Interaction of Coding and Multihop Transmission in Wireless Networks

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Interaction of Coding and Multihop Transmission in Wireless Networks Martin Haenggi, Marcin Sikora, J. Nicholas Laneman, Daniel J. Costello, Jr., and Thomas Fuja Department of Electrical Engineering University of Notre Dame Notre Dame, IN 46556 E-mail: [email protected]

Abstract— For multihop wireless networks, a fundamental question is whether it is advantageous to route over many short hops (short-hop routing) or over a smaller number of longer hops (long-hop routing). Short-hop routing has gained a lot of support, and its proponents mainly produce two arguments: higher SNR at every hop and smaller interference. This paper focuses on the first argument and examines different routing strategies from an informationtheoretic perspective, taking into account the rate penalty incurred by multihop transmission. We analytically derive the optimum number of hops for a linear network given the path loss exponent and the desired end-to-end rate for the case where the delay is not constrained. The analysis is based on the concept of asymptotic spectral efficiency. It turns out that routing over longer hops is competitive for higher rates; in particular, single-hop transmission outperforms two-hop transmission if the rate is larger than the path loss exponent. For delay-constrained transmission, we compare single-hop and two-hop transmission using a sphere-packing bound and simulations with practical codes, taking into account the reduced block length in the two-hop case. Index Terms—Wireless networks, routing, coding.

I. I NTRODUCTION

I

N the context of studying the advantages and disadvantages of layered architectures for wireless networks, cross-layer design has become a prevalent research topic [1]. Many examples suggest the potential benefits of limited interaction among algorithms at traditionally separate layers. These benefits include, for example, increased capacity, broader coverage, higher reliability, and fair and efficient resource allocation. Despite the growing popularity of cross-layer design, comparatively little research has been devoted to the basic and critical interactions between channel coding and routing for emerging wireless networks that employ multihop

transmission. These networks include, for example, the important classes of ad hoc [2–5], multihop cellular [6], and sensor networks [7–10]. We refer to these networks collectively as wireless multihop networks. The issues of routing and channel coding have largely been studied in isolation, at the network and physical layers, respectively, whereas this paper focuses on their interaction. The main result is that the benefits of multihop communication are eroded by bandwidth and delay constraints, in particular at high spectral efficiencies. Specifically, comparing single-hop and two-hop transmission, single-hop is generally preferable when the rate (bps/Hz) is larger than the path loss exponent. More generally, we will derive the optimum hop number for a desired rate using the concept of asymptotic spectral efficiency. The rest of the paper is organized as follows: In Section 2, we introduce the main issues and the system model. Section 3 considers the case of multihop transmission without a delay constraint, derives the optimum number of hops for this case, and introduces the notion of asymptotic spectral efficiency. In Section 4, we present analytical and simulation results for the delay-constrained case, and Section 5 concludes the paper. II. S YSTEM M ODEL

AND

K EY I SSUES

A. System model 1) Linear topology and AWGN channel model: We consider the communication system shown in Fig. 1. It consists of a source S and a destination D and N − 1 intermediate (relay) nodes placed equidistantly on the line SD. The distance SD is assumed to be normalized to 1. The objective is the reliable delivery of bits generated at S at a rate 1/Tb bits per second to the destination node using coded transmission. The resources available comprise a frequency band permitting a signaling rate 1/Ts

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0 S

1

2

2

N

... ...

D

N−hop routing D S single−hop routing Fig. 1. N -hop and single-hop transmission.

complex-valued symbols per second and a total transmit power PT . The received power for a transmission over distance l is given by PR = PT Gl−α ,

(1)

where: G is a constant depending upon antenna gain and wavelength; α is the path loss exponent; PT is the transmit power. In general, G  lα , but since the constants do not matter for our analysis, we will assume G = 1 for simplicity. We assume that the transmission is corrupted by additive white Gaussian noise (AWGN) with one-sided spectral density N0 , so that the average received signal-tonoise ratio (SNR) is given by [11] SNR =

PR , N0 W

(2)

where W is the transmission bandwidth. Assuming a path loss model with α > 2, proponents of multihop routing argue that more short hops are preferable to fewer long hops, because the minimum SNR along the route is larger for multihop. Specifically, if the nodes are equally spaced, the minimum SNR along the multihop route is a factor of N α larger, corresponding to a gain of 10α log10 (N ) dB. If the total power of the network is to be the same under single-hop and multihop transmission, the gain of multihop transmission reduces to 10(α − 1) log10 (M ) dB. 2) TDMA channel access: With perfect synchronization, coherent reception would be possible for the nodes in Fig. 1, and the channel could be interpreted as a Gaussian relay channel. In this case, the achievable data rates can always be increased by increasing the number of relays— at least as long as the power law (1) holds [12]. In most practical ad hoc networks, however, neither the synchronization nor the complexity required for such operation can be met, so attention must be directed to simpler but more robust transmission schemes. Therefore, in this paper, we focus on TDMA channel access. In a TDMA multihop system, the end-to-end transmission is split into N partial transmissions (hops) between neighboring nodes. At any time, at most one

