Cooperative Transmission in Poisson Distributed ... - IEEE Xplore

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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 5, NO. 10, OCTOBER 2006

Cooperative Transmission in Poisson Distributed Wireless Sensor Networks: Protocol and Outage Probability Liang Song, Member, IEEE, and Dimitrios Hatzinakos, Senior Member, IEEE

Abstract— We study cooperative wireless communications in the physical layer of a Poisson distributed wireless sensor network, where the spatial diversity of multiple relay nodes is utilized to improve the link performance. The tradeoff among network power consumption, spectral efficiency, outage probability, and sensor node density is discussed under the proposed Cooperative Transmission Protocol for Sensor Networks (CTPSN). CTP-SN is considered as a typical implementation of the two-phase cooperative transmission paradigm in wireless sensor networks. We derive an asymptotic upper bound for the capacity outage probability of CTP-SN. The bound is shown to be decreasing exponentially, when the sensor node density increases. Via the bound, we demonstrate that the cooperative protocol performs asymptotically much better than the non-cooperative direct transmission. Index Terms— Sensor networks, Poisson distributions, capacity outage probability, physical layer, spectral efficiency, power consumption.

I. I NTRODUCTION

W

IRELESS sensor networks are formed by a collection of battery-resource limited sensor nodes working together for a common task [1]–[3]. Under the limited wireless infrastructure, the network works autonomously like a distributed data collection and processing machine. It is desired that sensor networks have long lifetime after deployment. Hence, cooperative and energy efficient protocols for sensor nodes have attracted a great deal of interest in all layers of communication system design. Sensor networks have been considered under the same paradigm as classical ad hoc networks. On the network layer, the topology management of sensor networks is, however, decided by application specific cooperations. Due to the packets traffic pattern, clustering, [2], is naturally of interest in a data-gathering sensor network, while flat networks [3] can be favored in target detection and tracking. When the application is delay non-sensitive, Sensor Networks with Mobile Sinks (MSSN), [4], achieve the energy efficiency by converting the

Manuscript received September 1, 2004; revised February 2, 2005 and September 5, 2005; accepted November 28, 2005. The associate editor coordinating the review of this paper and approving it for publication was A. Scaglione. This work is supported by Canadian Natural Sciences and Engineering Research Council (NSERC). L. Song and D. Hatzinakos are with the Edward S. Rogers Sr. Department of Electrical and Computer Engineering, University of Toronto, 10 King’s College Road, Toronto, ON M5S 3G4, Canada (email: {songl, dimitris}@comm.utoronto.ca). Digital Object Identifier 10.1109/TWC.2006.04598.

multi-hop information propagations into single hop transmissions. With the corresponding medium access control (MAC) layer design, the network topology can decide the transport capacity. In general, for a flat peer to peer ad-hoc network, it has been concluded in[5] that the capacity per node decreases with the order of Θ √1N under a uniform traffic pattern, where N denotes the number of nodes in a constant area. This work was further extended in [6], [7], where mobility and delay were considered. Our work is focused on cooperative transmission protocols at the physical layer, where classical relay channels, [8], and their multiple terminal extensions are central to the study. In infrastructure networks, multiple access channels with internode cooperations have been studied in [9], [10]. On the other hand, it was shown in [11] that the multi-terminal coding cooperation allows a sizable gain in the throughput of a flat ad-hoc network. A general study on the Shannon capacity of wireless networks remains an open information theoretic problem even for the Gaussian case. From a practical perspective, the “Two-phase” cooperative transmission paradigm has been of interest [10], [12]–[17]. In the paradigm, the source node broadcasts the shared information to relays during Phase I, and relay nodes cooperatively transmit the information to the destination node during Phase II. Based on a Poisson node distribution model, we study how the two-phase cooperative transmission can reduce the link capacity outage probability. The tradeoff among network power consumption, spectral efficiency, outage probability, and sensor node density is investigated for the proposed Cooperative Transmission Protocol for Sensor Networks (CTP-SN). CTP-SN takes the form of a typical implementation of the two-phase cooperation paradigm in wireless sensor networks. The underlining philosophy is to convert the spatial diversity of multiple relay sensor nodes into the gain of lower wireless link outage probability. Compared to the existing literature, our study is based on the following three considerations. First, wireless sensor networks are energy limited. Instead of enforcing transmission power constraints on every individual node [10], [12], the total power consumption of the network should be considered [13]–[15]. Different from the more information theoretical studies in [13]–[15] where only the transmission power is discussed, we consider both the transmission and transceiver circuit power consumption. The latter has been shown to be non-negligible for wireless sensor networks, in [18], [19]. As it will be shown later, we find

