]:
∅Ya (ω ) = E E eiωYa |n nodes in Ra
(7)
After inserting Eq. (7) into Eq. (1), we get [6]:
∅Ya (ω ) =
∞ e−λt π a
2
λt π a2
m
m=0
E eiωYa |m nodes in Ra
m!
(8)
where λt = pλ is the average number of transmitting terminals per unit area over a given time slot. Furthermore, since Ya is the sum of i.i.d random variables, (8) can be presented as:
∅Ya (ω ) =
∞ e−λt π a
2
E e iω a
2
/r γ
=
λt π a2
m!
m=0
where
a
∞
e 0
0
iω α 2 rγ
m
E e iω a
2
/r γ
fr,α (r, α )dα dr
m
(9)
(10)
Please cite this article as: M.A. Khodeir et al., Performance of multihop CDMA ad hoc networks with diversity combining techniques over fading environments, Simulation Modelling Practice and Theory (2016), http://dx.doi.org/10.1016/j.simpat.2015.12.005
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If we assume that the fading channel is independent of the distance, γ = 4, and the fading attenuation of each interferer is i.i.d, then (10) can be written as:
E e
iω a2 r4
=
a
0
∞
2r iωγα2 e r fα (α )dα dr a2
0
(11)
where α has the κ −μ distribution and its PDF is given in (5). Inserting (11) into (9) and evaluating the integrals with the help of [24] yields ∞ e−λt π a
∅Ya (ω ) =
2
λt π a2
n
n!
n=0
{I1 + I2 − I3 }n = eλt π a
2
(I1 +I2 −I3 −1)
(12)
where,
√ iαˆ iωπ (1 + κ )3 [μ + 0.5] I1 = 1 F1 [μ + 0.5, μ, μκ ], μ [μ ] a2
μ (μα 4 (1 + κ ) ) μ2 α 4 κ ( 1 + κ ) I2 = 2 , 2 μ exp μα 4 (1 + κ ) − iω αˆ exp[μκ ] μα 4 (1 + κ ) − iω αˆ and
I3 = 2
∞
−1 j j!(2 j + 1 )
j=0
√
i iω a2
2 j+2
(αˆ )2 j+2 [μ + 0.5] × 1F1 [μ + j + 1, μ, μκ ] j+1 [μ ] exp [μκ ](μ(1 + κ ) )
.
Next, the region of interest is extended to investigate the infinite plane case, which can be done by extending the region of interest from a circle of radius a to an infinite network using ∅Y (ω ) = lima→∞ ∅Ya (ω ), as [6,21]: iπ
∅Y (ω ) = exp[−βI π 2 e− 4
ω /2
(13)
Here, when I = 1 is for the κ −μ fading channel, where:
β1 = αλ ˆ t
2(1 + κ )3 [μ + 0.5] 1F1 πμ [μ ]
1 2
μ + , μ, μκ
When μ = 1, β 1 reduces to the Rician case, where κ coincides with the Rician parameter, K. Moreover, for I = 2 is for Nakagami-m fading channel, where:
β2 = λt
2α 2 [m + 0.5] π m [m ]
and α 2 is the second moment in Nakagami-m PDF (i.e. α 2 = E[α 2 ]). Here, β 2 will represent the Rayleigh case when the parameter m = 1. Furthermore, applying the inverse Fourier transform yields the PDF and CDF of Y for κ −μ distribution and for Nakagami-m distribution as shown in Eqs. (14) and (15), respectively:
f y (y ) =
β1
π 3 / 2 2y
FY (y ) = er f c f y (y ) =
β2
2
4
/8y
y>0
β1 π 2 y>0
(14)
2 2y
π 3 / 2 2y
FY (y ) = er f c
e − β1 π
e − β2 π 2
4
/8y
y>0
β2 π 2 y>0
2 2y
(15)
3. Expected progress In this section we derive the expected progress per hop for the κ -μ fading distribution with SC and MRC diversity and without √ diversity. √ The performance of multihop CDMA ad hoc networks is measured in terms of expected progress per hop [5,6] as λZ = λζ R, where ζ is the one-hop throughput of a transmitter, and R is the distance between the transmitter and the receiver. Previous analysis in [6] has shown that ζ is presented as:
ζ = (1 − p) 1 − e−p Ps
Please cite this article as: M.A. Khodeir et al., Performance of multihop CDMA ad hoc networks with diversity combining techniques over fading environments, Simulation Modelling Practice and Theory (2016), http://dx.doi.org/10.1016/j.simpat.2015.12.005
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where the term (1 − p)(1 − e−p ) is the tendency for a given node to pair up with another node, and Ps is the probability of successful packet transmission, which is given in Eq. (3). Let N = λπ R2 , then N is the average number of nodes that are closer to the receiver than the transmitter. Then, the expected progress per hop is given as [6]:
√
λZ =
(N/π )(1 − p)(1 − e(−p) )Ps N/π (1 − p) 1 − e−p 1 − Fη (ηc )
(16)
3.1. The κ -μ fading channels without diversity The CDF of η conditioned on α is derived in [21, Eq. (13)] as:
F(η|α ) (η|α ) = 1 − FY
3Gα 2 R4 η
= 1 − er f c
√
βI π 2 R2 η √ 2α 6G
α2 > 0
(17)
where refc(.) is the complementary error function, I = 1, and I = 2 corresponding for κ −μ fading and Nakagami-m fading channels, respectively. Furthermore, the unconditional CDF of η can be derived by averaging the CDF of SNIR, F(η|α ) (η|α ), over the PDF of α as [6,21]:
Fη (η ) =
∞ 0
F(η|α ) (η|α ) fα (α )dα
(18)
Now, inserting Eqs. (5) and (17) into Eq. (18) yields, after replacing Iv (.) and erf(.) = 1-erfc(.) with their infinite series representations [24] and evaluating the integral with help of [24], the CDF of η for κ −μ fading as [21]:
Fη (η ) =
∞ z=0
∞ μ+2v−1 2z+1 D2z+1 A C √ π (2z + 1)! v=0 v! [μ + v]
B D2
14 (1+2z−2v−2μ) Kμ+v−z−0.5
√ 2D B
(19)
where:
2μ (1 + κ )κ μ (1 + κ ) A= , B= , C= , (μ−1 ) αˆ αˆ 2 αˆ (μ+1) κ 2 exp (μκ ) N pαˆ πη [μ + 0.5] D= 1F1 [μ + 0.5, μ, μκ ], and Kα (x ) is 2exp[μκ ] 3μG(1 + κ ) [μ ] 2μ ( 1 + κ )
(μ+1 ) 2
the modified Bessel function of the 2nd order. The CDF of η for the Nakagami-m fading case can be obtained as [21]: ∞ C1 2k+1 (C3 )2k+1 Fη (η ) = √ (2k + 1)! π k=0
C4
14 (1+2k−2m)
Km−k−0.5 2C3 C4
C3 2
Nπ pC
(20)
m+0.5] where C1 = [2m] ( m2 )m , C2 = 2παm [ , C3 = 2 2 6ηG , and C4 = m2 . [m] α α Finally, substituting Eqs. (19) and (20) for an arbitrary threshold, ηc , in Eq. (16) yields the expected progress per hop for the κ −μ and the Nakagami-m fading channels, respectively. 2
3.2. With SC route diversity The CDF of κ - fading is given as [25, Eq. (10)]:
Fα (α ) = 1 − Qμ
2κμ,
2(1 + κ )μ
α αˆ 2
(21)
where, Q. [., .] is the generalized Marcum Q-function. The CDF in Eq. (21) can be rewritten as [26, Eq. (7)]:
Fα (α ) =
−1 ∞ (−1 )i e−κμ Lμ (κμ ) α i + μ i μ (1 + κ ) 2 [μ + i + 1] αˆ
(22)
i=0
where, Lzi is the zth order generalized Laguerre polynomial of degree i. The CDF of the SNR at the SC output is given as:
Fα (α )|SC,κ −μ = (Fα (α ) )L
