Distributed MIMO channels in Pervasive network

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DEL COSO et al.: COOPERATIVE DISTRIBUTED MIMO CHANNELS IN WIRELESS SENSOR NETWORKS 403

layer multi-hop structure described in [13]. Multi-hop there is

considered as a multiple relay channel, with all source-relay,

relay-relay and relay-destination communications enabled in

order to maximize the data rate at the receiver node. It

constitutes the optimum way to share network resources under

Information Theory metrics. A second approach is based upon

the data flooding scheme proposed in [14], [15]. Cooperation

is considered there in an opportunistic way: the source nodestarts transmitting directly to the destination node and, as soon

as other network nodes are able to decode the broadcasted

data, they start transmitting a cooperating signal. Results

show that it significantly outperforms non-cooperative multi-

hop without any global coordination. Finally, a cluster-based

approach was presented in [16]. It considers all sensor nodes

grouped in collaborative sets, so-called cooperative clusters.

All cluster nodes cooperate to transmit and receive data from

any other cooperative group. Some advantages can be found in

the cluster-based strategy with respect to the others: 1) limiting

cooperation inside the clusters reduces synchronization and

resource management complexity, and 2) current multi-hoptheory and routing algorithms can be applied by considering

the cluster as the minimal subset within the network. However,

restricting cooperative relaying to a finite group of nodes (i.e.,

the cardinality of the cluster) makes the network capacity to

scale following results in [17], rather than [18] as the first

two approaches. Previous results concerning cluster design and

corresponding energy saving can be found in [19], [20], where

the relationship between the optimum number of clusters and

the long haul distance is derived.

In this paper we follow the last approach described, and

consider a WSN with sensor nodes grouped in cooperative

clusters. To efficiently group sensor nodes into clusters, anyof the distributed clustering algorithms in [21]–[24] may be

adopted. Such clustering algorithms assume no centralized

control and consider that every sensor node decides to join

a cluster based only on local information. Mostly, they are

based on iterative schemes where nodes join or create clusters,

depending on whether they can reach existing ones or not. In

particular, we assume a topology with one level hierarchy, with

disjoint clusters and no cluster leaders as in [25, pp. 288-292].

Moreover, routing in the network is assumed to be carried

out by layer-3 hierarchical protocols, as defined in [26]–[29].

However we consider routing and clustering as given by upper

layers, and we only focus on the physical layer transmissionwithin the clustered network.

In our approach, both transmit and receive cooperation

are enabled among cluster nodes, which operate under a

half-duplex constraint (i.e., they use orthogonal channels to

transmit and receive). The multi-hop transmission consists

of a concatenation of single cluster-to-cluster hops (see Fig.

1). Transmit diversity within every cluster-to-cluster hop is

exploited by considering a decode-and-forward, time-division

relaying scheme, based upon two consecutive channels (see

Fig. 2): 1) a Broadcast channel accounting for data shar-

ing among cluster nodes, and 2) a MIMO channel between

clusters, that allows cluster nodes to jointly transmit data tothe receiving cluster (that could be the intermediate cluster

of a multi-hop transmission or the destination cluster). We

assume different degrees of channel state information (CSI)

for both channels. For the broadcast channel, we consider

that every node has perfect and updated transmit and receive

CSI of its channel to all other nodes of the cluster (transmit

channel knowledge can be obtained via channel reciprocity).

For the MIMO channel, we assume no transmit CSI among

nodes of two independent clusters, since there is no channel

reciprocity in the proposed scheme. Hence, we propose the

use of Distributed Space Time Codes (DSTC) when cluster nodes jointly transmit data to the next cluster, aiming to

minimize the outage probability (insight on amplify-and-

forward and decode-and-forward DSTC design is found in

[30], [31], respectively). At the receiving cluster, we propose

a cooperative multiple antenna reception protocol based upon

a Selection Diversity receiver [32], where the cluster node

with highest received signal level acts as cluster coordinator

and decodes data. Such protocol obtains full receiver spatial

diversity at moderate complexity. We consider that every node

of the receiving cluster has updated receive CSI of its channel

with the transmitting nodes of the transmitter cluster. However,

each node is unaware of the channels of other receiving nodesof the cluster. As explained in Section II-A, selection diversity

can be implemented with such an individual CSI and without

any centralized control.

In this paper, the proposed clustered WSN is optimally

designed for minimum end-to-end outage probability, given a

per link energy constraint. First, we focus on the optimum

resource allocation within every cluster-to-cluster hop. As

shown in [33]–[35], time and power allocation drastically

change the performance of half-duplex relaying architectures

(a wide range of resource allocation schemes are analyzed

in [33]). We derive the optimum fraction of time dedicated

to the intracluster broadcast channel and to the intercluster MIMO transmission, and the optimum power allocation over

the two channels. Moreover, in order to obtain closed form

expressions, we propose a simplified suboptimum time alloca-

tion. Finally, we show that the cluster-to-cluster hop achieves

full transmit and receive spatial diversity. The remainder of

this paper is organized as follows: Section II describes the

cluster-based multi-hop protocol, the signal definition, and

states the problem. In Section III we study and derive the

optimum resource allocation on every cluster-to-cluster hop,

given a per link energy constraint. Additionally, we propose

the suboptimum time and power allocation. Section IV shows

that the proposed scheme achieves the full spatial diversityof the system, and Section V depicts the numerical results.

Finally, Section VI summarizes the conclusions.

II . SYSTEM MODEL

We consider a multi-hop WSN with sensors grouped in

cooperative clusters. Clustering of sensor nodes is carried out

following the above mentioned distributed algorithms while

routing is performed by layer-3 hierarchical protocols. When

a sensor node has data to transmit, it first shares it within

its cluster. Next, all cluster nodes jointly transmit data as a

distributed antenna array to the neighboring cluster. Multi-hopcommunication is then carried out by concatenating this single

hop structure, in what we refer to as cluster-based multi-hop

protocol.



404 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 25, NO. 2, FEBRUARY 2007

A. Cluster-based Multi-hop Protocol

The proposed multi-hop communication models each clus-

ter as a multiple antenna device composed of a set of sin-

gle antenna cooperative sensors. Every hop is defined as a

distributed MIMO channel between two consecutive clusters,

with cooperative transmission and reception of data at the

transmitter and receiver cluster, respectively. To illustrate the

basic idea on the construction of a distributed MIMO linkbetween two clusters, we focus independently on the two

sides of the communication. We describe how nodes of the

transmitter cluster optimally share the data to transmit, how

they jointly transmit it and how the nodes of the receiver

cluster implement the reception protocol (see Fig. 2).

