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Int. J. Electron. Commun. (AEÜ) 67 (2013) 910– 925

Contents lists available at SciVerse ScienceDirect

International Journal of Electronics andCommunications (AEÜ)

journa l h o me pa ge: www.elsev ier .com/ locate /aeue

Decentralized multiuser diversity with opportunistic packet transmission inMIMO wireless sensor networks

Hakkı Soya,∗, Özgür Özdemirb,d, Mehmet Bayrakc, Ridha Hamilad, Naofal Al-Dhahire

a Vocational School of Technical Sciences, Karamanoglu Mehmetbey University, Karaman, Turkeyb Department of Electrical and Electronics Engineering, Fatih University, Istanbul, Turkeyc Department of Electrical and Electronics Engineering, Mevlana University, Konya, Turkeyd Department of Electrical Engineering, Qatar University, Doha, Qatare Department of Electrical Engineering, The University of Texas at Dallas, USA

a r t i c l e i n f o

Article history:Received 2 September 2012Accepted 7 May 2013

Keywords:Multiuser diversityOpportunistic beamformingMIMOLinear combiningWSNs

a b s t r a c t

In this study, we consider a single-hop wireless sensor network where both the sensor nodes and the con-troller node have multiple antennas. We focus on single beam opportunistic communication and proposea threshold-based medium access control (MAC) scheme for uplink packet transmission which exploitsmultiuser diversity gain without feedback in a decentralized manner. Packet transfer from sensor nodesto the controller node is initiated when the channel quality of any node exceeds the predefined thresholdbased on the effective signal-to-noise ratio (ESNR) measurements at the sensor nodes through linearcombining techniques. The optimum threshold is determined to maximize the probability of successfulpacket transmission where only one sensor node transmits its packet in one time-slot. The proposedscheme trades the successful packet rate to increase the SNR of the successful packets assuming Rayleighfading and collision-based reception model. Computer simulations confirm that proposed scheme hashigher successful packet SNR compared to the simple time division multiple access (TDMA)-based MACscheme with round-robin fashion. The use of multiple antennas at the sensor nodes can also improve thethroughput of proposed scheme compared with our previous scheme without implementing the spatialdiversity at the SNs.

© 2013 Elsevier GmbH. All rights reserved.

1. Introduction

A wireless sensor network (WSN) consists of large number of spatially distributed sensor nodes (SNs) to collect physical or environmentalmeasurements. The SNs are typically equipped with a low-power radio transceiver and a low-cost microcontroller together with an energysource, usually a battery. In many practical applications, the SNs’ batteries cannot be easily replaced or recharged. Therefore, the SNs havea limited battery energy and finite lifetime whereas it is expected to operate for up to months [1]. The major source of energy consumptionfor WSNs is the radio transceiver [2] and the energy consumption of power-constrained transceiver includes both the transmission energyand the circuit energy [3]. In order to reduce the energy consumption due to transceiver operation, many packet transmission schemesdepend on the different MAC and routing protocols were reported in the literature [4–6].

In a single-hop case, there is a central controller node (CN) on top of the hierarchy as a common sink and the SNs can communicatedirectly with the CN. On the contrary, in a multi-hop case, the data packet to be transmitted from an SN is relayed by other SNs locatedbetween the sink and the source via radio transmitter. In multi-hop communication, it is feasible to achieve this energy saving in packettransmission, but each hop leads to extra energy spent on packet reception, processing, routing and forwarding. Furthermore, the multi-hoprouting also incurs more contentions, interference and delays [7]. Even though the multi-hop routing schemes are more energy-efficientwhen only packet transmission is considered, indeed the single-hop communication may be more effective when the circuit processingenergy is taken into account [8,9]. But due to the long distance in single-hop network, the transmission energy increases significantlywith the distance from the CN and dominates the total energy consumption. In order to improve the energy efficiency of transceiver,

∗ Corresponding author. Fax: +90 338 226 21 66.E-mail addresses: [email protected], [email protected] (H. Soy), [email protected] (Ö. Özdemir), [email protected] (M. Bayrak), [email protected]

(R. Hamila), [email protected] (N. Al-Dhahir).

1434-8411/$ – see front matter © 2013 Elsevier GmbH. All rights reserved.http://dx.doi.org/10.1016/j.aeue.2013.05.002


H. Soy et al. / Int. J. Electron. Commun. (AEÜ) 67 (2013) 910– 925 911

an alternative policy is to maximize throughput without consuming much more energy by improving link reliability. At this point, thediversity technique plays a key role in improving reliability of wireless links [10,11].

In WSNs, significant improvement in link reliability can be achieved through the multiple-input multiple-output (MIMO) system model,where multiple antennas are used at both the CN and the SNs. Naturally the circuit energy consumption is directly proportional to thenumber of active antennas [12]. In case of short range transmission as in multi-hop network, the circuit energy is dominant factor in theenergy efficiency and hence the SISO model is more energy-efficient than the MIMO model. In fact, although the MIMO system requires morecircuit energy at both ends of the link, it consumes much less transmission energy than single-input single-output (SISO) system [3,13].When the communication distance is extended over a fading channel, e.g., single-hop network, the SNR requirement for achieving successfulpacket transmission is reduced to save energy through multiple antennas [14]. By employing antenna diversity, spatially separated differentreplicas of the signal experience independent fading which makes transmission links more robust and thus it reduces transmission energyof SNs [15].

WSNs operate under stringent resource limitations due to wireless transmission characteristics. The channel conditions of SNs havetime-varying behavior which means that different SNs experience different channel gains at the same time-slot. This beneficial effect iscalled multiuser diversity (MUD) which arises in multiuser networks [16]. The presence of fading is crucial in order to realize the MUDbecause it increases the probability such that one of the SNs experiences a good channel gain. Especially in little scattering and/or slowfading environments with a sufficient number of SNs, the MUD gain is amplified by opportunistic beamforming (Opp-BF) method thatinduces larger and faster fluctuations on the fading channel [17,18]. Opportunistic scheduling (OS) exploits the MUD to maximize networkthroughput when the SN with favorable channel conditions is scheduled for packet transmission in each time-slot. In principle, it can beconsidered that the scheduler, located in the CN, decides which SN which is going to transmit or receive in each time-slot based on thechannel state information (CSI) [19–21].

