ad hoc wireless networks (a communication-theoretic perspective) || a communication-theoretic...

Chapter 2

A Communication-TheoreticFramework for Multi-hop Ad HocWireless Networks: Ideal Scenario

2.1 IntroductionAn ad hoc wireless network can be described as a collection of nodes, which have to interactamong themselves without any centralized authority [34]. The fact that (i) all nodes are atthe same hierarchical level and (ii) each of them has finite energy makes the design of anad hoc wireless network a challenging task. In particular, a minimum performance level hasto be guaranteed, and a meaningful definition of performance (that takes into account multiplecriteria) is difficult to find. In fact, many factors have to be considered, and the design mustsimultaneously involve multiple layers of the protocol stack, from physical to applicationlayer [2].

In the literature, a lot of attention has been devoted to multi-hop wireless networks.However, the concept of ad hoc wireless networks has gained a lot of popularity inthe last few years, since new communication scenarios, such as sensor networks, areemerging [34–36]. A lot of research activity has focused on routing protocols [4, 9]. Thishas probably been induced by the experience and knowledge gained with fixed wirednetworks, where the communication links between nodes are very reliable, and routing canbe studied independently of the physical layer characteristics. However, ad hoc wirelessnetworks do not admit this basic assumption, and thus it may not be possible to apply manyof the results derived in the literature for fixed networks. One of the major concerns inad hoc wireless networks is the fact that the energy at each node is limited [7, 9, 37–39].Specific medium access control (MAC) protocols, suitable for ad hoc wireless networkcommunication scenarios, also need to be studied. The relationship and interaction betweenMAC and physical layers will be the focus of Chapter 3.

The aim of this chapter (together with Chapter 3) is to provide a simple yet powerfulcommunication-theoretic framework for ad hoc wireless networks. In this first part, weconsider a wireless network communication scenario without inter-node interference (INI).Even if this represents an ‘idealized’ scenario, nonetheless it allows one to gain significantinsights, which are refined in Chapter 3 in order to account for INI and the particular MAC

Ad Hoc Wireless Networks: A Communication-Theoretic Perspective Ozan K. Tonguz and Gianluigi Ferrari© 2006 John Wiley & Sons, Ltd. ISBN: 0-470-09110-X

16 Chapter 2. A Communication-Theoretic Framework: Ideal Scenario

protocol. To that end, we present a logical approach for the derivation of a simple, yetmeaningful, expression for the bit error rate (BER) at the end of a multi-hop route in anad hoc wireless network. While binary modulation is considered in this chapter, the proposedapproach can be extended to higher-order modulations [40]. We consider both the cases ofuncoded and coded transmission. In this chapter, we consider a regular spatial distribution ofthe nodes to gain some fundamental insights into the problem.

The remainder of this chapter is organized as follows. In section 2.2, preliminaries,in terms of topology, route discovery and the average number of hops in a route, arepresented. In section 2.3, we provide the reader with simple communication-theoretic basics:in particular, we derive a simple expression for the BER at the end of a route and we derive asimple expression for the link signal-to-noise ratio (SNR). In section 2.4, we analyze the BERperformance of an ad hoc wireless networking scenario characterized by binary transmissionover an additive white Gaussian noise (AWGN) channel with free-space loss and without INI.Based on the insight gained in terms of BER performance, in section 2.5 we characterize thenetwork behavior from a communication-theoretic perspective. In particular: (i) we introduce(and quantify) the minimum spatial energy density necessary to guarantee connectivity; (ii)we introduce the concept of the average sustainable number of hops as a simple indicator ofconnectivity; and (iii) we derive a simple estimate of the lifetime of a node, in terms of majornetwork parameters. Concluding remarks are given in section 2.6.

2.2 Preliminaries

2.2.1 Topology

In general, no specific assumption should be made regarding the topology of a multi-hopad hoc wireless network. In fact, the nodes could be placed randomly inside the networksurface, as shown in Figure 2.1 (a). In this chapter, however, we focus on an ad hoc wirelessnetworking scenario with regular topology, which can provide valuable insights into the mainsystem parameters that affect the performance of ad hoc wireless networks. Moreover, weassume that the nodes are static. The proposed communication-theoretic framework can beextended to account for random topology (see [41]) and/or node mobility (see Chapter 6).

We assume that N nodes are placed inside a planar surface of area A and are regularlydistributed at the vertices of a square lattice: this scenario is shown in Figure 2.1 (b).We denote the distance between any pair of neighboring nodes (i.e. a link length) as dlink.We define by ρS � N/A the node spatial density of the network. Considering the regularlattice distribution in Figure 2.1 (b), each node has four neighboring nodes at distance dlink,while all other nodes are at a larger distance. For simplicity, in Figure 2.1 we assume that thenetwork surface is circular, but the qualitative discussion that follows is not necessarily basedon this premise. In fact, assuming that the square ‘tiles’ of side dlink (one of them is shownin Figure 2.1 (b)) cover more or less the entire network surface (neglecting border effects), itfollows that, for each surface shape, one can write

dlink = �

(1√ρS

)(2.1)

2.2. Preliminaries 17

��

��

S1

S2

D2

D1

S3

D3

S2

D2

S3

D3

S1

D1

(a) (b)

dlink

side dlink

Tile of

Figure 2.1 Possible topologies: (a) random and (b) regular lattice. In each case, possiblemulti-hop routes are shown.

where the notation y = �(x) indicates that y is around x, i.e. there exists ε1, ε2 > 0 suchthat x − ε1 ≤ y ≤ x + ε2.1 Expression (2.1) for dlink quantifies the intuitive observation thatfor increasing node spatial density the length of a link between nearest neighbors decreases.

In (2.1), we have introduced the notation �(·). This is a key characteristic of the simpleframework proposed in this chapter (and in Chapter 3). In fact, most of the results which willbe presented are valid ‘on the order’, i.e. trendwise. In other words, if the topology is ‘almost’regular and the border effects are negligible, then dlink is ‘almost’ equal to 1/

√ρS.

2.2.2 Route Discovery

Discovery of a multi-hop route from a source to a destination is a crucial phase in awireless networking scenario with flat architecture. It is usually based on broadcast of a‘search’ message from the source node, and it involves a non-negligible exchange of controlmessages [4]. The focus of this chapter (and of Chapter 3) is on the characterization ofthe behavior of on-going peer-to-peer (P2P) multi-hop communications. Therefore, we willassume that a route between the source and the destination exists. Moreover, in the case ofregular lattice topology, we will assume that each link of a route is between nearest neighbors:in other words, a node in a multi-hop route forwards the received information to one of itsfour nearest nodes in the square lattice. This is the case, for instance, for the multi-hop routesshown in Figure 2.1 (b). We note that this routing strategy may not correspond to the usualshortest path (SP) routing [5,42], in the sense that a node, rather than connecting to the farthestnode within its transmission range, connects to the geographically nearest node. However, anearest neighbor routing strategy is energy saving and, therefore, attractive for ad hoc wirelessnetworks where nodes might have limited battery energy. The advantages/disadvantages of

1The meaning of the notation �(·) is very similar to that of �(·) used, in the realm of algorithms, to describethe asymptotic functional relationship between functions of time [17]. More precisely, the notation f (n) = �(g(n))

means that there exists an n0 such that for n ≥ n0, ∃c1 ∈ (0, 1), c2 > 1 such that c1g(n) ≤ f (n) ≤ c2g(n). Thenotation �(·), however, is not a functional relationship, and, as such, it does not represent asymptotic behavior.


this strategy are analyzed in more detail in Chapter 9. Finally, in this chapter we will assumethat a multi-hop route (formed by consecutive hops between nearest neighbors) lies ‘around’the straight line connecting source and destination.