1) Transmitting Cluster: Let a transmitter cluster (TxC)

be composed of a set of N t cooperative sensor nodes, com-

municating with a receiver cluster (RxC) composed of a set

of N r cooperative sensor nodes. In order to transmit to the

neighboring cluster, the TxC implements two functions: 1)

broadcasting of data within the cluster, so that all active nodes

can decode the data to relay during the MIMO transmission(in general, the set of active nodes nt is a subset of the total

cluster nodes N t), and 2) the transmission of the data via

a nt × N r MIMO channel. Due to half duplex limitations,

both functionalities are carried out in two orthogonal channels,

assumed as time division (TD) channels. These two TD

channels are referred to as Intracluster (ITA) channel and

Intercluster (ITE) channel, used for broadcasting and MIMO

transmission, respectively. Transmissions are allocated into

two consecutive time slots: ITA slot and ITE slot.

• Intracluster (ITA) Slot : During this slot, the data to

transmit is broadcasted within the cluster with power p1during a fraction of time α. The set of nodes falling into

the broadcast capacity region decode data and cooperate

in the ITE slot (this set is henceforth called decoding

set ). Updated transmit CSI is considered at the source

of the broadcast (i.e., the source knows its channel with

all nodes of the cluster), as well as receive CSI at the

receiving nodes within the cluster. Notice that the number

of nodes belonging to the decoding set depends upon the

selection of α and p1 and, due to the transmit channel

knowledge, it is known and controlled (by properly

allocating resources) by the source.

• Intercluster (ITE) Slot : During the relay period 1 − α,

the subset of nt nodes (consisting of the source of

the broadcast plus the decoding set ) jointly transmit

data, with power p2, to the RxC. We assume: i) sym-

bol synchronization between cluster nodes and ii) no

intercluster channel knowledge at the transmitters. The

transmission scheme is based upon Gaussian DSTC with

the transmitted power per each active node equal to p2/nt

[31]. (As mentioned earlier, nt is known at the source of

the ITA slot, and is sent to the decoding set).

2) Receiving Cluster: During the 1 − α time interval of

the ITE slot, the RxC receives data through an nt × N r

MIMO channel. In order to obtain full receiver diversity,it runs a distributed multiple antenna decoding algorithm.

For that purpose, a reception protocol based upon Selection

Diversity (SD) over the N r parallel MISO channels (each of

nt transmitting antennas from TxC) at the RxC is adopted. In

other words, the sensor node of the RxC with highest signal-

to-noise ratio (SNR) becomes cluster coordinator, decodes

data, and broadcasts it during the ITA slot of the next hop.

As previously mentioned, we assume that received CSI for the

distributed MIMO channel is not complete. Every sensor node

has updated knowledge of its channel with the nt antennas of

TxC, but it is not aware of the individual receive CSI of theother nodes. To implement the Selection Diversity receiver

with such individual CSI, we make use of a distributed node

selection algorithm [36]. The sketch of the algorithm is as

follows: first, every node i in RxC measures its own received

SNRi and initializes a deterministic timer T i = 1

SNRi

to start

the broadcast of the ITA slot of next hop. When the timer has

run out to zero, node i starts transmitting the decoded data

within the cluster, unless another sensor has started before.

Thus, the node with min {T i}, (or equivalently max {SNRi}),

is the node selected for transmission in the broadcast, and thus

becomes the cluster coordinator. Hereafter, we assume that

the time spent in the node selection algorithm is negligiblecompared with the time scheduled for the cluster-to-cluster

hop (i.e., min {T i} << α).

Finally, since the power budget is the main constraint within

sensor networks, we assume that the energy consumption in

the cluster-to-cluster hop is limited to E t, thus

E t = αp1 + (1 − α) p2 . (1)

Notice that the transmission performance can be optimized

from a judicious choice of p1, p2 and α.

B. Signal Definition

Let a multihop communication be composed of M − 1hops connecting cluster 1 (source cluster) with cluster M (destination cluster) through clusters 2,...,M − 1 (routing

clusters). We consider each hop being split into ITA and ITE

Slots, where slot duration α and power allocation (p1, p2) are

independently designed for each cluster-to-cluster hop. In any

given cluster m, we assume that the total number of available

nodes when transmitting (N t) equals the total number of

available nodes when receiving (N r), i.e., N t = N r = N m.

The distance between the center of consecutive clusters

is dITE for any intercluster link m, and intracluster and

intercluster channels are modelled with a path loss exponent

δ and frequency flat Rayleigh fading. Taking into account all

the considerations above, the received signal at sensor κ of

cluster m + 1, during the ITE slot of hop m, corresponds to

the multiple-input-single-output (MISO) channel:

yκ,m+1 (t) = Γ · hT κ,m · xm (t) + nκ,m+1 (t) , (2)

where Γ = d−δ/2ITE is the intercluster path loss, hκ,m =

am1,κ, . . . , am

nt,κ

T the channel vector and xm (t) =

[x1,m (t) , . . . , xnt,m (t)]T the transmitted vector at time t ∈(α, 1]. am

i,κ ∼ CN (0, 1) is a unitary power, Rayleigh fading

coefficient between node i of cluster m and node κ of cluster m + 1. We assume invariant channels during the entire

frame duration and independent, identically distributed (i.i.d)

entries on the channel vector hκ,m ∼ CN (0, Int). Finally,




Fig. 2. Cluster-to-cluster transmission scheme.

nκ,m+1 (t) ∼ CN

0, σ2o

is additive white Gaussian noise

(AWGN) at sensor κ. Furthermore, considering a transmitted

space-time codeword of length1 s, the (s × 1) received vector

signal at sensor κ is:

yT κ,m+1 = Γ · hT κ,m · Xm + nT κ,m+1, (3)

being Xm = [xm (1) , . . . ,xm (s)] ∈ C nt×s the transmitted

codeword with RXm= 1

s · Xm · XH m = p2/nt · Int , and

nκ,m+1 ∈ C s×1.

Decoding at the receiver cluster is based on SelectionDiversity (SD) over the N m+1 MISO channels of the RxC.

Therefore, taking into account the model in (3), the instanta-

neous SNR at the SD receiver of cluster m + 1 is computed

as:

γ SDm+1 =η2nt

maxκ

|hκ,m|2, (4)

where η2.

=p2

σ2od−δITE

.