(23)
The PDF at the output of SC can be found by differentiating Eq. (23) with respect to α as: Please cite this article as: M.A. Khodeir et al., Performance of multihop CDMA ad hoc networks with diversity combining techniques over fading environments, Simulation Modelling Practice and Theory (2016), http://dx.doi.org/10.1016/j.simpat.2015.12.005
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fα (α )|SC,κ −μ = L fα (α )(Fα (α ) )L−1
(24)
where L is the number of diversity branches. Substituting Eqs. (5) and (22) in Eq. (24) yields:
fα (α )|SC,κ −μ = LA
α μ
αˆ
2 e[−Bα ] Iμ−1 [C α ]
−1 ∞ (−1 )i e−κμ Lμ (κμ ) i × [μ + i + 1]
i=0
α 2 μ α 2 i μ (1 + κ ) 2 μ (1 + κ ) 2 αˆ αˆ
L−1 (25)
The infinite series in Eq. (25) can be rewritten as [24]:
∞
2 i L−1 ∞ 2 i ρ = γi α i α i=0 i=0
(26)
where,
i −1 (−1 )i Lμ (κμ ) μ(1 + κ ) i ρi = , 2 [μ + i + 1] αˆ 2
γf =
f 1
f ρ0
(iL − f )ρi γ f −i ,
and
γ0 = (ρ0 )L−1 =
i=1
μ−1
(κμ ) (μ + 1 )
L0
L−1
After inserting Eq. (26) in Eq. (25) then Eq. (25) in Eq. (18) and evaluating the integral by following the same process used to derive Eq. (19), we obtain the CDF of η with SC diversity over κ −μ fading channel as:
Fη (η )|SC,κ −μ
i α 2 μ L−1 ∞ μ (1 + κ ) = e γi μ (1 + κ ) 2 2 αˆ αˆ π αˆ 2 i=0 ∞ ∞ 2 j+1 μ +2 n −1 ‘ 2 j (D ) 1 C × Fη (η )|SC,κ −μ n!(μ + n ) 2 2 j + 1 )!! ( n=0 j=0 2LA
μ √
−κμ
where,
F`η (η )|SC,κ −μ =
∞
0
α exp −Bα − z
2
D 2 α
dα =
B −((z+1)/4) D2
(27)
√ K0.5+0.5z 2D B
and z = (2μL + 2n + 2i − 2 j − 2). Setting L = 1 in Eq. (27), we obtain Eq. (19). Substituting Eq. (27) for an arbitrary threshold, ηc , in Eq. (16) yields the expected progress per hop for the κ −μ fading channels with SC diversity. Moreover, the CDF of η with SC diversity over Rician fading channel can be found by setting μ = 1 in Eq. (27) and replacing κ with the Rician parameter, K, as:
Fη (η )|SC,Rician
i α 2 L−1 ∞ (1 + K ) e (1 + K ) γi R 2 αˆ 2 αˆ 2 i=0 ∞ ∞ 2 j+1 2 n 2 j ( DR ) ‘ 1 CR × Fη (η )|SC,Rician n!(1 + n ) 2 2 j + 1 )!! ( n=0 j=0
2LAR = √ αˆ π
where,
−K
−((zR +1)/4) D 2 BR α zR exp −BR α 2 − R dα = K 2 DR BR , 0 . 5+0 . 5 z R α 0 DR 2 ( 2L + 2n + 2i − 2 j − 2 )
F`η (η )|SC,Rician
=
zR
=
AR
=
DR
=
(28)
∞
2 (1 + K ) (1 + K ) A|μ=1&κ K = 2 , BR = B|μ=1&κ K = , = = αˆ exp (K) αˆ 2 Npαπ ˆ η D|μ=1&κ K = F1 [1.5, 1, K ] = 4 exp [K ] 3G(1 + K )
CR = C |μ=1&κ K = =
2
(1 + K )K αˆ
1
Moreover,
ρi R = ρi |μ=1&κ = K
i f (−1 )i L0i (K ) (1 + K ) 1 = , γ = (iL − f )ρi R γ f −i R 2 fR [i + 2] f γ0 αˆ 2 i=1
and
γ0 R = (ρ0 R )L−1 = 1 Please cite this article as: M.A. Khodeir et al., Performance of multihop CDMA ad hoc networks with diversity combining techniques over fading environments, Simulation Modelling Practice and Theory (2016), http://dx.doi.org/10.1016/j.simpat.2015.12.005
and
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As a special case, the PDF at the output of the combiner for the Nakagami-m case is given by [27, Eq. (5)] as:
m
m
fα (α )|SC,Nak. = 2
α2
Lα 2m−1
( [m] )L
exp
−mα 2
α2
L−1 mα 2 γ m, α2
(29)
where, γ [., .] is the incomplete gamma function, given by [27, Eq. (6)]:
γ [s, x] =
∞ (−1 )n xn+s n! ( n + s )
(30)
n=0
Inserting Eq. (30) in Eq. (29), one gets:
fα (α )|SC,Nak. = L
m
m
α2
⎧ n+m ⎫L−1 n mα 2 ⎪ ⎪ ⎨ ⎬ ∞ (−1 ) Lα −mα α2 exp α2 ⎪ ( [m] )L ⎩n=0 n!(n + m) ⎪ ⎭
2m−1
2
(31)
Using [24], one can rewrite the power series in Eq. (31) as:
mα 2
m L−1
α2 where, n =
∞
L−1
n α
2n
=
n=0
(−1 )n (
m
)n
νn α 2 n
(32)
n=0
, ν f = f 1ν 0
α2 n! ( n+m )
∞
f " i=1
(iL − f )i ν f −i , and ν0 = (0 )L−1 = ( m1 )L−1
Now, by inserting Eq. (32) in Eq. (31), then using Eq. (18), one can get the CDF of η for SC over Nakagami-m fading channel, which is given as follows:
Fη (η )|SC,Nak.
2LC1 = √
π
where, F`η (η )|SC,Nak. =
#∞ 0
C4 m [m ]
L−1
∞
νn
n=0
∞ 2 j (C3 )2 j+1 F`η (η )|SC,Nak. (2i + 1)!!
(33)
j=0
2 α q exp[−C4 α 2 − ( Cα3 ) ] dα = ( CC42 )−((q+1)/4) K0.5+0.5q [2C3 C4 ] 3
and
q = 2(mL + n − j − 1). Furthermore, by setting L = 1 in Eq. (32), this will make ν0 = 1, ν f = 0 ∀ f ≥ 1 and q = 2(m − j − 1 ). Again, Eqs. (33) and (20) are equivalent when L = 1. Finally, one can find the expected progress per hop in Nakagami-m fading environments with SC by evaluating the expression of Fη (η) in Eq. (33) for an arbitrary threshold, ηc , and then substituting that in Eq. (16). 3.3. With MRC route diversity The PDF of the MRC output in κ −μ fading is given as [25, Eq. (14)]:
fα (α )|MRC,κ −μ = A1 α Lμ exp −B1 α 2 Iμ−1 [C1 (Lμ+1 )
where A1 =
(L(αˆ ))
2Lμ(1+κ ) 2 (Lμ−1 )
Lμ+1 2
κ
2
exp(Lμκ )
α]
2μ κ) , B1 = μ(α1+ , and C1 = ˆ2
(34) √
L (1+κ )κ . αˆ
Inserting Eq. (34) in Eq. (18), and evaluating the integral by following the same steps used to obtain Eq. (19) yields the CDF of η with MRC diversity over κ −μ fading channel as:
C μ+2n−1 ∞ 2i (D )2i+1 1 F`η (η )|MRC,κ −μ (2i + 1)!! π n=0 n!(μ + n ) 2 i=0
∞ 2A1 Fη (η )|MRC,κ −μ = √
where,
F`η (η ) =
∞ 0
1
(35)
D 2 B −(z1 +1/4) 1 α z1 exp −B1 α 2 − dα = K0.5+0.5z1 2D B1 2 α D
and
z1 = μ ( L + 1 ) + 2n − 2i − 2. Again, setting L = 1 in Eq. (35) yields Eq. (19) as expected. Substituting Eq. (35) for an arbitrary threshold, ηc , in Eq. (16) yields the expected progress per hop for the κ -μ fading channels with MRC diversity. Furthermore, when μ = 1, Eq. (35) reduces to the CDF of η with MRC diversity over Rician fading channels as:
Fη (η )|MRC,Rician =
∞ 2A1 R √
π
n=0
1 C1 R n!(n + 1 ) 2
2 n ∞ 2i (DR )2i+1 F`η (η )|MRC,Rician (2i + 1)!! i=0
(36)
Please cite this article as: M.A. Khodeir et al., Performance of multihop CDMA ad hoc networks with diversity combining techniques over fading environments, Simulation Modelling Practice and Theory (2016), http://dx.doi.org/10.1016/j.simpat.2015.12.005
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Fig. 1. Expected progress per hop versus probability of transmission, p, over κ −μ fading channel when μ = 1.1.
