i ei
k1 k2 k3 k4
Figure 2.7: Sensor i broadcasting a message to its neighbours k1, k2, k3,k4
Cross-layer reception and diversity schemes
The techniques based on diversity schemes consist of sending two or more instances of a message signal through dierent communication channels so as to improve the reliability of the decoding process at the reception. A particular type of diversity, called space diversity, nds application in the multiple-input and multiple-output (MIMO) systems in radio telecommunica-tion, where arrays of spatially separated antennas are used at the transmitter and the receiver to generate phase-shifted versions of the transmitted signal.
Space diversity is achieved in wireless sensor networks when an identical message signal is transmitted to a receiving sensor by two or more sensor nodes (typically: a transmitting sensor and any neighbouring sensor(s) hearing and forwarding the signal). Since the various signal instances originate from dier-ent locations, they are likely to experience dierdier-ent levels of noise and fading.
The multiplicity of the message signal is exploited at the reception device by combining the signal instances into a unique, clearer signal. Frequently discussed combining diversity techniques include equal-gain combining, where the phases of the signals are aligned with each other and the signal instances added coherently leading to constructive interference, and maximal-ratio com-bining , where the signal instances are weighted and aligned in function of their respective SNRs and added to each so that the SNR of the weighted sum signal is maximised and equal to the sum of the individual SNRs of the incoming signals. Phase alignment in wireless sensor networks can be di-cult to achieve as it requires accurate CSI knowledge by the sensors, and thus prior CSI acquisition in fast-fading conditions. Alternatively, phase alignment may be asymptotically approached by distributed iterative algorithms based on successive corrections demanding little communication overhead.
2.3. Cooperative transmissions 29
i j
k
mi(t)
mk(t)
Figure 2.8: Sensor i transmitting a message to j with relaying node k
2.3.2 Cooperative transmission techniques
The examples of cooperative transmission techniques discussed in this section result from combinations of broadcasts and space diversity schemes9.
Single-relay cooperation
The principle of single-relay cooperation is depicted in Fig. 2.8, where infor-mation is to be sent from a sensor i to another sensor j located at a remote distance. The power consumption at the transmitting node can be reduced with the help of a third sensor k serving as a relay between i andj. Letm(t) denote a signal symbol to be transmitted. Sensor i broadcasts mi(t) (an am-plied version of m(t)), which is received as rk(t) by the relay k. Sensor k then forwards mk(t) to j, where the received signal rk(t) is the sum of the signals radiated by iand k. The single-relay cooperative transmission ofm(t) from i to j via k can be modelled as
mi(t) = aim(t),
rk(t) = hikmi(t) +nk(t), mk(t) = akrˆk(t),
rj(t) = hijmi(t) +hkjmk(t) +nj(t),
(2.22)
whereai andak are complex amplication gains, hik, hij andhkj are the com-plex gains of the channels(i, k), (i, j) and(k, j), respectively, nk(t) andnj(t) are random noise processes, andrˆk(t)is a function ofrk(t)normalised to unit energy and specied by the type of cooperative coding strategy in use. In amplify-and-forward (AF) strategies, the signal rk(t) is simply amplied at the sensor k and one has rˆk(t) = rk(t)/|rk(t)|. In decode-and-forward (DF)
9It seems fair to regard the integration to wireless sensor networks of cooperative transmission techniques less as a mature discipline as an active topicstill at the experimental levelof the networking research. The implementability and eciency of cooperative transmissions in wireless sensor networks remains strongly context-dependent and subject to discussion and dissension among the networking community.
strategies, however, the sensor k rst attempts to decode the signal rk(t) be-fore transmitting rˆk(t) = m(t) on the condition of successful decoding at k. The values ofai andak can be optimised so as to satisfy, with minimum over-all power consumption or maximum bit rate, the SNR constraint (2.3) at the decoding node(s), namely node j in the AF case and nodes k and j in the DF case.
A question raised by the model given in (2.22) is the choice of the relay sensor if several relay candidates are available. Relay selection for single-relay cooperation can be stated as an optimisation problem for which various selection criteria have been proposed, such as selecting the sensor with min-imum distance or maxmin-imum SNR to the receiving node, or the sensor which minimises a cost function related to the overall power consumption or to the network lifetime. The relay selection process sometimes relies on a (local) re-lay contention protocol of the type RTS/CTS/ACK where the potential rere-lay nodes estimate themselves their aptness in assisting the transmission.
Considering single-relay cooperation in wireless sensor networks increases the number of possible paths available for routing information. The optimal value of energy- or lifetime-related objective functions of network optimisation problems can be further improved by integrating single-relay transmissions into the routing process. This is done in Example 2.9, where the problem formulated in Example 2.3 is modied so as to include single-relay cooperation.