C. Problem Statement

Energy is the most limiting factor in WSN. Cooperative

diversity aims at reducing transmitted power while maintain-ing a reliability level for the link, determined by its outage

probability for a given outage capacity. In multihop networks,

the probability of outage of an M − 1 hop communication

P out is evaluated in terms of the probability of outage of every

single hop P mout as

P out = 1 −M −1m=1

(1 − P mout) . (5)

In our approach, considering that Gaussian DSTC are used

for transmission, and that instantaneous received power at the

1Further details on space-time code design are out of the scope of thispaper. Codeword length s can be considered as arbitrary, being s = 1−α

∆ ,with ∆ de symbol length.

Selection Diversity of cluster m + 1 is given by (4), the single

hop outage probability is:

P mout = Pr

η2nt

maxκ

|hκ,m|2 < 2Cout1−α − 1

=

N m+1κ=1

Pr

η2nt

|hκ,m|2 < 2Cout1−α − 1

, (6)

where C out (in bps/Hz) is the outage capacity, scaled by

1 − α according to the proposed TD system. The second

equality follows from the cumulative density function (cdf) of the maximum of i.i.d channels. Notice that, constraining the

per-hop energy as in (1), the single-hop outage probability

(6) depends upon the power allocated in the ITA slot (p1)

and in the ITE slot (p2), as well as on the time duration

(α). Therefore, the outage performance of the cooperative

distributed MIMO channel can be optimized as:

P mout = min

(η1,η2,α)

N m+1κ=1

Pr

η2nt

|hκ,m|2 < 2Cout1−α − 1

(7)

s.t. αη1 + (1 − α) η2 = SNR

where the constraint in (1) has been normalized as power at the

receiving cluster by defining SNR.

= E tσ2o

d−δITE

and ηi.

= piσ2o

d−δITE

for i ∈ {1, 2}.In the following sections, optimum time and power al-

location for each cluster-to-cluster transmission are derived.

Intuitively, we can observe the following tradeoff: as we allo-

cate more resources for the ITE transmission, less resources

are available for the ITA transmission. This in turn reduces

the number of nodes in the decoding set, which reduces the

diversity of ITE transmission.

III . OPTIMUM DESIGN FOR MINIMUM OUTAGE

PROBABILITY

The outage probability of the distributed MIMO channel

strongly depends on the power and time allocation for the




ITA and ITE Slots [33]. In this section we derive the resource

allocation that optimizes the proposed transmission scheme.

As mentioned earlier, we consider the outage probability P out

as the performance metric and we optimize independently each

cluster-to-cluster hop according to (7). First hop, intermediate

hops, and final hop require a separate analysis.

A. First Hop

The first hop connects cluster 1 (i.e., the cluster that contains

the source node) to cluster 2 and starts the communication.

1) ITA Slot: Let node 1 of cluster 1 be the source node of

the multihop communication; it uses the ITA slot to broadcast

its data to nodes {2, · · · , N 1} of cluster 1. During this time

slot, node i ∈ {2, · · · , N 1} is expected to correctly decode

the broadcast data (and thus to be able to cooperate during

the ITE Slot) if and only if the broadcast rate RBC is below

the node 1 to node i channel capacity:

RBC ≤ α log2 (1 + η1ξ i) , (8)

where ξ i = |a1,i|2

dITEd1,i

δ

is the source-relay path gain

(known at the source node). d1,i is the distance between source

and node i, and a1,i ∼ CN (0, 1) corresponds to the Rayleigh

fading coefficient (assumed invariant during the communica-

tion). Nevertheless, in degraded relay channels2 the source-

relay rate (i.e., the broadcast rate) cannot be lower than the

relay-destination rate (i.e., the MIMO rate) [9, Theorem 1] [13,

Theorem 1]. Therefore, being the intercluster communication

rate set to C out, the capacity region of node 1 to node i is

constrained to:

C out ≤ RBC . (9)

Thereby, node i is guaranteed to decode during the broadcast

slot and to cooperate during the MIMO transmission if and

only if

η1 ≥Ψ (C out, α)

ξ i, (10)

with Ψ (R, α).

= 2Rα − 1. Furthermore, by ordering the

instantaneous path gains for all receiver nodes as

ξ 2 ≥ ... ≥ ξ i ≥ ... ≥ ξ N 1 (11)

we derive the deterministic relationship3 between the number

nt of active nodes during the ITE Slot (i.e., source node plus

decoding set) and the pair (η1, α) as in (12) on the top of the

following page. Notice that nt = 1 means that only the source

transmits within the ITE slot, while for nt = n > 1 there are

n − 1 relays that cooperate to transmit during the ITE slot.

(We make use of slack variable ξ N 1+1 = 0).

2The proposed cooperative scheme is a degraded relay channel since thesignal at the receiver cluster is only statistically dependent upon the received

and transmitted signal at the relay nodes, and it does not depend upon thetransmitted signal by the source [9].3Notice that η1, α and ξn with n = 2, · · · ,N 1 are known values at

the transmitter node. Therefore, nt is also known, and controlled, at thetransmitter side following (12).

2) ITE Slot: In this interval, the nt nodes jointly transmit

data with transmission rate C out [bps/Hz] to destination

cluster 2. The outage probability of the distributed multiple

antenna link is given by (7), where the number of cooperating

transmitters nt follows (12) and |hκ,m|2 ∼ X 22nt(i.e., chi-

square distributed R.V. with 2nt degrees of freedom). Since

the cdf of the X 22ntis the regularized incomplete Gamma

function4

, i.e., F X 22nt (b) = γ (nt, b), the optimization (7) canbe rewritten as:

P 1out = min

(η1,η2,α)

γ

nt, Ψ(Cout,(1−α))η2/nt

N 2. (13)

s.t. αη1 + (1 − α) η2 = SNR

Moreover, since nt in (12) is constant over N 1 regions in

(η1, η2, α), the minimization of (13) can be carried out by

first minimizing the objective function on every region and

then selecting the minimum of the minima. Every region is

interpreted as the subset in (η1, η2, α) that makes the number

of active sensors during the ITE slot constant and equal to n.