Fig. 2. Expected progress per hop versus probability of transmission, p, in κ −μ fading channel when κ = 0.2.
where
F`η (η ) =
∞ 0
α
z1 R
exp −B1 R α − 2
D 2 R
α
dα =
B1 R D2R
−((z1 R +1)/4)
K0.5+0.5z1 2DR
B1 R
z1 R = ( L + 1 ) + 2n − 2i − 2, A1 R = A1 |
μ=1&κ =K
=
2L ( 1 + K )
(L+1 ) 2
, B1 R = B1 |
=
L+1 μ=1&κ = K (L−1 ) 2 L αˆ K 2 exp (LK )
and
(1 + K ) , αˆ 2
C1 R = C1 |
μ=1&κ = K
=
2 L(1 + K )K
αˆ
.
whenκ → 0, Eq. (35) reduces to the CDF of η with MRC diversity over Nakagami-m fading channel. 4. Numerical results In this section, some numerical examples are presented to illustrate the expected progress per hop over κ −μ fading channels with SC and MRC and without diversity. The values of the parameters κ , μ and m were selected for the examples below as integers and non-integers values to illustrate that the new results in this paper are valid for both cases. 4.1. The κ -μ fading channels √ Figs. 1 and 2 plot the expected progress per hop, λZ, as a function of the probability of transmission, p, for different values of κ and μ, respectively. Here, GP = 64, ηc = 0.44 dB, γ = 4, and N = 15. Fig. 3 shows the same variations used in Please cite this article as: M.A. Khodeir et al., Performance of multihop CDMA ad hoc networks with diversity combining techniques over fading environments, Simulation Modelling Practice and Theory (2016), http://dx.doi.org/10.1016/j.simpat.2015.12.005
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Fig. 3. Expected progress per hop versus probability of transmission, p, over Nakagami-m fading channel with N = 15.
Fig. 4. Expected progress per hop versus N over κ −μ fading channel with p = 0.27.
Fig. 5. Expected progress per hop versus N over Nakagami-m fading channel with p = 0.27.
√ Figs. 1 and 2 for the√Nakagami-m fading case. It is noted in Figs. 1 and 2 that λZ improves with κ while μ is constant. √ On the other hand, λZ decreases as μ increases while κ kept constant. Fig. 3 shows that increasing m enhances λZ. All 3 figures show that the best √ performance is achieved when p ≈ 0.18. Figs. 4 and 5, which are parameterized by κ and μ, and m, respectively, show λZ versus N, where p = 0.27. The two figures show that there is an√optimum value of N that maximizes the expected progress per hop, for different values of κ , μ, and when N √ m. Fig. 5 reveals that λZ is maximized √ is about 5 for all values of m. It is also noted that the performance of λZ improves with m. However, λZ improvement Please cite this article as: M.A. Khodeir et al., Performance of multihop CDMA ad hoc networks with diversity combining techniques over fading environments, Simulation Modelling Practice and Theory (2016), http://dx.doi.org/10.1016/j.simpat.2015.12.005
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Fig. 6. Expected progress per hop versus the parameter m for different values of N with p = 0.27.
Fig. 7. Expected progress per hop versus probability of transmission, p, with SC route diversity over κ −μ fading channel.
√ diminishes as m gets √ larger (i.e., fading gets less severe). Fig. 6 plots λZ versus the parameter m for different values of N. √ Again, as with Fig. 5, λZ has the best performance when N = 5 for all m. Furthermore, λZ improvement √ diminishes for all N as m increases, which lessens the fading severity. Furthermore, it has been noticed in Figs. 4–6 that λZ decreases after reaching the optimum value of N due to the aggregated interference caused by higher number of users after the optimum value of N. 4.2. The SC route diversity √ Fig. 7 shows λZ with SC versus p for various orders of L and the optimum values of N, where μ = 0.55, and κ =0.95. It can be clearly seen that the best performance can be achieved when p ≈ 0.57. furthermore, the figure shows that SC √ √ diversity has improved the performance of λZ. Fig. 8 plots λZ with SC versus p for various values of N, where L = 1, μ = 1, and κ −→ 0 (the Rayleigh fading case). √ Figs. 9 and 10 present λZ with SC versus L for different values of κ and μ, respectively, where p = 0.27. From Fig. 9 it can be seen that any increase in κ while μ is constant leads to decrease the range of L. Moreover, Fig. 10 shows that any increase in μ results in a decrease in the range of L when κ is constant. 4.3. The MRC route diversity √ Fig. 11 shows λZ with MRC versus p for various values of L and N over κ −μ fading channel. It is clear from the Figure that the best performance is√obtained when p ≈ 0.27. Figs. 12 and 13 present λZ versus L-branch MRC for different values of κ and μ, respectively, where p = 0.18. Fig. 12 shows that when μ is constant, any increase in κ leads to improvement in both the expected progress per hop and the range of L. However, increasing μ in Fig. 13, with κ is constant, leads to decrease the range of L in addition the expected progress per hop for L > 2. Please cite this article as: M.A. Khodeir et al., Performance of multihop CDMA ad hoc networks with diversity combining techniques over fading environments, Simulation Modelling Practice and Theory (2016), http://dx.doi.org/10.1016/j.simpat.2015.12.005
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Fig. 8. Expected progress per hop versus probability of transmission, p, with SC route diversity over κ −μ fading channel with μ = 1 and κ −→0.
Fig. 9. Expected progress per hop versus L over κ −μ fading channel using SC route diversity, when μ = 0.5 and N = 10.
Fig. 10. Expected progress per hop versus L over κ −μ fading channel with SC route diversity, when κ = 1.25 and N = 7.
Please cite this article as: M.A. Khodeir et al., Performance of multihop CDMA ad hoc networks with diversity combining techniques over fading environments, Simulation Modelling Practice and Theory (2016), http://dx.doi.org/10.1016/j.simpat.2015.12.005
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Fig. 11. Expected progress per hop versus probability of transmission, p, with MRC route diversity over κ −μ fading channel with μ=1 and κ =0.5.
Fig. 12. Expected progress per hop versus L over κ −μ fading channel with MRC route diversity and μ = 2.25
Fig. 13. Expected progress per hop versus L over κ −μ fading channel with MRC route diversity and κ = 0.95.
Please cite this article as: M.A. Khodeir et al., Performance of multihop CDMA ad hoc networks with diversity combining techniques over fading environments, Simulation Modelling Practice and Theory (2016), http://dx.doi.org/10.1016/j.simpat.2015.12.005
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5. Conclusion This paper analyzed the expected progress per hop with SC and MRC diversity systems and without diversity over the generalized κ − μ fading distribution, where the expected progress was adopted as a performance measurement of multihop CDMA ad hoc network. Results revealed that the best expected progress can be achieved when the probability of transmission p ≈ 0.18 in the case of a single branch used (no diversity). On the other hand, the best expected progress is achieved when p ≈ 0.57 in the case of SC used, and p ≈ 0.27 when MRC is utilized. Moreover, analysis shows that, in the case of no diversity, the expected progress increases as the parameter κ increases while μ is constant, and it decreases as the parameter μ increases with κ kept constant. Furthermore, in the SC case, any increase in κ with μ is constant or any increase in μ with κ is constant leads to decrease in the range of L. Moreover, in the MRC case, increasing κ with μ is constant leads to improvement to both the expected progress per hop and the range of L, whereas increasing μ with κ is constant leads to a decrease in the range of L. Additionally, the expected progress per hop decreases for L > 2. Appendix The two tables below contain some abbreviations and notations that were used in this paper. Tables A1 and A2 Table A1 Abbreviation
Description
MRC SC LOS CDMA PDF CDF SNIR SNR MAI DS-BPSK i.i.d AWGN
Maximal ratio combining Selection combining Line of sight Code division multiple access Probability density function Cumulative density function Signal-to-noise and interference ratio Signal-to-noise ratio Multiple access interference Direct sequence binary phase shift keying independent and identically distributed Additive white Gaussian noise
Table A2 Notation
Description
γ
Path loss exponent Transmission probability Average density of nodes per unit area SNIR at the receiver SNIR at the receiver in the absence of MAI The total received interference power The processing gain The received signal power Unconditioned probability of successful transmission An arbitrary threshold of η The transmitting power The channel fading amplitude The rms value of α in κ −μ PDF The modified Bessel function of the first kind Average density of nodes per unit area over a time slot The second moment in Nakagami-m PDF The number of diversity paths The one-hop throughput of a transmitter Average number of nodes are closer to the Rx than Tx Generalized Marcum Q−function Generalized Laguerre polynomial of zth order and degree i
p
λ η η0 Y Gp Pr Ps
ηc PT
α αˆ
Iz (.)
