Content uploaded by Enzo Baccarelli
Author content
All content in this area was uploaded by Enzo Baccarelli
Content may be subject to copyright.
Routing for Multi-Antenna Wireless Mesh Network
Backhaul
Enzo Baccarelli, Mauro Biagi, Cristian Pelizzoni, Nicola Cordeschi
{enzobac, biagi, pelcris, cordeschi }@infocom.uniroma1.it
Abstract—The goal of this contribution is to show which
practical routing approach, for Wireless Mesh Network (WMN)
backhaul, is the best for systems presenting a constraint for
the total power to be emitted be network nodes by modeling
rounting as a power allocation problem. We consider at first
Multiple-Input Multiple-Output (MIMO) point-to-point capacity-
achieving codes, and nodes cooperation. Furthermore we take
care of a WMN backhaul with a single source node, a single
destination node, and N-1 intermediate nodes placed equidis-
tantly on a line between them. Additionally we consider also
an interference cancellation able to enhance throughput and we
compare the single-hop paradigm with multi-hop one.
Keywords—Wireless Mesh Networks, MIMO, Multi User
Interference, Routing, Power Allocation.
I. INTRODUCTION AND GOAL OF THE WORK
Broadband Wireless Access (BWA) has recently gained
much interest both from the industrial and research commu-
nities for providing high-speed data and multimedia services
[1]. It offers attractive features such as easy and fast deploy-
ment, fast realization of revenues, and low infrastructure cost,
compared to other alternative broadband access technologies,
including xDSL, Passive Optical Networks (PON), and Hybrid
Fiber Coax (HFC). For these reasons, broadband radio access
standards and architectures are currently the focus of serious
efforts in the scientific community. In the spring of 1997
ETSI established a standardization project for Broadband
Radio Access Networks (BRAN). The project target was to
output standards for providing broadband (25 Mbit/s or more)
wireless access to wire-based networks in both private and
public environments, operating in either licensed or unlicensed
spectrum. ETSI BRAN currently produced specifications for
three major Standard Areas: HiperLAN2, HIPERACCESS,
HIPERMAN. However, none of these standards have ongoing
activities to define extensions supporting mesh networking [2-
4].
The IEEE Standardization Bodies have undertaken standard-
ization of BWA beginning in July 1999, with the purpose of
supporting the development of fixed wireless access systems.
Currently, the IEEE Committees have planned several standard
specifications for broadband wireless access in metropolitan
Enzo Baccarelli, Mauro Biagi, Cristian Pelizzoni and Nicola Cordeschi are
with INFO-COM Dept., University of Rome ”La Sapienza”, via Eudossiana
18, 00184 Rome, Italy. Ph. no. +39 06 44585466 FAX no. +39 06 4873330.
This work has been partially supported by the Italian National project:
”Wireless 8O2.16 Multi-antenna mEsh Networks (WOMEN)” under grant
number 2005093248.
area Networks. In particular, the following standards have been
released:
• IEEE802.11 is an umbrella that contains several standard
committees that have developed technologies for the
WLAN environment [5] operating at either the 2.4GHz or
5GHz frequency bands. The efforts of the standardization
activities lead to the availability of highly interoperable
802.11-based standards providing higher speeds (more
than 100 Mbps), QoS support, faster handoffs, and several
additional capabilities. Relevant to the mesh-networking
paradigm is the extension under development by the
P802.11s Extending Service Set (ESS) Mesh Networking
Task Group (TG). The scope of this TG is to extend
the IEEE 802.11 architecture and protocol for providing
the functionality of an Extended Service Set Mesh, i.e.,
access points able of establishing wireless links among
each other that enable an automatic topology learning and
dynamic path configuration.
• IEEE802.16 specifies the air interface of a fixed (station-
ary) point-to-multipoint broadband wireless access sys-
tem providing multiple services in a wireless metropolitan
area network (MAN) operating between 10 and 66 GHz,
using long distance (up to 20 Km) Line Of Sight (LOS)
wireless links [6]. IEEE 802.16a is an amendment to this
standard, to support 2-11 GHz using also Non-Line Of
Sight (NLOS) communications [7]. Other enhancements
currently under definition by different task groups are: the
IEEE802.16e standard, intended for supporting combined
fixed and mobile operations in licensed bands, and the
IEEE 802.16d standard (aimed at revising and integrating
802.16 and 802.16a). Starting from the 802.16a version,
the IEEE802.16 WGs has undertaken the specification of
an optional ”mesh mode”. The 802.16 mesh in the cur-
rent standard draft has several limitations: it has limited
scalability due to the adoption of centralized scheduling
strategies; the mesh is based on a connectionless MAC,
so QoS of real-time services is difficult to guarantee; and
no interference between nodes that are two hops away is
considered.