Therefore, we may rewrite:

P 1out = min

1≤n≤N 1

min

(η1,η2,α)

γ

n, Ψ(Cout,(1−α))η2/n

N 2

(14)

s.t. αη1 + (1 − α) η2 = SNR

η1 ≥ Ψ(Cout,α)ξn

The second constraint in (14) follows from (12), where it is

shown that the link has n active transmitter nodes if and only

if η1 ≥ Ψ(Cout,α)

ξn(notice that we set ξ 1 = ∞). Maximization

on every region is carried out as in Appendix I. Thus, the

outage probability of the first link is obtained as:

P 1out = min

1≤n≤N 1

γ

n, n·Ψ(Cout,(1−αn))η2n

N 2(15)

=

γ

τ, τ ·Ψ(Cout,(1−ατ ))η2τ

N 2

being nt = τ the optimum number of cooperating (active)

nodes within the ITE slot and

αn = arg maxαon≤α≤1

ξ n · SNR − αΨ (C out, α)(1 − α) Ψ (C out, (1 − α))

η1n =Ψ (C out, αn)

ξ nη2n =

SNR − αnη1n

1 − αn, (16)

where αon is described in (I-3). Therefore, the optimum power

and time allocation for the first hop is:

α = ατ η1 = η1τ η2 = η2τ . (17)

Notice here that for n = 1 (i.e., just the source of the broadcast

is active during the ITE slot) we made use of slack variable

ξ 1 = ∞, therefore (η11, η21, α1) = (0, SNR, 0).

4The regularized incomplete Gamma function is defined as γ (nt,b) =1

(nt−1)!· b0xnt−1e−xdx




nt = 1 η1 < Ψ(Cout,α)ξ2

nt = n Ψ(Cout,α)ξn

≤ η1 < Ψ(Cout,α)ξn+1

, for 2 ≤ n ≤ N 1 . (12)

B. Intermediate Hops

The cooperative strategy for an intermediate hops is slightly

different than for the first hop. In an intermediate (routing)

hop m, all nodes belonging to the transmitter cluster have

previously received a copy of data through the ITE slot of hop

m − 1. Every sensor node has received the data with different

instantaneous power and, according to the Selection Diversity

algorithm, at least the sensor with the most favourable channel

condition has fully decoded the message (otherwise the com-

munication would be in outage). This decoder sensor uses

the ITA Slot of hop m to broadcast the decoded data within

the transmitter cluster. Nevertheless, since all cluster nodes

already have a degraded copy of the data, the broadcasting

should only provide the differential of mutual information that

allows them to decode the codeword free of errors. We define

this communication as a differential broadcast channel (DBC).1) ITA Slot: In this differential broadcast channel, we

assume that node 1 of cluster m is the decoder sensor (i.e.,

the coordinating node, selected to decode in hop m − 1according to the Selection Diversity algorithm) and nodes

{2, · · · , N m} of cluster m are the receiver nodes. Every

node i ∈ {2, · · · , N m} has previously received a copy of

the data (during hop m − 1) with spectral efficiency I i =1s I (yi,m;Xm−1) (being I (·; ·) the mutual information). We

assume that I i is known at the source node via feedback.

During the ITA slot, the coordinating node broadcasts withrate RDBC and, as previously, node i ∈ {2, · · · , N m} is

expected to correctly decode data at this rate if and only if:

RDBC ≤ α log2 (1 + η1ξ i) + I i , (18)

being ξ i the path gain between coordinating node and sensor i.

Notice that sensor i accumulates the received mutual informa-

tion during current hop, i.e., α log2 (1 + η1ξ i), and previous

hop, i.e, I i. However, to guarantee that node i decodes data

and retransmits during the ITE Slot, its decoding rate RDBC

cannot be lower than the relay-destination rate [9, Theorem 1]

[13, Theorem 1]:

C out ≤ RDBC . (19)

Therefore, by defining C i = max {0, C out − I i}, a node ibelongs to the decoding set only if:

η1 ≥Ψ (C i, α)

ξ i. (20)

Now, considering that there are N m − 1 receiver nodes of the

DBC channel that for convenience are ordered as

Ψ(C2,1)ξ2

≤ ... ≤ Ψ(Ci,1)ξi

≤ ... ≤Ψ(CN m ,1)

ξN m

, (21)

then, we may fairly approximate the deterministic relationship

between the number of active nodes nt during the ITE slot

and η1 and α as in (22) on the top of the next page.

2) ITE Slot: The analysis of the nt × N m+1 MIMO

transmission between cluster m and m + 1 is equivalent tothe analysis carried out for the first hop. Nevertheless, here

nt depends upon η1 and α according to (22). Therefore, by

adapting the optimization (14), the outage probability remains:

P mout = min

1≤n≤N m

min

(η1,η2,α)

γ

n,Ψ(Cout,(1−α))

η2/n

N m+1

(23)

s.t. αη1 + (1 − α) η2 = SNR

η1 ≥ Ψ(Cn,α)ξn

with C 1 = 0. Similarly to optimization for the first hop,

and using results on the Appendix I, we derive the outageprobability of m-th link:

P mout = min

1≤n≤N m

γ

n, n·Ψ(Cout,(1−αn))η2n

N m+1

(24)

=

γ

τ, τ ·Ψ(Cout,(1−ατ ))η2τ

N m+1

with nt = τ being the optimum number of active nodes in the

transmitter cluster and


ξ n · SNR − αΨ (C n, α)

(1 − α) Ψ (C out, (1 − α))

η1n =Ψ (C n, αn)

ξ nη2n =

SNR − αnη1n

1 − αn, (25)

where αon comes from (I-3). Finally, the optimum time and

power allocation for the intermediate hop is:

α = ατ η1 = η1τ η2 = η2τ . (26)

Considerations over n = 1 are the same as in the first hop.

C. Final Hop

The optimization of the final hop is similar to that for the intermediate hops. Nevertheless, in this hop the receiving

cluster contains the destination node, which introduces one

modification in the reception protocol. As before, we assume

that the destination cluster receives the data through an nt ×N M MIMO channel and decodes it according to the Selection

Diversity criterion previously described. Therefore, if the link

is not in outage, the sensor node with highest instantaneous

SNR decodes the data addressed to the destination node.