λt α2 L
ζ N Q[.,.] Lzi
References [1] [2] [3] [4]
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[6] M. Souryal, B. Vojcic, R. Pickholtz, Ad hoc, multihop CDMA etworks with route diversity in a Rayleigh fading channel, IEEE Proceedings of MILCOM, vol. 2, Oct. 2001, pp. 1003–1007. [7] N. Kim, B. An, D. Kim, Y. Lee, Wireless ad-hoc networks using cooperative diversity-based routing in fading channel, in: Proceedings of IEEE PACRIM, Victoria, Canada, Aug. 2007. [8] M.B. Pursley, The role of spread spectrum in packet radio networks, Proc. IEEE 75 (1) (Jan. 1987) 116–134. [9] T.J. Shepard, A channel access scheme for large dense packet radio networks, ACM SIGCOMM, Stanford Univ., Aug. 1996 pp. 219–230. [10] M. Rego, R. Ramos, R. Filho, C. Almeida, CDMA Ad hoc networks in multipath fading channels, in: IEEE Ninth International Symposium on Spread Spectrum Techniques and Application, 2006, pp. 406–409. [11] J. Andrews, S. Weber, M. Haenggi, Ad hoc networks: to spread or not to spread, IEEE Commun. Mag. 45 (12) (Dec. 2007) 8491. [12] D. Brennan, Linear diversity combining techniques, in: Proceedings of the IRE, 47, June 1959, pp. 1075–1102. [13] M.K. Simon, M.-S. Alouini, Digital Communications over Fading Channels: A Unified Approach to Performance Analysis, John Wiley & Son, Inc, New York, 20 0 0. [14] G.L. Stüber, Principles of Mobile Communication, 2nd ed, Kluwer, Norwell, MA, 2001. [15] M. Zorzi, S. Pupolin, Optimum transmission ranges in multi-hop packet radio networks in the presence of fading, IEEE Trans. Commun. 43 (7) (Jul. 1995) 2201–2205. [16] Alaa Muqattash and Marawan Krunz,“CDMA-Based MAC protocol for wireless Ad hoc networks,” Proceedings of the 4th ACM International Symposium on Mobile ad hoc Networking & Computing (MobiHoc ’03), pp. 153-164. [17] S. Weber, X. Yang, G. Veciana, J. Andrews, Transmission capacity of CDMA ad hoc networks, in: IEEE Eighth International Symposium on Spread Spectrum Techniques and Applications, 2004, pp. 245–249. [18] A. Hunter, J. Andrews, S. Weber, The transmission capacity of ad hoc networks with spatial diversity, IEEE Trans. Wirel. Commun. 7 (12) (Dec. 2008) 5058–5071. [19] L.L. Zhang, B.H. Soong, Interference distribution in the Nakagami fading wireless CDMA ad hoc networks with multi-code multi-packet transmission (MCMPT), in: Proceedings of the IEEE LCN, 2005, pp. 84–90. [20] C. Chen, C. Yin, D. Li, G. Yue, Expected density of progress for wireless ad hoc networks with Nakagami-m fading, in: Proceedings of IEEE ICC, June. 2011, pp. 1–5. [21] MahmoudA. Khodeir, Sari Khatalin, SaharAl Ahmad, Performance of multihop CDMA ad hoc networks over κ -μ fading channels, in: Proceedings of the Sixth International Conference on Information and Communication Systems (ICICS 2015), Apr 2015, pp. 07–09. [22] M.D. Yacoub, The κ −μ distribution and the η−μ distribution, IEEE Antennas Propag. Mag. 49 (1) (Feb 2007) 68–81. [23] M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing, Dover, New York, 1972. [24] I.S. Gradshteyn, I.M. Ryzhik, Table of Integrals, Series, and Products, 7th ed., Academic Press, San Diego, CA, 2007. [25] M. Milisi, M. Hamza and M. Hadziali, “outage performance of L-branch Maximal-Ratio Combiner for Generalized κ -μ fading”, [26] S. Andras, A. Baricz, Y. Sun, The generalized Marcum Q-function: an orthogonal polynomial approach, Acta Univ.Sapientiae, Mathmatica, 2011, pp. 60– 70. [27] C.M. Lo, W.H. Lam, Approximate BER performance of generalized selection combining in nakagami-m fading, IEEE Commun. Lett. 5 (6) (JUNE 2001).
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Simulation Modelling Practice and Theory 0 0 0 (2016) 1–14
Contents lists available at ScienceDirect
Simulation Modelling Practice and Theory journal homepage: www.elsevier.com/locate/simpat
Performance of multihop CDMA ad hoc networks with diversity combining techniques over fading environments Mahmoud A. Khodeir∗, Sari Khatalin, Sahar Al Ahmad Department of Electrical Engineering, Jordan University of Science and Technology, Irbid 22110, Jordan
a r t i c l e
i n f o
Article history: Available online xxx Keywords: MRC SC Expected progress per hop κ -μ fading CDMA Multihop Ad hoc networks
a b s t r a c t This study investigates the impact of the diversity combining techniques on the performance of wireless code division multiple access (CDMA) ad hoc networks operating over κ -μ fading channels, where the performance criterion in this study is in terms of the expected progress per hop. The probability density function (PDF) and the cumulative density function (CDF) of the interference power are obtained in κ -μ fading conditions. Furthermore, expression for the unconditional CDF of the signal-to-noise-and-interference ratio (SNIR) is derived, which is utilized to derive expressions for the expected progress per hop with selecting combining (SC) and maximal ratio combining (MRC) diversity. Corresponding expressions for Nakagami-m and Rician fading are presented in this paper as special cases of κ -μ fading. Numerical results are presented for illustration purposes. © 2016 Elsevier B.V. All rights reserved.
1. Introduction An ad hoc wireless network comprises of a collection of wireless nodes that can dynamically self-configure to form a temporary wireless network without the help of any inherent infrastructure. Such a network is established to provide wireless communication services in areas where communication infrastructure is too costly or impractical to build. [1–3]. Multihop routing strategy encompasses transmitting packets from the source to the destination via multiple nodes (i.e., multiple short paths) rather than utilizing one long path. It is well-known that most of the nodes in ad hoc networks are limited in terms of power. Therefore, multihop routing strategy can be utilized in ad hoc networks in order to maximize reliability and throughput while maintain power consumption to minimum since transmitting packets via multiple short paths may consume less power than that of long paths [4–7]. One of significant performance metrics in multihop ad hoc networks is the expected progress per hop, which considers both the throughput per hop and the number of hops [5,6]. The expected progress per hop increases as the probability of successful packet transmission increases, and decreases as the number of hops increases. CDMA allows several users to transmit data simultaneously over a single channel by utilizing spread spectrum technology. CDMA has several desirable properties including capacity improvement, network security, multipath resistance and interference rejection, which makes it appropriate to be utilized in ad hoc wireless networks [8–11]. Wireless ad hoc networks have several applications including military wireless applications (e.g., communication services can be established during battlefields or at sea between ships), rescue missions applications (e.g., wireless communication networks can be formed before or after natural disasters such as earthquakes, tornadoes or hurricanes to coordinate emergency disaster relief efforts), business meetings and interactive lectures applications [4,6,11].
∗
Corresponding author. Tel.: +962 272010 0 0; fax: +962 (0) 27095123. E-mail addresses: [email protected] (M.A. Khodeir), [email protected] (S. Khatalin).
http://dx.doi.org/10.1016/j.simpat.2015.12.005 S1569-190X(15)00176-8/© 2016 Elsevier B.V. All rights reserved.