2.2.3 Average Number of HopsIn general, a multi-hop route could have any number of hops nh. In order to derive anaverage network performance, an intuitive approach could be that of considering a routewith an average number of hops, denoted as nh. To this end, one has to identify the maximumnumber of hops nmax

h and a probability distribution for the number of hops nh, which isobviously an integer random variable assuming values between 1 and nmax

h . Assuming thatsource and destination nodes lie at opposite ends of a diameter over a circular surface andthat the deviation from a straight line is limited, it follows that

nmaxh =

⌊diametersurface

dlink

⌉=⌊

2√π

√N

⌉= �(

√N) (2.2)

where the notation �∗� denotes the integer value closest to ∗ (this notation will be usedthroughout the chapter).

Considering any symmetric probability distribution for the number of hops, it can beimmediately concluded that nh = nmax

h /2 = �(√

N). This is the case, for example, of aprobability mass function for nh given by a modified binomial distribution with parameter0.5, without any probability mass concentrated in nh = 0. In other words, P {nh = i} =(nmax

hi

)/(2nmax

h −1), i = 1, . . . , nmaxh . This represents a good approximation for the distribution

of the number of hops in a realistic scenario, since very long or very short routes are muchless likely than routes with an average number of hops.2 In the case of a circular surface onehas nh = �√N/π�.

2.3 Communication-Theoretic BasicsAt this point, we introduce communication-theoretic basics for the analysis of an ad hocwireless networking scenario. We first compute a simple expression for the BER at the endof a multi-hop route, and then we find an expression for the link SNR.

2.3.1 Bit Error Rate at the End of a Multi-hop RouteConsidering the communication between two nodes at a distance dlink, we denote the linkBER by BERlink. This probability depends, of course, on the SNR at the receiving nodeof the link, modulation, possible channel coding, channel characteristics (e.g. the presenceof frequency selective or non-selective fading), etc. Considering a route consisting of nhconsecutive links, we would like to find a simple expression for the BER at the final node ofthe multi-hop route, i.e. at the destination. Assuming no outstanding routing problem (this isconsistent with our previous assumption that a multi-hop route has been created and reservedby the source node), in this subsection we focus only on data transmission (i.e. payload).Considering a worst-case scenario approach (in this sense, the derived expression for the

2Our simulation results show that the realistic probability distribution of nh in a network with square gridtopology is a sort of binomial distribution slightly shifted to the left, i.e. the actual average value is slightly lowerthan nmax

h /2, but still of the order of√

N .

2.3. Communication-Theoretic Basics 19

BER at the end of a multi-hop route will be an upper bound for the true BER, as will beconfirmed by simulation results), we assume that bit errors in consecutive hops accumulate.Therefore, the BER at the end of the nhth link, denoted by BER(nh)

route, can be expressed as

BER(nh)route = 1 − (1 − BERlink)

nh . (2.3)

Should consecutive links be characterized by different values of BERlink, expression (2.3)could be straightforwardly modified as

BER(nh)route, general = 1 −

nh∏j=1

(1 − BERlink j ). (2.4)

However, the assumption of identical BER over consecutive links allows for more immediate,yet meaningful, insights into the network performance. A scenario with different link lengthscalls for semi-analytical performance evaluation, as considered in [41] for analyzing ad hocwireless networks with random topology.

Denoting by BERroute the BER at the end of a route with an average number of hops,from (2.3) one obtains

BERroute = BER(nh)route = 1 − (1 − BERlink)

nh = 1 − (1 − BERlink)�(

√N). (2.5)

In the case of a circular network surface, expression (2.5) becomes

BERroute = 1 − (1 − BERlink)�√

N/π�. (2.6)

Expression (2.6) shows the dependence of the BER at the end of a multi-hop route with anaverage number of hops, on the number of nodes N and the link BER. In particular, the linkBER depends on the considered channel model, transmission scheme and, obviously, linkSNR, which will be analyzed in more detail in the following subsection.

Note that the BER expression at the end of a route with an average number of hops is notexactly equal to the average BER, i.e. the value obtained by averaging over the number ofhops. In fact, taking into account the quasi-binomial distribution of the number of hops, theaverage BER can be computed as follows:

Enh

[BER(nh)

route

]=

nmaxh∑

i=1

(nmaxhi

)2nmax

h − 1

[1 − (1 − BERlink)

i]

= 1 − (2 − BERlink)nmax

h − 1

2nmaxh − 1

. (2.7)

Comparing (2.5) with (2.7), it is clear that the two BER expressions are different. In orderto better understand the difference, we make use of the fact that, even for a small numberof hops, to achieve a route BER lower than or equal to 10−3, BERlink has to be very low(for instance, lower than 10−5). Considering a first-order Taylor series expansion of the term(2 − BERlink)

nmaxh for small values of BERlink, one obtains

Enh

[BER(nh)

route

]� 1 − 2nmax

h − nmaxh 2(nmax

h −1)BERlink − 1

2nmaxh − 1

� nmaxh

2BERlink. (2.8)


r(t)Source

Node

Destination

Node

s(t)Attenuation

sa(t)

f(t) wthermal(t)

Figure 2.2 Link communication model in an ideal (no INI) network communicationscenario.

On the other hand, a first-order Taylor series expansion of (2.5) leads to the followingapproximate expression for the route BER:

BERroute = BER(E[nh])route � nhBERlink = nmax

h

2BERlink. (2.9)

Obviously, the final BER expressions in (2.9) with (2.8) coincide. Therefore, although theanalytical results derived in the following refer to the BER at the end of a multi-hop routewith an average number of hops, they will also be meaningful for the average route BER.

2.3.2 Link Signal-to-Noise RatioThe link model in a wireless communication scenario without INI and with frequency non-selective (flat) fading is shown in Figure 2.2. In particular, the received signal r(t) can bewritten as

r(t) = f (t)sa(t) + wthermal(t) (2.10)

where sa(t) is an attenuated version of the transmitted signal s(t), f (t) is the fading process,and wthermal(t) is an AWGN process.

The absence of the INI assumption here corresponds to considering a multiple accessscheme with no collisions. Clearly, this is a simplifying assumption which may not be realisticand such a decision is the subject matter of MAC design in ad hoc wireless networks.The impact of the MAC protocol characteristics on the physical layer performance will beconsidered in the next chapter for a few interesting cases. Nevertheless, in this chapter we usethis simplifying assumption to gain some basic insight into the performance of these networksunder somewhat ideal conditions.