c 2006 IEEE 1536-1276/06$20.00

SONG and HATZINAKOS: COOPERATIVE TRANSMISSION IN POISSON DISTRIBUTED WIRELESS SENSOR NETWORKS

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that the transceiver circuit power consumption is decisive on whether or not the cooperation is beneficial. Second, our study for energy efficiency exploits the tradeoff between the network power consumption and the capacity outage probability [20]. Note that in [13]–[15], the energy efficiency is defined in terms of energy consumption per bit, where the analysis tools of [21] and the ergodic capacity of [22] are utilized. To achieve ergodic capacity, sophisticated long channel coding is required for averaging over the channel fading states, which can be difficult for implementation in wireless sensor networks. Capacity outage probability, on the other hand, examines the tradeoff between a fixed rate and the probability that the rate is achievable over the fading channel, which is much simpler and therefore more favorable. Note that the capacity outage probability is also used as the figure of merit in [10], [12]. Third, in [21], it has been suggested that wireless communications should operate under low spectral efficiency (wide band) regime in order to achieve high energy efficiency. In consequence, the results of [13]–[15] apply only in the low Signal to Noise Ratio (SNR) and low spectral efficiency regime. In practice, however, narrow band and high spectral efficiency might be required, as it has been demonstrated in [16]. In this paper and for our study, we show that the two conditions (high/low spectral efficiency) are equivalent. The major contribution of our work is on a statistical framework in the performance analysis, where the Poisson probabilistic node distribution model is adopted. The practical importance of the contribution relies on the following two facts. First, in a large scale dense sensor network, having individual nodes locally manage the information of network neighborhood might not be feasible, since the neighborhood can consist of a large number of nodes. Second, the sensor nodes might frequently fall into sleep, or suffer from node failures and location shifts. Also, there will be new sensor nodes joining the network over time due to new deployments. To the best of our knowledge, it is the first time that the impact of statistical node density is introduced in the performance limit analysis of wireless relay networks. The two main results of the paper are the following. • By means of physical layer cooperation, it is shown that the link capacity outage probability decreases asymptotically at least exponentially, when the sensor node density increases. (Theorem 2, Section IV-E) • With the same network power consumption, the outage probability of the non-cooperative direct transmission decreases asymptotically inversely with the node density, at the most. (Theorem 3, Section IV-E) In what follows, the system modelling is described in Section II. The CTP-SN protocol is described in Section III. The performance analysis is given in Section IV. Numerical results are provided in Section V to validate the performance analysis. Discussions from various perspectives appear in Section VI. Concluding remarks are given in Section VII.

modelled as a homogeneous Poisson process, [23], with the node density λ. That is, given an area of the size |A| in the field, the number of nodes in the area, μ(A), follows the Poisson distribution with the parameter λ|A|, i.e.

II. M ODEL A SSUMPTIONS A. Network Model We consider a large number of sensor nodes uniformly and randomly deployed over a 2-D field. The distribution can be

Pon,t (k) = (1 + α) · Pt (k) + Pc,t , Pon,r (k) = Pc,r .

P rob(μ(A) = m) = e−λ·|A| · m = 0, 1, . . . , ∞ .

(λ·|A|)m m!