The goal of this work is to analyze the impact of multi-antenna
equipped transceiver on routing when a linear backhaul of a
wireless mesh network is considered. We consider the problem
of power allocation as a routing and access problem. In fact,
in order to perform routing we need a cost function (e.g., the
rate) so to study the impact of both space-diversity and channel
reuse. These two last parameters are strictly linked to access
scheme since the channel reuse and diversity are techniques
usually employed to reduce multi-user interference, that is
MAC collisions. Different schemes, in terms of achievable
rate, are considered so to underline what are the possibilities
for linear routing, that is, if it is worth to make all nodes
transmit or if it is better to excluding nodes (this is equivalent
to consider a network composed by a less number of nodes).
A. Previous related works
The problem of finding bounds on the transmission rates
achievable over wireless networks has been studied in [8]-
[14] under various assumptions on the network topologies
and node capabilities. In [8],[9] and [10] considered planar
networks with nodes and multiple source-destination pairs
and characterized attainable transport rates (in bit-meters per
second) for finite and infinite. For networks with a single
source and destination, perfect channel state information (CSI)
at all nodes, and perfect synchronization, in [11] it is showed
that the achievable rate is logarithmic if the distance from each
relay node to the source and destination is lower-bounded, and
without such a bound it can grow linearly in [9]. Furthermore,
a more general problem of rates achievable over a multiple-
relay channel was considered in [12] and [13], [14]. Routing is
an important special case of relaying in the sense of [8]-[13].
In this paper, we analyze, under different conditions, various
routing schemes for wireless mesh network backhaul based
on capacity-achieving point-to-point codes are best suited in
the power constrained application scenario. The same linear
network model is used in several works. We require that
the multihop transmission be based entirely on point-to-point
coding. This implies that each node fully decodes the original
message based on the signal received from the preceding
node, re-encodes it, and forwards it to the following node.
The decoding operation must rely on the decoder, and all
interference from all nodes transmitting simultaneously with
the preceding node is regarded as additional Gaussian noise.
As more difficult to implement, but still practical, we also
consider canceling a known interference from the received
signal before decoding. Among the techniques that are fre-
quently encountered in the literature but cannot be integrated
into the above framework are synchronous cooperation and
sliding-window decoding. Synchronous cooperation (used in,
e.g., [9]-[11]), which is analogous to beamforming performed
by several transmit antennas controlled by a single transmitter,
gives the transmitter full control over how the signals add
up at the receiver’s antenna. This includes maximum ratio
transmission, in which several copies of the same signal
transmitted from different antennas add up in amplitude at the
receiver, providing large power savings, and active interference
cancellation (IC), in which two signals sent from two different
nodes add up to zero at a selected receiver, and neither of
them interferes with that receiver. However, whenever the
antennas attempting to synchronously cooperate are controlled
by separate transmitters, providing precise timing and phase
synchronization between them is extremely difficult. Sliding-
window decoding, which was originally introduced by Carleial
[15] and then extended in [16] for the MIMO case, involves
determining the most likely message using not only the signal
from the preceding node received in the current slot, but also
from the upstream transmissions received in past slots.
B. Organization of the work
The paper is organized as follows. After giving the system
model description in Sect.II, we present the afforded problem
in Sect.III by considering different scenarios by changing the
antennas configuration. Then in Sect.IV we give some insight
about interference mitigation so to evaluate the performance
in Sect.V.
About the adopted notation, bold letters stand for vectors
while bold capital letters represent matrices. Furthermore 1
denotes a vector composed by all ones, exp represents the
exponential function while Trastands for matrix trace. Finally
E{.} denotes expectation, (.)
T
transposition, (.)
∗
conjugation
and (.)
†
transposed conjugation.
II. S
YSTEM MODEL
We consider the problem of packet forwarding for a wireless
network where the backhaul is characterized by several nodes
able to detect and forward the information from a source to a
destination. From this point of view we consider a cooperation
between nodes. In detail, we assume, according to the work in
[17], that the backhaul is composed by a source node S that is
the node generating information to be carried to the destination
node D (see Fig.1) at a distance equal to d, and N −1 nodes
(F
1
, ..., F
N−1
) are present between the source and destination
and disposed over a line with equal distances d/N. The nodes
Fig. 1. Network system model.
share a band of radio frequencies allowing for a signaling
rate of W complex-valued symbols per second. The objective
of the system is the reliable delivery of bits generated at the
source node S at a bandwidth-normalized rate of R bits per
second per hertz (i.e., RW bits per second) to the destination
node using coded transmission and consuming the least pos-
sible total transmission power P
budget
. No constraints on how
this total power is allocated among nodes are considered. The
nodes comprising the system operate in half-duplex, i.e., they
are not capable of simultaneous transmission and reception.
Similarly to [17] we can consider two different sets where the
first one collects the node able to transmit, that is
S
T
⊂{S, F
1
,F
2
, ..., F
N−1
} (1)
while the second one gathers the nodes able to receive that is
S
R
⊂{F
1
,F
2
, ..., F
N−1
,D} (2)
and each of the above defines set contains N nodes while the
link is composed by N +1 nodes
1
. Each node is assumed to
be equipped with N
T
transmitting antennas and N
R
receive
ones. The analytical expression describing the hop from the
node F
l
to the (contiguous) node F
l+1
is given by
Y
(l)
=
√
aH
(l)
Φ
(l)
+ V
(l)
(3)
where Y
(l)
is the (T × N
R
) matrix collecting the received
samples, a is a power loss term given by a =(d/N)
−α
, the
(T ×N
T
) matrix Φ
(l)
is the general space-time block codeword
transmitted by node F
l
, the (T × N
R
) matrix V
(l)
collects
the interference and noise samples at the node F
l+1
, while
the (N
R
× N
T
) matrix H
(l)
describes the MIMO channel.