As indicated in Figure 2.2, the channel attenuates the transmitted signal, and we assumethat there is simply free-space loss [43] – the extension of the proposed approach to otherpropagation models is straightforward. Hence, according to Friis’ free-space formula [43],the received signal power can be written as

E{s2a (t)} = Pr = αPt

d2link

where Pt = E{s2(t)} is the transmit power (common for all nodes) and

α = GtGrc2

(4π)2flf 2c

2.3. Communication-Theoretic Basics 21

where Gt and Gr are the transmitter and receiver antenna gains, fc is the carrier frequency,c is the speed of light and fl ≥ 1 is a loss factor. The bit energy can be written as

Ebit = Pr

Rb(2.11)

where Rb is the transmission data-rate (dimension [b/s]). Recalling the concept of the noisefigure F of a receiver [43], the power spectral density of the (white) thermal noise is

Ethermal = FkT0 (2.12)

where k = 1.38 × 10−23 J/K is Boltzmann’s constant and T0 is the room temperature (T0 =300 K).

In general, the fading process f (t) can be written as

f (t) = a(t)ejθ(t)

where a(t) is the fading amplitude process and a(t) is the fading phase process. Under theassumption of a slow fading process, it is possible to assume that the fading amplitude andphase are constant over a symbol interval. In other words, over the nth symbol interval Ts onecan assume that f (t) = ane

jθn, nTs ≤ t ≤ (n + 1)Ts. Denoting by a the fading amplitude ina generic symbol interval, the instantaneous link SNR can be written as

� a2Ebit

Ethermal. (2.13)

In other words, the instantaneous link SNR is the ratio between the product of the bit energywith the square fading amplitude and the thermal noise energy. The (average) link SNR, withrespect to the fading amplitude, will be denoted as SNRlink and can be written as

SNRlink � E[a2]Ebit

Ethermal. (2.14)

Using expressions (2.11) and (2.12), one can finally express the link SNR as follows:

SNRlink = E[a2]αPt

FkT0Rbd2link

. (2.15)

At this point, according to the statistics of the fading process, one can evaluate the averagelink SNR. We now outline three possible scenarios which can be considered as typical ofwireless communications: (i) strong line-of-sight (LOS) communication; (ii) strong multipathfading (no LOS); (iii) multipath and LOS.

Strong LOS

In this case, the fading process disappears. In other words, the transmitted signal is affectedby attenuation only, and the received signal in (2.10) can be written as

r(t) = sa(t) + wthermal(t). (2.16)

This scenario corresponds to AWGN communications. In this case, the average link SNR canbe written as

SNRAWGNlink = αPt

FkT0Rbd2link

= αPtρS

FkT0Rb. (2.17)


Strong Multipath Fading

In this case, the fading amplitude can be characterized by the following Rayleigh probabilitydensity function (PDF) [44]:

p(a) = 2a

σ 2fad

exp

(− a2

σ 2fad

)U(a)

where U(·) is the unit step function and σ 2fad = E[a2]. Consequently, it can be shown that

the instantaneous link SNR has the following chi-squared distribution with two degrees offreedom [33, 44, 45]:

p() = 1

exp

(−

SNRRayleighlink

)U()

where

SNRRayleighlink = σ 2

fadEbit

Ethermal. (2.18)

In the following, we will assume3 that σ 2fad = 1.

Multipath and LOS

There are many radio channels which are basically LOS communication links with multipathcomponents arising from secondary reflections from surrounding terrain and buildings.In this case, the square fading amplitude a2 has a non-central chi-squared distribution withnon-centrality parameter a2

0 [33, 44]. It can be shown that the fading amplitude a can becharacterized with the following Rice PDF [44]:

p(a) = a

σ 2fad

exp

(−a2 + 2a2

0

2σ 2fad

)I0

(√2a0 a

σ 2fad

)U(a) (2.19)

where a20 can be interpreted as the mean power in the non-fading component and σ 2

fad isthe mean power in the fading component. By defining the Rice factor K � a2

0/σ 2fad, which

quantifies the ratio between the non-fading (direct LOS) and fading components, one canrewrite (2.19) as

p(a) = a

σ 2fad

exp

[−(

a2

σ 2fad

+ K

)]I0

(2√

K a

σfad

)U(a).

In this case, the average link SNR has the following expression:

SNRRicelink = (1 + K)

σ 2fadEbit

Ethermal. (2.20)

3Note that this assumption does not affect the generality of the approach. Should the parameter σ 2fad be larger,

this would correspond to a communication scenario with a stronger multipath component.

2.4. BER Performance Analysis 23

The PDF of the instantaneous link SNR can be given the following expression [45]:

p() = 1 + K

SNRRicelink

exp(−K) exp

[− (1 + K)

SNRRicelink

]I0

[2

√K(1 + K)

SNRRicelink

]U().

In order to make a direct comparison with the strong LOS communication scenario, weassume a0 = 1. In this case, it follows that σ 2

fad can be written as 1/K and the average linkSNR (2.20) can be equivalently expressed as

SNRRicelink = 1 + K

K

Ebit

Ethermal. (2.21)

It is possible to show that the limiting cases for K = ∞ and K = 0 correspond to thescenarios with strong LOS and strong multipath fading, considered in the two previoussubsubsections, respectively.

2.4 BER Performance Analysis2.4.1 Uncoded TransmissionAt this point, assuming that the channel fading is sufficiently slow that the phase θ can beestimated without error, we assume coherent detection of the received signal. In the caseof binary phase shift keying (BPSK) signaling, should the attenuation a due to fading beconstant, the link BER could be written as [33]

BERlink() = Q(√

2)

= 1√2π

∫ ∞√

2

e−x2/2 dx (2.22)

where Q(x) � 1√2π

∫ +∞x e−y2/2 dy, is the instantaneous link SNR defined in (2.13) and

we have shown the dependence of the link BER on the instantaneous link SNR. In order toobtain the average link BER, assuming that the statistics of a and, consequently, of areknown at the receiver, one can write

BERlink =∫ ∞

0BERlink() p() d

where p() depends on the particular fading environment. We now outline the expressionsof the link BER in the three scenarios considered at the end of subsection 2.3.2.

Strong LOS

In the case of strong LOS, i.e. of transmission over an AWGN channel, the link BER can bewritten as [33]

BERAWGNlink = Q

(√2 SNRAWGN

link

)= Q

(√2

Ebit

Ethermal

)= Q

(√2aρSPt

FkT0Rb

). (2.23)

Therefore, the route BER in (2.6) becomes

BERroute = 1 −[

1 − Q

(√2αρSPt

FkT0Rb

)]�√N/π �

. (2.24)


Equation (2.24) allows one to explore the relationship between the BER at the end of a multi-hop route with an average number of hops and very important quantities, such as the nodespatial density ρS, the transmit power Pt, the data-rate Rb and the number of nodes in thenetwork N .