,

(1)

We study the performance of one point-to-point wireless link between a source node s and a destination node d. All sensor nodes in the neighborhood can serve as potential relay nodes. In the developing of CTP-SN, we also make the following two realistic assumptions: 1) Channel State Information (CSI) is not available on transmitting nodes. 2) Sensor nodes are half-duplex, i.e. unable to transmit and receive simultaneously. B. Wireless Channel Model The wireless channel assumes a frequency non-selective Rayleigh fading model with additive Gaussian noise. More specifically, let Lk denote the location of node k. Given two nodes k1 and k2 , with the inter-node distance Dk1 ,k2 = Lk1 − Lk2 , the channel gain βk1 ,k2 between k1 and k2 is an exponential random variable [24], with the expected value as, 1 E (βk1 ,k2 ) = G · n , (2) Dk1 ,k2 where G is a constant determined by antenna gains. The parameter n is the path loss component [25], which takes the value of ‘2’ in the ideal free space, and it is between ‘2’ and ‘5’ in more complicated propagation environments. Moreover, let W denote the channel bandwidth, and N0 denote the additive Gaussian noise power density. The channel noise power is denoted by σn2 = N0 · W . Let Pt (k1 ) denote the transmission power on node k1 . Given βk1 ,k2 , the instantaneous channel capacity from k1 to k2 is [8], C(γk2 |k1 ) = W · log(1 + γk2 |k1 ), where γk2 |k1 =

Pt (k1 ) · βk1 ,k2 , σn2

(3) (4)

denotes the instantaneous SNR at node k2 , given that only k1 is transmitting. C. Energy Consumption Model In determining the energy consumption, for the node k, we consider both the transmission power Pt (k), and the circuit powers Pc,t , Pc,r at the transmitter and the receiver, respectively. Note that the circuit power consumption is assumed to be uniform over all the nodes. We adopt the model in [26], where the active power of transmitter, Pon,t (k), and receiver, Pon,r (k), are respectively, (5)

The parameter α in Eq. (5) is a constant decided by the drain efficiency of the RF power amplifier.

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be neglected. Once the relay nodes know that the packet is available on d, Phase II is cancelled, otherwise the relay nodes proceed to Phase II. Compared with the protocol without feedback, the protocol with feedback will also force the destination node to join Phase I. 3) Direct Transmission: Simple direct transmission will be employed in the sequel for the performance comparisons with CTP-SN. In this case, the source s transmits the packet directly to the destination d. No node in the neighborhood would act as a cooperative relay node participating in the transmission. B. Parameter Constraints In direct transmissions, given Ptot , there is Pon,t (s) + Pon,r (d) = Ptot . Then, from Eq. (5), we find that, Ptot − Pt,c − Pt,r . (6) 1+α Conditions are more complicated in CTP-SN. Let ΔT1 and ΔT2 denote the duration of Phase I and Phase II per packet, respectively. And let P1,all and P2,all denote the average power consumption in Phase I and Phase II, respectively. Then the following equations hold, Pt (s) =

Fig. 1.

CTP-SN Protocol.

III. CTP-SN T RANSMISSION P ROTOCOL Consider the scenario in Fig. 1, the source and destination nodes are denoted by s and d, respectively. Ds,d is the distance between the two. Let Ptot denote the average network power consumption in transmission, and let R denote the desired average spectral efficiency in bit/s/Hz. We develop the CTPSN protocol in accordance with the two-phase cooperative transmission paradigm. A. Protocol Outline 1) CTP-SN Protocol without Feedback: The CTP-SN protocol is divided into two phases. In Phase I, the source s broadcasts the packet to the relay nodes with the transmission power P1 and the spectral efficiency R1 (transmission rate W · R1 ). Further, we define two parameters D1 and D2 , satisfying D1 +D2 > Ds,d , which are the radius of two circles centered on s and d respectively. We impose the restriction that only the nodes within the overlapping region of the two circles are permitted to serve as relay nodes, which is depicted in Fig. 1 as the shaded area. Ar and |Ar | denote this shaded area and the size of this area, respectively. Let Π1 and Nr,1 denote the set and the number of potential relay nodes respectively, where Nr,1 = |Π1 |. Define Π2 ⊆ Π1 as the subset of relay nodes which can successfully decode the packet transmitted by s. Also let Nr,2 denote the number of nodes in Π2 , Nr,2 = |Π2 |. In Phase II, all the Nr,2 nodes in Π2 transmit space-time coded packets to the destination d, with the spectral efficiency R2 (transmission rate W · R2 ), and the transmission power P2 per node. Thus, the spatial diversity of the order Nr,2 is achieved by means of distributed space-time coding. More detailed comments are given in Section III-C.2. 2) CTP-SN Protocol with Feedback: Consider that after Phase I, the destination d can send a short acknowledge (ACK) packet to all the relay nodes, which indicates whether the desired packet has been successfully decoded in d. The ACK packet needs to contain only one bit yes/no information, hence, the energy and time consumption of the ACK packet can