In centralized OS, the SNs continuously estimate their own CSI based on the common pilot signal broadcast from CN in the downlink andthen feed it back to the CN via the uplink. When the CN has accurate CSI for all SNs, the network throughput can be maximized with the helpof the perfect scheduling by giving priority to only one SN with the best CSI. But in case of the SNs send feedback to the CN periodically, theenergy consumption of the system increases considerably [22,23]. Besides, the MUD is well known to require a large number of SNs, andthus feedback overhead is significant challenge in multiuser systems [24]. Eventually, the idea of using centralized OS in multiple accessis too energy consuming and operationally inconvenient for WSNs [25]. On the contrary, in distributed OS [26], each SN has only its ownchannel state, but unaware of the channel states of other SNs. This phenomenon is also known as the decentralized CSI (D-CSI) [27–29]. Inthis way, the decentralized MUD gain can be obtained by enabling each SN to transmit packet when the measured channel gain exceeds apredefined threshold.

In this study, we propose a novel distributed MAC scheme for MIMO WSNs in which packet transmission decision is based on channelgain threshold. Here, each SN is authorized to send a packet autonomously when its CSI is above the threshold as specified in [22]. Thedesign criterion for the channel gain threshold is the maximization of the successful packet transmission probability. We further analyzethe expected value of successful packet SNR and the theoretical throughput performance of the proposed scheme. The CSI measurementsare based on effective SNR (ESNR) metric after diversity-combining through selection combining (SC) or maximum ratio combining (MRC)techniques. The obtained results are compared against our previous scheme with single antenna SNs [30] and the simple TDMA scheme,called round-robin (RR) scheme which provides the highest short term fairness and the equal resource allocation [31].

The remainder of the paper is organized as follows. Section 2 describes the system model. Proposed opportunistic MAC scheme isexplained in Section 3. The description of the threshold optimization problem is also explained in this section. In Section 4, the simulationresults of the proposed scheme are presented based on MATLAB, in terms of the successful packet SNR and throughput. Finally, the paperis concluded in Section 5.

2. System model

We consider a WSN with star topology in which the CN is equipped with M antennas serving K SNs which has L antennas provided thatL ≤ M. We assume that the CN has unconstrained power supply and the SNs have enough signal processing capacity. The SNs are randomlydeployed in an open field and connect into a WSN. In order to simplify the analysis, we assume that channel statistics of all the SNs are thesame. The channel is divided into non-overlapping time-slots and the slot duration is long enough to transmit one packet. Since SNs donot know when the convenient channel conditions will exist for packet transmission, their radio transceivers should be kept turned on atall time-slots without any power-saving mechanism that operates alternately in sleep and awake modes. The SN with strong channel gaintransmits its own packet, while the SNs with bad channel condition stay in an idle state until their channel conditions become favorableto transmit packet.

Designed scheme consists of two iterative operations following one after another in time, i.e., pilot signal broadcast in downlink (DL)phase and packet transmission in uplink (UL) phase, respectively. The DL and UL phases share the same frequency band with alternatingtime-slots in time division duplex (TDD) system. Therefore, we assume the channels identical in two directions. Slotted timing structurerequires time synchronization to align the slot limits in scheme under consideration. A pilot signal (PS) is placed at the beginning of eachslot for synchronization as well as channel estimation. As shown in Fig. 1, the pilot signal broadcast time duration of the blocks of NDL pilotsymbols is usually smaller than the packet transmission time duration of the blocks of NUL data symbols, where N = NDL + NUL is the numberof symbols in one slot.

We assume classical frequency-flat, block-fading (quasi-static) channel and the channels of the SNs are assumed to be constant over afixed number of time-slots called one frame and change between different frames independently. Each row of the L × M channel matrixHk = [hk,a . . . hk,L]H is given by M × 1 channel gain vector hk,l from all antennas of the CN to the lth antenna of the kth SN and each entryof the channel gain vector hk,l,m is the channel gain from the mth antenna of the CN to the lth antenna of the kth SN for m = 1, . . ., M, l = 1,. . ., L and k = 1, . . ., K. Here, channel gain coefficients are modeled to be independent and identically distributed (i.i.d.) adopting circularlysymmetric complex Gaussian distribution, hk,l,m∼CN(0, �). It is also assumed that the coherence time of the channel is large enough toconsider the channel characteristics of the DL and the UL being equal to each other.


912 H. Soy et al. / Int. J. Electron. Commun. (AEÜ) 67 (2013) 910– 925

Fig. 1. Slotted timing structure of proposed scheme.

2.1. Pilot signal broadcast in the downlink (DL) phase

In pilot signal broadcast operation, randomly generated beamforming vector is sent out directly over the deployment area and each ofthe SNs measures its own channel gain. The DL system model of the proposed scheme is shown in Fig. 2. During a particular time-slot p,the CN forms the single beam by choosing the M × 1 beamforming vector wp = [wp

1 . . . wpM]

T. The distribution of wp is the same as hk,l but

it is normalized to 1 to keep the power fixed wp ∼ h/||h|| so that (wp)H(wp) = 1.The pilot signal x(n) with power εx is transmitted from CN to SNs. The received signal yp

k(n) = [yp

k,1(n) . . . ypk,L

(n)]T

at the kth SN is writtenas

ypk(n) = (Hkwp)x(n) + zp

k(n), n = 1, . . . , NDL (1)

where zpk(n) = [zp

k,1(n) . . . zpk,L

(n)]T

represents additive white Gaussian noise (AWGN) term which is modeled as i.i.d. CN(0, �2). As seen in(1), observed channel state of the SNs is (Hkwp) and instant CSIs of the SNs change continuously as a result of beamforming vector changedby the CN. It is assumed that each SN can obtain its own channel Hk perfectly from training signals. In this case, the received signal at thelth antenna of the kth SN can be expressed as

ypk,l

(n) = (hHk,lw

p)x(n) + zpk,l

(n). (2)

Beamformer

Tra

inin

g Se

quen

ce

CS

I Estim

ation

CS

I Estim

ation

CS

I Estim

ation

Combiner

1s t SN

kth SN

K th SN

Σ

Σ

Σ

1

2

M

1

1

1

L

L

L

)(, nzpLk

)(1, nz pk

)(, nzp LK

)(1, nzpK

)(,1 nz pL

)(1,1 nz p pa 1,1

pLa ,1

pka 1,

pLka ,

pKa 1,

pLKa ,

pw1

pw 2

pMw

)(nx

)(1 nv

)(nvk

)(nvK

)(1,1 ny p

Combiner

)(,1 nypL

)(1, nypk

)(, nypLk

Combiner

)(1, nypK

)(, nyp LK

CN

Fig. 2. The DL system model of the WSN based on proposed scheme.