Once accomplished this step, the modification is introduced:

the decoder sensor forwards the data to the destination node

through a dedicated differential broadcast channel. In our

results, we assume that power and time used in this final step isnegligible, compared with the total time and energy allocated

for the cluster-to-cluster communication; thus it is not taken

into account on computations.




nt = 1 η1 < Ψ(C2,α)ξ2

nt = n Ψ(Cn,α)ξn

≤ η1 < Ψ(Cn+1,α)ξn+1

, for 2 ≤ n ≤ N m. (22)

D. Suboptimum Resource Allocation

Optimum time allocation αn, as derived previously, has no

closed form expression. Additionally, the optimization may

be computationally intensive. In this subsection, we propose a

closed form for suboptimal time allocation with a negligible

performance loss. To do so, we use the results in Appendix II,

where high SNR approximations are derived for the optimum

resource allocation. Such approximations yield the following

suboptimal allocation when used in the low SNR regime:

αson =

R ln (2)

W0 1e SNRξn−K K + ln (2)

ηso1n =

Ψ (R, αson )

ξ nηso2n =

SNR − αson ηso

1n

1 − αson

, (27)

with R = C out for the first hop and R = C n for the rest

of hops; K = 2Cout−12Cout(ln(2)Cout−1)+1

, and W0 (κ) is defined

as the branch 0 of the Lambert W function evaluated at

κ [37]. By definition, the proposed suboptimum allocation

(27) asymptotically converges (in the high SNR regime) to

optimum solutions in (16) and (25).

IV. SPATIAL DIVERSITY OF THE DISTRIBUTED MIMO

LIN K

In previous section we derived optimum and suboptimum

resource (time and power) allocation for the cluster-to-cluster

links. It was shown that the per-hop minimum outage prob-

ability is obtained by searching for the optimum number

of cooperating nodes nt during the ITE Slot. Notice that

the optimum set of active nodes can be a subset of the

total number of antennas available for cooperation at the

transmitter cluster (i.e., nt ≤ N t). In this section we show

that, under the proposed cluster-to-cluster cooperative scheme,

the spatial diversity of the link converges to the product of

the total number of antennas available for cooperation at both

transmitter and receiver clusters.The spatial diversity of the mth cluster-to-cluster link is

defined as [38, Definition 1]:

dm = − limSNR→∞

log P mout (SNR)

log SNR(28)

where P mout (SNR) is defined in (24) for m > 1 and in (15)

for m = 1. The definition above, when applied to classical

MIMO channels, results in a deterministic value. However,

when applied to distributed MIMO links the probability of

outage P mout (SNR) is a random variable that depends upon

the source-relay random (but known at the transmitter side)

channels {ξ 2, . . . , ξ i, . . . ξ N m}. Therefore, dm is also a random

variable, defined as the limit of a random sequence. The

behaviour of this limit is established by the following theorem.

Theorem 1: The spatial diversity of a cooperative dis-

tributed MIMO channel converges almost surely to the productof the total number of antennas at the transmitter cluster and

at the receiver cluster, i.e.,

d1 = − limSNR→∞

log P 1out (SNR)

log SNR

a.s.→ N 1 · N 2, (29)

where transmit diversity is exploited via time-division multi-

plexing of a broadcast channel and a distributed space-time

coded MIMO channel, and receive diversity via the Selection

Diversity criterion.

Proof : see Appendix III-A.

Corollary 1: The previous theorem also applies if thebroadcast channel is replaced by a differential broadcast

channel:


log P mout (SNR)

log SNR

a.s.→ N m · N m+1 (30)

Proof : see Appendix III-B.

Corollary 2: The end-to-end diversity gain d of the multi-

hop transmission converges almost surely to the minimum of

the single hop diversity gains, i.e. as shown in (31) on the top

of the following page.

Proof : Using Theorem 1 and Corollary 1 with results in

[39, Section IV-B-2], we derive (31).

V. SIMULATION RESULTS

The outage probability of the multi-hop cooperative WSN is

evaluated here, following results derived in previous sections.

The simulation setup is as follows: first, all cooperative

clusters have the same number of sensor nodes, located within

circles of radius RITA. Cluster nodes are randomly placed

within each cluster, according to a uniform distribution over

the circle. Furthermore, the distances covered by all hops of

the multi-hop transmission are equal and normalized so that

dITE = 1. Finally, fading coefficients among all network nodesare modelled as unitary power, complex Gaussian random

variables, and the path loss exponent is set to δ = 3 for both

intercluster and intracluster propagation.

The outage probability of the multi-hop transmission is

computed from the single hop outage probability as in (5).

Cluster-to-cluster outage probability is derived in (15) for

the first hop and in (24) for all other hops. However, as

previously mentioned, cluster-to-cluster outage probability is

a random variable that depends upon source-relay channels

in the transmitter cluster (notice that intracluster channels

are random, but known, at the transmitter cluster). Results

are obtained for the mean outage probability, averaged over the intracluster channels. Fig. 3 shows the (mean) outage

probability of the first cluster-to-cluster hop, with three sensor

nodes per cluster and for C out = 1.4 bps/Hz. Results for




d = − limSNR→∞

log P out (SNR)

log SNR

a.s.→ min

(1≤m≤M −1)N m · N m+1 (31)

0 5 10 15 2010

10

10

10

10

10

100

SNR(dB)

P o u t

1

Non Cooperative single HopDist. 3x3 MIMO R

ITA/d

ITE= 0.1

Dist. 3x3 MIMO RITA

/dITE

= 0.3

Dist. 3x3 MIMO RITA

/dITE

= 0.5

D

Fig. 3. Single cluster-to-cluster outage probability vs. SNR for differentRITAdITE

ratios. N 1 = 3, N 2 = 3 and Cout = 1.4 bps/Hz. The performances

of a non-cooperative link and a Space-Time Coded 3×3 MIMO channel withSelection Diversity at the receiver side are depicted for comparison.

different RITA

dITEratios are shown. For comparison, the outage

performance of a space-time coded 3 × 3 MIMO system with

Selection Diversity at the receiver side and no CSI at the

transmitter is also plotted. From the figure it can be seen thatthe system achieves full spatial diversity of the link: (N 1 · N 2).

However, there is a constant SNR loss between conventional

and distributed MIMO, corresponding to the fraction of power

and time used to broadcast data in the ITA Slot. Those losses

are, of course, greater when increasing the cluster radius RITA

with respect to the intercluster distance dITE. Moreover, notice

that the distributed MIMO channel outperforms the conven-

tional MIMO channel in the low SNR regime (∼ [−5, 2] dB).

This is explained due to the channel hardening effect analyzed

in [40]–[42]: in those works, authors claim that it is not always

worthwhile to increase the number of transmitting antennas

in space-time coded systems with Selection Diversity at the

receiver end (i.e., systems serving the best channel). Notice

that the conventional MIMO system has a fixed number of

active antennas N t, while the distributed MIMO system selects

the optimum number of transmit antennas nt = τ ≤ N t from

optimization (14). Therefore, we conclude that in the low SNR

regime the channel hardening effect causes the transmitter with

N t antennas to underperform τ -antenna transmitter.