ΔT1 · R1 = ΔT2 · R2 = (ΔT1 + ΔT2 ) · R , P1,all · ΔT1 + P2,all · ΔT2 = Ptot · (ΔT1 + ΔT2 ) .

(7)

From Eq. (7), it is straightforward to obtain, R= and, P1,all ·

R1 · R2 , R1 + R2

R R + P2,all · = Ptot . R1 R2

(8)

(9)

Furthermore, by using the model of Eq. (5), we have, P1,all = Pon,t (s) + E k∈Π1 Pon,r (k) (10) = P1 · (1 + α) + Pt,c + E(Nr,1 ) · Pr,c , and, P2,all

= =

Pon,r (d) + E k∈Π2 Pon,t (k) (P2 · (1 + α) + Pt,c ) · E(Nr,2 ) + Pr,c .

(11)

By combining Eqs. (9,10,11), we get, R + [P1 · (1 + α) + E(Nr,1 ) · Pr,c + Pt,c ] · R1 [(P2 · (1 + α) + Pt,c ) · E(Nr,2 ) + Pr,c ] · RR2 = Ptot .

(12) In summary, Eqs. (8) and (12) describe the parameter constraints in CTP-SN. C. Comments 1) Comments on the CTP-SN Protocol: We consider CTPSN as a typical implementation of the two-phase cooperative transmission paradigm in wireless sensor networks, where CSI is not available at the transmitters. The explicit definitions of D1 , D2 are beneficial for our study in deriving performance limits. In practice, the relay nodes can be chosen by the broadcasting of two wake-up beacons from the source and the destination, respectively. The wake-up beacons can possibly be sent by secondary low power wake-up radio, which controls the broadcasting range of D1


and D2 , respectively. A similar case can be found in [3]. The nodes with the receptions of both two beacons participate the transmission as relay nodes. The definitions of the two parameters D1 and D2 can be conceptually understood by noting the followings. A node k, with a distance to the source Ds,k > D1 , does not participate in the cooperation protocol, because the probability that the node k can decode the packet correctly, i.e. P rob(k ∈ Π2 |k ∈ Π1 ), is too small, compared with the energy cost of participating in Phase I, given by Pon,r (k) · ΔT1 . On the other hand, a node k, with a distance to the destination Dk,d > D2 , does not participate in the cooperation protocol, because the contribution of node k in Phase II is too small, compared with the energy cost of participating in Phase II, i.e., Pon,t (k)·ΔT2 . 2) Comments on the Distributed Space-time Coding: In the Phase II of CTP-SN, an implementation issue is to design space-time codes able to provide the spatial diversity, under the condition that the knowledge of Nr,2 is not available at individual relay nodes. The requirement can be achieved by space-time block codes [27]. More specifically, assume that the pre-designed space-time block code is of the size M , and each relay node randomly picks up one of the M codes. Then, the diversity order Nr,2 can be achieved approximately, when M is much larger than E(Nr,2 ). Note that a similar idea, with predetermined instead of randomly-picked codes, has been considered in [10], [28]. Note that the same diversity order can also be achieved by repetitive coding, if the Nr,2 nodes in Π2 can transmit on orthogonal channels, e.g. assuming a wideband scenario [29]. Compared with the space-time coded approach, the tradeoff for the simplicity of repetitive coding can be a waste of spectral resources. 3) Comments on Parameter Constraints: By Eqs. (8,12), given Ptot and R, there are two degrees of freedom on the P1 1 system design, which are R R2 and P2 , respectively. We note that Eq. (12) is exact only for the CTP-SN protocol without feedback. It should be modified when the feedback protocol is considered. Note that the destination node may be active in Phase I, even if it is not within Ar . Also there is a probability that the energy consumption in Phase II is zero. However, in the sequel we neglect this small difference, and compare the protocols with and without feedback side by side under the same setup of the parameters, P1 , P2 , R1 , R2 . 4) Comments on the Degrees of Freedom: Overall, the degrees of freedom in the considered protocol are, D1 , D2 , R1 P1 R2 , and P2 . The impact of D1 and D2 on the protocol performance is numerically discussed in Section V. The choice P1 1 for R R2 and P2 introduces a problem of optimal resource allocation, which is discussed is Section VI-B. However, all of them are independent from the main performance analysis results of the paper, which are found in the next section. IV. P ERFORMANCE A NALYSIS In this section, we provide the performance analysis of CTP-SN by deriving the link capacity outage probability, and compare the performance of CTP-SN with that of the direct transmission protocol. In presenting the results, we define that the notation “f (x) ≈ g(x), when x → 0” indicates