The signals received from all antennas of each SN are linearly combined to improve the ESNR. SNs combine the received signals ypk,l

(n)

by multiplying the L × 1 weighting vector apk

= [apk,1 . . . ap

k,L]T. The SNR of the kth SN on the lth antenna is given by

�pk,l

= εx

�2E[|hH

k,lwp|2] (3)

and the received signal at the kth SN after linear combining can be expressed as

vpk(n) = (ap

k)H

ypk(n) =

L∑l=1

(apk,l

)∗yp

k,l(n). (4)

In SC, the kth SN selects the l*th antenna with the highest received SNR where l* is determined as

l∗ = arg max1≤l≤L

(�pk,l

) (5)

and the entries of the weighting vector apk

is given by

apk,l

={

1 l = l∗

0 otherwise.(6)

Consequently, the received signal at the kth SN is then given by

vpk(n) = (ap

k,l∗ )∗yp

k,l∗ (n) = (hHk,l∗ wp)x(n) + zp

k,l∗ (n) (7)

and the ESNR of the kth SN in SC can be found as [32]

�pk

= εx

�2E[|hH

k,l∗ wp|2] = �pk,l∗ = max

1≤l≤L{�p

k,l}. (8)

In the above formulation of the SC technique, we chose the antenna with the best SNR. Alternatively, the MRC compensates for thephases and amplitudes of the signals from the different antenna branches according to their SNR and obtains the weighting vector ap

kthat

maximizes the ESNR. The ESNR is maximum when apk

is linearly proportional to observed channel state of SNs (Hkwp), but it is normalizedto keep the power fixed in UL phase. Then, the weighting vector ap

kis selected as

apk

= Hkwp

‖Hkwp‖ . (9)

Hence, the received signal at the kth SN is given by

vpk(n) =

(Hkwp

‖Hkwp‖)H

[(Hkwp)x(n) + zpk(n)] (10)

and the ESNR of the kth SN in MRC can be found as [32]

�pk

= εx

�2E[∥∥Hkwp

∥∥2] =

L∑l=1

�pk,l

. (11)

2.2. Packet transmission in the uplink (UL) phase

In packet transmission operation, SNs compare the calculated ESNR based on the SC/MRC technique with the predefined threshold thatis discussed in Section 3. Under the simplified collision model assumption, there will be three different channel states for SNs in the ULphase, namely successful packet transmission, collision and idle listening. If only one of the SNs transmits during a particular time-slot,successful packet transmission occur whereas simultaneous transmissions of more than one SN in the same time-slot results in collisionas shown in Fig. 3. Otherwise, none of the SNs exceed the threshold, an idle listening eventuates and the CN will not receive any packetuntil next transmission period. In our single-hop scenario, we consider that the energy consumption of the SNs in the idle listening periodmay be significantly lower than the energy consumption in the packet transmission period.

The UL system model of the proposed scheme is shown in Fig. 4. If the ESNR of any SN exceeds the threshold, the SN transmits itspacket to the CN hoping that none of the other SNs transmit during the same time-slot. The CN uses same beamforming vector in pilottransmission and in packet reception. Forming single beam results in the effective channel being identical in the DL phase where the SNslisten to the CN and in the UL phase where the SNs do packet transmission. This is essential for the proposed scheme to work properly andthis would not be the case if spatial multiplexing through multiple beams were used. According to the channel reciprocity principle, theSNs under the formed beam have also a good CSI in UL channel for considered TDD system.

Assume that only the kth SN has ESNR over the threshold, the data packet is denoted by spk(n) and it is transmitted with the same energy

on the pilot signal broadcast phase (εs = εx). The received signal at the CN is denoted tp(n) = [tp1(n) . . . tp

M(n)]T

and it is written as

tp(n) = (HHk ap

k)sp

k(n) + zp(n), n = 1, . . . , NUL (12)



Fig. 3. Packet transmission operation in a WSN with single-hop infrastructure.

where zp(n) = [zp1(n) . . . zp

M(n)]T

represents additive white Gaussian noise (AWGN) term which is modeled as i.i.d. CN(0, �2). In this case,the received signal at the mth antenna of the CN is expressed as

tpm(n) = (hH

k,mapk)sp

k(n) + zp

m(n) (13)

where hk,m represents the L × 1 channel gain vector from all antennas of the kth SN to the mth antenna of the CN and the noise at the mthantenna of the CN is denoted as zp

m(n).The signals received from all antennas of the CN are combined to improve the ESNR of transmitted packet. Hereby, the CN provides

antenna-array gain to increase in the received power due to receive diversity and packet transmission energy can be reduced in this way.

Dig

ital

Sig

nal P

roce

ssin

g (D

SP)

Packet T

ransmission

Packet T

ransmission

Packet T

ransmission

1s t SN

kth SN

K th SN

1

2

M

)(1 nz p

pa 1,1

pLa ,1

pka 1,

pLka ,

pKa 1,

pLKa ,

pw1

pw 2

pMw

)(nr

)(1 ns

)(nsk

)(nsK

)(1 nt p

Beamformer

1

L

1

L

1

L

)(2 nz p

)(nzpM

)(2 nt p

)(nt pMCombiner

Beamformer

BeamformerCN

Fig. 4. The UL system model of the WSN based on proposed scheme.



The CN combines the received signals tpm(n) by multiplying the weighting vector wp. The combined signal at the CN is the weighted sum

of the received signals at each diversity branch and is given by

rp(n) = (wp)Htp(n) =M∑

m=1

(wpm)

∗tpm(n). (14)

Eventually, the ESNR of the obtained signal is determined as

�p = εs

�2E[||HH

k apk||2] (15)

which is the same as in DL.

3. Proposed packet transmission scheme

In this section, we derive the optimum threshold in order to make decisions to transmit or not in a given time-slot. Threshold optimizationis based on the maximization of the probability that only one of the SNs transmits during packet transmission period. This optimizationreduces the rate of collisions and hence the energy which is required to send data to the CN. Optimum threshold is determined dependingon whether SC or MRC is used for linear combining at the SNs.