Strong Multipath Fading

In this case, the link BER can be written as follows [33]:

BERRayleighlink = 1

2

1 −√√√√ SNRRayleigh

link

1 + SNRRayleighlink

= 1

2

(1 −

√σ 2

fad Ebit

Ethermal + σ 2fad Ebit

)

= 1

2

(1 −

√αρSPt

FkT0Rb + αρSPt

)(2.25)

where, in the last passage, we have used the assumption that σ 2fad = 1. Similar to the previous

case with strong LOS, in this case as well the evaluation of the route BER is straightforward.

Multipath and LOS

In this case, the link BER can not be given a closed-form expression. However, acomputationally efficient expression is the following [45]:

BERRicelink =

∫ π/2

0

(1 + K) sin2(θ)

SNRRicelink + (1 + K) sin2(θ)

· exp

[− KSNRRice

link

SNRRicelink + (1 + K) sin2(θ)

]dθ. (2.26)

Since the integration region in the link BER expression in (2.26) is the finite interval [0, π/2],it can be easily evaluated numerically.

Numerical Results

In the remainder of this chapter, we will assume, without loss of generality, the followingfixed values for some of the network parameters:

Gt = Gr = fl = 1, fc = 2.4 GHz, F = 6 dB.

The gains Gt and Gr are related to the effective areas of the antennas used at the transmitterand at the receiver, respectively – assuming that the antenna gains are equal to one isequivalent to considering omnidirectional antennas, which is reasonable in a wireless networkwith a flat architecture. However, the use of antennas with gains larger than one or the use ofdirectional antennas could be an interesting option for improving network performance [46].We do not pursue this possibility further, since it is beyond the scope of this book.


The assumption that fl = 1 corresponds to the assumption that there are no system lossesunrelated to propagation – extending the proposed approach (and results) to a scenariowith fl > 1 is straightforward. The value considered for the noise figure F is reasonableaccording to measurements conducted for commercial devices [43, 47]. The choice of thecarrier frequency fc = 2.4 GHz corresponds to the carrier frequency considered in the IEEE802.11b-based wireless local area networks (WLANs) [48].

The following significant scenarios can be distinguished on the basis of the transmitpower:

• WLAN4 (Pt = 0.5 W) [47, 48];

• sensor network (Pt = 15 mW) [36];

• smart dust-like network (Pt = 0.2 µW) [49].

The BER performance, for the three indicated transmit power values and for Rb = 2 Mb/s, isshown, as a function of the node spatial density, in Figure 2.3. The considered communicationmodel is the one with strong LOS, i.e. AWGN, and the cases with N = 102, 103 and 105

nodes are considered. In particular, besides the BER behavior predicted by our analysis, inthe figure we also show simulation results obtained as follows. We consider the number ofhops in a route as a quasi-binomially distributed random variable. For each realization of thisrandom variable, we assume the transmission of a packet of L = 1000 randomly generatedbits. At each link transmission, the final node of the link receives the bits transmitted by theinitial node of the link and corrupted by randomly generated independent Gaussian noisesamples with variance Ethermal – the corresponding link SNR is equal to (2.14). Each nodeperforms threshold detection, and retransmits the decided bits over the following links, untilthe bits reach the destination node, where a final decision is made. The route BER resultsare obtained by averaging over a sufficiently large number of route realizations in order forthe Monte Carlo simulation to be reliable – for instance, large enough that at least 200 biterrors are collected over all route realizations. Observe that there is an excellent agreement,for BER values of interest (i.e. lower than 10−3), with the results predicted by the analysis,confirming that the assumption of no correction, in the subsequent links, of the bit errorsmade in a link, is a valid assumption at sufficiently low BER values. We note that while theuse of simulations might be more accurate, the simple route BER expression (2.5) providesseveral additional insights into the network performance, as will be shown later.

It is important to emphasize that the fact that the route BER becomes lower and lower indenser networks is due to the considered no interference (no INI) assumption. In Chapter 3,it will be shown that this is not true in a realistic scenario where communication is affectedby INI, which increases as the network becomes denser.

One might suspect that Pt = 0.2 µW is a very low transmit power. However, observethat in this chapter we assume that the propagation is simply affected by free-space loss.From Figure 2.3, we notice that a route BER lower than 10−3 is obtained; for example, ifρS = 2 × 10−2 m−2. This value of node spatial density corresponds to a link length dlink �7.1 m. This implies that the receiver sensitivity of the receiving node of the link has to be−94 dBm. Should the sensitivity of a given node be higher or the channel attenuation stronger

4In [47], the authors show that the amount of power spent transmitting a given packet has two components: a fixedpower consumption and a power consumption proportional to the packet size. The values considered in this chaptercorrespond to the proportional component, and are used to derive reasonable results. In [47], it is also observed thatreceiving is less costly, in terms of power, than transmitting.


10-10

10-8

10-6

10-4

10-2

100

ρS [m

-2]

10-5

10-4

10-3

10-2

10-1

100

BERroute

N=102 (Analysis)

N=103 (Analysis)

N=105 (Analysis)

N=103 (Simulation)

Pt=0.2 µW

Pt=0.5 W

Pt=15 mW

Figure 2.3 Route BER versus node spatial density for a multi-hop ad hoc wireless networkwith strong LOS communications (i.e. AWGN). The data-rate is Rb = 2 Mb/s. Various valuesof the number of nodes N are considered, and the performance is evaluated both analyticallyand through simulations. (Reproduced by permission of © 2003 IEEE.)

(for example, the path loss exponent is 4, rather than 2, in the case of a fully obstructedpropagation path), then the transmit power should be increased in order to guarantee thesame final route BER. However, the proposed framework remains unchanged, and can bedirectly applied to that case as well.

Note that the route BER performance does not strongly depend on the number of nodes,and, consequently, on the number of hops. Observe also that the BER curves in Figure 2.3have a marked ‘waterfall’ shape. This is due to what could be defined as a cumulative effect, interms of BER, of a multi-hop route. The BER curve of a multi-hop route, therefore, looks likea steeper version of the link BER curve. In other words, there is a sort of threshold, in termsof node spatial density, above which the network is supposed to work and below which thenetwork basically stops working – a precise expression for this critical node spatial densitywill be derived in the following.