“limx→0 of x.

f (x) g(x)

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= 1”, where both f (x) and g(x) are functions

A. Direct Transmission Performance In direct transmission, the SNR at the destination node d, γd|s , is, βs,d · Pt (s) . (13) γd|s = σn2 The outage probability, Pout,D , is a function of Ptot and R, that is, Pout,D =(a) P rob C(γ d|s ) < W · R β ·P (s) λ0 , where A is a deterministic positive constant. V. N UMERICAL R ESULTS The numerical analysis is performed under the system parameters listed in Table 1. These parameters are considered typical in wireless sensor networks under a high spectral efficiency scenario, such as the IEEE 802.15.4 [31]. We generate the set of Poisson points Π3 , according to the density given by Eq. (27). The outage probabilities of CTP-SN with f and Pout,C , are obtained by and without feedback, Pout,C averaging over 50000 Monte-Carlo runs. The results of the direct transmission, Pout,D , under the same network power consumption Ptot in Eq. (12), is also provided for comparP1 1 isons. In all simulations, we fix R R2 = 1 and P2 = 1. TABLE I S YSTEM PARAMETERS Parameter n G 2 σn Pt,c Pr,c α R

Unit dB dBm dBm dBm bit/s/Hz

Value 4 −31 −92 10 10 2 1

The first set of results examine the variation of the outage probability versus P1 . Fig. 2 to Fig. 4 show the results when the source-destination distance Ds,d = 30, 50, 80 meters, respectively. Furthermore, in the figures, we fix D1 = D2 = Ds,d and λ = 0.003(1/m2). We first examine the behavior of the upper bound according to Theorem 2. When P1 increases,

Outage Probability

Pout,D >

−2

10

−4

10

−6

10

Pout,C −8

f

10

Pout,C P

out,D

Bound on P

out,C

−10

10

4

6

8

10

12 P1 (dBm)

14

16

18

20

Fig. 4. Capacity outage probability vs. P1 (λ = 0.003/m2 , Ds,d = D1 = D2 = 80m).

ε1 converges to zero. κ will then converge to |Ar |·λ according to Eq. (35). Thus, in Fig. 2, the Pout,C bound tends to exhibit a flat error floor. This floor is, however, not obvious in Fig. 3 and Fig. 4, because the limit is not reached due to a much larger |Ar |. In all three figures, CTP-SN contributes a significant performance gain over the direct transmission protocol, which is generally 10 ∼ 20dB, in the region 10−3 ∼ 10−2 of practical interests. When Ds,d = 30m, the CTP-SN with feedback offers an additional significant performancegain, because according to Eq. (29), the P rob C(γd|s ) ≥ R1 provides a nontrivial reduction on the outage probability. When Ds,d increases, shown in Fig. 3 and Fig. 4, this gain becomes smaller because P rob C(γd|s ) ≥ R1 approaches zero. The upper bound is shown to be loose in this set of results, f . In however, it demonstrates the trend of Pout,C and Pout,C Fig. 2, where the bound is flat, the outage probability of CTP-SN, in general, decays at the same rate as in the direct transmission protocol. When the bound is steep, as in Fig. 3 and Fig. 4, the performance of CTP-SN increases much faster