Please cite this article as: M.A. Khodeir et al., Performance of multihop CDMA ad hoc networks with diversity combining techniques over fading environments, Simulation Modelling Practice and Theory (2016), http://dx.doi.org/10.1016/j.simpat.2015.12.005
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Fading phenomena, which is caused by either multipath propagation or shadowing, can drastically impact the performance of wireless communication systems. Hence, fading mitigation techniques are needed. Diversity combining has long been known as an effective scheme for combating fading [12–14]. Two of the most prevalent diversity combining schemes are MRC and SC. In MRC, the diversity branches are co-phased, individually weighted and then combined. MRC diversity is the optimal combining technique, but this comes at the expense of system complexity. On the other hand, SC, in which the combiner selected the branch with the highest SNR, is less complicated system to implement than MRC. However, SC yields sub-optimal performance relative to MRC [13]. Several studies have been published on wireless multipath ad hoc networks including [5,6,10,15–21]. In [5] the optimum transmission ranges were obtained in multihop CDMA networks over an additive white Gaussian noise (AWGN) channel in order to maximize the expected progress per hop. The authors in [15] have extended the channel model to study the optimum transmission ranges under the effect of fading and shadowing. Expressions of the expected progress per hop in multipath CDMA ad hoc networks were derived in [6] over Rayleigh fading channels with SC diversity and without diversity. In [16] the authors proposed a CDMA-based power controlled medium access protocol for wireless ad hoc networks, which takes into consideration the multiple access interference (MAI). This proposed protocol addresses the near-far problem, which results in a network throughput improvement. The impact of multipath diversity on the performance of wireless CDMA ad hoc networks was studied in [10]. It was shown that by utilizing the rake receiver approach for the CDMA systems, multipath diversity can mitigate the multipath fading effect resulting in network capacity improvement. The transmission capacity of CDMA ad hoc networks was considered in [17] for three different models. In [18] the authors analyzed the outage probability and transmission capacity of ad hoc networks in fading and non-fading environments, where multiple antenna diversity schemes were employed. The distribution of MAI power was derived in [19] for the CDMA ad hoc wireless networks operating in Nakagami-m fading environment. The authors in [19] proposed a multi-code multi-packet transmission technique for the derived distribution to improve the expected forward progress. Closed formed expressions for the expected density of progress of ad hoc networks were obtained in [20] over Nakagami-m fading. The three next hop relay receiver selection approach was considered in [20] to derive those expressions for the expected density of progress. Finally, authors in [21] have studied the expected progress per hop of multihop CDMA ad hoc wireless networks operating over κ -μ fading channels. The aim of this paper is to extend the study in [21] by studying the impact of diversity combining techniques on the performance (i.e., in terms of the expected progress per hop) of multihop CDMA ad hoc networks operating over κ -μ fading channels. The κ −μ fading distribution has been proposed in [22] as a general fading distribution that describes the small scale variations of a fading signal in the presence of line of sight (LOS) components. As a general model, κ −μ includes the Rayleigh, Nakagami-m, Nakagami-n (Rician), and one-sided Gaussian distributions as special cases. In particular, expressions for the PDF and the CDF of the interference power operating over κ −μ fading channels are derived in this paper by employing the characteristic function approach. Furthermore, the unconditional CDF of the SNIR is obtained for κ −μ fading, which is used to derive expressions for the expected progress per hop with SC and MRC, and without diversity. The remainder of this paper is organized as follows. Section 2 describes the system model. Expressions of the expected progress per hop are derived in Section 3. Numerical results and discussions are offered in Section 4. Finally, the conclusion is given in Section 5. 2. System model The system model considered in this paper is comprised of a multihop direct sequence CDMA ad hoc network operating under heavy traffic conditions. The system uses slotted ALOHA as a channel access method, which is an improvement to the original ALOHA protocol (i.e., pure ALOHA) in terms of throughput. More details on slotted ALOHA can be found in [3]. In this system model each terminal transmits its packets independently of others with transmission probability p and at the same fixed transmission power. Assuming the nodes are randomly distributed according to a two dimensional Poisson point process model, and the traffic matrix is uniform. Thus, the probability that there are n nodes within a region, RA , with area A is presented as [6,10,15]:
e−λA (λA ) n!
n
Pr (n nodes in RA ) =
(1)
where λ is an average density of nodes per unit area. Assuming the system model considered here is operating in an asynchronous Direct Sequence-Binary Phase Shift Keying (DS-BPSK) transmission with a rectangular chip pulse, then the SNIR,η,can be expressed as [15]:
η=
1 1
η0 +
Y 3GPR
(2)
where η0 denotes the received SNIR in the absence of MAI, Y represents the total MAI power, which is assumed to be a Gaussian random variable), G is the processing gain, and PR is the received signal power. The unconditioned probability of successful packet transmission, Ps , can presented as [6]:
Ps = 1 − Fη (ηc )
(3)
where Fη (.) signifies the CDF of the random variable η, and ηc signifies an arbitrary threshold value of η. Please cite this article as: M.A. Khodeir et al., Performance of multihop CDMA ad hoc networks with diversity combining techniques over fading environments, Simulation Modelling Practice and Theory (2016), http://dx.doi.org/10.1016/j.simpat.2015.12.005
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The channel attenuation considered in this paper is a combination of large scale path loss with a path loss exponent, γ , and the generalized κ −μ fading distribution. Our analysis assumes the fading is slow enough in which the channel remains roughly constant over one time slot. The received power, PR , from a transmitter located at a distance r from the receiver is given as [6]:
α2
PR = γ PT r
(4)
where PT is the transmission power, α is the channel fading amplitude, which is κ −μ random variable with a PDF given by [22]:
αˆ κ
(μ−1 ) 2
α μ
(μ+1 )
2μ ( 1 + κ )
f α (α ) =
2
exp (μκ )
αˆ
e
( )
−μ(1+κ ) ααˆ
2
Iμ−1 2μ
α κ (1 + κ ) αˆ
(5)
where Iv (. ) is the modified Bessel function of the first kind of the order v, and αˆ is the root mean square (rms) value of the fading signal envelope. The parameter μ > 0 denotes the number of multipath clusters, and the parameter κ > 0 denotes the ratio between the total power of the dominant components and the total power of the scattered waves. It should be noted that when κ → 0, the κ −μ distribution in Eq. (5) reduces to the Nakagami-m distribution after utilizing the relation v−1 for the small argument of Bessel function Iv−1 (z ) ≈ [(z/2 ) ]/(v ) [23], where μ coincides with the Nakagami-m parameter m. The Rayleigh and the one-side Gaussian distributions can be obtained by setting m = 1 and m = 0.5, respectively, in the Nakagami-m distribution. Furthermore, for μ = 1, Eq. (5) becomes the Rician distribution, where κ coincide with the Rician parameter K = n2 . Now, we consider the derivation of the distribution of the total MAI power, Y, at a given node for the case of κ −μ fading. First of all, we consider only the interferers located within a distance a from a node located at the center of a circle with a radius a. Then, the total interference power caused by all the interferers can be presented as [6]:
Ya =
Yi
(6)
i in Ra
where Yi is the interference power of the ith interferer. Since we assume that all nodes transmit power equally. Then, one γ can normalize the transmit power as Yi = αi2 /ri , where α i is the κ −μ distributed amplitude attenuation due to fading between the ith interferer and the given node, and ri is the distance between the ith interferer and the given node. Given that the attenuation for each interferer is assumed to be independent and identically distributed (i.i.d), then Ya is the sum of i.i.d random variables. Furthermore, providing that the node and the interferer are positioned within a circle of radius a, then the probability that the interferer is positioned within another circle of smaller radius, r, (i.e., r < a) equals to the ratio between the area of the smaller circle to that of the larger circle. Accordingly, the CDF and PDF of r are given, respectively, as [6]:
Fr (r ) = Pr [r < a] =
r2 , a2
0
and
f r (r ) =
dFr (r ) 2r = 2, dr a
0
Next, the characteristic function of Ya is defined as [6]:
∅Ya (ω ) = E eiωYa
which can be written, after utilizing the identity E[x] = E[E[x|y]], as [6]:
∅Ya (ω ) = E E eiωYa |n nodes in Ra
(7)
After inserting Eq. (7) into Eq. (1), we get [6]:
∅Ya (ω ) =
∞ e−λt π a
2
λt π a2
m
m=0
E eiωYa |m nodes in Ra
m!
