On the Optimum Number of Hops in Linear Wireless Networks
Marcin Sikora, J. Nicholas Laneman, Martin Haenggi, Daniel J. Costello, Jr., and Thomas Fuja
Department of Electrical Engineering, University of Notre Dame
275 Fitzpatrick Hall, Notre Dame, IN 46556
msikora@nd.edu
Abstract — We consider a wireless communication
system with a single source node, a single destina-
tion node, and multiple relay nodes placed equidis-
tantly between them. We limit our analysis to the
case of coded TDMA multihop transmission, i.e., the
nodes do not cooperate and do not try to access the
channel simultaneously. Given a global constraint on
bandwidth, we determine the number of hops that
achieves a desired end-to-end rate with the least to-
tal transmission power. Furthermore, we examine
how the optimum number of hops changes when an
end-to-end delay constraint is introduced using the
sphere-packing bound and computer simulations. The
analysis demonstrates that the optimum number of
hops depends on the end-to-end rate and the path-
loss exponent. Specifically, we show the existence
of an asymptotic per-link spectral efficiency, which is
the preferred sp ectral efficiency in TDMA multihop
transmission.
I. Introduction
Layered architectures for communication networks have led
to significant progress and technological advances. For emerg-
ing wireless networks such as ad hoc, multihop cellular, and
sensor networks, momentum is growing behind multihop rout-
ing at the network layer, distributed channel access at the
link layer, and powerful channel coding at the physical layer.
These advances have been studied largely in isolation, whereas
this paper focuses on their interaction, especially in delay-
constrained scenarios.
To illustrate the main concepts, we consider a linear wire-
less network of N + 1 nodes. Transmission between the end
no des can occur in a single hop, or up to N hops. Proponents
of multihop routing argue that more short hops are preferable
to fewer long hops, because the minimum signal-to-noise ratio
(SNR) along the route is larger for multihop. As indicated
in [1] and [2] this observation does not take into account the
imp ortant practical issues of resource allocation, end-to-end
delay, error propagation, and interference induced by extra
transmissions. Consider the following motivating observations
and questions:
• Because wireless transceivers cannot both receive and
transmit at the same time on the same frequencies,
multi-hop requires excess bandwidth when compared to
single-hop. Do the costs of this excess bandwidth out-
weigh the benefits of SNR gain due to multihop?
• In a delay-constrained environment, the accumulated
delay incurred when coded packets are decoded and re-
enco ded at every hop can become unacceptable. For a
given end-to-end delay, is it better to decode/re-encode
1
This work was supported in part by NASA Grant NAG5-12792,
NSF Grant CCR02-05310, and NSF Grant ECS03-29766.
weaker codes (each with small delay) over many short
hops or to use stronger codes (each with large delay)
over fewer long hops?
To address these issues, we compare single hop and multi-
hop transmission under bandwidth and delay constraints.
First, we examine the impact of bandwidth constraints alone
on channel capacity. Then we incorporate delay constraints
by employing the sphere-packing bound. Finally, we illustrate
the performance of multihop coding schemes based upon con-
volutional and turbo codes and compare them to theoretical
p erformance predictions.
Our results indicate that the benefits of multihop are
ero ded by bandwidth constraint, especially for high spectral
efficiency (a similar conclusion was reported independently in
[3]), but the impact of a delay constraint is much less severe
than anticipated. We further show the existence of an asymp-
totic per-link spectral efficiency, which is the preferred spectral
efficiency in a time division multiple access (TDMA) multi-
hop transmission. Choosing the number of hops for which the
required per-link spectral efficiency is closest to this asymp-
totic value is an optimal strategy for most end-to-end rates.
In the simple case of deciding between one-hop and two-hop
transmission, one hop is generally preferred when the rate (in
b/s/Hz) is larger than the path-loss exponent.
II. System Description
A System model
The communication system under consideration is illus-
trated in Figure 1. It consists of a source node S and a
destination node D separated by a distance L, and N − 1
intermediate relay nodes F
i
, i = 1, ..., N − 1, placed equidis-
tantly on a line from S to D. The objective of the system
is the reliable delivery of bits generated at the source node
at a rate of 1/T
b
bits per second to the destination node us-
ing coded transmission. The resources available for this task
comprise a band of radio frequencies allowing for a signaling
rate of 1/T
s
complex-valued symbols per second and a total
transmit power P
T
.