3.1. Proposed scheme with selection combining (SC) technique

In case of using SC, the linear combiner chooses and processes the branch with the highest SNR. In this case, the ESNR is the maximumof L exponentially distributed random variables and its probability density function (PDF) is expressed as [33]

f SC� (�) = L

�(1 − e−�/� )

L−1e−�/� (16)

where � is the average SNR at the output of each of the L antennas. By using the binomial expansion, we can rewrite (16) as

f SC� (�) = L

�

L−1∑j=0

(−1)j

(L − 1

j

)e−((1+j)�)/� (17)

where

(L − 1

j

)= (L − 1)!

(L − j − 1)!j!denotes the binomial coefficient. In a homogeneous WSN, each SN transmits a packet with the same

probability distribution. Packet transmission probability is expressed as the probability of SN having an SNR over the threshold. Accordingto PDF expression in (16), the probability that one of the SNs has an ESNR above the threshold ̌ is obtained as follows

PSC (� > ˇ) =∫ ∞

ˇ

L

�(1 − e−�/� )

L−1e−�/� d� = 1 − (1 − e−ˇ/� )

L. (18)

For successful packet transmission, the probability that only one SN having an ESNR above the threshold (and all the others below thethreshold) is written in a compact form as

PSC (ˇ) = K[1 − (1 − e−ˇ/� )L][(1 − e−ˇ/� )

L]K−1

. (19)

Then, the optimum threshold ˇSC which maximizes the probability in (19) can be derived as

ˇSC = −� ln

(1 − L

√K − 1

K

). (20)

The PDF of the successful packet ESNR for proposed scheme with SC technique can be expressed as follows

f�́SC(�) = L(1 − e−�/� )

L−1e−�/�

�[1 − (1 − e−ˇSC /� )L]u(� − ˇSC ) = L

�[1 − (1 − e−ˇSC /� )L]

L−1∑j=0

(−1)j

(L − 1

j

)e−((1+j)�)/� u(� − ˇSC ) (21)

where �́ denotes the ESNR of a successful packet. Accordingly, the average SNR of successful packets in proposed opportunistic scheme isfound as follows

�OppSC =

∫ ∞

0

�f�́SC(�)d� = �L

[1 − (1 − e−ˇSC /� )L]

L−1∑j=0

(−1)j

(L − 1

j

)(1 + j)2

�c

(2,

(1 + j)ˇSC

�

)(22)



where �c(˛, x) =∫ ∞

xe−t t˛−1dt is the complementary incomplete Gamma function [34]. Appendix A gives a brief derivation of this equiv-

alent finite-series form of the successful packet SNR expression. Besides the average SNR of successful packets in RR scheme with SCtechnique is determined by

�RRSC =

∫ ∞

0

�f�SC (�)d� = �L

L−1∑j=0

(−1)j

(L − 1

j

)(1 + j)2

. (23)

The system ESNR which is denoted by �̂ is defined to be the ESNR of the single SN above the threshold for successful packet transmissionotherwise the system ESNR is zero. As shown in the Appendix B, the PDF of system ESNR can be found as follows

f�̂SC(�) = [1 − PSC (ˇSC )]ı(�) +

(PSC (ˇSC )

1 − (1 − e−ˇSC /� )L

)L

�(1 − e−�/� )

L−1e−�/� u(� − ˇSC ) (24)

where ı(.) stands for the Dirac delta function and u(.) stands for the unit step function. Note that, for idle listening/collision �̂SC is equal tozero with the probability of [1 − PSC(ˇSC)]ı(�). Due to the Shannon capacity [35], the achievable throughput of the proposed opportunisticscheme is obtained by

COppSC =

∫ ∞

0

log2(1 + �)f�̂SC(�)d�

= PSC (ˇSC )L

ln 2[1 − (1 − e−ˇ/� )L]

L−1∑j=0

(−1)j

(L − 1

j

)1 + j

[ln(1 + ˇSC )e−((1+j)ˇSC )/� + e(1+j)/� E1

((1 + ˇSC )(1 + j)

�

)](25)

where E1(x) =∫ ∞

xt−1e−tdt is the exponential–integral function of first order [34] and the evaluation of the throughput expression is also

derived in the Appendix C. In a non-opportunistic RR scheme with SC technique where each user is scheduled in turn therefore all packetsare transmitted successfully, the throughput can be found by [33]

CRRSC =

∫ ∞

0

log2(1 + �)f SC� (�)d� = L

ln 2

L−1∑j=0

(−1)j

(L − 1

j

)e(1+j)/� E1((1 + j)/�)

(1 + j)(26)

where f SC�k

(�) is defined in (16).

3.2. Proposed scheme with maximum ratio combining (MRC) technique

In optimum MRC combiner the ESNR is the sum of the SNRs of each individual diversity branch. In case of using MRC, the PDF of theESNR is expressed as [33]

f MRC� (�) = �L−1e−�/�

�L(L − 1)!. (27)

Then, the probability in the case of one of the SNs has an ESNR above the threshold ̌ is given by

PMRC (� > ˇ) =∫ ∞

ˇ

�L−1e−�/�

�L(L − 1)!d� =

�c(L, ˇ�

)

(L − 1)!. (28)

Accordingly, when there is only one SN with ESNR above the threshold, the probability expression may be written as

PMRC (ˇ) = K

(∫ ∞

ˇ

�L−1e−�/�

�L(L − 1)!d�

)(∫ ˇ

0

�L−1e−�/�

�L(L − 1)!d�

)K−1

. (29)

Hence, as shown in the Appendix D, the optimum threshold ˇMRC which maximizes the probability in (29) can be found by settingdPMRC

dˇ= 0. Based on the Leibniz integral rule, optimum threshold ˇMRC is obtained from the solution of the PMRC (�k > ˇ) − 1

K = 0 equationand it reduces to

�c(L, ˇMRC/�)(L − 1)!

= 1K

. (30)

By doing so, ˇMRC calculated numerically since the complicated form of the (30) makes it difficult to find a closed form solution.Accordingly, the PDF of successful packet ESNR for proposed scheme with MRC technique can be expressed as follows

f�́MRC(�) =

(1�

)�L−1e−�/�

�L(L − 1)!u(� − ˇMRC ) (31)



where

� = e−ˇMRC /�

L−1∑j=0

(ˇMRC/�)j

j!. (32)

As shown in Appendix E, the average SNR of successful packets in proposed scheme is found by

�OppMRC =

∫ ∞

0

�f�́MRC(�)d� = 1

�

∫ ∞

ˇMRC

�L−1e−�/�

�L(L − 1)!d� = �

�(L − 1)!�c

(L + 1,

ˇMRC

�

), (33)

whereas the average SNR of successful packets in RR scheme with MRC technique is determined as

�RRMRC =

∫ ∞

0

�f�MRC (�)d� = 1

�L(L − 1)!