node is transmitting, hence there is no interference at any receiver. Since by assumption all the internode distances are equal, all the hops are identical and should be assigned equal portions of the resources (channel time and power). Thus each hop can utilize 1/(N TS ) channel uses per second, each involving symbols with transmitted energy Es = PT Ts . Note that TDMA is the most efficient channel access scheme for small N . Define the signal-tonoise-and-interference ratio (SIR) as the ratio of The minimum SIR required for reception most likely only allows the simultanous transmission at every fourth1 to eighth hop (depending on the path loss exponent, but independent of the hop length), intra-route spatial reuse is only possible if N is at least 8 (for each link, there must be at least on other link that can be used simultaneously). Most current practical networks such as sensor networks restrict the number of hops to a small number2 , so often TDMA (without intra-route spatial reuse) is indeed the optimum channel access scheme. For single-hop transmission the received symbol energy equals the transmitted symbol energy. For multiple hops, however, the distance between neighboring nodes is 1/N , so the received energy is Es N α . The SNR per hop is thus Es N α /N0 . B. Key issues 1) Channel Coding: If channel codes are employed at each hop, then decoding delay accumulates. As illustrated in Figure 2, for the same end-to-end delay, the system can employ either more hops with less powerful codes, or fewer hops with more powerful codes. (In Fig. 2, L is code length and R is code rate.) A variety of tradeoffs between the number of hops and decoding delay need to be characterized. For conventional codes such as block and convolutional codes, block- and constraint-lengths may suffice to parameterize decoding delay. For more powerful codes, such as turbo and low density parity-check (LDPC) codes with iterative decoding, the number of iterations also needs to be considered. 2) Multihop Transmission: Multihop transmission can be viewed as communication over several channels in cascade. From this perspective, the most fundamental limit on the rate of reliable communication, i.e., the channel capacity, of multihop transmission is well-known [13] and often motivates the use of multihop. To illustrate this idea, 1 If every third link were active at the same time, the resulting SIR would be smaller than 2−α , which may not be sufficient for an acceptable packet error rate. 2 See, e.g., the experimental testbeds used in DARPA projects.

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3 R

mined using simulators that are based on simplistic models of the wireless channels, not taking into account channel coding and accumulation of bit or packet error rates and delay.

L 2R

2R

C. Performance measures L/2

L/2

Fig. 2. Single-hop vs. multihop with limited delay.

let Ci,j denote the capacity of the channel between nodes i and j. Then multihop transmission has capacity CMH =

min

i=1,2,...,N

Ci−1,i

(3)

and offers higher rates if CMH > C0,N . That this holds in many wireless situations is due to the path loss characteristics of the wireless medium. In our case, for the complex-valued channel with additive white Gaussian noise (AWGN), the channel capacity is given by Ci,j = log2 (1 + SNRi,j ) , where SNRi,j is of the form (2). Thus, from a capacity standpoint, the SNR improvement of multihop over single-hop (cf. Fig. 1) translates into a higher capacity. The analysis above is accurate if nodes have individual resources in terms of power, bandwidth, and delay at their disposal; however, in many wireless settings, bandwidth, delay, and power, must be allocated across the entire network. 3) Routing: The problem of whether routing over many short hops or fewer long hops leads to better performance was raised in the early days of packet radio [14] but never thoroughly and rigorously addressed. Indeed, 15 years later, in [15], it is still pointed out that (when we replace a larger number of short hops by a smaller number of long hops) “It is far from clear what happens to the overall transmission energy, since to implement a nearestneighbor policy, significantly augmented overhead control traffic will be required to coordinate the establishment of the routing paths and access control protocols across the entire network.” Even without considering overhead traffic, but including delay and reliability, the tradeoffs are not clear-cut. Although the routing problem in multihop wireless networks is well-studied, most of the work in routing is focused on single-path near(est)-neighbor schemes [2, 16, 17]. The performance of these algorithms is usually deter-