The channel is modeled to attenuate the transmitted signal
and corrupt it with additive white Gaussian noise (AWGN)
with a one-sided spectral density N
0
. The attenuation de-
p ends on the distance l between the transmitter and the re-
ceiver (for neighboring nodes l = L/N) according to
P
R
= P
T
cl
−α
, (1)
where P
T
is the transmitted signal power, P
R
is the received
signal power, α is the path loss exponent (typically taking
values between 2 and 4), and c is a constant. This model usu-
ally holds only for distances l for which cl
−α
¿ 1. However,
since the constant terms in (1) do not affect the analysis that
follows in terms of relative performance, we will assume that
c = 1 and L = 1 in order to simplify the notation.
Figure 1: Single-hop and multi-hop communication
systems.
B TDMA Access Mode
If all nodes in the system were p erfectly synchronized and
could coherently receive all transmitted signals, the system
presented in Figure 1 could be interpreted as a multiple Gaus-
sian relay channel. The data rates achievable in such a channel
can always be increased by increasing the number of relays—
at least as long as the distances L/N between neighboring
no des stay in a range in which the power law (1) holds [5]. In
most practical ad hoc networks, however, neither the synchro-
nization nor the complexity required to achieve such rates can
b e met. As a result attention must be directed to simpler, but
more robust, operating modes.
In a TDMA multihop system, the end-to-end transmis-
sion is split into N partial transmissions, called hops, be-
tween neighboring nodes. At any p oint in time only one
no de is transmitting, and there is no interference at any re-
ceiver. Since, by assumption, the distances between neighbor-
ing nodes are equal, all hops are identical and should be as-
signed equal portions of the available resources (channel time
and power). Hence, each of the N hops can utilize (N T
s
)
−1
channel uses per second, each involving symbols with trans-
mitted energy E
s
= P
T
T
s
.
For single-hop transmission the received symbol energy is
the same as the transmitted energy due to (1) and the as-
sumptions we made about c and L. For multiple hops, how-
ever, the distance between neighboring nodes is L/N, and so
the received energy is E
s
N
α
. The SNR per hop can thus be
expressed as SNR = E
s
N
α
/N
0
.
C Performance Measures
In order to fairly compare the performance of systems in-
volving different numbers of hops, the performance measures
must be chosen carefully. To measure the bandwidth effi-
ciency, we will use the bandwidth-normalized rate R defined
as R = T
s
/T
b
bits p er channel use. Note that this refers to
end-to-end, not node-to-node, transmission; the spectral ef-
ficiency at each hop equals NR due to the N-fold reduction
in channel uses in TDMA operation. Power efficiency will be
evaluated in terms of E
∗
b
/N
0
, where the total energy per bit
E
∗
b
is defined as the sum of the energies per bit spent over all
N hops, i.e., the energy spent to deliver one bit from node S
to node D. Hence, E
∗
b
= N E
s
/(NR) = E
s
/R.
The performance of a TDMA linear ad hoc network with
N hops will be characterized by the highest achievable band-
width-normalized rate R for a given E
∗
b
/N
0
. Without delay
constraint, the achievable rate is understood as the highest
rate for which an arbitrarily small bit error rate (BER) can
b e obtained using forward error correction coding. When the
end-to-end delay is limited, a rate is achievable if there exists a
co ding scheme with appropriate latency operating with a BER
or block error rate (BLER) not exceeding some prescribed
value.
III. Number of Hops Without Delay
Constraints
When no delay constraint is imposed on the system, the
highest achievable end-to-end transmission rate is the channel
capacity. For a single hop, this can be simply expressed using
Shannon’s well-known formula [4]
R = log
2
³
1 +
E
s
N
0
´
. (2)
Switching from a single hop to N time-shared hops has the
following consequences. The transmitted energy per symbol
remains unchanged, but the received energy at each hop is
N
α
times higher due to reduced attenuation over a smaller
distance. At the same time, each of the hops must accom-
mo date the transmission of the same number of information
bits in 1/N-th of the channel uses available in the single hop
mo de—thus increasing the required per-hop spectral efficiency
N-fold. The first of the two effects gives multihop an improve-
ment over single hop transmission at low SNR, while the latter
p enalizes multihop at high SNR. Hence we can write
R =
1
N
log
2
³
1 +
E
s
N
0
N
α
´
. (3)
The curves corresponding to (2) and (3), with E
∗
b
= E
s
/R,
for several values of N are presented in Figure 2 for the cases
α = 2 and α = 4. It can be observed that no single curve
provides the best achievable rate for all E
∗
b
/N
0
, but every
curve dominates in some range of E
∗
b
/N
0
values.
It might be tempting to ask for the value of E
∗
b
/N
0
for
which an N -hop system performs best. Note, however, that
if we consider an analogous plot for a system in which the
distance L is different from 1, the curves in Figure 2 will
b e shifted on the E
∗
b
/N
0
scale by −10α log
10
L dB. Hence, it
makes more sense to determine the ranges of the end-to-end
bandwidth-normalized rate R for which each curve dominates.