In Figure 2.4, the performance in a scenario with N = 100 nodes is shown, for thethree values of the transmit power previously considered. For each value of the transmitpower, three possible communication scenarios are considered: (i) strong LOS (i.e. AWGNlink channel model, as considered in Figure 2.3); (ii) strong multipath (i.e. link channel modelwith Rayleigh fading); and (iii) an intermediate scenario with LOS and multipath (i.e. linkchannel model with Rice fading with Rice factor K = 10 dB). As expected, while the routeBER curves in a scenario with strong LOS have the typical waterfall behavior (as a functionof the node spatial density), in a scenario with Rayleigh fading they decrease linearly (on adouble logarithmic scale) for increasing values of the node spatial density. In other words,


10−12

10−10

10−8

10−6

10−4

10−2

100

ρS [m

−2]

10−5

10−4

10−3

10−2

10−1

100

BERroute

AWGNRayleigh fading

Rice fading (K=10 dB)

Pt = 0.5 WPt = 0.2 µW

Pt = 15 mW

Figure 2.4 Route BER versus node spatial density for a multi-hop ad hoc wireless networkwith N = 100 nodes and various values of the transmit power. The data-rate is Rb = 2 Mb/s.For each value of the transmit power, the performance in scenarios with (i) strong LOS(AWGN), (ii) strong multipath (Rayleigh fading) and (iii) an intermediate communicationscenario (Rice fading with Rice factor K = 10 dB) is shown.

in order to support the same route BER, the presence of a strong multipath requires that thenetwork is denser.

In Figure 2.5, the BER is shown, as a function of the data-rate Rb, for increasing valuesof the transmit power Pt, for a node spatial density ρS = 10−3 m−2, in both the cases withN = 102 nodes and N = 103 nodes. As shown in Figure 2.5, in this case as well the routeBER strongly depends on the node spatial density and, in a weak manner, on the specificnumber of nodes.

In Figure 2.6, the route BER is shown as a function of the data-rate, for two values ofthe transmit power (Pt = 10−6 W and Pt = 10−4 W, respectively). For each value of thetransmit power, the three possible communication scenarios (strong LOS, strong multipath,LOS with multipath) are considered. In this case as well, the presence of multipath degradesthe performance significantly. In fact, in order to support the same route BER, the data-rate ina scenario with strong multipath has to be significantly lower than the corresponding data-ratenecessary in a scenario with strong LOS.

2.4.2 Coded TransmissionIn order to evaluate the impact of channel coding on the network performance we considera serially concatenated convolutional code (SCCC) [50] with rate 1

4 , constituted by arate- 1

2 outer non-recursive non-systematic convolutional code (NRNSCC) with four statesand generators (in octal notation) G1 = 7 and G2 = 5, and rate- 1

2 inner recursive


103

104

105

106

107

108

109

Rb [b/s]

105

104

103

102

101

100

BERroute

N=103

N=102

Pt=104 W

Pt=105 W

Pt=106 W

Pt=107 W

Pt=108 W

Figure 2.5 Route BER versus data-rate for a multi-hop ad hoc wireless network. The nodespatial density is fixed to ρS = 10−3 m−2. Various values of the number of nodes N andtransmit power Pt are considered.

systematic convolutional code (RSCC) with four states and generators G1 = 7 and G2 = 5.The two component codes are concatenated through a bit interleaver of length 1024. Thechosen code allows one to gain some basic understanding of the performance of an ad hocwireless network when forward error correction (FEC) is used. The SCCC is chosen sinceconcatenated convolutional codes have become part of several standards for next-generationwireless communication systems, such as the universal mobile telecommunications system(UMTS) [51]. Hence, a concatenated code might be considered as a possible choice forad hoc wireless networks as well. The proposed analysis can be extended by considering anycode, such as block codes (like low-density parity-check codes). However, the conclusionsderived by using the SCCC considered above are quite general. In order to evaluate theroute BER in (2.6), we simply use numerical tables, for the considered code, where thelink BER is given as a function of the link SNR. The BER curves for the considered SCCCare shown in Figure 2.7 in a communication scenario with strong LOS. In particular, theperformance of the coded scheme is compared to the corresponding performance of theuncoded scheme.

Obviously, the use of FEC techniques improves the overall performance in terms ofroute BER. More precisely, the minimum node spatial density required to guarantee anacceptable – from the perspective of higher network layers – route BER (e.g. 10−3) decreases(i.e. improves) with the use of coding. This means that a sparse network, which would notwork with uncoded transmission, might work with the use of coding. For example, withN = 103 nodes and a data-rate Rb = 2 Mb/s, the minimum necessary node spatial densitywith uncoded transmission is approximately equal to 10−2 m−2. When using the consideredSCCC, the link SNR gain, with respect to the uncoded case, is approximately 6 dB, i.e. 4 on a

2.5. Network Behavior 29

103

104

105

106

107

108

109

Rb [b/s]

105

104

103

102

101

100

BERroute

AWGNRayleigh fadingRice fading (K=10 dB)

Pt=104 W

Pt=106 W

Figure 2.6 Route BER versus data-rate for a multi-hop ad hoc wireless network. The nodespatial density is fixed to ρS = 10−3 m−2. Two values of the transmit power Pt are considered.For each value of the transmit power, the performance in scenarios with (i) strong LOS(AWGN), (ii) strong multipath (Rayleigh fading) and (iii) an intermediate communicationscenario (Rice fading with Rice factor K = 10 dB) is shown.

linear scale. In other words, the use of this SCCC in an ad hoc wireless network would allowone to sustain communications in scenarios with node spatial density equal to one-fourth ofthat sustainable in the uncoded case.

Several additional considerations, regarding the use of channel coding, can be made bytaking into account the fact that the total energy of a node is limited (each node has finiteenergy) and that delay in transmitting a packet is a major concern. These considerationsshow that the robustness towards a sparse topology comes at the expense of other importantnetwork quantities: (i) from a receiver perspective, the use of error correction coding requiressignal processing at each node, thus increasing delay and power consumption [35]; (ii)from a transmitter perspective, the use of a code adds redundancy in transmitting a givennumber of information bits and, thus, bandwidth expansion or, if bandwidth is limited, data-rate reduction. However, from a node (or network) lifetime longevity viewpoint, it mightbe worthwhile to pursue coding further. In particular, coding seems very attractive in non-real-time data communication scenarios. Moreover, coding could be very useful for securitypurposes as well.

2.5 Network BehaviorAs briefly mentioned in discussing the results in Figure 2.7, previous communication-theoretic analysis lends itself naturally to a physical layer-oriented quality of service (QoS)constraint in terms of maximum tolerable BER at the end of a route with an average numberof hops. In other words, denoting this maximum tolerable value as BERmax

route, we assume that


104

103

102

101

100

S [m 2]

105

104

103

102

101

100

BERroute

N=103

N=105

N=108

Pt=0.23 W

Rb=2 Mb/s Rb=11 Mb/s

Pt=0.06 W

SCCC

Uncod.

Figure 2.7 Performance comparison, in terms of BER at the end of a multi-hop route withan average number of hops, between the cases with uncoded and coded transmission (usinga SCCC), respectively. Various values of the number of nodes N , data-rate Rb, and transmitpower Pt are considered. The link channel model is with strong LOS.

an ad hoc wireless network is ‘fully connected’ (in an average sense) if the following keycondition is satisfied:

BERroute ≤ BERmaxroute. (2.27)

In the following, by simply elaborating on (2.27), we gain important insights in terms of theminimum transmit power necessary to guarantee proper network operation. We also introducethe novel concept of minimum spatial energy density, which quantifies the basic observationthat a minimum level of energy per unit area should be present in the network to guaranteecommunication among the nodes. Finally, we derive a simple expression for the node lifetime.