To fairly compare the performance of systems involving different numbers of hops, the performance measures need to be chosen carefully. To measure the bandwidth efficiency, we will use the bandwidth-normalized rate R = Ts /Tb . Note that this is the end-to-end rate—the spectral efficiency at each hop equals N R due to the N -fold reduction in channel uses in TDMA. Power efficiency will be evaluated in terms of Eb∗ /N0 , where the total energy per bit Eb∗ is defined as the sum of all the transmit energies per bit spent over all N hops, i.e., the energy spent to deliver one bit from S to D. Hence, Eb∗ = N Es /(N R) = Es /R. The performance of the TDMA linear ad hoc network with N nodes will be characterized by the highest achievable bandwidth-normalized rate R for a given Eb∗ /N0 . Without a delay constraint, the highest achievable rate is understood to be the highest rate for which an arbitrarily small BER can be obtained using forward error correction. If the end-to-end delay is limited, a rate is achievable if there exists a coding scheme with appropriate latency operating with a BER or block error rate (BLER) not exceeding some prescribed value. III. E ND - TO - END R ATES AND A SYMPTOTIC S PECTRAL E FFICIENCY Without a delay constraint, the highest achievable transmission rate is the channel capacity. For a single-hop, this is simply   Es R = log2 1 + . (4) N0 With N hops, the transmitted energy per symbol remains unchanged, but the received energy is N α times higher due to the reduced attenuation. At the same time, each hop must accommodate the transmission of the same number of information bits in 1/N -th of the channel uses, which requires an N -fold increase in the per-hop spectral effiency. These two effects, the SNR gain and the rate penalty, need to be traded off against each other. We have for the N -hop case   Es α 1 log2 1 + N R= . (5) N N0 The curves for (5) are plotted in Fig. 3 for N = 1, . . . , 6 and α = 2, 4 (with Eb∗ = Es /R). It can be observed

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R0 = ∞, the optimum hop number for an end-to-end rate R is given by

4 3.5 N=1

RN 6 R < RN −1

N=2

Achievable rate

3 2.5

= (N + 1)−α (2(N +1)RN − 1)

R =1.58 1

N=6

R2=0.93

(N + 1)−α (2RN )N +1 − N −α (2RN )N

0.5

−5

0 Eb/N0 (in dB)

5

6 N=2

5

N=3

R =3.91

Achievable rate

4

1

3 N=6

R =2.29 2

2 R =1.63 3

1

0 −25

−20

−15

−10

−5 0 E /N (in dB) b

5

10

+ N −α − (N + 1)−α = 0

10

(a) α = 2 N=1

15

(7)

or

R3=0.66

0 −10

(6)

Es /N0 = N −α (2N RN − 1)

2

1

Nopt (R) = N .

Equating Es /N0 for N hops and N + 1 hops, we have

N=3

1.5

⇐⇒

20

(8)

as the defining equation for RN . For N = 1, we find R1 = log2 (2α − 1). So, if the desired end-to-end spectral efficiency exceeds α, the transmission from S to D should be performed in a single hop. In general, 2RN is a root of a polynomial of order N + 1. In the Appendix, the properties of these polynomials are discussed, and it is shown that there exists exactly one positive solution of (8) which tends to zero as N grows. Fig. 4 presents the optimum number of hops Nopt for a given normalized end-to-end rate R and the corresponding link spectral efficiencies RNopt at each hop. Of particular interest is the behavior of N RN , which converges to a positive constant that we denote as the asymptotic spectral efficiency. In the spectral efficiency plots (Fig. 4), this asymptotic spectral efficiency is indicated by the dashed line.

0

(b) α = 4 Fig. 3. End-to-end spectral efficiencies achievable with one to six hops: (a) α = 2, (b) α = 4. The values Ri are the crossover spectral efficiencies, i.e., the rates where the curves for N = i and N = i + 1 intersect.

Proposition 1 (Asymptotic spectral efficiency.) There exists an asymptotic spectral efficiency S(α) : = lim N RN = lim RNopt (R) N →∞

= that no single curve is optimum for all rates, but every curve dominates in some range of Eb∗ /N0 values. It may be of interest to know for which value of Eb∗ /N0 an N hop system performs best. Note, however, that if the total distance SD is L rather than 1, the Eb∗ /N0 axis is shifted by −10α log10 L dB. So it makes more sense to determine the ranges of the end-to-end spectral efficiency R for which each curve dominates, or, given a desired R, what the optimum number of hops is. The value of R where the curves for N and N + 1 intersect will be referred to as the crossover spectral efficiency or crossover rate and be denoted RN . If we define

R→0

W(−αe−α ) + α , ln 2

(9) (10)

where W(·) denotes the (principal branch of the) Lambert W function3 . A tight upper bound is given by S(α) /

α(1 − e−α ) . ln 2

(11)

Proof: See the Appendix. From S(α), we can derive a very good approximation for the optimum number of hops given a certain rate R. From N RN < S(α) < N RN −1 , it follows that ∃RN < R∗ < RN −1 3

s.t.

N R∗ = S(α) ,

(12)

Sometimes called the Product Log function, e.g., in Mathematica.