Fig. 4 shows the performance of a multi-hop communication

for different number of total hops. Again, three sensor nodes

per cluster are considered (N m = 3), and the cluster radius

is set to RITA

dITE= 0.3 for all clusters. The non-cooperative

multi-hop case is also plot as reference. In the figure, themulti-hop outage probability of the non-cooperative system is

kept constant for all number of total hops, as we consider that

the network designer fixes the end-to-end outage probability

0 5 10 15 2010

10

10

10

10

10

100

P o u t

Π m = 1

H O P S ( 1 P

o u t

m

)

SNR x Hops (dB)

Non Cooperative Multihop; Hops = {1,2,3,4} Dist. 3×3 MIMO; Hops = 1

Dist. 3×3 MIMO; Hops = 2



Fig. 4. Multihop outage probability vs. overall SNR, varying the number

of cluster-to-cluster hops and assuming 3 nodes per cluster.RITAdITE

= 0.3.

For any given number of hops, the outage capacity of cooperative and non-cooperative systems are equal.

independently of the number of hops in between. Moreover,

overall power consumption (i.e., per hop SNR× Number

of hops) is also kept constant. Thus, when increasing the

number of hops, the outage capacity decreases. For any given

number of hops, the outage capacity of cooperative and non-

cooperative system are equal. Numerical analysis shows a

substantial advantage of the distributed MIMO system with

respect to the non-cooperative multi-hop system. Moreover,

when increasing the number of hops (i.e., when decreasing

the outage capacity) the energy savings also increase.

Fig. 5 compares the optimum and suboptimum resource

allocations (proposed in section III-D) for the first cluster-

to-cluster hop. We consider four sensor nodes per cluster,

C out = 1.4 bps/Hz and we obtain results for two cluster

radii, RITA

dITE= 0.1 and RITA

dITE= 0.2. Simulations show

that the performance of resource allocation based upon thelarge SNR approximation is close to that of the optimum

solution, which is due to the asymptotic convergence of

both solutions. Additionally, we also note that, in the low

SNR regime, they have the same performance. This can

be explained noting that, for very low SNR, the transmit

diversity degenerates into a single source transmission (i.e.,

the optimum number of transmitters in (15) is nt = τ = 1due to channel hardening) and therefore, no time or power

allocation is necessary. Indeed, in the low SNR regime, the

system benefits from receiver diversity only, which is equal

for both solutions. Finally, Fig. 6 shows the energy savings of

the first cluster-to-cluster hop over a single non-cooperativehop for different number of sensor nodes per cluster. These are

measured for a link working at C out = 1.4 bps/Hz and P 1out =

10−3, and for a WSN with a constant density of sensors




0 1 2 3 4 5 6 7 8 9 1010

10

10

10

10

10

100

SNR(dB)

P o u t

1

Non Cooperative Single HopDist. 4×4 MIMO; Optimum Resource Allocation R

ITA/d

ITE=0.1

Dist. 4×4 MIMO; Optimum Resource Allocation RITA

/dITE

=0.2

Dist. 4×4 MIMO; Suboptimum Resource Allocation RITA

/dITE

=0.1

Dist. 4×4 MIMO; Suboptimum Resource Allocation RITA

/dITE

=0.2

4× D

Fig. 5. Single cluster-to-cluster outage probability vs. SNR, for RITAdITE

= 0.1

andRITAdITE

= 0.2. Optimum (solid line with marker) and suboptimum (dash-

dotted line with marker) resource allocation are plotted, considering N 1 = 4,N 2 = 4 and Cout = 1.4 bps/Hz. Space-time Coded 4 × 4 MIMO channelwith Selection Diversity at the receiver side is shown as comparison.

ρ

sensors

d2ITE

. In the non-cooperative case, to communicate

with the fixed pair

C out, P 1out

, a mean received signal-to-

noise ratio SNRnc (dB) is needed. However, for the cluster

approach, the necessary SNR to support the same rate and

probability of outage is SNRco (dB). Fig. 6 shows the relative

saving

∆E (%) = 100 ·SNRnc − SNRco

SNRnc. (32)

In the simulation, assuming a number of sensor nodes per

cluster N , the radius of the cluster is computed from:

ρ =N

π · R2ITA

→ RITA =

N

ρ · π. (33)

The relative energy savings for different values of ρ are

plotted. For comparison, the relative energy gain obtained

when using an N × N space-time coded MIMO channel

with Selection Diversity is also considered. Results show

that considerable energy savings are obtained when clusteringnodes in cooperative groups. Moreover, it is shown that gains

saturate when increasing the number of nodes per cluster, i.e.,

it is not worthwhile to increase indefinitely the number of

nodes per cluster, since with only 5 cooperating sensors the

energy savings of the system is 80% − 90%. Additionally,

we may note that cooperative clustering not only performs

very well for highly populated WSN (ρ = 1000) but also

for sparsely populated ones (ρ = 10). Nevertheless, as sensor

density increases, the performance of the distributed MIMO

system approaches that of the space-time coded MIMO system

with Selection Diversity.

VI. CONCLUSIONS

In this paper we proposed a clustered topology to introduce

cooperative diversity in multi-hop wireless sensor networks

2 4 6 8 10 12 140

10

20

30

40

50

60

70

80

90

100

Number of Sensors per Cluster

E n e r g y S a v i

n g s ; ∆ E ( % )

Dist. MIMO;ρ = 10

Dist. MIMO;ρ = 100

Dist. MIMO;ρ = 1000

N× D

Fig. 6. Energy savings (in %) with respect to a non-cooperative system, for a single cluster-to-cluster link with P

1out = 10−3 and Cout = 1.4 bps/Hz

vs. the number of sensor nodes per cluster. Results for different densities of

relays ρsensors

d2ITE

are plotted. Space-time Coded N ×N MIMO channel

with Selection Diversity at the receiver side is shown as reference.