2.5.1 Minimum Spatial Energy Density and Minimum Transmit Powerfor Full Connectivity

Since the link BER is a decreasing function of the link SNR, in order to obtain, at the end ofa multi-hop route with an average number of hops, a BER lower than BERmax

route, the link SNRmust be larger than a minimum required value SNRmin

link, which depends on BERmaxroute and the

number of hops nh (hence, the number of nodes N). More precisely, from expression (2.24),it follows that SNRmin

link must satisfy the following relationship:

Q

(√2 SNRmin

link

)= 1 − (

1 − BERmaxroute

)�√π/N �

. (2.28)


Denoting by Q−1(x) the inverse5 function of Q(x), the minimum link SNR can be written as

SNRminlink = 1

2

{Q−1

[1 − (

1 − BERmaxroute

)⌊√ πN

⌉]}2

. (2.29)

Recalling the expression for SNRlink in (2.15), from (2.27) the following fundamentalrelationship can be derived:

PtρS

Rb≥ ρmin

energy � FkT0

αSNRmin

link. (2.30)

The quantity ρminenergy (dimension [J/m2]) is the minimum spatial energy density required to

guarantee full connectivity (in an average sense) in the network. Inspection of (2.30) revealsthat ρmin

energy in an ad hoc wireless network is a function of SNRminlink, transmitter and receiver

antenna gains Gt and Gr, carrier frequency fc, and system loss parameter fl. In other words,ρmin

energy is a function of the key parameters in an ad hoc wireless network. The quantityPtρS/Rb, on the left-hand side of (2.30), can be interpreted as the ‘actual’ spatial energydensity in the network. In addition, note that the effect of channel coding, which is that ofreducing the minimum required link SNR, can be equivalently interpreted as that of reducingthe minimum spatial energy density required for full connectivity. In Figure 2.8, the behaviorof ρmin

energy as a function of BERmaxroute is shown for various values of N in a communication

scenario with strong LOS, i.e. with an AWGN link channel model.It is then possible to provide the following conditions which guarantee full connectivity

(for the strong LOS case).

• Given Pt and ρS, the maximum acceptable data-rate can be written as

Rmaxb = PtρS

ρminenergy

. (2.31)

• Given Rb and ρS, then the minimum required transmit power for full connectivity canbe written as

P mint = ρmin

energyRb

ρS. (2.32)

Considering a mobile ad hoc network, condition (2.32) implies that for a fixednumber of nodes, if the nodes spread over a larger surface due to their motion,i.e. the node spatial density decreases, then the minimum transmit power has toincrease proportionally to preserve full connectivity in the network. Alternatively,condition (2.32) implies that when the node spatial density decreases, if the transmitpower per node needs to be kept at a constant value, due to node (or network) lifetimeconcerns, then the data-rate has to be reduced as well. In Figure 2.9, it is shown that, fora fixed node spatial density, the minimum required transmit power for full connectivitydepends in a very limited way on the number of nodes N and on the maximumtolerable route BER, i.e. BERmax

route. This particular, and somehow counterintuitive,limited dependence of the minimum transmit power on N and BERmax

route is to be

5The inverse function of Q(x) exists, but it does not have a closed-form expression. Hence, it has to benumerically evaluated.


10-7

10-6

10-5

10-4

10-3

10-2

BERroute

10-15

10-14

ρenergy

[J/m2]

N=104

N=103

N=102

max

min

Figure 2.8 Minimum spatial energy density required for full connectivity in ad hoc wirelessnetworks as a function of the maximum tolerable BER at the end of a multi-hop route with anaverage number of hops. The node spatial density is ρS = 10−3 m−2. Various values of thenumber of nodes N are considered. The considered link channel model is with strong LOS.(Reproduced by permission of © 2003 IEEE.)

attributed to the typical bimodal behavior of an ad hoc wireless network, which willbe analyzed in detail in the next subsection. In other words, either the network worksperfectly (i.e. it is connected) or it does not work at all. Therefore, for a given nodespatial density, there is a critical minimum transmit power, such that if the transmitpower is larger than this value, then the network can support any required physicallayer QoS, given by BERmax

route.

• Finally, if Rb and Pt are fixed, the critical minimum node spatial density for fullconnectivity can be written as

ρminS = ρmin

energyRb

Pt. (2.33)

Note that the existence of a critical node spatial density above which the networkis fully connected is also predicted in the context of percolation theory [32, 52] andrandom graph theory [53]. The main advantage of the proposed communication-theoretic framework is the fact that it allows one to derive an explicit expression forthe critical minimum node spatial density for full connectivity (i.e. ρmin

S ) in terms ofmajor physical network parameters.

An additional observation is worth making at this point. Considering a single link withAWGN, the Shannon–Hartley formula predicts that the capacity of the link, in b/s/Hz, has the


1010

108

106

104

102

S [m2]

108

106

104

102

100

102

Ptmin

[W]

N=104

N=103

N=102

BERroute

=106max

maxBER

route=10

3

Figure 2.9 Minimum transmit power required for full connectivity as a function of thenode spatial density. The data-rate is fixed to Rb = 1 Mb/s and there are strong LOScommunications. Various values of the number of nodes N and the maximum tolerable BERat the end of a multi-hop route with an average number of hops, i.e. BERmax

route, are considered.

following expression [15]:

Clink = log2(1 + SNRlink).

In order to transmit 1 b/s/Hz, as in the case of uncoded BPSK signaling, it follows that

SNRlink ≥ 1. (2.34)

Condition (2.34) can be equivalently rewritten as

PtρS

Rb≥ FkT0

α. (2.35)

Comparing (2.30) with (2.35), it is possible to conclude that condition (2.30) can beconsidered as an extension, to a multi-hop scenario with a route BER QoS, of an information-theoretic condition valid for a single link communication. In fact, in order to have fullconnectivity, on average, in the considered multi-hop network communication scenario, aminimum value of the link SNR equal to SNRmin

link � 1, is required.In Chapter 3, it is shown how the concept of minimum spatial energy density can be

extended to a (realistic) ad hoc wireless networking scenario with INI.In order to understand the impact of the channel propagation model on the minimum

spatial energy density, in Figure 2.10 the behavior of the node spatial density is shown in


10-7

10-6

10-5

10-4

10-3

10-2

BERroute

10-16

10-15

10-14

10-13

10-12

10-11

10-10

10-9

ρenergy

[J/m2]

AWGNRayleigh fading

Rice fading(K=10 dB)

max

min

Figure 2.10 Minimum spatial energy density required for full connectivity in ad hocwireless networks as a function of the maximum tolerable BER at the end of a multi-hoproute with an average number of hops. The node spatial density is ρS = 10−3 m−2. Thenumber of nodes is fixed to N = 100 and the considered scenarios are characterized by (i)strong LOS (AWGN), (ii) strong multipath (Rayleigh fading) and (iii) LOS with multipath(Rice fading with Rice factor K = 10 dB).

the three scenarios with (i) strong LOS, (ii) strong multipath and (iii) LOS with multipath.As one can observe from the figure, for a more stringent QoS condition, i.e. for a lower valueof BERmax

route, the minimum spatial energy density in the presence of strong multipath increasesdramatically. On the other hand, the minimum spatial energy density required in a scenariowith strong LOS is basically independent (with respect to the scenario with strong multipath)of BERmax

route.In Figure 2.11, the minimum transmit power is shown, as a function of the node spatial

density, in a network with N = 100 nodes, in the two cases with (a) BERmaxroute = 10−3 and

(b) BERmaxroute = 10−6. In both cases, the behavior in the three cases with strong LOS, strong

multipath and LOS with multipath is shown. As expected from the results in Figure 2.10, thepresence of strong multipath dramatically increases, with respect to a scenario with strongLOS communications, the minimum transmit power necessary to guarantee the desired QoS.