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5 8 numerical approximation

Best number of hops Nopt(R)

i.e., Nopt (R∗ ) = S/R∗ . Together with N < S(α)/RN < N + 1, this implies that S(α)/R is a smooth approximation of Nopt (R) and that

7 6 5 4

∃  1 s.t. for RN (1 + ) 6 R 6 RN −1 (1 − ) :

3 2

Nopt (R) = [S(α)/R] , (13)

IV. O PTIMUM N UMBER OF H OPS C ONSTRAINT

WITH A

D ELAY

The predictions of the optimum number of hops made in the previous section were based on the assumption that the block lengths used by the channel codes can be arbitrarily large. In many applications, however, there is a limit on the tolerable delay. The relationship between channel coding and routing strategies under delay constraints is addressed in this section. The main consequence is that, due to the accumulation of processing time, transmission time, and coding and decoding delay, the block lengths used in N -hop transmission can be only 1/N of the block length of single-hop transmission (see Fig. 2). Suppose again the source produces information bits at a rate of 1/Tb bits per second and the available channel bandwidth allows transmission of 1/Ts symbols (channel uses) per second. The maximum tolerable end-to-end delay is denoted by D, so the information bits must be recovered at the receiver after D seconds. The number of channel uses is Ds = D/Ts , and we assume that by choosing the block length (in terms of coded symbols) for an N -hop system as n = Ds /N , the delay constraint is satisfied. Since this reduction in block length by a factor of N is accompanied by an N -fold increase in the code rate (due to the TDMA scheme), we can think of coding for an N -hop system as squeezing a fixed number of information bits (namely RDs ) into Ds /N channel symbols. Error-correcting codes with a limited block length cannot achieve arbitrarily low error rates. Hence, for a

Spectral efficiency per hop (in b/s/Hz)

0.5

1

0

0.5

1

1.5

2

2.5

3

3.5

1.5 2 2.5 Bandwidth−normalized rate (in b/s/Hz)

3

3.5

3.5

3

2.5

2

1.5

(a) α = 2 8 (R)

is accurate for large intervals of R and never off by more than 1. Since S(α) lies in between N RN and (N + 1)RN , we have RN ≈ S(α)/(N + 1/2) as a simple approximation for RN .

0

Numerical Approximation

7

opt

(14)

Best number of hops N

Nopt (α, R) ≈ [S(α)/R]

6 5 4 3 2 1

Spectral efficiency per hop (in b/s/Hz)

where [x] denotes the nearest positive integer to x. Consequently, the approximation (the dashed line in the upper plots of Figs. 4 (a) and (b))

1

0

1

2

0

1

2

3

4

5

6

7

8

6

7

8

8 7 6 5 4 3

3 4 5 Bandwidth−normalized rate (in b/s/Hz)

(b) α = 4 Fig. 4. Optimum number of hops and corresponding per-hop spectral efficiencies: (a) α = 2, (b) α = 4. For R → 0, the optimum hop number tends to infinity, but the product RNopt (R) tends to a constant, the so-called asymptotic spectral efficiency S(α) that only depends on the path loss exponent (dashed line in the spectral efficiency plots). The dashed approximation in the upper plots shows the optimum hop numbers based on Nopt ≈ [S(α)/R].

rate R to be achievable in a multihop system, a coding scheme must exist for which the end-to-end block error rate (BLER) falls below some tolerable value Pe . Since a block error at any of the N hops will result in a block error at the destination (except for the unlikely event of several block errors canceling each other), we can require that the BLER at each hop does not exceed Pe /N . Hence, in a system with a delay constraint, increasing the number of hops is penalized by requiring coding schemes that operate at a higher rate, with a smaller block length, and

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requiring a lower BLER, while, at the same time, enjoying an SNR increase by a factor N α .

2 1.8 R1

1−hop/2−hop

1.6

A. Analysis based on the sphere packing bound

Turbo code

(15)

where the function Qn depends on the signal amplitude parameter A, an angle Θ, and the block length n (in realvalued channel symbols). The angle Θ is the half-angle of the n-dimensional cone encompassing a fractional solid angle 2−rn and is related to the block length n and code rate r (in bits per real-valued symbol) by the approximation sinn−1 Θ (16) 2−rn ≈ √ 2πn cos Θ for large n. This equation needs to be solved for Θ. (15) does not have a closed-form solution, so we will use the following large n approximation
n 2 G sin Θe(−A −AG cos Θ)/2 , Qn (Θ, A) ≈ √ √ nπ 1 + G sin Θ(AG sin2 Θ − cos Θ) (17) where p
1 G(Θ, A) = A cos Θ + A2 cos2 Θ + 4 . (18) 2 The variables used in the above expressions are given by A2 = 2Es N α /N0 , n = 2Ds /N , and 2r = RN , where the factor 2 in the last two formulas accounts for the fact that we are using a complex-valued channel. We will write Q(Ds /N, RN, Es N α /N0 ) to denote Qn (Θ, A), computed from (17), where G is given by (18) and Θ is obtained from (16). We compute the crossover points RN by numerically searching for the solutions of the equations D Es α  Pe s N =Q , RN, (19) N N N0   D Pe Es s (N + 1)α (20) =Q , R(N + 1), N +1 N +1 N0 for Pe = 10−4 . The results of this search are shown in Fig. 5 for α = 2 and α = 4. The plots indicate that the crossover points remain quite stable for most practical values of the delay constraint Ds , which means that the simpler predictions about RN based on the capacity formulas in the previous section remain fairly accurate.