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0

0

10

10

Pout,C f Pout,C −1

−1

10

Pout,D

10

Bound on P

out,C

Outage Probability

Outage Probability

−2

−2

10

−3

10

10

−3

10

−4

10

Pout,C Pf

−4

10

−5

out,C

10

Pout,D Bound on P

out,C

−6

−5

10

10

2

2.5

3

3.5

4

4.5

λ (1/m2)

5

0.5

1

1.5

Fig. 5. Capacity outage probability vs. λ (P1 = 16dBm, Ds,d = D1 = D2 = 30m).

2

2.5

3

3.5

λ (1/m2)

−3

x 10

−3

x 10


0

10

0

10 −1

10

−1

−2

10

Outage Probability

Outage Probability

10

−3

10

Pout,C −4

10

−2

10

−3

10

f out,C

Pout,C

P

−4

Pout,D

f

10

Pout,C

Bound on P

out,C

−5

10

0.5

P

1

1.5

out,D

2 λ (1/m2)

2.5

3

3.5 −3

x 10


than that of the direct transmission. The second set of results, on the other hand, examine the variation of outage probability versus λ. Similar to the first set of results, Fig. 5 - Fig. 7 are obtained when Ds,d = 30, 50, 80 meters, respectively. Once again there is D1 = D2 = Ds,d , and P1 equals 16dBm. The numerical results validate Theorems 2 & 3. More specifically, the outage probability of CTPSN decreases exponentially with the network density λ, while the outage probability of the direct transmission decreases only with rate λ1 . More careful observation reveals that the curves are steeper at larger distance Ds,d , which is due to fact that we have fixed D1 = D2 = Ds,d . A larger Ds,d suggests a larger |Ar |, and hence a higher κ in Eq. (35). Furthermore, there is a constant gap between the upper bound and the real outage probability for CTP-SN, because in deriving the bound we have replaced all Dk,d with D2 , where k ∈ Π2 . Finally, we look into the impacts of the protocol parameters D1 , D2 on the outage probabilities. The third set of

Bound on P

out,C

−5

10

25

30

35

40

45 50 D1 (m)

55

60

65

70

Fig. 8. Capacity outage probability vs. D1 (P1 = 12dBm, λ = 0.003/m2 , Ds,d = 50m, D2 = D1 ).

results, Fig. 8 and Fig. 9, are obtained when the power P1 equals 12dBm and 8dBm, respectively, and λ is fixed at 0.003(1/m2). The distance Ds,d is set to be 50m, and we also set D1 = D2 . As it is shown in the figures, the bound initially decreases with D1 and then starts increasing. This is due to the interaction of the two factors in κ, which are |Ar | and e−ε1 , respectively, in Eq. (35). While the former increases polynomially with D1 , the latter decreases exponentially with D1 . The Pout,C decreases monotonically with D1 , because more relay nodes are available with a higher D1 . The upper bound is close to the real outage probability Pout,C , when D2 is small, and diverges when D2 is large. This is due to the approximation that we have made on the distance Dk,d in deriving the bound, where k ∈ Π2 . In practice, large coefficients D1 , D2 suggest large interference, and the setup of D1 = D2 = Ds,d generally suffices.

SONG and HATZINAKOS: COOPERATIVE TRANSMISSION IN POISSON DISTRIBUTED WIRELESS SENSOR NETWORKS 0

10

B. On the Optimal Resource Allocation

Outage Probability

−1

10

−2

10

P

out,C f

Pout,C Pout,D Bound on Pout,C

−3

10

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25

30

35

40

45 50 D (m)

55

60

65

70

1

Fig. 9. Capacity outage probability vs. D1 (P1 = 8dBm, λ = 0.003/m2 , Ds,d = 50m, D2 = D1 ).