∫ ∞

0

�Le−�/� d� = �L. (34)

As shown in the Appendix F, the PDF of system ESNR which is denoted by �̂MRC can be found as follows

f�̂MRC(�) = [1 − PMRC (ˇMRC )]ı(�) +

(PMRC (ˇMRC )

�

)�L−1e−�/�

�L(L − 1)!u(� − ˇMRC ). (35)

Hence, the achievable throughput of proposed opportunistic scheme is obtained by

COppMRC =

∫ ∞

0

log2(1 + �)f�̂MRC(�)d�. (36)

Due to the complicated form of (36), the network throughput COppMRC is calculated numerically. In a non-opportunistic RR scheme with

MRC technique, the throughput is obtained as

CRRMRC =

∫ ∞

0

log2(1 + �)f MRC� (�)d� = 1

ln 2

L−1∑�=0

(−1/�)L−�−1

(L − � − 1)!

[e1/� E1

(1�

)+

L−�−1∑p=1

(p − 1)!(−�)p

](37)

where f MRC�k

(�) is defined in (27). The derivation details are described in the Appendix G.

3.3. Proposed scheme without combining

In special case, when the WSN consists of single-antenna SNs, the channel quality of any node can be determined based on the SNRmeasurements. The SNR metric has an exponential distribution therefore the PDF expression is given by [33]

f SNR� (�) = 1

�e−�/� (38)

where � is the average SNR at the SNs. For successful packet transmission, the probability for which only one SN has an SNR above thethreshold may be written as follows

PSNR(ˇ) = Ke−ˇ/� (1 − e−ˇ/� )K−1

. (39)

Thus, the optimum threshold ˇSNR which maximizes the successful packet transmission probability given in (39) can be found by settingits first derivative to zero and also by setting L = 1 in (20) or (30)

ˇSNR = � ln(K). (40)

The PDF of successful packet SNR for proposed scheme without combining can be expressed as follows

f�́SNR(�) = e−�/�

�e−ˇSNR/�u(� − ˇSNR). (41)

Accordingly, the average SNR of successful packets in the proposed scheme without combining becomes

�OppSNR = �

e−ˇSNR/��c

(2,

ˇSNR

�

)(42)

and it can be easily verified by setting L = 1 in (22) or (33). From (23) or (34), the average SNR of successful packets in the RR scheme withoutcombining is equal to � . Similarly, the system SNR �̂SNR is defined to be the SNR of the successful packet and the PDF of system SNR can beeasily found as follows

f�̂SNR(�) = [1 − PSNR(�)]ı(�) +

(PSNR(ˇSNR)

�e−ˇSNR/�

)e−�/� u(� − ˇSNR). (43)

The achievable throughput of the proposed scheme without combining is obtained as

COppSNR = PSNR(ˇSNR)

ln 2

[ln(1 + ˇSNR) + e(1+ˇSNR)/� E1

(1 + ˇSNR

�

)](44)



0 25 50 75 100 125 1500

1

2

3

4

5

6

7

8

Number of SNs (K)

Opt

imum

thre

shol

d (β

)

4×2 MRC2×2 MRC4×2 SC2×2 SC4×1 SNR2×1 SNR

Fig. 5. Variation of the threshold with M and K values for L = 2 and � = 0 dB.

and it can be verified by setting L = 1 in (25). Finally, by setting L = 1 in (26) or (37), the throughput of the non-opportunistic RR schemewithout combining is given by [36]

CRRSNR = e1/�

ln 2E1

(1�

)(45)

where the equality of �c(0, 1/�) = E1(1/�) [34] is used for simplification.

4. Simulation results

In this section, the system performance of the proposed scheme was evaluated analytically and through the use of statistical (MonteCarlo) simulation studies for several scenarios. Here, the number of frames is chosen as 1000 to achieve an acceptable convergence and thenumber of time slots in each frame is equal to at least the number of SNs. We consider all the transmit-receive antennas as independent andthere is no correlation among them. At the beginning of each frame, the channels of the SNs are randomly generated and kept constant ineach frame, whereas the beamforming vector is generated randomly in each time-slot. It is considered that SNs know optimum thresholdof the current system and each SN always has a packet to transmit in each time-slot.

4.1. Optimum threshold analysis

The variation of the optimum threshold ̌ versus the number of SNs K is shown in Fig. 5 for the different number of antennas at the CN(M = 2, 4) when the number of antennas at the SNs L = 2 and the average SNR � = 0 dB. Similarly, Fig. 6 shows the variation of the optimumthreshold ̌ versus the number of SNs K for the different number of antennas at the SNs (L = 2, 4) when the number of antennas at the CN isM = 4 and the average SNR is � = 0 dB. As seen in these figures, M does not have an effect on the threshold and as L increases the thresholdalso increases. The threshold also increases for larger number of SNs, meaning that the packet transmission can be made more selective.However, the MRC technique has higher threshold than the SC technique for the same K value.

4.2. Successful packet SNR analysis

In order to investigate the superiority of our approach, the average SNR of successfully received packets with two different antennaconfigurations, namely 4 × 2 and 4 × 4, is provided. Comparison to our previous scheme with single-antenna SNs, 4 × 1 configuration, isalso provided. Fig. 7 shows the successful packet SNR versus the number of SNs K when M = 4, L = 2 and � = 0 dB. As seen from this plot, theproposed opportunistic scheme has an important advantage over the non-opportunistic RR scheme. As expected, the RR scheme provides

0 25 50 75 100 125 1500

2

4

6

8

10

12

Number of SNs (K)

Opt

imum

thre

shol

d (β

)

4×4 MRC4×2 MRC4×4 SC4×2 SC4×1 SNR

Fig. 6. Variation of the threshold with L and K values for M = 4 and � = 0 dB.