1.2 1

Convolutional code

R

2

2−hop/3−hop

0.8

3−hop/4−hop

R3

5−hop/6−hop

R5

0.6 0.4 0.2 0 2

4

6

8

10 12 14 Block length log2 n

16

18

20

(a) α = 2 5 4.5 4

1−hop/2−hop

R1

2−hop/3−hop

R

3−hop/4−hop

R3

4−hop/5−hop

R4

3.5 Crossover rate

Pe > Qn (Θ, A) ,

1.4 Crossover rate

A useful tool that relates the BLER to the code rate, the SNR, and the block length for the AWGN channel is the sphere-packing bound introduced by Shannon [18]. Following the notation in [19], the probability of a block error on a real-valued AWGN channel is lower bounded by

3

2.5

2

2

1.5 1 0.5 0 2

4

6

8

10 12 14 Block length log2 n

16

18

20

(b) α = 4 Fig. 5. Crossover rates for delay-constrained transmission: (a) α = 2, (b) α = 4. For large block lengths, the crossover rate tends to the value RN .

B. Simulation results In order to obtain simulation results with practical codes, we focus on two transmission strategies, a singlehop strategy and a two-hop strategy. 1) Crossover rate: The sphere-packing bound allows the estimation of the achievable performance of channel codes under limited transmission power and block length. In a practical system, there is one more constraint, namely the code complexity. Therefore we repeat the process of finding the first crossover rate R1 using the simulated BLER vs. Eb /N0 performance obtained for two common families of codes: convolutional codes (CC) and turbo

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TABLE I

2.5

S IMULATED CODING SCHEMES .

0.5 1 2

4

Convolutional codes (decoding delay: 100 bits) QPSK + CC R=1/4, memory 10 [20] QPSK + CC R=1/2, memory 10 [20] 8PSK + TCM R=2/3, memory 7 [20] 32CROSS + TCM R=4/5, memory 6 [20]

Turbo codes (decoding delay: 1000 bits) QPSK + Turbo code, R=1/4 [21] QPSK + Turbo code R=1/2 [21] 16QAM + Turbo code, R=1/2 [21] 32CROSS + Turbo TCM R=4/5 [22]

codes. Each family includes codes with increasing spectral efficiencies and decreasing block lengths. The parameters of the codes used are listed in Table 1. The Eb /N0 values needed to achieve a BLER of 10−4 have been established for each code and plotted vs. the spectral efficiency in Fig. 6—once assuming they were used in a single-hop system and once for a two-hop system for α = 2.4 The crossover rates obtained from these curves are also included in Fig. 5(a) with their corresponding block lengths. These rates fall below the values predicted by the sphere-packing bound. Although the method we used in determining the crossover rates is sensitive to the choice of representative codes, this result suggests that in practical scenarios there are additional incentives to use a lower number of hops that cannot be assessed by just comparing the capacities. 2) Waterfall curves: To satisfy the constraints in the single-hop system, we can use a code with block length LSH = T /Ts channel symbols and a spectral efficiency RSH = Ts /Tb bits per symbol. For the two-hop system, we will assume time division multiplexing, i.e., half of T is devoted to sending information bits from the originating node to the intermediate node, and the other half to sending them from the intermediate node to the final node. For each of the two links we employ a code with block length LMH = T /(2Ts ) = (1/2)LSH symbols and code rate RMH = 2Tb /Ts = 2RSH . Hence, each time we compare single-hop and two-hop, we shall pick one code for the single-hop scenario, and another code with twice the rate and half the block length for the two-hop case. The bit-error rate (BER) vs. Eb /N0 curve for the for4

The rates of the simulated codes were not high enough to yield intersections for α = 4.