(8)
where λt = pλ is the average number of transmitting terminals per unit area over a given time slot. Furthermore, since Ya is the sum of i.i.d random variables, (8) can be presented as:
∅Ya (ω ) =
∞ e−λt π a
2
E e iω a
2
/r γ
=
λt π a2
m!
m=0
where
a
∞
e 0
0
iω α 2 rγ
m
E e iω a
2
/r γ
fr,α (r, α )dα dr
m
(9)
(10)
Please cite this article as: M.A. Khodeir et al., Performance of multihop CDMA ad hoc networks with diversity combining techniques over fading environments, Simulation Modelling Practice and Theory (2016), http://dx.doi.org/10.1016/j.simpat.2015.12.005
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If we assume that the fading channel is independent of the distance, γ = 4, and the fading attenuation of each interferer is i.i.d, then (10) can be written as:
E e
iω a2 r4
=
a
0
∞
2r iωγα2 e r fα (α )dα dr a2
0
(11)
where α has the κ −μ distribution and its PDF is given in (5). Inserting (11) into (9) and evaluating the integrals with the help of [24] yields ∞ e−λt π a
∅Ya (ω ) =
2
λt π a2
n
n!
n=0
{I1 + I2 − I3 }n = eλt π a
2
(I1 +I2 −I3 −1)
(12)
where,
√ iαˆ iωπ (1 + κ )3 [μ + 0.5] I1 = 1 F1 [μ + 0.5, μ, μκ ], μ [μ ] a2
μ (μα 4 (1 + κ ) ) μ2 α 4 κ ( 1 + κ ) I2 = 2 , 2 μ exp μα 4 (1 + κ ) − iω αˆ exp[μκ ] μα 4 (1 + κ ) − iω αˆ and
I3 = 2
∞
−1 j j!(2 j + 1 )
j=0
√
i iω a2
2 j+2
(αˆ )2 j+2 [μ + 0.5] × 1F1 [μ + j + 1, μ, μκ ] j+1 [μ ] exp [μκ ](μ(1 + κ ) )
.
Next, the region of interest is extended to investigate the infinite plane case, which can be done by extending the region of interest from a circle of radius a to an infinite network using ∅Y (ω ) = lima→∞ ∅Ya (ω ), as [6,21]: iπ
∅Y (ω ) = exp[−βI π 2 e− 4
ω /2
(13)
Here, when I = 1 is for the κ −μ fading channel, where:
β1 = αλ ˆ t
2(1 + κ )3 [μ + 0.5] 1F1 πμ [μ ]
1 2
μ + , μ, μκ
When μ = 1, β 1 reduces to the Rician case, where κ coincides with the Rician parameter, K. Moreover, for I = 2 is for Nakagami-m fading channel, where:
β2 = λt
2α 2 [m + 0.5] π m [m ]
and α 2 is the second moment in Nakagami-m PDF (i.e. α 2 = E[α 2 ]). Here, β 2 will represent the Rayleigh case when the parameter m = 1. Furthermore, applying the inverse Fourier transform yields the PDF and CDF of Y for κ −μ distribution and for Nakagami-m distribution as shown in Eqs. (14) and (15), respectively:
f y (y ) =
β1
π 3 / 2 2y
FY (y ) = er f c f y (y ) =
β2
2
4
/8y
y>0
β1 π 2 y>0
(14)
2 2y
π 3 / 2 2y
FY (y ) = er f c
e − β1 π
e − β2 π 2
4
/8y
y>0
β2 π 2 y>0
2 2y
(15)
3. Expected progress In this section we derive the expected progress per hop for the κ -μ fading distribution with SC and MRC diversity and without √ diversity. √ The performance of multihop CDMA ad hoc networks is measured in terms of expected progress per hop [5,6] as λZ = λζ R, where ζ is the one-hop throughput of a transmitter, and R is the distance between the transmitter and the receiver. Previous analysis in [6] has shown that ζ is presented as:
ζ = (1 − p) 1 − e−p Ps
Please cite this article as: M.A. Khodeir et al., Performance of multihop CDMA ad hoc networks with diversity combining techniques over fading environments, Simulation Modelling Practice and Theory (2016), http://dx.doi.org/10.1016/j.simpat.2015.12.005
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where the term (1 − p)(1 − e−p ) is the tendency for a given node to pair up with another node, and Ps is the probability of successful packet transmission, which is given in Eq. (3). Let N = λπ R2 , then N is the average number of nodes that are closer to the receiver than the transmitter. Then, the expected progress per hop is given as [6]:
√
λZ =
(N/π )(1 − p)(1 − e(−p) )Ps N/π (1 − p) 1 − e−p 1 − Fη (ηc )
(16)
3.1. The κ -μ fading channels without diversity The CDF of η conditioned on α is derived in [21, Eq. (13)] as:
F(η|α ) (η|α ) = 1 − FY
3Gα 2 R4 η
= 1 − er f c
√
βI π 2 R2 η √ 2α 6G
α2 > 0
(17)
where refc(.) is the complementary error function, I = 1, and I = 2 corresponding for κ −μ fading and Nakagami-m fading channels, respectively. Furthermore, the unconditional CDF of η can be derived by averaging the CDF of SNIR, F(η|α ) (η|α ), over the PDF of α as [6,21]:
Fη (η ) =
∞ 0
F(η|α ) (η|α ) fα (α )dα
(18)
Now, inserting Eqs. (5) and (17) into Eq. (18) yields, after replacing Iv (.) and erf(.) = 1-erfc(.) with their infinite series representations [24] and evaluating the integral with help of [24], the CDF of η for κ −μ fading as [21]:
Fη (η ) =
∞ z=0
∞ μ+2v−1 2z+1 D2z+1 A C √ π (2z + 1)! v=0 v! [μ + v]
B D2
14 (1+2z−2v−2μ) Kμ+v−z−0.5
√ 2D B
(19)
where:
2μ (1 + κ )κ μ (1 + κ ) A= , B= , C= , (μ−1 ) αˆ αˆ 2 αˆ (μ+1) κ 2 exp (μκ ) N pαˆ πη [μ + 0.5] D= 1F1 [μ + 0.5, μ, μκ ], and Kα (x ) is 2exp[μκ ] 3μG(1 + κ ) [μ ] 2μ ( 1 + κ )
(μ+1 ) 2
the modified Bessel function of the 2nd order. The CDF of η for the Nakagami-m fading case can be obtained as [21]: ∞ C1 2k+1 (C3 )2k+1 Fη (η ) = √ (2k + 1)! π k=0
C4
14 (1+2k−2m)
Km−k−0.5 2C3 C4
C3 2
Nπ pC
(20)
m+0.5] where C1 = [2m] ( m2 )m , C2 = 2παm [ , C3 = 2 2 6ηG , and C4 = m2 . [m] α α Finally, substituting Eqs. (19) and (20) for an arbitrary threshold, ηc , in Eq. (16) yields the expected progress per hop for the κ −μ and the Nakagami-m fading channels, respectively. 2
3.2. With SC route diversity The CDF of κ - fading is given as [25, Eq. (10)]:
Fα (α ) = 1 − Qμ
2κμ,
2(1 + κ )μ
α αˆ 2
(21)
where, Q. [., .] is the generalized Marcum Q-function. The CDF in Eq. (21) can be rewritten as [26, Eq. (7)]:
Fα (α ) =
−1 ∞ (−1 )i e−κμ Lμ (κμ ) α i + μ i μ (1 + κ ) 2 [μ + i + 1] αˆ
(22)
i=0
where, Lzi is the zth order generalized Laguerre polynomial of degree i. The CDF of the SNR at the SC output is given as:
Fα (α )|SC,κ −μ = (Fα (α ) )L
(23)