0 25 50 75 100 125 150−1

0

1

2

3

4

5

6

7

8

9

10

Number of SNs (K)

Ave

rage

SN

R (

dB)

4×2 MRC−Proposed4×2 SC−Proposed4×1 SNR−Proposed4×2 MRC−RR4×2 SC−RR4×1 SNR−RR

Fig. 7. Variation of the average SNR of successful packets with K values for M = 4, L = 2 and � = 0 dB. Markers are obtained by analytical results, while solid lines are obtainedby simulation (� for MRC, ◦ for SC and ♦ for SNR in proposed scheme, ∗ for MRC, × for SC and • for SNR in RR scheme).

a constant average SNR, whatever the number of SNs since it does not take advantage of the MUD gain. On the contrary, by using proposedscheme (with/without combining), we observe an important increment of the successful SNR curve when the number of SNs increases.Consequently, packet transmission reliability can be increased significantly by using our opportunistic approach. It is also obvious that theMRC technique has higher successful packet SNR than the SC technique for the same K value.

Similarly, Fig. 8 shows the successful packet SNR versus number of SNs K when M = 4, L = 4 and � = 0 dB. Results from this plot show thatthe average SNR of successfully received packets increases with the increasing L values for both the SC and the MRC techniques. However,the increment of the average SNR arises from the MRC is more powerful than the SC based scheme. On the other hand, the proposed schemewith diversity-combining has great advantage compared with our previous scheme (4 × 1) without combining, as described in [30]. Notethat, successful packet SNR of proposed scheme for K = 1 is equal to the RR scheme’s successful packet SNR.

In our optimization, we find an optimum threshold to maximize the successful packet transmission probability. Accordingly, if we tryto maximize the average SNR of successfully received packets, we could choose higher threshold than the threshold of our optimization.But this would obviously cause a decrease in successful packet rate. At this point, there is a tradeoff between the successful packet rateand the SNR of the successful packets. The successful packet rate which is defined as the ratio of the number of successful transmissionsover the number of all time slots. The variation of the successful packet rate versus the number of SNs K for different L values when M=4and � = 0 dB is shown in Fig. 9. As seen from this plot, successful packet rate of proposed scheme stays about 37% for all of the consideredconfigurations. The RR scheme on the other hand has 100% packet success rate. Hence, the proposed scheme trades the successful packetrate to increase the successful packet SNR.

4.3. Throughput analysis

Here, we demonstrate the throughput performance of the proposed scheme with two different antenna configurations, 4 × 2 and 4 × 4.Fig. 10 presents the achieved throughput versus number of SNs K for proposed scheme as well as the RR scheme when M = 4, L = 2 and� = 0 dB. The results indicate that as the number of SNs K in the WSN increases the network throughput also increases for proposed schemedue to the MUD gain. Besides, the MRC technique has a better throughput than the SC technique for the same K value. But the throughputis low in our previous scheme (4 × 1) without combining [30] compared with the proposed MIMO WSN model as seen in presented plot.Note that the analytical results obtained by using (25) and (36) perfectly matches with the simulation results.

Similarly, Fig. 11 shows the throughput versus the number of SNs K for proposed scheme when M = 4, L = 4 and � = 0 dB. It is obviousthat, as the number of antennas at the SNs L increases the achieved throughput also increases for proposed scheme due to the increasingreceive-diversity gain. Note also that the proposed scheme is capable of outperforming the RR scheme only when the SNs have a single

0 25 50 75 100 125 150−1

0

1

2

3

4

5

6

7

8

9

10

11

12

Number of SNs (K)

Ave

rage

SN

R (

dB)


Fig. 8. Variation of the average SNR of successful packets with K values for M = 4, L = 4 and � = 0 dB. Markers are obtained by analytical results, while solid lines are obtainedby simulation (� for MRC, ◦ for SC and ♦ for SNR in proposed scheme, ∗ for MRC, × for SC and • for SNR in RR scheme).



0 25 50 75 100 125 1500

10

20

30

40

50

60

70

80

90

100

Number of SNs (K)

Suc

cess

ful p

acke

t rat

e (%

)

4×4 MRC4×2 MRC4×4 SC4×2 SC4×1 SNR

Fig. 9. Variation of the packet success rate with K values for M = 4 and � = 0 dB.

0 25 50 75 100 125 1500.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

Number of SNs (K)

Thr

ough

put (

bps/

Hz)


Fig. 10. Variation of the throughput with K values for M = 4, L = 2 and � = 0 dB. Markers are obtained by analytical results, while solid lines are obtained by simulation (� forMRC, ◦ for SC and ♦ for SNR in proposed scheme, ∗ for MRC, × for SC and • for SNR in RR scheme).

antenna (4 × 1) as the number of SNs are larger than 25. But when we use multiple antennas at the SNs, the RR scheme outperforms theproposed scheme for moderate number of SNs in the WSN. When comparing the throughput performance of the SC and the MRC techniquesfor 4 × 4 configuration, it can be observed that the MRC technique provides approximately 25% more gain with respect to the SC technique.Besides, the MRC technique provides approximately 50% gain over the previous scheme without combining (4 × 1), whereas for the SCtechnique it is approximately 20%.

Furthermore, in order to see the effect of the different average SNR values (� = −3, 0, 3 dB) over the throughput performance, weextent our analysis beyond the practical number of SNs by using theoretical calculations. Fig. 12 shows the throughput versus the numberof SNs for proposed scheme assuming that 4 × 2 configuration. Results from this plot show that the network throughput increases withthe increasing average SNR values for both proposed scheme and also the RR scheme. But the throughput increase for the RR schemeis higher than the proposed scheme as there are no packet collision and idle listening cases. Hence, the efficiency of proposed schemedecreases compared to the RR scheme when average SNR in the environment is high. For instance, when � = −3 dB, both the SC and theMRC techniques outperform the RR scheme when the number of SNs is greater than 100 and 700 for 4 × 2 configuration, respectively. Onthe other hand, when � = 3 dB, the MRC technique cannot outperform the RR scheme for the scale considered.

0 25 50 75 100 125 1500.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

Number of SNs (K)

Thr

ough

put (

bps/

Hz)


Fig. 11. Variation of the throughput with K values for M = 4, L = 4 and � = 0 dB. Markers are obtained by analytical results, while solid lines are obtained by simulation (� forMRC, ◦ for SC and ♦ for SNR in proposed scheme, ∗ for MRC, × for SC and • for SNR in RR scheme).



Fig. 12. Variation of the throughput with K and � values for M = 4, L = 2.