The value of R for which E
∗
b
/N
0
is the same for both N
hops and N + 1 hops will b e referred to as the crossover rate
and denoted R
N
. By equating E
s
/N
0
in (3) for N and N + 1,
R
N
can be obtained as a solution to the polynomial equation
(N + 1)
−α
(2
R
N
)
N+1
− N
−α
(2
R
N
)
N
+ N
−α
− (N + 1)
−α
= 0.
(4)
For N = 1 the above equation yields R
1
= log
2
(2
α
−1). Hence,
as a simple rule, if the required end-to-end rate exceeds α, then
the transmission from S to D should be performed in a single
hop. For the calculation of crossover rates for higher N, we
resort to numerical methods.
Figure 3 presents the optimal number of hops N
opt
(R) for a
given bandwidth-normalized end-to-end rate R, as well as the
corresp onding per-hop spectral efficiencies RN
opt
(R) at each
hop. The latter plots are particularly interesting, since they
suggest the existence, for any value of α, of an asymptotic
p er-hop sp ectral efficiency
S(α) = lim
R→0
RN
opt
(R) (5)
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Figure 2: End-to-end bandwidth-normalized rates achievable with zero to five intermediate nodes for a) α = 2 and
b) α = 4.
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Figure 3: Optimal numb ers of hops N
opt
(R) and the corresponding per-hop spectral efficiencies for a) α = 2 and b)
α = 4.
preferred by the system. Moreover, an accurate prediction of
the optimal number of hops can be obtained from
N
opt
(R) ≈ [S/R] , (6)
where by [·] we denote the nearest positive integer. However,
we were unable to obtain from (4) a closed form expression for
S(α). The values obtained numerically are plotted in Figure 4
for a range of path loss exponents.
IV. Number of Hops with a Delay Constraint
The predictions about the preferred number of hops made
in the previous section were based on the assumption that the
blo ck lengths used by channel codes can be arbitrarily large.
In many applications, however, there is a strict limit on the
tolerable transmission delay, or end-to-end latency. There are
several sources of latency in a communication system:
• Waiting for the data source to emit enough bits to form
a block of a desired length (for channel coding);
• Processing delay caused by encoding the information
bits for transmission;
• Transmission and reception of the whole encoded mes-
sage;
• Processing delay caused by decoding.
If the communication system involves multiple hops, the latter
three elements are repeated several times, increasing overall
latency. To compensate for this, shorter block lengths must
b e used—at a cost of reduced error-correcting capabilities at
each link.
In this paper we will consider only the delay caused by the
need to receive the whole block of symbols before decoding
can proceed. We also assume that the symbol duration T
s
is much larger than the actual propagation delay. We denote
the largest tolerable end-to-end delay as D, which corresponds
to n
s
= D/T
s
channel uses. In what follows we assume that
by choosing n = n
s
/N as the block length (in term of coded
symbols) in an N-hop system, the delay constraint is satis-
fied. Since this reduction in block length by the factor 1/N
Figure 4: Asymptotic per-hop spectral efficiencies S for
a range of path loss exponents.
is accompanied by an N-fold increase in code rate due to the
TDMA mode of op eration, we can code for an N-hop sys-
tem by squeezing a fixed number Rn
s
of information bits into
n
s
/N channel symbols.
Error-correcting codes with a limited block length cannot
achieve arbitrarily low error rates. Hence, for a rate R to be
achievable in a multi-hop system, a coding scheme must exist
for which the end-to-end block error rate (BLER) falls below
some tolerable value P
e
. Since a block error at any of the N
hops will result in a blo ck error at the destination, except for
the unlikely event of several block errors canceling each other
out, we can instead require that the BLER at each hop does
not exceed P
e
/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 achieving a lower BLER, while at the same time
enjoying an SNR increase by a factor N
α
.
The sphere-packing bound introduced by Shannon [6] is a
useful tool that relates the BLER to the co de rate, the SNR,
and the block length for the real-valued AWGN channel. Fol-
lowing the notation in [7], the probability of a block error on
the real-valued AWGN channel has a lower bound
P
e
≥ Q
n
(θ, A) (7)
dep ending on the signal amplitude parameter A, an angle θ,
and the block length n in real-valued channel symbols. The
angle θ is the half-angle of the n-dimensional cone encompass-
ing a fractional solid angle 2
−rn
and relates to the block length
n and code rate r in bits per real-valued symbol according to
Ω
n
(θ ) = 2
−rn
, (8)
so that Ω
n
must be inverted in order to obtain θ. Since the
functions Q
n
(θ, A) and Ω
n
(θ ) appearing in (7) and (8) do
not have closed-form expressions, we will use their asymptotic
forms, which are accurate for n > 20 [7]:
Ω
n
(θ ) ≈
sin
n−1
θ
√
2πn cos θ
(9)
Table 1: Simulated coding schemes.