(WSN). We defined multi-hop transmission as the concate-

nation of single cluster-to-cluster hops, where cooperative

transmission and reception of data are enabled among cluster

nodes in what we referred to as a Cooperative Distributed

MIMO channel. We proposed a time-division relaying scheme

to exploit transmit diversity, based upon two slots: the ITA slot,

that accounts for data sharing among cluster nodes, and the

ITE slot that allows for joint transmission of data to the neigh-

bor cluster using distributed space-time codes. At the receivingcluster a distributed multiple antenna reception protocol is

devised based upon a Selection Diversity technique. The

proposed multi-hop WSN have been optimally designed for

minimum end-to-end outage probability by properly allocating

time and power resources, independently on every single

cluster-to-cluster link. A closed form, suboptimum resource

allocation was also obtained. Both optimum and suboptimum

allocation were compared, showing very small performance

losses of the proposed suboptimum approach. Numerical anal-

ysis over the proposed cooperative WSN showed that: 1) full

transmit-receive (i.e., N t × N r) diversity is obtained on every

cluster-to-cluster link with small SNR losses, allowing for significant energy savings, 2) performance degrades as the

cluster size increases with respect to the hop length, and

3) when fixing end-to-end outage probability, energy savings

increase for growing number of hops.

APPENDIX I

OPTIMIZATION PROBLEM

In this Appendix, we analyze the minimization problem:

P = min(η1,η2,α)

γ

⎛⎝

n,n·

2Cout1−α−1

η2

⎞⎠

(I-1.1)

s.t. αη1 + (1 − α) η2 = SNR (I-1.2)

η1 ≥2Rα − 1

ξ (I-1.3)




αη1 ≤ SNR (I-1.4)

0 ≤ α ≤ 1 (I-1.5)

where n,R, C out and ξ are fixed constants, and γ (n, b) =1

(n−1)! · b0 xn−1e−xdx, the regularized incomplete Gamma

function. We consider R = C out for the first hop and R = C nfor the rest of hops. The first two constraints (I-1.2) and (I-

1.3) are explicit constraints. The others, (I-1.4) and (I-1.5),are implicit constraints, forcing η2 to be positive and α to be

positive and not greater than one, respectively.

The first step in the optimization is the analysis of the

feasible set: constraint (I-1.3) establishes that η1 has to be

at least 2Rα−1

ξ , while constraint (I-1.4) forces the product αη1to be lower than SNR. Therefore, all α for which:

α ·2Rα − 1

ξ > SNR (I-2)

do not belong to the feasible set. Thereby, since:

α ·2Rα−1

ξ ≤ SNR ⇔α ≥ αo = − ln(2)·R

W−1

− ln(2)·RSNR·ξ ·e

− ln(2)·RSNR·ξ

+ ln(2)·R

SNR·ξ

(I-3)

only α ≥ αo has to be considered in the minimization.

W−1 (κ) is defined as the branch −1 of the Lambert W

function [37].

Second step is the concatenation of the minimization pro-

cess:

P = minαo≤α≤1

⎧⎨⎩

min(η1,η2)

γ

⎛

⎝n,

n ·

2Cout1−α − 1

η2

⎞

⎠

⎫⎬⎭

(I-4)

s.t. αη1 + (1 − α) η2 = SNR

η1 ≥ 2Rα−1

ξ

αη1 ≤ SNR

From (I-4), can be seen that (for fixed α) the goal function

is a decreasing function with η2 (in the feasible set) and

independent of η1. Then the minimum is given at the point

where η2 is maximum, and due to constraint (I-1.2), where η1is minimum:

η∗1 =2Rα − 1

ξ → η∗2 =

SNR − α · η∗11 − α

. (I-5)

Therefore, minimization in (I-4) reduces to:

P = minαo≤α≤1

γ

⎛⎝n, n · ξ ·

(1−α)·

2Cout1−α −1

ξ·SNR−α·

2Rα −1

⎞⎠ (I-6)

Moreover, taking into account that the regularized incomplete

Gamma function satisfies that the minimum over b of γ (n, b)is given for the minimum value of b, the optimum time

allocation α∗ will be:

α∗ = arg maxαo≤α≤1

ξ · SNR − α ·

2Rα − 1

(1 − α) · 2

Cout1−α − 1 (I-7)

Now, defining f (R, α) = α ·

2Rα − 1

with R ∈ R+

constant, and noting that: i) f (R, α) ≥ 0 and f (R, α) ≥ 0,

and ii) for the feasible set, SNR·ξ ≥ f (R, α) due to constraint

(I-1.4). Then (by computing the second derivative) it is readily

shown that maximization in (I-7) is a convex optimization

problem and therefore α∗ exists and may be found using

numerical methods.

APPENDIX II

HIG H SNR APPROXIMATIONS

In the Appendix, we obtain high SNR approximations for

the optimum time and power allocation derived in previous

sections. We consider first the time allocation.

A. Time Allocation

Defining P (α) = SNRξ n − α

2Rα − 1

and Q (α) =

(1 − α)

2Cout1−α − 1

, we rewrite the optimum time allocation

in (16) and (25) as:


P (α)

Q (α) , (II-1)

with R = C out for the first hop and R = C n for the rest of

hops. As previously stated, maximization in (II-1) is convex.

Therefore, the maximum will be given at the point ddα

P (α)Q(α) =

0, and thus the optimum time allocation satisfies:

P (αn)

P (αn)=

Q (αn)

Q (αn)(II-2)

which comes out directly when making the first derivative

of the quotient equal to zero. Furthermore, making the as-

sumption that the optimum time allocation αn is sufficiently

small when SNR → ∞, the right hand side of the equationis approximated by the quotient of the evaluation of Q and

Q

in 0. As we show later, αn → 0 when the SNR tends to

∞, validating the assumption made. Then, rewriting (II-2) we

obtain:

P (αn)

P (αn)≈

Q (0)

Q (0), (II-3)

which can be evaluated Q (0) /Q

(0) =2Cout−1

2Cout (ln(2)Cout−1)+1= K . Moreover, noting that

P

(α) = 2Rα ln(2) R

α − 1 + 1, we may extend (II-3)

as:

(SNRξ n − K ) + αn ≈ 2Rαn

ln (2) K

R

αn− K + αn

,(II-4)

where, considering again that αn is sufficiently small, we may

rewrite:

(SNRξ n − K ) ≈ K · 2Rαn

ln (2)

R

αn− 1

, (II-5)

approximating the optimum time allocation, when solving over

αn:

αn ≈R ln(2)

W0

1e

SNRξn−K K

+ ln (2)

, (II-6)




where W0 (β ) is the branch 0 of the Lambert W function,

i.e., the function that satisfies W 0 (β ) eW 0(β) = β [37]. Two

remarks should be done over the obtained approximation: i) it

is shown that αn in (II-6) tends to 0 for high SNR, validating

the assumption made, ii) the solution is only valid (obviously)

for ξ n = 0.