Example 2.9 (Routing with single-relay cooperation) In this example we wonder how single-relay transmissions can be integrated into the routing problem (2.10). For clarity of illus-tration, the power consumption at the receiving sensors during direct transmissions will not be considered. Let Es denote the set of all the triples (i, k, j) such that a single-relay transmission is feasible fromi to j with nodek as the relaying node. Consider now the single-relay cooperation model depicted in Figure 2.8, where the sensoritransmits tojviakwith packet ow denoted byx˙ikj. The expected quantities of energy consumed by the sourcei, relay k and destinationj in the trans-mission of one packet are respectively denoted bye¯iikj, ¯ekikj, and¯ejikj. These quantities are assumed to be constant in this example and can be computed by optimisation of the parameters in (2.22). If x= (⃗x, ⃗x) denotes the ow vector of the direct transmissions (cf. Example 2.3), andx˙ that of the single-relay transmissions, a simple model for the total power consumed byiis given by
pi(x,x) =˙ ∑
j:(i,j)∈E
¯
eiij⃗xij+ ∑
j:(j,i)∈E
¯
eiij ij⃗x + ∑
(j,k):
(i,j,k)∈Es
¯
eiijkx˙ijk+∑
(j,k):
(k,i,j)∈Es
¯
eikijx˙kij+ ∑
(j,k):
(j,k,i)∈Es
¯
eijkix˙jki, (2.23)
where the rst term is the power consumed in direct transmissions, and the three remaining terms successively model the total powers consumed byiduring single-relay transmissions as a transmitter, relay and receiver. Integrating the single-relay transmissions into (2.10)-(2.12) yields the following
2.3. Cooperative transmissions 31
i j
k1 k2 k3
Figure 2.9: Multiple relay transmission from i toj via k1, k2, k3.
routing problem minimise
x≥0,x≥0˙ f(x,x˙)
subject to ∑
j:(i,j)∈E
[⃗xij− ji⃗x ] + ∑
j:(j,i)∈E
[ ⃗xij−⃗xji] + ∑
(j,k):
(i,k,j)∈Es
˙
xikj− ∑
(j,k):
(j,k,i)∈Es
˙
xjki=si, ∀i∈N
⃗ xij+∑
k:
(i,k,j)∈Es
˙
xikj+∑
k:
(k,i,j)∈Es
˙
xkij+∑
k:
(i,j,k)∈Es
˙
xijk ≤x¯ij, ∀(i, j)∈E
⃗xij+∑
k:
(i,k,j)∈Es
˙
xikj+∑
k:
(k,i,j)∈Es
˙
xkij+∑
k:
(i,j,k)∈Es
˙
xijk ≤x¯ij, ∀(j, i)∈E pi(x,x)˙ ≤p¯i, ∀i∈N
(2.24)
where the rst constraint is the ow conservation constraint, the second and third constraints are capacity constraints limiting the total ow of information on the communication channels, and the last constraint is a limitation on the power consumptions of the sensors withpi(x,x˙)given by (2.23).
The generalisation from single-relay transmissions schemes to multiple-relay cooperation is straightforward. In the multiple-multiple-relay transmission il-lustrated in Figure 2.9, the sensor i transmits to the destination j with the support of several relaying nodes. For such transmissions, the maximum ac-cessible bit rate is a function of the SNR at the destination node. It is therefore rewarding to optimise the selected combination of candidate relays and the design of the transmission scheme. Note that a multiple-relay transmission may be successful even if the signal emitted by the source node does not reach the destination node, where decoding can be based on the relayed signals only.
This particular scenario is treated in Section 2.3.3 and usually referred to as cooperative beamforming.
2.3.3 Cooperative beamforming
The idea of exploiting space diversity in wireless sensor networks has inspired various cooperative transmission schemes taking dierent forms and shapes depending on the contexts and the models used for the communication chan-nels and sensors. The technique we now consider aims at transmitting infor-mation between two sensors located at a relatively long distance from each other, and can be seen as the combination of two ingredients: a broadcast di-rectly followed by cooperative beamforming. As depicted in Figure 2.10, the
i
1 2 3 4
j
Figure 2.10: Transmission from i to j via broadcast to 1, 2, 3, 4 and cooperative beamforming
transmitting sensor rst broadcasts its message to close neighbours, which in turn exploit the space diversity property to `beamform' the message to the destination sensor. The succession of broadcasting and beamforming is in-tended to lead to energy savings provided that the total power consumed by the transmitting sensor and the relays is less than that required by direct transmissions.
In Example 2.10, transmissions of the type broadcast-beamforming are in-corporated as additional paths into a network lifetime maximisation problem.
The coupled optimisation of the routing policy and of the transmission power allocation of the sensors taking part in broadcast-beamforming transmissions is a dicult problem which is not considered here. Instead, the transmission powers of each potential cooperative beamforming scheme are xed to the val-ues minimising the total power consumption for the considered transmission.
An optimal routing policy is then sought among all the possible paths.