VI. D ISCUSSIONS A. On the Impact of Circuits Power Consumption By incorporating more than two nodes (the source and the destination) in a single wireless link, CTP-SN spends more power in circuit power consumption, compared with the direct transmission. Further considerations show that the circuit power consumption is crucial in deciding whether or not the cooperation is beneficial. Consider the following extreme example. Let ∗ = Ptot

[E(Nr,1 ) · Pr,c + Pt,c ] · [Pt,c · E(Nr,2 ) + Pr,c ] ·

R R1 + R R2 .

(37)

This is equivalent to Eq. (12) with P1 = P2 = 0. Thus, the f = 1. outage probability of CTP-SN is strictly Pout,C = Pout,C On the other hand, Pout,D is strictly less than 1, since there is Pt (s) > 0 according to Eq. (6). Therefore, Pout,D < Pout,C , which shows that, under the extreme condition of Eq. (37), the direct transmission protocol performs strictly better than CTP-SN. Furthermore, we note that in the first set of numerical results, Fig. 2 to Fig. 4, when P1 is low, the direct transmission outperforms CTP-SN. This can also be explained by realizing that the circuit energy consumptions, Pt,c and Pr,c , dominate Ptot under such conditions in CTP-SN. Classically, Shannon information theory only considers the transmission power consumption, which is denoted as Pt (k) for the node k in the paper. The introduction of hardware considerations, such as circuit power consumption, generally remains unexplored in information theory. These considerations, however, are practically interesting. In [21], for example, it was found that the maximum energy efficiency is achieved when SNR approaches zero, which later becomes the basic consideration in [13]–[15]. However, when circuit power consumption is taken under consideration, it can be shown that the condition “SNR Approaching Zero” is neither achievable nor optimal. These studies are subject to future research.

In the paper, the way of resource allocation is strictly R1 1 suboptimal, i.e., based on the choice of P P2 and R2 . The optimal choice of these two parameters in minimizing the outage probability can be found by means of numerical tools. R1 1 The absence of closed forms on P P2 and R2 , which is mathematically difficult to obtain, makes the implementation of optimal resource allocation difficult in dynamic environments. Note that a similar optimization problem is found in [15] under a different criterion, where the optimal resource allocation problem under a single relay node condition can be solved only by exhaustive numerical search. Furthermore, in the paper, all relay nodes in the set Π2 assume a uniform transmission power P2 , which is also suboptimal. It is intuitively attractive to allocate more power to those relay nodes with better channel statistics, i.e. the nodes closer to the destination node d. This problem is considered under the centralized multiple antenna condition in [32], where different antenna elements assume different fading statistics. Since the optimal allocation could only be found by exhaustive numerical search in [32], we consider that it will be difficult to achieve the optimality in our problem of developing distributed power allocation schemes. Therefore, in both of the above problems, suboptimal alternatives for resource allocation are of more value for future research. C. On the Cross-layer Design From the network perspective, the cooperative protocol in the physical layer, such as CTP-SN, leads to the crosslayer communication network design of link and physical layers [3]. The underlying reason is that more than two nodes (the source and the destination) are involved in one single wireless link, without the coordination from upper network layer. Specifically, in our analysis, we neglect the co-channel interference in the performance analysis. In other words, the arriving radio signal at the relay nodes in Phase I is only a function of the transmitted sequence from the source and the additive noise. This assumption can be justified by regulating the MAC layer. Besides the point to point communications studied in this paper, the related interesting research also includes cooperative energy efficient broadcasting [29], [33]. By means of inter-node cooperations, the compound link and physical layer can provide better wireless link performances to upper layers, i.e., in terms of lower outage probability, higher energy efficiency, and longer network hop distance. VII. C ONCLUSION In this paper, we discuss a typical cooperative transmission protocol CTP-SN in the physical layer of wireless sensor networks. The performance analysis of the protocol is presented under the Poisson node distribution assumption. For the first time, the impact of relay node statistical density is introduced in the research of wireless relay networks. An asymptotic upper bound on the capacity outage probability of CTP-SN is derived, which is shown to be decreasing exponentially with the node density λ. For this reason, CTP-SN performs asymptotically much better than the simple direct transmission