B. Power Allocation

To approximate the power allocation during the ITE slot,

we rewrite η2n in (16) and (25) as:

η2n =SNR

1 − αn

1 +

αnη1n

SNR

, (II-7)

and we analyze αnη1nSNR

when SNR → ∞. We remind that,

from (I-5), η1n = 2Rαn −1

ξn, with αn approximated in (II-6).

Therefore, defining β = 1e

SNRξn−K K , we compute the limit

as (II-8) below. Furthermore, noting that for high SNR we may

state that W 0 (β ) ln(2) and that eW 0(β) 1, the limit in(II-8) simplifies to:

limSNR→∞

αnη1n

SNR= lim

SNR→∞

R ln(2)

ξ n

eW 0(β)

SNRW 0 (β )

= limSNR→∞

R ln(2)

ξ n

e2W 0(β)

SNRW 0 (β ) eW 0(β)

= limSNR→∞

R ln(2)

ξ n

e2W 0(β)

SNRβ

W 0 (β )

W 0 (β )

2

= lim

SNR→∞

R ln(2)

ξ n

β

SNR

1

W 0 (β )2

(II-9)

where second equality follows from the multiplication up-

and-down by eW 0(β), third equality from the definition of

the Lambert W function and from the multiplication of the

limit by

W 0(β)W 0(β)

2, and fourth again from the Lambert W

function definition. Now, considering that β , as defined before,

is proportional to SNR, and that limSNR→∞W 0 (β ) = ∞, the

limit remains:

limSNR→∞

αnη1n

SNR= 0 (II-10)

Therefore, from (II-7), at high SNR regime the power allocatedduring the ITE slot is:

η2n ≈SNR

1 − αn. (II-11)

APPENDIX II I

PROOF OF THEOREM 1 AND COROLLARY 1

A. Theorem 1

We consider the first hop. Demonstration of the Theorem

1 follows two steps: 1) we show that, assuming ξ N 1 = 0 (i.e,

each relay may be reached from the source), the diversity d1

has sure convergence to N 1 · N 2, and 2) showing that the

probability P [ξ N 1 = 0] = 0, then almost surely convergence

is demonstrated.

Let us assume that ξ N 1 = 0 and make use of results on

Appendix II. First, for high SNR regime, the time allocation

in (16) is approximated as (II-6), being R = C out. Further-

more, the power allocated during the ITE slot is computed

with (II-11), and the regularized incomplete Gamma function

approximated as γ (n, b) ≈ 1n! · bn when b → 0. Therefore,

at the high SNR regime, we may approximate regularized

incomplete Gamma functions in (15) as:

γ

n,

n · Ψ (C out, (1 − αn))

η2n

≈

1

n!

2

Cout1−αn − 1

n

SNR/n1−αn

n (III-1)

Moreover, noting that for SNR → ∞ and assuming ξ n = 0,

αn in (II-6) has sure convergence to zero:

limSNR→∞

αn = 0 for n ∈ [1, N 1] , (III-2)

then, we derive from (15), and making use of (III-1) and (III-

2), the equalities shown on the top of the following page.

Reducing the spatial diversity derivation in (28) to:

d1 = − limSNR→∞

log P 1out (SNR)

log SNR= N 1 · N 2 . (III-3)

This shows that, assuming ξ N 1 = 0, the spatial diversity of

the proposed distributed MIMO channel sure converges to the

multiplication of the total number of transmit antennas at the

transmitter cluster and the total number of receive antennas at

the receiver cluster. Therefore, considering that for Rayleigh

distributed channels the probability P [ξ N 1 = 0] = 0, this

concludes the proof.

B. Corollary 1

From Appendix II, the time allocation for the intermediate

hops is approximated, at the high SNR regime, again by (II-

6), with R = C n < C out. Therefore, equations (III-1)- (III-3)

hold, deriving the diversity order of the intermediate hop m(with almost sure convergence) as:


log P mout (SNR)

log SNR

a.s.→ N m · N m+1 (III-4)

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Aitor del Coso was born in Madrid, Spain, in 1980.He received M.Sc. degree in electrical engineeringfrom Universidad Politecnica de Madrid (UPM),Madrid, in 2003, and he is currently with the Cen-tre Tecnologic de Telecomunicacions de Catalunya(CTTC), Barcelona, Spain, pursuing his Ph.D. onSignal Theory and Communications at the Univer-sitat Politecnica de Catalunya (UPC), Barcelona.

Prior to joining the CTTC, he was a researchassistant at the Microwave and Radar Group of theUPM, and junior research staff at Indra Sistemas,

Madrid, working on signal processing algorithms for remote sensing systems.

During Fall 2005 he was a visiting research assistant at the Politecnico diMilano, Milano, Italy, where he worked on cooperative schemes for wirelesssensor networks. Later, in Spring 2006, he visited the Center for Commu-nications and Signal Processing Research at the New Jersey Institute of Technology (NJIT), Newark, NJ, where he carried out research on cooperativemulticasting schemes.

He has been granted by the Generalitat de Catalunya and by the CentreTecnologic de Telecomunicacions de Catalunya PhD Fellowship Programs.His current research interests are wireless cooperative communications, multi-terminal information theory, MIMO channels, convex optimization and oppor-tunistic approaches.

Umberto Spagnolini (SM’03) received theDott.Ing. Elettronica degree (cum laude) fromPolitenico di Milano, Milan, Italy, in 1988. Since1988, he has been with the Dipartimento diElettronica e Informazione, Politecnico di Milano,where he is Full Professor in Telecommunications.His general interests are in the area of statisticalsignal processing. The specific areas of interestinclude channel estimation and space-timeprocessing for wireless communication systems,parameter estimation and tracking, signal processing

and wavefield interpolation applied to UWB radar, geophysics, and remotesensing. Dr. Spagnolini serves as an Associate Editor for the IEEETransactions on Geoscience and Remote Sensing.

Christian Ibars received degrees in electrical engi-neering from Universitat Politecnica de Catalunya,Barcelona, Spain, and Politecnico di Torino, Torino,Italy, in 1999, and a Ph. D. degree in electrical engi-neering from the New Jersey Institute of Technology,Newark, NJ, in 2003. During 2000, he was a visitingstudent at Stanford University. Since 2003 he hasbeen with the Centre Tecnologic de Telecomuni-cacions de Catalunya, Castelldefels, Spain, in thearea of Access Technologies. His current researchinterests include signal processing, channel coding,

and access protocols for wireless multiuser communications.

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