5. Conclusion

In this study, we proposed and analyzed an opportunistic packet transmission scheme for MIMO WSNs to exploit limited MUD gaindue to decentralized CSI exploitation in an autonomous manner. The driving motivation of this study is to investigate the benefits ofopportunistic scheduling with limited MUD gain due to a lack of feedback. The proposed threshold-based MAC scheme does not needany centralized scheduler and it allows to transmit only when the SNs have a strong channel gain. Predetermined threshold value isoptimized based on successful packet reception probability at the CN. Thus, the system performance is improved by using consideredoptimization approach. We extend analysis to derive closed-from throughput expressions for proposed scheme with both the SC andthe MRC techniques and compare it to the throughput of the RR scheduler. To test the proposed scheme, experimental analysis werealso performed. Essentially, compared with the existing non-opportunistic RR scheme, our proposed opportunistic scheme increases theaverage SNR of the received successful packets. When compared to our previous scheme without combining, proposed scheme is often moreefficient in terms of throughput and packet transmission reliability. Furthermore, when the number of SNs is large enough, the numericalsimulation shows that the throughput performance of the proposed unscheduled strategy surpasses the centralized RR scheduler. Anotherimportant attribute of the proposed MAC scheme is the scalability to the change in network size. The time-varying nature of the WSNsdoes not affect the performance of the proposed scheme due to the preferred optimization framework. The proposed distributed time-slotassignment strategy is specially tailored for applications that more reliable packet delivery, high-level scalability and autonomy.

Acknowledgement

This work is supported by Qatar National Research Fund (QNRF), Grant NPRP 09-062-2-035.

Appendix A. Derivation of the average SNR of successful packets for proposed scheme with SC technique

In this appendix, we describe the derivation of the average SNR of successful packets for proposed opportunistic scheme in (22). Thisexpression can be written as

�OppSC =

∫ ∞

0

�f�́SC(�)d� =

∫ ∞

0

�L

�[1 − (1 − e−ˇSC /� )L]

L−1∑j=0

(−1)j

(L − 1

j

)e−((1+j)�)/� u(� − ˇSC )

= L

�[1 − (1 − e−ˇSC /� )L]

L−1∑j=0

(−1)j

(L − 1

j

)∫ ∞

ˇSC

�e−((1+j)�)/� d�. (A.1)

By using the identity∫ ∞

uxv−1e−�xdx = �−v�c(v, �u) [34], we get the desired expression as

�OppSC = �L

[1 − (1 − e−ˇSC /� )L]

L−1∑j=0

(−1)j

(L − 1

j

)(1 + j)2

�c

(2,

(1 + j)ˇSC

�

). (A.2)

Appendix B. Derivation of the system ESNR PDF for proposed scheme with SC technique

Noting that the system ESNR will be equal to zero with probability [1 − PSC(ˇSC)], the system ESNR PDF can be written as

f�̂SC(�) = [1 − PSC (ˇSC )]ı(�) + GSC

L

�(1 − e−�/� )

L−1e−�/� u(� − ˇSC ) (B.1)



where GSC is scaling factor to satisfy∫ ∞

−∞ f�̂SC(�)d� = 1. Therefore, the area under the second term in (B.1) should be equal to PSC(ˇSC).

PSC (ˇSC ) =∫ ∞

−∞GSC

L

�(1 − e−�/� )

L−1e−�/� u(� − ˇSC )d� = GSC

L

�

∫ ∞

ˇSC

(1 − e−�/� )L−1

e−�/� d� = GSC [1 − (1 − e−ˇSC /� )L]. (B.2)

Then, the GSC is found as follows

GSC = PSC (ˇSC )

1 − (1 − e−ˇSC /� )L

. (B.3)

Substitute (B.3) into (B.1), we can rewrite the system ESNR PDF as

f�̂SC(�) = [1 − PSC (ˇSC )]ı(�) +

(PSC (ˇSC )

1 − (1 − e−ˇSC /� )L

)L

�(1 − e−�/� )

L−1e−�/� u(� − ˇSC ). (B.4)

Appendix C. Derivation of the throughput for proposed scheme with SC technique

In this appendix, we briefly describe the derivation of the throughput for proposed scheme in (25). We can write the throughputexpression as

COppSC =

∫ ∞

0

log2(1 + �)f�̂SC(�)d� =

∫ ∞

0

log2(1 + �)

([1 − PSC (ˇSC )]ı(�) +

(PSC (ˇSC )

1 − (1 − e−ˇSC /� )L

)L

�(1 − e−�/� )

L−1e−�/� u(� − ˇSC )

)

= PSC (ˇSC )L

� ln 2[1 − (1 − e−ˇSC /� )L]

L−1∑j=0

(−1)j

(L − 1

j

)∫ ∞

ˇSC

ln(1 + �)e−((1+j)�)/� d�. (C.1)

By using the identity∫

ln(1 + x)e−�xdx = − 1� [ln(1 + x)e−�x + e�E1(�(x + 1))] [34], we get the desired expression as

COppSC = PSC (ˇSC )L

ln 2[1 − (1 − e−ˇ/� )L]

L−1∑j=0

(−1)j

(L − 1

j

)1 + j

[ln(1 + ˇSC )e−((1+j)ˇSC )/� + e(1+j)/� E1

((1 + ˇSC )(1 + j)

�

)]. (C.2)

Appendix D. Derivation of the optimum threshold for proposed scheme with MRC technique

In this appendix, we provide the details of the derivation of (30). The probability of successful packet transmission can be formulatedas follows

PMRC (ˇ) = K

(∫ ∞

ˇ

� (L−1)e−�/�

�L(L − 1)!d�

)(∫ ˇ

0

� (L−1)e−�/�

�L(L − 1)!d�

)K−1

. (D.1)

With the help of the Leibniz integral rule ddx

∫ h(x)g(x)

f (t)dt = f (h(x))h′(x) − f (g(x))g′(x) [34], the optimum threshold which maximizes (B.1)

can be found by setting first derivative to zero,

dPMRC

dˇ= K

[d

dˇ

(∫ ∞

ˇ

� (L−1)e−�/�

�L(L − 1)!d�

)]︸︷︷︸

♣

(∫ ˇ

0

� (L−1)e−�/�

�L(L − 1)!d�

)K−1

+ K

(∫ ∞

ˇ

� (L−1)e−�/�

�L(L − 1)!d�

)⎡⎣ d

dˇ

(∫ ˇ

0

� (L−1)e−�/�

�L(L − 1)!d�

)K−1⎤⎦

︸︷︷︸♠

.