The PDF at the output of SC can be found by differentiating Eq. (23) with respect to α as: Please cite this article as: M.A. Khodeir et al., Performance of multihop CDMA ad hoc networks with diversity combining techniques over fading environments, Simulation Modelling Practice and Theory (2016), http://dx.doi.org/10.1016/j.simpat.2015.12.005
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fα (α )|SC,κ −μ = L fα (α )(Fα (α ) )L−1
(24)
where L is the number of diversity branches. Substituting Eqs. (5) and (22) in Eq. (24) yields:
fα (α )|SC,κ −μ = LA
α μ
αˆ
2 e[−Bα ] Iμ−1 [C α ]
−1 ∞ (−1 )i e−κμ Lμ (κμ ) i × [μ + i + 1]
i=0
α 2 μ α 2 i μ (1 + κ ) 2 μ (1 + κ ) 2 αˆ αˆ
L−1 (25)
The infinite series in Eq. (25) can be rewritten as [24]:
∞
2 i L−1 ∞ 2 i ρ = γi α i α i=0 i=0
(26)
where,
i −1 (−1 )i Lμ (κμ ) μ(1 + κ ) i ρi = , 2 [μ + i + 1] αˆ 2
γf =
f 1
f ρ0
(iL − f )ρi γ f −i ,
and
γ0 = (ρ0 )L−1 =
i=1
μ−1
(κμ ) (μ + 1 )
L0
L−1
After inserting Eq. (26) in Eq. (25) then Eq. (25) in Eq. (18) and evaluating the integral by following the same process used to derive Eq. (19), we obtain the CDF of η with SC diversity over κ −μ fading channel as:
Fη (η )|SC,κ −μ
i α 2 μ L−1 ∞ μ (1 + κ ) = e γi μ (1 + κ ) 2 2 αˆ αˆ π αˆ 2 i=0 ∞ ∞ 2 j+1 μ +2 n −1 ‘ 2 j (D ) 1 C × Fη (η )|SC,κ −μ n!(μ + n ) 2 2 j + 1 )!! ( n=0 j=0 2LA
μ √
−κμ
where,
F`η (η )|SC,κ −μ =
∞
0
α exp −Bα − z
2
D 2 α
dα =
B −((z+1)/4) D2
(27)
√ K0.5+0.5z 2D B
and z = (2μL + 2n + 2i − 2 j − 2). Setting L = 1 in Eq. (27), we obtain Eq. (19). Substituting Eq. (27) for an arbitrary threshold, ηc , in Eq. (16) yields the expected progress per hop for the κ −μ fading channels with SC diversity. Moreover, the CDF of η with SC diversity over Rician fading channel can be found by setting μ = 1 in Eq. (27) and replacing κ with the Rician parameter, K, as:
Fη (η )|SC,Rician
i α 2 L−1 ∞ (1 + K ) e (1 + K ) γi R 2 αˆ 2 αˆ 2 i=0 ∞ ∞ 2 j+1 2 n 2 j ( DR ) ‘ 1 CR × Fη (η )|SC,Rician n!(1 + n ) 2 2 j + 1 )!! ( n=0 j=0
2LAR = √ αˆ π
where,
−K
−((zR +1)/4) D 2 BR α zR exp −BR α 2 − R dα = K 2 DR BR , 0 . 5+0 . 5 z R α 0 DR 2 ( 2L + 2n + 2i − 2 j − 2 )
F`η (η )|SC,Rician
=
zR
=
AR
=
DR
=
(28)
∞
2 (1 + K ) (1 + K ) A|μ=1&κ K = 2 , BR = B|μ=1&κ K = , = = αˆ exp (K) αˆ 2 Npαπ ˆ η D|μ=1&κ K = F1 [1.5, 1, K ] = 4 exp [K ] 3G(1 + K )
CR = C |μ=1&κ K = =
2
(1 + K )K αˆ
1
Moreover,
ρi R = ρi |μ=1&κ = K
i f (−1 )i L0i (K ) (1 + K ) 1 = , γ = (iL − f )ρi R γ f −i R 2 fR [i + 2] f γ0 αˆ 2 i=1
and
γ0 R = (ρ0 R )L−1 = 1 Please cite this article as: M.A. Khodeir et al., Performance of multihop CDMA ad hoc networks with diversity combining techniques over fading environments, Simulation Modelling Practice and Theory (2016), http://dx.doi.org/10.1016/j.simpat.2015.12.005
and
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As a special case, the PDF at the output of the combiner for the Nakagami-m case is given by [27, Eq. (5)] as:
m
m
fα (α )|SC,Nak. = 2
α2
Lα 2m−1
( [m] )L
exp
−mα 2
α2
L−1 mα 2 γ m, α2
(29)
where, γ [., .] is the incomplete gamma function, given by [27, Eq. (6)]:
γ [s, x] =
∞ (−1 )n xn+s n! ( n + s )
(30)
n=0
Inserting Eq. (30) in Eq. (29), one gets:
fα (α )|SC,Nak. = L
m
m
α2
⎧ n+m ⎫L−1 n mα 2 ⎪ ⎪ ⎨ ⎬ ∞ (−1 ) Lα −mα α2 exp α2 ⎪ ( [m] )L ⎩n=0 n!(n + m) ⎪ ⎭
2m−1
2
(31)
Using [24], one can rewrite the power series in Eq. (31) as:
mα 2
m L−1
α2 where, n =
∞
L−1
n α
2n
=
n=0
(−1 )n (
m
)n
νn α 2 n
(32)
n=0
, ν f = f 1ν 0
α2 n! ( n+m )
∞
f " i=1
(iL − f )i ν f −i , and ν0 = (0 )L−1 = ( m1 )L−1
Now, by inserting Eq. (32) in Eq. (31), then using Eq. (18), one can get the CDF of η for SC over Nakagami-m fading channel, which is given as follows:
Fη (η )|SC,Nak.
2LC1 = √
π
where, F`η (η )|SC,Nak. =
#∞ 0
C4 m [m ]
L−1
∞
νn
n=0
∞ 2 j (C3 )2 j+1 F`η (η )|SC,Nak. (2i + 1)!!
(33)
j=0
2 α q exp[−C4 α 2 − ( Cα3 ) ] dα = ( CC42 )−((q+1)/4) K0.5+0.5q [2C3 C4 ] 3
and
q = 2(mL + n − j − 1). Furthermore, by setting L = 1 in Eq. (32), this will make ν0 = 1, ν f = 0 ∀ f ≥ 1 and q = 2(m − j − 1 ). Again, Eqs. (33) and (20) are equivalent when L = 1. Finally, one can find the expected progress per hop in Nakagami-m fading environments with SC by evaluating the expression of Fη (η) in Eq. (33) for an arbitrary threshold, ηc , and then substituting that in Eq. (16). 3.3. With MRC route diversity The PDF of the MRC output in κ −μ fading is given as [25, Eq. (14)]:
fα (α )|MRC,κ −μ = A1 α Lμ exp −B1 α 2 Iμ−1 [C1 (Lμ+1 )
where A1 =
(L(αˆ ))
2Lμ(1+κ ) 2 (Lμ−1 )
Lμ+1 2
κ
2
exp(Lμκ )
α]
2μ κ) , B1 = μ(α1+ , and C1 = ˆ2
(34) √
L (1+κ )κ . αˆ
Inserting Eq. (34) in Eq. (18), and evaluating the integral by following the same steps used to obtain Eq. (19) yields the CDF of η with MRC diversity over κ −μ fading channel as:
C μ+2n−1 ∞ 2i (D )2i+1 1 F`η (η )|MRC,κ −μ (2i + 1)!! π n=0 n!(μ + n ) 2 i=0
∞ 2A1 Fη (η )|MRC,κ −μ = √
where,
F`η (η ) =
∞ 0
1
(35)
D 2 B −(z1 +1/4) 1 α z1 exp −B1 α 2 − dα = K0.5+0.5z1 2D B1 2 α D
and
z1 = μ ( L + 1 ) + 2n − 2i − 2. Again, setting L = 1 in Eq. (35) yields Eq. (19) as expected. Substituting Eq. (35) for an arbitrary threshold, ηc , in Eq. (16) yields the expected progress per hop for the κ -μ fading channels with MRC diversity. Furthermore, when μ = 1, Eq. (35) reduces to the CDF of η with MRC diversity over Rician fading channels as:
Fη (η )|MRC,Rician =
∞ 2A1 R √
π
n=0
1 C1 R n!(n + 1 ) 2
2 n ∞ 2i (DR )2i+1 F`η (η )|MRC,Rician (2i + 1)!! i=0
(36)
Please cite this article as: M.A. Khodeir et al., Performance of multihop CDMA ad hoc networks with diversity combining techniques over fading environments, Simulation Modelling Practice and Theory (2016), http://dx.doi.org/10.1016/j.simpat.2015.12.005
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Fig. 1. Expected progress per hop versus probability of transmission, p, over κ −μ fading channel when μ = 1.1.