Bandwidth−normalized rate (bps/Hz)

Crossover 1−2 hops

Spectral efficiency

2

1.5

(theoretical)

Crossover 1−2 hops for turbo codes

1 Crossover 1−2 hops for convolutional codes

0.5

0 −4

−2

0

2 Eb/N0 (in dB)

4

6

8

Fig. 6. Comparison of the crossover spectral efficiencies for α = 2.

mer scheme characterizes the performance of single-hop transmission. For the two-hop case, we need to normalize the BER curve to account for multihop transmission: scale Eb /N0 by a factor of 2α−1 and scale the BER by a factor of two. (Here, 2α−1 comes from a path attenuation advantage of 2α obtained by halving the distance and a factor of 1/2 that is included because the rate is doubled, and so every channel symbol corresponds to twice as many information bits. The BER scaling is due to the fact that each hop can cause independent errors, approximately doubling their number.) The scenarios that were simulated had information bit delays T /Tb of 50 and 1024, and code rates Ts /Tb equal to 1/2, 1, and 2 bits per channel symbol. For the delay of 50 bits we used convolutional codes (for BPSK and QPSK modulation) and trellis coded modulation (TCM) schemes (for higher order modulation). For T /Tb = 1024 bits we used the 3GPP turbo code (http://www.3gpp.org/) punctured to rates 1/4 and 1/2 mapped onto QPSK and 16QAM signal constellations, and a turbo TCM scheme [22] for higher spectral efficiencies (see Table 1 for an overview). In all the simulations, a path loss exponent of α = 2 was used. The comparisons in Figure 7 clearly indicate that twohop transmission’s advantage due to reduced path loss deteriorates with increasing spectral efficiency—an experimental verification of the capacity-motivated conclusion in the previous section. In both cases, when the singlehop scheme has a bandwidth-efficiency of 2 bits/sec/Hz, the performance of the competing two-hop scheme is inferior to that of the single-hop scheme. Given the trend seen

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V. C ONCLUSIONS

0

10

R = 1/2 bit/sym (QPSK, CC 1/4) SH R = 1 bit/sym (QPSK, CC 1/2) MH RSH = 1 bit/sym (QPSK, CC 1/2) RMH = 2 bit/sym (8PSK, TCM 2/3) RSH = 2 bit/sym (8PSK, TCM 2/3) RMH = 4 bit/sym (32CROSS, TCM 4/5)

−1

10

−2

Bit Error Rate

10

−3

10

−4

10

0.1 dB

2.2 dB

−5

10

1.7 dB

−6

10

−2

−1

0

1

2

3 4 Eb/No (in dB)

5

6

7

8

9

For wireless multihop networks, such as ad hoc and sensor networks, a fundamental question is whether it is advantageous to route over many short hops or over a smaller number of longer hops. The proponents of shorthop routing mainly produce two arguments, namely the SNR gain and the reduction in interference. This paper addresses the first point and shows that, depending on the path loss exponent, the SNR gain may be offset by the necessary increase in the spectral efficiency. In particular, single-hop routing outperforms two-hop routing for rates larger than the path loss exponent. For delay-constrained transmission, this rate threshold (above which single-hop is better) is somewhat lower.

(a) 0

A PPENDIX : A SYMPTOTIC S PECTRAL E FFICIENCY

10

R = 1/2 bit/sym (QPSK, Turbo 1/4) SH R = 1 bit/sym (QPSK, Turbo 1/2) MH RSH = 1 bit/sym (QPSK, Turbo 1/2) RMH = 2 bit/sym (16QAM, Turbo 1/2) RSH = 2 bit/sym (16QAM, Turbo 1/2) RMH = 4 bit/sym (32CROSS, Turbo TCM)

−1

10

Consider the sequence RN given by (8). q := 2RN is given by N α (q N +1 − 1) − (N + 1)α (q N − 1) = 0 .

−2

Bit Error Rate

10

The trivial solution q = 1 (RN = 0) always exists and can be factored out to yield the N -th order monic polynomial

−3

10

−1  N + 1 α  NX xk . pN (x) = x + 1 − N

−4

10

N

2.0 dB

−5

10

−3

(21)

−2

−1

0

0.7 dB

1

3.0 dB

2 3 Eb/No (in dB)

4

5

6

7

8

(b) Fig. 7. Bit-error rate (BER) vs. Eb /N0 for varying transmission rates: (a) convolutional codes and trellis coded modulation (TCM) and (b) turbo codes and turbo TCM. Solid curves correspond to single-hop transmission, and dashed curves correspond to two-hop transmission. The path loss exponent is α = 2. These results suggest single-hop transmission gains significant advantages at high spectral efficiency.

in Figure 7, it is clear that this advantage for single-hop would increase at higher spectral efficiencies. These results indicate that multihop transmission— and, more specifically, nearest-neighbor multihop transmission—may not be the best strategy for networks operating at high spectral efficiency. Conversely, nearestneighbor multihop may well be optimal for networks operating at low spectral efficiency.