Example 2.10 (Network lifetime maximisation with beamforming) This example inte-grates cooperative beamforming transmissions into the routing problem (2.10). Again, power con-sumption at the receivers is not considered. We denote by Eb the set of all the sensor pairs (i, j) such that a broadcast-beamforming transmission is feasible fromi toj. Let (i, j)∈Eb be any such transmission as depicted in Figure 2.10. The set of the sensors which participate in the beamforming phase of the transmission as relays is denoted by r(i, j). The energy consumed by the broadcasting nodei during the transmission of one packet is denoted bye¯iij, whilee¯kij represents the energy con-sumed by a beamforming nodek∈r(i, j) during the same transmission. If the ow of information communicated by nodeito nodej through the broadcast-beamforming transmission is denoted by˚xij, andx= (⃗x, ⃗x)denotes the ow vector of the direct transmissions, the total power consumed by iis modelled by
pi(x,˚x) = ∑
j:(i,j)∈E
¯
eiij⃗xij+ ∑
j:(j,i)∈E
¯
eiij ij⃗x + ∑
j:(i,j)∈Eb
¯
eiij˚xij+ ∑
(j,k)∈Eb:i∈r(j,k)
¯
eijk˚xjk, (2.25) where the four terms model the total power consumed by i in the role of the transmitter of direct transmissions, as the broadcasting sensor in broadcast-beamforming transmissions, and as a
beam-2.3. Cooperative transmissions 33
forming sensor. A lifetime-maximising routing problem is given by minimise
x≥0,x≥0,˘˙ x≥0 x˘
subject to ∑
j:(i,j)∈E
[⃗xij− ji⃗x ] + ∑
j:(j,i)∈E
[ ⃗xij−⃗xji] + ∑
j:(i,j)∈Eb
[˚xij−˚xji] =si, ∀i∈N
⃗xij+ ∑
k:(i,k)∈Eb,j∈r(i,k)
˚xik+ ∑
k:(k,j)∈Eb,i∈r(k,j)
˚xkj≤x¯ij, ∀(i, j)∈E
⃗xij+ ∑
k:(i,k)∈Eb,j∈r(i,k)
˚xik+ ∑
k:(k,j)∈Eb,i∈r(k,j)
˚xkj≤x¯ij, ∀(j, i)∈E pi(x,˚x)≤p¯i, ∀i∈N pi(x,˚x)−bix˘≤0, ∀i∈N
(2.26)
where the variable x˘ stands for the inverse network lifetime (see Example 2.4), and the rst four constraints model the ow conservation, a (double) channel capacity constraint, and a limitation on the power consumptions of the sensors withpi(x,˚x)given by (2.25).
The problem (2.26) was formulated and solved in [BAR11]. In the same study, comparisons were made between the optimal lifetimes of networks where broadcast-beamforming transmissions are permittedthe network life-time is then the optimal value of (2.26), and the lifelife-times of networks with-out cooperative transmissions. The allocation of the transmission powers in the broadcast-beamforming transmissions, i.e. the parameters e¯iij and e¯kij, were chosen in accordance with the cooperative beamforming model10 used in [KAMZ03, MMMZ09]. In low trac conditions (i.e. comparatively large components for the parameters x¯ and p¯), simulations realised on randomly-generated networks with 25 nodes showed that the introduction of broadcast-beamforming transmissions lead to network lifetime improvements in 65% of the simulated networks, and to network lifetime extensions by more than50%
for 41% of the networks. Although these statistics are specic to the cho-sen transmission models, they could be reproduced, by following a similar approach, for any framework achieving energy savings by exploitation of the multicast advantage and the spatial diversity of wireless sensor networks.
10The framework proposed in [KAMZ03] considers cooperative transmissions models such that a sensoriis able to transmit a message to a remote sensorjwith the help ofqrelaying sensors, denoted by 1, ..., q and proceeding in a decode-and-forward fashion. The relaying sensors are assumed to phase-align and the transmitted signals are received coherently at the destinationj. Ifm(t)denotes a symbol transmitted byi, then decoded and forwarded by the relays, the signal received atj is modelled by
rj(t) =∑q
k=1|hkj||ak|m(t) +nj(t), (2.27) where nj(t) is a random, independent noise process with mean power Pn, and for k = 1, ..., q, ak is the complex amplication gain at sensor k, andhkj the complex gain of the channel (k, j).
The amplication gains are chosen so as to minimise the total power consumption for the relaying nodes∑q
k=1|ak|2subject to the constraint[∑q
k=1|hkj||ak|]2/SNRminj ≥Pnwhich guarantees correct decoding atj in accordance with (2.3). The optimal power allocation yields
|ak|= |hkj|
∑q
k=1|hkj|2
√
SNRminj Pn, k= 1, ..., q. (2.28) It can be veried that the total power consumption of the relaying sensors is then equal to the