2.5.2 Connectivity: Average Sustainable Number of HopsIn section 2.2.1, it has been shown that, for a network with regular topology, the averagenumber of hops is �(

√N). However, given the network area A, the data-rate Rb, and the

transmit power Pt, the results previously derived imply that there is a critical threshold valuefor the number of nodes, denoted by Nmin, such that the corresponding node spatial densityρmin

S = Nmin/A is the minimum value required to guarantee the desired BER at the end


10-10

10-8

10-6

10-4

10-2

ρS [m

-2]

10-8

10-6

10-4

10-2

100

102

104

106

Ptmin

[W]

AWGNRayleigh fading


(a)

10-10

10-8

10-6

10-4

10-2

ρS [m

-2]

10-8

10-6

10-4

10-2

100

102

104

106

Ptmin

[W]

AWGNRayleigh fading


(b)

Figure 2.11 Minimum transmit power required for full connectivity as a function of the nodespatial density, in a scenario with N = 100 nodes: (a) BERmax

route = 10−3 and (b) BERmaxroute =

10−6. The data-rate is fixed to Rb = 1 Mb/s. In both cases, the three scenarios with strongLOS, strong multipath and LOS with multipath are considered.

of a multi-hop route with an average number of hops. In order to find this value, from thegeneral BER expression (2.3), it is possible to derive the maximum number of hops whichcan be sustained by the network. More precisely, the maximum sustainable number of hops,


0 200 400 600 800N

0

4

8

12

16

20

24

Hops

(N/π)1/2

BERroute=10−2

BERroute=10−3

Rb=104 b/s

Pt=10−7

W

Pt=10−6

W

Rb=5x104 b/s

Pt=10−7

W

Rb=5x104 b/s

max

max

Figure 2.12 Maximum sustainable number of hops in an ad hoc wireless network withregular topology and no INI versus the number of nodes N . The network area is A = 106 m2

and there are links with strong LOS. Two possible values of the maximum tolerable BER atthe end of a route with an average number of nodes are considered, along with various valuesof the transmit power Pt and the data-rate Rb.

denoted as nmaxsh , corresponding to a maximum prescribed final BER value equal to BERmax

route,can be written as

nmaxsh =

⌊log(1 − BERmax

route)

log(1 − BERlink)

⌉� BERmax

route

BERlink. (2.36)

In Figure 2.12, the maximum sustainable number of hops nmaxsh is plotted as a function of

the number of nodes N , in a scenario with strong LOS communication. The area A is equalto 106 m2 (i.e. 1 km2) and the network surface is assumed to be circular (nh = �√N/π�).For various values of the significant system parameters (the transmit power Pt and the data-rate Rb), the maximum sustainable number of hops is shown in the cases with BERmax

route =10−2 (dotted lines) and BERmax

route = 10−3 (dashed lines) – we refer to these types of curvesas ‘max-hop-curves’. For comparison, the curve corresponding to an average number ofhops required by a communication (solid line), i.e.

√N/π , is shown6 – we refer to this

curve as ‘average-hop-curve’. Observe that the max-hop-curves cross the average-hop-curveat a certain threshold value in terms of N . As expected, for fixed values of the parametersPt, Rb and A, the threshold value decreases for higher values of BERmax

route. From the previous

6To be more precise, the average number of hops is �√N/π�, i.e. it is a piecewise line around the continuouscurve

√N/π . For purposes of comparison, we simply consider the continuous version.


derivation, it is easy to realize that the threshold value, in terms of number of nodes, is

Nmin = ρminS A = ρmin

energyRb

PtA (2.37)

where the minimum node spatial density is given by (2.33). To be more precise, since ρminenergy

is a function of the number of nodes N , (2.37) is a nonlinear equation in N . However, ascan be observed from Figure 2.8, ρmin

energy is slightly dependent on N (for a given value ofBERmax

route), so that (2.37) can be used directly to derive an estimate of the threshold valueNmin.

In Figure 2.13, the maximum sustainable number of hops is shown, in a networkwith N = 100 nodes, considering (i) strong LOS communications, (ii) strong multipathcommunications and (iii) LOS communications with multipath. In all cases, the transmitpower is set to Pt = 1 µW and the data-rate is set to Rb = 5 × 104 b/s. While in the twoscenarios with strong LOS and LOS with multipath the maximum sustainable number of hopsbecomes larger than the average number of hops nh, in the scenario with strong multipath themaximum sustainable number of hops is always lower than the average number of hops.In other words, for the network parameters considered in Figure 2.13 connectivity is absentin the presence of strong multipath.

The average sustainable number of hops, denoted by nsh, is defined as the minimumbetween the maximum sustainable number of hops nmax

h , given by (2.36), and the averagenumber of hops nh. In other words, we define

nsh � min{nmax

sh , nh} =

⌊

log(1 − BERmaxroute)

log(1 − BERlink)

⌉N ≤ Nmin

�(√

N) N > Nmin.

The average sustainable number of hops is related to the connectivity of the network, sinceit quantifies the effective distance which can be covered, through a multi-hop transmission,with an acceptable final BER. Based on the developed framework, it is possible to show thatthe average sustainable number of hops is related to the concept of the minimum necessarynumber of neighbors needed for full connectivity [54, 55]. More details on the application ofthe proposed communication-theoretic approach to this scenario are given in [56, 57].

In Figure 2.14, the typical behavior of the average sustainable number of hops, as afunction of the number of nodes for a fixed network area, is shown. It is evident that thenode spatial density is a crucial parameter for an ad hoc wireless network. In fact, there existsa sharp threshold, given by Nmin/A, such that if the node spatial density falls below it, thenthe network basically cannot support any communication (in this case, nsh = nmax

sh < nh).This is equivalent to saying that connectivity is lost if the node spatial density falls belowa certain value. At the other extreme, if N is above the critical value Nmin, the maximumsustainable number of hops is large (nmax

sh > nh and nsh = nh), and a possible routing strategyshould take this specific information into account – in fact, from a physical layer perspective,communication over long routes is feasible. This bimodal behavior of the average sustainablenumber of hops related to the connectivity in ad hoc wireless networks, is also predicted inthe realm of percolation theory [32].