(22)

k=0

The first twelve polynomials are displayed in Fig. 8 for α = 2 and α = 4. All but the leading coefficient are negative, so there exists exactly one positive root5 , which must be larger than 1 since pN (1) < 0 for α > 1. (For α = 1, pN (1) = 0 for all N .) This positive root q > 1 is the one we are interested in since RN = log2 (q) > 0. For N = 1, we find R1 = log2 (2α − 1), as noted in Section 3. For N = 2, we have √
R2 = −1 − α + log2 3α − 2α + 9α + 2 · 6α − 3 · 4α . √ In particular, for α = 2, R2 = log √ 2 (5+ 105)−3 ≈ 0.93 and for α = 4, R2 = log2 (65+ 8385)−5 ≈ 2.29. While R3 and R4 can still be determined analytically, the higher Ri , i > 4, must be found numerically. Of interest is the behavior of q for N → ∞. Rearranging pN (q) = 0 by summing the geometric series PN −1 qk k=0 q N yields 1 1 − qN = , N q (1 − q) (1 + 1/N )α − 1

5

This is due to Descartes’ Rule of Signs.

(23)

IEEE INFOCOM 2005

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N RN = log2 (q N ), this implies for RN that

3 2.5

0 < lim N RN < ∞ . N →∞

2 1.5

Note that N RN denotes the spectral efficieny per hop, so we have proven the existence of an asymptotic spectral efficiency defined as follows:

p

12

p(x)

1 0.5

p

3

S(α) = lim RN N

0

N →∞

−0.5

= lim RN (N + 1) = lim RN −1 N ,

p2

N →∞

−1 −1.5

p

1

−2 −1

−0.5

0

0.5 x

1

1.5

2

(a) α = 2 2

p

12

0

p

3

p(x)

−2

N →∞

S(α) = lim RNopt (R) . R→0

−8

−10

p

1

−0.5

0

0.5

1

1.5

2

2.5

3

3.5

Since limN →∞ (1 + 1/N )N = e, we have for large N 4

x

2RN (N +1) − 1 , 2RN N − 1

(29)

α 2RN N (2RN − 1) = . N 2RN N − 1

(30)

eα/N ≈

(b) α = 4 Fig. 8. Polynomials pN (x) for N = 1, . . . , 12: (a) α = 2, (b) α = 4. Note that there is a single minimum for all N , and that there is only one real root for odd orders.

where the right side diverges for N → ∞ since limN →∞ (1 + 1/N )α = 1. So, limN →∞ q = 1 on the left side, or limN →∞ RN = 0, as expected. On the other hand, q N cannot converge to one since limN →∞ pN (1) = 1 − α, and hence the coefficients do not add up to zero (except for α = 1). Also, q N cannot diverge since that would lead to a contradiction. Assuming it diverges, the q −N term in 1 1 − q −N = q−1 (1 + 1/N )α − 1

(27)

To find this limit, we need to solve (8) in the regime N → ∞ or, equivalently, RN → 0. Rearranging yields   N + 1 α 2RN (N +1) − 1 = . (28) N 2RN N − 1

p2

−6

(26)

This is illustrated in Fig. 4, which presents the optimum number of hops Nopt for a given normalized end-to-end rate R and the corresponding link spectral efficiencies RNopt at each hop. The dashed line in the spectral efficiency plot indicates the asymptotic spectral efficiency. In (26), RN N approaches S from below, and RN (N + 1) approaches S from above (see Fig. 4). So, RN N < S < RN (N + 1) for all finite N . Together with (6), this implies that the asymptotic spectral efficiency can also be expressed as

−4

−12 −1

(25)

(24)

vanishes, resulting in q = limN →∞ = (1 + 1/N )α , and limN →∞ q N = eα , which contradicts the assumption. So, the limN →∞ q N exists, and it is bigger than one. Since

which tends to one. So the first order Taylor expansion ln(x) ≈ x − 1 is accurate, yielding

Now, substituting S = RN N we obtain α(1 − 2−S ) = S

2RN − 1 . RN

(31)

Noting that the right side goes to S ln 2 for RN → 0 (this is the same expression as the Shannon limit), we have α S = (1 − 2−S ) (32) ln 2 as the defining equation6 for S. Rearranging yields (S ln 2 − α)eS ln 2−α = −αe−α , 6

(33)

We are interested in the positive solution of this equation. Of course, the trivial solution S = 0 exists for all α.