(D.2)

In the above formulation

♣ = d

dˇ

(∫ ∞

ˇ

� (L−1)e−�/�

�L(L − 1)!d�

)= −ˇL−1e−ˇ/�

�L(L − 1)!(D.3)

and

♠ = d

dˇ

(∫ ˇ

0

� (L−1)e−�/�

�L(L − 1)!d�

)K−1

= (K − 1)

[d

dˇ

(∫ ˇ

0

� (L−1)e−�/�

�L(L − 1)!d�

)]︸︷︷︸

�

(∫ ˇ

0

� (L−1)e−�/�

�L(L − 1)!d�

)K−2

. (D.4)



By using the identity ddx

∫ x

af (t)dt = f (x) [34],

� = d

dˇ

(∫ ˇ

0

� (L−1)e−�/�

�L(L − 1)!d�

)= ˇ(L−1)e−ˇ/�

�L(L − 1)!. (D.5)

Finally, substituting (D.3), (D.4) and (D.5) in (D.2),

dPMRC

dˇ= K

(−ˇ(L−1)e−ˇ/�

�L(L − 1)!

)(∫ ˇ

0

� (L−1)e−�/�

�L(L − 1)!d�

)K−1

+K

(∫ ∞

ˇ

� (L−1)e−�/�

�L(L − 1)!d�

)(K−1)

(ˇ(L−1)e−ˇ/�

�L(L − 1)!

)(∫ ˇ

0

� (L−1)e−�/�

�L(L − 1)!d�

)K−2

(D.6)

and after some simplifications

dPMRC

dˇ= K

(ˇ(L−1)e−ˇ/�

�L(L − 1)!

)(∫ ˇ

0

� (L−1)e−�/�

�L(L − 1)!d�

)K−1⎡⎣−1 + (K − 1)

(∫ ∞

ˇ

� (L−1)e−�/�

�L(L − 1)!d�

)(∫ ˇ

0

� (L−1)e−�/�

�L(L − 1)!d�

)−1⎤⎦ , (D.7)

dPMRC

dˇ= 0 ⇒

∫ ˇ

0

� (L−1)e−�/�

�L(L − 1)!d� = (K − 1)

(∫ ∞

ˇ

� (L−1)e−�/�

�L(L − 1)!d�

), (D.8)

K

∫ ∞

ˇ

� (L−1)e−�/�

�L(L − 1)!d� =

∫ ˇ

0

� (L−1)e−�/�

�L(L − 1)!d� +

∫ ∞

ˇ

� (L−1)e−�/�

�L(L − 1)!d�︸︷︷︸∫

f�̂MRC(�)d�=1

, (D.9)

∫ ∞

ˇ

� (L−1)e−�/�

�L(L − 1)!d� = PMRC (�k > ˇ) = 1

K. (D.10)

Hence, we conclude that

�c(L, ˇMRC/�)(L − 1)!

= 1K

. (D.11)

Appendix E. Derivation of the average SNR of successful packets for proposed scheme with MRC technique

In this appendix, we briefly describe the derivation of the average SNR of successful packets for proposed opportunistic scheme in (33).We can write this expression as

�OppMRC =

∫ ∞

0

�f�́MRC(�)d� =

∫ ∞

0

�(

1�

)�L−1e−�/�

�L(L − 1)!u(� − ˇMRC ) (E.1)

where � is defined in (32). After that, further simplifications gives

�OppMRC = 1

��L(L − 1)!

∫ ∞

ˇMRC

�Le−�/� d�. (E.2)

By using the identity∫ ∞

uxne−�xdx = �−(n+1)�c(n + 1, �u) [34], we get the desired expression as

�OppMRC = �

�(L − 1)!�c(L + 1,

ˇMRC

�). (E.3)

Appendix F. Derivation of the system ESNR PDF for proposed scheme with MRC technique

Similarly to the proof of (A.1), the PDF given by (35) can be rewritten as

f�̂MRC(�) = [1 − PMRC (ˇMRC )]ı(�) + GMRC

� (L−1)e−�/�

�L(L − 1)!u(� − ˇMRC ) (F.1)

where GMRC is scaling factor. It can be easily seen that

PMRC (ˇMRC ) =∫ ∞

−∞GMRC

� (L−1)e−�/�

�L(L − 1)!u(� − ˇMRC )d� = GMRC

�L(L − 1)!

∫ ∞

ˇMRC

� (L−1)e−�/� d� (F.2)

and by using identity∫ ∞

uxv−1e−�xdx = �−v�c(v, �u), we can simplify the probability expression as

PMRC (ˇMRC ) = GMRC

(L − 1)!�c(L,

ˇMRC

�). (F.3)



Using the series expansion of �c(˛, �x) = ( ̨ − 1)!e−�x∑˛−1

j=0(�x)j

j! , (F.3) is replaced by the its equivalent finite sum,

PMRC (ˇMRC ) = GMRCe−ˇMRC /�

L−1∑j=0

( ˇMRC�

)j

j!(F.4)

then, the GMRC is found as follows

GMRC = PMRC (ˇMRC )�

(F.5)

where

� = e−ˇMRC /�

L−1∑j=0

(ˇMRC/�)j

j!. (F.6)

Plugging (F.5) into (F.1), the system ESNR PDF can be obtained as follows

f�̂MRC(�) = [1 − PMRC (ˇMRC )]ı(�) +

(PMRC (ˇMRC )

�

)�L−1e−�/�

�L(L − 1)!u(� − ˇMRC ). (F.7)

Appendix G. Derivation of the throughput for Round Robin scheme with MRC technique

In this appendix, we briefly describe the derivation of the analytical solution in (37). We can write the throughput expression as

CRRMRC =

∫ ∞

0

log2(1 + �)f MRC� (�)d� = 1

ln 2�L(L − 1)!

∫ ∞

0

ln(1 + �)�L−1e−�/� d�. (G.1)

By using the variable transform x = �/� , the throughput expression is obtained as

CRRMRC = 1

ln 2(L − 1)!

∫ ∞

0

ln(1 + �x)xL−1e−xdx. (G.2)

Then, by using the identity∫ ∞

0ln(1 + ax)xe−xdx =

∑�=0

!(−�)!

[(−1)−�

a−� e1/aE1( 1a ) +

∑−�p=1 (p − 1)!

(− 1

a

)−�−p]

[34], we get the desired

expression as

CRRMRC = 1

ln 2

L−1∑�=0

(−1/�)L−�−1

(L − � − 1)!

[e1/� E1

(1�

)+

L−�−1∑p=1

(p − 1)!(−�)p

]. (G.3)

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