0 200 400 600 800 1000 1200 1400N

0

4

8

12

16

20

24

Hops (N/π)1/2

BERroute=10−2

BERroute=10−3max

max

AWGN

Rayleigh fading

Rice fading

K=10 dB

Figure 2.13 Maximum sustainable number of hops in an ad hoc wireless network withregular topology and no INI versus the number of nodes N . The network area is A = 106 m2,the transmit power is Pt = 1 µW and the data-rate is Rb = 50 kb/s. Two possible valuesof the maximum tolerable BER at the end of a route with an average number of nodes areconsidered: for each of these values, the behavior in the three cases with strong LOS, strongmultipath, LOS and multipath, is shown.

Based on condition (2.30), it is possible to derive the following alternative expression forthe average sustainable number of hops:

nsh =

�(

√N) Rb ≤ Rmax

b⌊log(1 − BERmax

route)

log(1 − BERlink)

⌉Rb > Rmax

b

where Rmaxb is given in (2.31). A representative case is shown in Figure 2.15, for a network

with a circular surface of area A = 106 m2 and considering various values of the transmitpower Pt. As one can see, if the data-rate is above the maximum value Rmax

b , the averagesustainable number of hops rapidly drops to zero. For values of the data-rate lower than Rmax

b ,the maximum number of sustainable hops is far larger than the average number of hops, sothat nsh = nh = �(

√N).

In Figure 2.16, the average sustainable number of hops is shown, in the case with Pt =10−5 W, in the three possible propagation scenarios: (i) strong LOS (no fading); (ii) strongmultipath (presence of Rayleigh fading) and (iii) LOS with multipath (presence of Rice fadingwith K = 10 dB). The results shown in Figure 2.16 confirm once more that the presenceof strong multipath has a deleterious effect on network connectivity. In fact, the average


nsh

Strong LOS Strong multipath

N

LOS and multipath

Figure 2.14 Typical behavior of the average sustainable number of hops, as a function ofthe number of nodes N , in a wireless networking scenario without INI.

104

105

106

107

108

Rb

0

4

8

12

16

20

nsh

Pt=10-7

W

Pt=10-6

W

Pt=10-5

W

Figure 2.15 Average sustainable number of hops, as a function of the data-rate, in an ad hocwireless network without INI. The number of nodes is N = 103, the network area is A =106 m2 and the maximum tolerable BER at the end of a multi-hop route with an averagenumber of hops is BERmax

route = 10−3. Various values of the transmit power are considered.Communication links are characterized by strong LOS.

sustainable number of hops drops to zero for much lower values of the data-rate, with respectto a scenario with strong LOS. The reader should also observe that the ‘shape’ of the averagesustainable number of hops curve is similar to that of the link BER.


1×103

1×104

1×105

1×106

1×107

1×108

Rb

0

4

8

12

16

20

nsh

AWGNRayleigh fadingRice fading (K=10 dB)

Figure 2.16 Average sustainable number of hops, as a function of the data-rate, in an ad hocwireless network without INI. The number of nodes is N = 103, the network area is A = 106

m2, the transmit power is Pt = 10−5 W and the maximum tolerable BER at the end ofa multi-hop route with an average number of hops is BERmax

route = 10−3. Three scenarios,corresponding to (i) strong LOS, (ii) strong multipath and (iii) LOS with multipath linkcommunications are considered.

2.5.3 Lifetime of a NodeWe now evaluate, on the basis of the proposed approach, the lifetime of a node, assuming thateach node in the network has finite energy Ebattery. Once the battery energy is depleted, thenode can be considered ‘dead’. Hence, given the data-rate Rb and the transmit power Pt, thelifetime of a continuously active node would be

τlife = Ebattery

Pt. (2.38)

Provided that the node spatial density is fixed and cannot be controlled, the only parametersthat can be practically modified in a node are the transmit power and the data-rate.For example, assuming that the data-rate is fixed, the minimum necessary transmit poweris given by (2.32). Hence, the lifetime of a node can be upperbounded as

τlife ≤ EbatteryρS

Rbρminenergy

. (2.39)

Inspection of (2.39) reveals that a reduction of the data-rate has a significant beneficial effecton the lifetime of a node. Hence, a possible approach to the maximization of the overallnetwork lifetime could be to adaptively adjust the data-rate of each node. For example, ifa node is in a sparse zone, it might be more convenient to reduce the data-rate rather thanincreasing the power to reach some other node farther away. In the presence of mobile nodes,the considered node should wait, in order to increase its data-rate, for a topology changeleading to an increased node spatial density around itself.

2.6. Concluding Remarks 41

2.6 Concluding RemarksIn this chapter, a novel communication-theoretic framework for ad hoc wireless networks hasbeen introduced. The proposed approach takes the viewpoint that in ad hoc wireless networksthe ultimate network performance heavily depends on the capabilities and limitations of thephysical layer. Such a ‘bottom-up’ approach raises several interesting design questions as wellas providing basic insight into the capabilities and limitations of ad hoc wireless networking.The proposed approach allows one to derive simple relationships between fundamentalnetwork quantities (e.g. inequality (2.30)) such as the data-rate, the node spatial density, theBER at the end of a multi-hop route, the noise figure, the carrier frequency, the antenna gainsand the number of nodes. The main findings of this chapter can be summarized as follows.

• The BER performance at the end of a multi-hop route is strictly related to other majornetwork quantities (data-rate, transmit power, noise figure, node spatial density andcarrier frequency). Simple mathematical expressions derived in this chapter can beused as design guidelines (as ‘rules of thumb’) for guaranteeing a prescribed BER atthe end of a multi-hop route with an average number of hops, i.e. for guaranteeing aphysical layer-oriented QoS constraint.

• The use of channel coding can be extremely helpful in sustaining communications insparse networks. The improved robustness towards sparsity comes at the expense ofdelay in the transmission and increased signal processing at each node.7 An attractivealternative could be the use of dynamic adaptation of the data-rate Rb of a single node.The impact of the use of retransmission techniques on the network performance isanalyzed in detail in [58].

• We have introduced the novel concept of minimum spatial energy density, whichquantifies the intuition that a minimum level of energy has to exist in the networkto guarantee full connectivity.

• We have shown that the average sustainable number of hops in an ad hoc wirelessnetwork is a key characteristic for physical layer performance as well as networkperformance (in terms of routing). The existence of a critical threshold value for thenumber of nodes (and then in terms of node spatial density), below which connectivityis basically lost, has been verified, and an expression which relates this threshold to themajor network parameters has been derived.

• A simple expression has been derived that shows how the lifetime of a node can berelated to the transmit power Pt, the data-rate Rb and the minimum spatial energydensity ρmin

energy.

7The power consumption related to the processing associated with decoding seems negligible, compared to thepower consumption involved with the transmission and reception [35].

ad hoc wireless networks (a communication-theoretic perspective) || a communication-theoretic...

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