IEEE INFOCOM 2005

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6.5

asymptotic spectral efficiency

6 5.5 5 4.5 4 3.5 3 S(α) exp. bound linear bound

2.5 2 2

2.5

3

α

3.5

4

4.5

Fig. 9. Asymptotic spectral efficiency and two upper bounds.

which can be solved using the (principal branch of the) Lambert W function W(·) which is defined as W(x)eW(x) = x, resulting in S=

W(−αe−α ) + α . ln 2

(34)

This concludes the proof of Proposition 1. We know that, for (the hypothetical case) α = 1, S = 0 since q N = 1. This is indeed the case, since W(−1/e) = 1, which confirms that for α = 1, multihop transmission would never be efficient. For increasing α, e−α tends to zero, and W(x) / x for small x, so we obtain an increasingly tight upper bound S/

α(1 − e−α ) . ln 2

(35)

A looser linear upper bound, that also gets tight, is obviously α/ ln 2. S(α) and the two bounds are displayed in Fig. 9. The numerical values for α = 2 and α = 4 are S = 2.299 and S = 5.656, respectively. R EFERENCES [1] Vikas Kawadia and P. R. Kumar, “A Cautionary Perspective on Cross-Layer Design,” IEEE Wireless Commun. Mag., 2003, Submitted for publication. Available at http://black.csl.uiuc.edu/˜prkumar/ps_ files/cross-layer-design.pdf. [2] Charles E. Perkins, Ed., Ad Hoc Networking, Addison Wesley, 2000, ISBN 0-201-30976-9. [3] Magnus Frodigh, Per Johansson, and Peter Larsson, “Wireless ad hoc networking – The art of networking without a network,” Ericsson Review, , no. 4, pp. 248–263, 2000. [4] C-K. Toh, Ad Hoc Mobile Wireless Networks - Protocols and Systems, Prentice-Hall, 2002, ISBN 0-13-007817-4.

[5] Andrea J. Goldsmith and Stephen B. Wicker, “Design Challenges for Energy-Constrained Ad Hoc Wireless Networks,” IEEE Wireless Communications, vol. 9, no. 4, pp. 8–27, Aug. 2002. [6] Ying-Dar Lin and Yu-Ching Hsu, “Multihop Cellular: A New Architecture for Wireless Communications,” in IEEE INFOCOM’00, 2000, vol. 3, pp. 1273–1282. [7] I. F. Akyildiz, W. Su, Y. Sankarasubramaniam, and E. Cayirci, “A Survey on Sensor Networks,” IEEE Communications Magazine, vol. 40, no. 8, pp. 102–114, Aug. 2002. [8] Edgar H. Callaway, Wireless Sensor Networks: Architectures and Protocols, Auerbach Publications, 2003, ISBN 0-8493-1823-8. [9] “Sensor Networks and Applications,” IEEE Proceedings, vol. 91, no. 8, Aug. 2003. [10] M. Ilyas (Ed.), Handbook of Sensor Networks: Compact Wireless and Wired Sensing Systems, CRC Press, 2004. [11] Theodore S. Rappaport, Wireless Communications – Principles and Practice, Prentice Hall, 2nd edition, 2002, ISBN 0-13042232-0. [12] L. Xie and P. R. Kumar, “A Network Information Theory for Wireless Communication: Scaling Laws and Optimal Operation,” IEEE Transactions on Information Theory, vol. 50, no. 5, pp. 748–767, May 2004, Available at http://black.csl.uiuc.edu/˜prkumar/ps_ files/net_inf_theory.pdf. [13] Thomas M. Cover and Joy A. Thomas, Elements of Information Theory, John Wiley & Sons, Inc., New York, 1991. [14] Barry M. Leiner, Donald L. Nielson, and Fouad A. Tobagi, “Issues in Packet Radio Network Design,” Proceedings of the IEEE, vol. 75, no. 1, pp. 6–20, Jan. 1987. [15] Anthony Ephremides, “Energy Concerns in Wireless Networks,” IEEE Wireless Communications, vol. 9, no. 4, pp. 48–59, Aug. 2002. [16] Elizabeth M. Royer and C-K. Toh, “A Review of Current Routing Protocols for Ad-Hoc Mobile Wireless Networks,” IEEE Personal Communications, vol. 6, no. 2, pp. 46–55, Apr. 1999. [17] C-K. Toh, “Maximum Battery Life Routing to Support Ubiquitous Mobile Computing in Wireless Ad Hoc Networks,” IEEE Communications Magazine, vol. 39, no. 6, pp. 138–147, June 2001. [18] C. E. Shannon, “Probability of error for optimal codes in a gaussian channel,” Bell Syst. Tech. J., vol. 38, pp. 611–656, 1959. [19] S. Dolinar, D. Divsalar, and F. Pollara, “Code Performance as a Function of Block Size,” Tech. Rep., TMO Progress Report, May 1998. [20] S. Lin and D. J. Costello, Jr., Error Control Coding, 2nd Ed., Prentice Hall, 2004. [21] “3GPP2. C.S0002-B, Physical layer standard for cdma2000 spread spectrum systems, Release B, ver. 1.0,” Apr. 2002. [22] P. Robertson and T. W¨orz, “Bandwidth Efficient Turbo Trellis Coded Modulation Using Punctured Component Codes,” IEEE Journal on Selected Areas of Telecommunications, vol. 16, no. 2, pp. 206–218, Feb. 1998.