Fig. 2. Expected progress per hop versus probability of transmission, p, in κ −μ fading channel when κ = 0.2.
where
F`η (η ) =
∞ 0
α
z1 R
exp −B1 R α − 2
D 2 R
α
dα =
B1 R D2R
−((z1 R +1)/4)
K0.5+0.5z1 2DR
B1 R
z1 R = ( L + 1 ) + 2n − 2i − 2, A1 R = A1 |
μ=1&κ =K
=
2L ( 1 + K )
(L+1 ) 2
, B1 R = B1 |
=
L+1 μ=1&κ = K (L−1 ) 2 L αˆ K 2 exp (LK )
and
(1 + K ) , αˆ 2
C1 R = C1 |
μ=1&κ = K
=
2 L(1 + K )K
αˆ
.
whenκ → 0, Eq. (35) reduces to the CDF of η with MRC diversity over Nakagami-m fading channel. 4. Numerical results In this section, some numerical examples are presented to illustrate the expected progress per hop over κ −μ fading channels with SC and MRC and without diversity. The values of the parameters κ , μ and m were selected for the examples below as integers and non-integers values to illustrate that the new results in this paper are valid for both cases. 4.1. The κ -μ fading channels √ Figs. 1 and 2 plot the expected progress per hop, λZ, as a function of the probability of transmission, p, for different values of κ and μ, respectively. Here, GP = 64, ηc = 0.44 dB, γ = 4, and N = 15. Fig. 3 shows the same variations used in Please cite this article as: M.A. Khodeir et al., Performance of multihop CDMA ad hoc networks with diversity combining techniques over fading environments, Simulation Modelling Practice and Theory (2016), http://dx.doi.org/10.1016/j.simpat.2015.12.005
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Fig. 3. Expected progress per hop versus probability of transmission, p, over Nakagami-m fading channel with N = 15.
Fig. 4. Expected progress per hop versus N over κ −μ fading channel with p = 0.27.
Fig. 5. Expected progress per hop versus N over Nakagami-m fading channel with p = 0.27.
√ Figs. 1 and 2 for the√Nakagami-m fading case. It is noted in Figs. 1 and 2 that λZ improves with κ while μ is constant. √ On the other hand, λZ decreases as μ increases while κ kept constant. Fig. 3 shows that increasing m enhances λZ. All 3 figures show that the best √ performance is achieved when p ≈ 0.18. Figs. 4 and 5, which are parameterized by κ and μ, and m, respectively, show λZ versus N, where p = 0.27. The two figures show that there is an√optimum value of N that maximizes the expected progress per hop, for different values of κ , μ, and when N √ m. Fig. 5 reveals that λZ is maximized √ is about 5 for all values of m. It is also noted that the performance of λZ improves with m. However, λZ improvement Please cite this article as: M.A. Khodeir et al., Performance of multihop CDMA ad hoc networks with diversity combining techniques over fading environments, Simulation Modelling Practice and Theory (2016), http://dx.doi.org/10.1016/j.simpat.2015.12.005
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Fig. 6. Expected progress per hop versus the parameter m for different values of N with p = 0.27.
Fig. 7. Expected progress per hop versus probability of transmission, p, with SC route diversity over κ −μ fading channel.
√ diminishes as m gets √ larger (i.e., fading gets less severe). Fig. 6 plots λZ versus the parameter m for different values of N. √ Again, as with Fig. 5, λZ has the best performance when N = 5 for all m. Furthermore, λZ improvement √ diminishes for all N as m increases, which lessens the fading severity. Furthermore, it has been noticed in Figs. 4–6 that λZ decreases after reaching the optimum value of N due to the aggregated interference caused by higher number of users after the optimum value of N. 4.2. The SC route diversity √ Fig. 7 shows λZ with SC versus p for various orders of L and the optimum values of N, where μ = 0.55, and κ =0.95. It can be clearly seen that the best performance can be achieved when p ≈ 0.57. furthermore, the figure shows that SC √ √ diversity has improved the performance of λZ. Fig. 8 plots λZ with SC versus p for various values of N, where L = 1, μ = 1, and κ −→ 0 (the Rayleigh fading case). √ Figs. 9 and 10 present λZ with SC versus L for different values of κ and μ, respectively, where p = 0.27. From Fig. 9 it can be seen that any increase in κ while μ is constant leads to decrease the range of L. Moreover, Fig. 10 shows that any increase in μ results in a decrease in the range of L when κ is constant. 4.3. The MRC route diversity √ Fig. 11 shows λZ with MRC versus p for various values of L and N over κ −μ fading channel. It is clear from the Figure that the best performance is√obtained when p ≈ 0.27. Figs. 12 and 13 present λZ versus L-branch MRC for different values of κ and μ, respectively, where p = 0.18. Fig. 12 shows that when μ is constant, any increase in κ leads to improvement in both the expected progress per hop and the range of L. However, increasing μ in Fig. 13, with κ is constant, leads to decrease the range of L in addition the expected progress per hop for L > 2. Please cite this article as: M.A. Khodeir et al., Performance of multihop CDMA ad hoc networks with diversity combining techniques over fading environments, Simulation Modelling Practice and Theory (2016), http://dx.doi.org/10.1016/j.simpat.2015.12.005
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Fig. 8. Expected progress per hop versus probability of transmission, p, with SC route diversity over κ −μ fading channel with μ = 1 and κ −→0.
Fig. 9. Expected progress per hop versus L over κ −μ fading channel using SC route diversity, when μ = 0.5 and N = 10.
Fig. 10. Expected progress per hop versus L over κ −μ fading channel with SC route diversity, when κ = 1.25 and N = 7.
Please cite this article as: M.A. Khodeir et al., Performance of multihop CDMA ad hoc networks with diversity combining techniques over fading environments, Simulation Modelling Practice and Theory (2016), http://dx.doi.org/10.1016/j.simpat.2015.12.005
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Fig. 11. Expected progress per hop versus probability of transmission, p, with MRC route diversity over κ −μ fading channel with μ=1 and κ =0.5.
Fig. 12. Expected progress per hop versus L over κ −μ fading channel with MRC route diversity and μ = 2.25
Fig. 13. Expected progress per hop versus L over κ −μ fading channel with MRC route diversity and κ = 0.95.
Please cite this article as: M.A. Khodeir et al., Performance of multihop CDMA ad hoc networks with diversity combining techniques over fading environments, Simulation Modelling Practice and Theory (2016), http://dx.doi.org/10.1016/j.simpat.2015.12.005
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5. Conclusion This paper analyzed the expected progress per hop with SC and MRC diversity systems and without diversity over the generalized κ − μ fading distribution, where the expected progress was adopted as a performance measurement of multihop CDMA ad hoc network. Results revealed that the best expected progress can be achieved when the probability of transmission p ≈ 0.18 in the case of a single branch used (no diversity). On the other hand, the best expected progress is achieved when p ≈ 0.57 in the case of SC used, and p ≈ 0.27 when MRC is utilized. Moreover, analysis shows that, in the case of no diversity, the expected progress increases as the parameter κ increases while μ is constant, and it decreases as the parameter μ increases with κ kept constant. Furthermore, in the SC case, any increase in κ with μ is constant or any increase in μ with κ is constant leads to decrease in the range of L. Moreover, in the MRC case, increasing κ with μ is constant leads to improvement to both the expected progress per hop and the range of L, whereas increasing μ with κ is constant leads to a decrease in the range of L. Additionally, the expected progress per hop decreases for L > 2. Appendix The two tables below contain some abbreviations and notations that were used in this paper. Tables A1 and A2 Table A1 Abbreviation
Description
MRC SC LOS CDMA PDF CDF SNIR SNR MAI DS-BPSK i.i.d AWGN
Maximal ratio combining Selection combining Line of sight Code division multiple access Probability density function Cumulative density function Signal-to-noise and interference ratio Signal-to-noise ratio Multiple access interference Direct sequence binary phase shift keying independent and identically distributed Additive white Gaussian noise
Table A2 Notation
Description
γ
Path loss exponent Transmission probability Average density of nodes per unit area SNIR at the receiver SNIR at the receiver in the absence of MAI The total received interference power The processing gain The received signal power Unconditioned probability of successful transmission An arbitrary threshold of η The transmitting power The channel fading amplitude The rms value of α in κ −μ PDF The modified Bessel function of the first kind Average density of nodes per unit area over a time slot The second moment in Nakagami-m PDF The number of diversity paths The one-hop throughput of a transmitter Average number of nodes are closer to the Rx than Tx Generalized Marcum Q−function Generalized Laguerre polynomial of zth order and degree i
p
λ η η0 Y Gp Pr Ps
ηc PT
α αˆ
Iz (.)
λt α2 L
ζ N Q[.,.] Lzi
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Please cite this article as: M.A. Khodeir et al., Performance of multihop CDMA ad hoc networks with diversity combining techniques over fading environments, Simulation Modelling Practice and Theory (2016), http://dx.doi.org/10.1016/j.simpat.2015.12.005
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