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protocol. Some important related research directions are then presented on the information theoretic energy efficiency, the optimal/suboptimal resource allocation, and the cross-layer network design, respectively. A PPENDIX I O N THE I MPLICATIONS OF THE A SYMPTOTIC A PPROXIMATIONS In the high spectral efficiency regime, it is obvious that the approximations ε1 , ε2 → 0 are equivalent to a high SNR approximation. In the low spectral efficiency regime, i.e., R1 , R2 → 0, the ε1 and ε2 , defined in Eq. (30) and Eq. (31) respectively, can be written as, ln 2 · R1 · W · N0 · D1n , (38) ε1 = P1 · G and ln 2 · R2 · W · N0 · D1n . (39) ε2 = P1 · G Thus, given the parameters {W, R1 , R2 } as {W h , R1h , R2h } for the high spectral efficiency regime, and {W l , R1l , R2l } for the low spectral efficiency regime, the two scenarios are equivalent in our study under the following conditions, (2 and,

Rh 1

h

− 1) · W = ln 2 ·

R1l

l

·W ,

(40)

(2R2 − 1) · W h = ln 2 · R2l · W l .

(41)

h

It is evident from Eqs. (40,41) that it is more energy efficient to operate in the low spectral efficiency regime, given the transmission rate constraints W · R1 and W · R2 . A PPENDIX II P ROOF OF L EMMA 1 pK (x) denote the probability density function of Let K k=1 ζk , there is [24], K−1 x −x , x≥0 (K−1)! e pK (x) = . (42) 0, x 0, ∀ k; (c) is based on the observation that (b) depends only on the size of Poisson points in Π2 , Nr,2 ; (d) is due to Lemma 1 and ε2 → 0; (e) is straightforward. A PPENDIX IV P ROOF OF T HEOREM 2 Due to the definition of Ar , we have Ds (x, y) ≤ D1 , ∀ [x, y] ∈ Ar . Thus, from Eq. (22), we obtain, (2R1 −1)·σn2 ·D1n − P1 ·G λΠ2 (x, y) ≥ λΠ1 (x, y) · e −ε1 = λ (x, y) · e (45) Π1 −ε λ · e 1 , [x, y] ∈ Ar = . 0, [x, y] ∈ Ar By means of Eqs. (24,45), we get, E(Nr,2 ) ≥ = Then, Pout,C

≤(a) =(b) Ds,d , without loss of generality, D we assume D1 > 2s,d . The condition, εD → 0, is proven by, εD

=(a) (b)

< =(c)

n (2R −1)·(1+α)·σn2 ·Ds,d

G·(Ptot −Pt,c −Pr,c ) (2R1 −1)·σn2 ·D1n 2 · G·P1 2 n · ε1 . n

(48)

In Eq. (48), (a) is by the definition in Eq. (16); (b) is obtained D P −Pt,c −Pr,c due to the fact, D1 > 2s,d , R < R1 , and P1 < tot 1+α ; (c) is obtained directly from the definition of ε1 in Eq. (30). Given εD → 0, from Eq. (17), it is, Pout,D

> =

2 n (2R −1)·(1+α)·σn ·Ds,d G·Ptot A1 Ptot ,

(49)

where A1 is defined as a positive constant. From Eq. (12), we also obtain, Ptot < E(Nr,1 ) · A2 + A3 ,

(50)

where A2 , A3 are also positive constants. Combining Eqs. (49,50), and by Eq. (19), we have, Pout,D

> =

A1 E(Nr,1 )·A2 +A3 A1 |Ar |·λ·A2 +A3 .

(51)

Then, the statement in Theorem 3 is proven by defining A < A1 A2 , in straightforward.


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