Sp ectral Convolutional codes Turbo co des
efficiency (deco ding delay (decoding delay
(b/s/Hz) 100 bits) 1000 bits)
0.5 QPSK + CC R=1/4, QPSK + Turbo
memory 10, [8] co de R=1/4, [9]
1 QPSK + CC R=1/2, QPSK + Turbo
memory 10, [8] co de R=1/2, [9]
2 8PSK + TCM R=2/3, 16QAM + Turbo
memory 7, [8] co de R=1/2, [9]
4 32CROSS + TCM 32CROSS + Turbo
R=4/5, memory 6, [8] TCM R=4/5, [10]
and
Q
n
(θ, A) ≈
h
G sin θe
−
(
A
2
−AG cos θ
)
/2
i
n
√
nπ
√
1 + G sin θ
£
AG sin
2
θ − cos θ
¤
, (10)
where
G(θ , A) =
1
2
³
A cos θ +
p
A
2
cos
2
θ + 4
´
. (11)
The variables used in the above formulas relate to the
variables introduced earlier as follows: A
2
= 2E
s
N
α
/N
0
,
n = 2n
s
/N, and 2r = R/N, with the factor of 2 in
the last two formulas accounting for the fact that we are
using a complex-valued AWGN channel. We will write
Q(n
s
/N, R/N, E
s
N
α
/N
0
) to denote Q
n
(θ, A) combined with
an inverse of (9) and with the above substitutions.
We computed the crossover points R
N
by numerically
searching for the solutions to the following set of equations:
½
P
e
N
= Q
¡
n
s
N
, RN,
E
s
N
0
N
α
¢
P
e
N+1
= Q
¡
n
s
N+1
, R(N + 1),
E
s
N
0
(N + 1)
α
¢
(12)
with P
e
= 10
−4
. The results of this search are shown in Fig-
ure 5 for α = 2 and α = 4. The plots indicate that the
crossover rates remain quite stable for most practical values
of the delay constraint D. This means that the simpler predic-
tions about R
N
based on the capacity formulas in the previous
section remain fairly accurate.
V. Simulation Results
The sphere-packing bound allows us to estimate the achiev-
able performance of channel codes under limited transmission
p ower and block length constraints. In a practical system one
more constraint is present: code complexity. Therefore we
rep eated the process of finding the first crossover rate R
1
us-
ing the simulated BLER vs. E
∗
b
/N
0
p erformance obtained for
two common families of codes: convolutional codes and turbo
co des. Each family includes codes with increasing spectral ef-
ficiencies and decreasing block lengths. The parameters of the
simulated codes are listed in Table 1.
The E
∗
b
/N
0
values needed to achieve a BLER of 10
−4
have
b een established for each code and plotted vs. the spectral ef-
ficiency in Figure 6—once assuming they were used in a single
hop system and once in a two-hop system (with α = 2). The
crossover rates obtained from these curves are also included
in Figure 5a at their corresponding block lengths. These
rates fall somewhat below the values predicted by the sphere-
packing bound. Although the method we used in determining
a)
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Figure 5: Crossover rates for delay-constrained transmission for a) α = 2 and b) α = 4.
the crossover rates is very sensitive to the choice of represen-
tative codes, this result suggests that in practical scenarios
there are additional incentives to using a smaller number of
hops that cannot be assessed by just comparing the capacities.
VI. Conclusion
For wireless multihop networks, such as ad hoc, multi-
hop cellular, and sensor networks, a fundamental question is
whether it is advanta-geous to route over many short hops or
over a smaller number of longer hops. The benefits of short-
hop routing include SNR gain and the reduction in interfer-
ence. This paper addresses the first point and shows that, de-
p ending on the path loss exponent, the SNR gain may be offset
by the required increase in spectral efficiency. In particular,
for the one-dimensional scenario studied here, single-hop rout-
ing outperforms two-hop routing for bandwidth-normalized
rates larger than the value of the path loss exponent. For
delay-constrained transmission, this rate threshold for single
hop to be preferable is somewhat lower.
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Figure 6: Crossover p oint between single-hop (solid
lines) and two-hop (dashed lines) systems for practical
coding schemes (α = 2).
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