The general element of H
(l)
, e.g. h
(l)
ji
, models the path gain
from the transmit antenna i to the receive one j as a complex
zero mean unit-variance proper random variable (r.v.) and,
for sufficiently spaced apart antennas, these gains {h
ji
∈
C
1
, 1 ≤ j ≤ N
R
, 1 ≤ i ≤ N
T
} may be considered mutually
independent. Accuracy of this assumption fast improves when
the antennas spacing becomes large compared to the RF
wavelength λ of radiated signals. Specifically, for outdoor
applications links sited in urban and suburban environments
may require antenna spacings over 10λ at the base stations
and 3λ at the mobile units [18]. However, several measures
and theoretical contributions support the conclusion that per-
formance loss induced by signal correlation among receiving
antennas is not dramatic even for correlation coefficients as
high as 0.5-0.6 [18]. This is equivalent to assume that the
coherence-time T
coh
of the MIMO forward channel of Fig.1
equates the length T of the transmitted packets (e.g., T
coh
=
T ). The resulting ”block-fading” model generally gives rise to
adequate representations of several TDMA, frequency-hopping
and packet-based interleaved systems of practical interest,
where each transmitted frame is detected independently.
III. T
HE AFFORDED PROBLEM
At first we suppose that all nodes are active (and in
transmission) from source to the (N − 1)-th node where this
last represents the last hop. So, the general node F
l
receives
information by node F
l−1
and, after decoding, it forwards
information to node F
l+1
. The node labeled as F
l+1
can
use the signal received during a time slot so to recover the
information message. The receiver structure, as detailed in
the next sections, may include an interference cancellation
module so to take into account for the possibility to mitigate
interference contribution due to the messages sent by the other
nodes. Furthermore, considerations based on Central Limit
Theorem allow us to assume the interference as spatially
and temporal Gaussian [19]. Differently from [17] where two
different parameters play a key role in the optimization of
the system and these are the total available power and the
1
For sake of simplicity S can be considered as F
0
.
reuse term K that measures the fraction of time the node
transmits, in this work we consider the number of transmit
and receive antennas too. So, the power constraint is given by
the following expression
Tra
B
S
+
N−1
l=1
B
l
= Tra
E
φ
(S)
φ
(S)
†
+
N−1
l=1
E
φ
(l)
φ
(l)
†
≤ P
budget
, (4)
where the general B
l
matrix takes care of how the power
is allocated on different antennas for the l-th relay and φ
(l)
is the block vector collecting the elements of Φ
(l)
. So, let
us start by defining the rate (objective) function that we take
into account in order to evaluate linear routing strategy. We
consider the following function for the generic l-th hop [19]
R
l
=
1
K
E{log det{I
N
R
+
1
N
T
K
(l)
v
−1/2
H
(l)T
B
l
H
(l)∗
K
(l)
v
−1/2
}
(5)
where the matrix K
(l)
v
is the covariance matrix of the dis-
turbing samples as in eq.(1). In order to detail the problem
in depth, similarly to [17], we consider, according to the
interference model proposed in [19] and references therein,
that the interference event occurs for the l-th node if and only
if a node s belongs to the set I
l
defined according to the
following relationship
I
l
≡{s ∈{0, ..., N −1}|s = land|l − s|
K
=0} (6)
being the expression |l −s|
K
=0equivalent to ”K divides l −
s”. The above expression is valid for omnidirectional antennas
while, if we assume to employ directional ones, the definition
for set in (6) changes in the following way
2
I
l
≡{s ∈{0, ..., N −1}|s = land|l − s|
K
=0,l>s}.
(7)
Furthermore, in order to give an analytical model and expres-
sion for the interference covariance matrix K
v
we have (see
[19] and references therein)
K
(l)
v
≡
W N
0
+
I
l
1
|1+s − l|
−α
a
E{χ
2
s
}
1+k
s
P
(s)
I
N
R
+
I
l
1
|1+s − l|
−α
a
k
s
E{χ
2
s
}
N
T
(1 + k
s
)
11
T
B
s
1
T
1, (8)
where E{χ
2
s
} takes care for shadowing effect and k
s
takes into
account for Rice propagation factor. A closed form solution
for the problem of maximization of eq.(5) is not possible
since it requires to solve it with respect to each value of
H and, more, it is not a convex problem since the allotted
power concurs to generate interference. For this reasons, at
first, we specialize the analysis for some particular cases. The
first one is the case when the number of transmit antennas
N
T
approaches infinity (N
T
→∞) for a finite value of N
R
2
We consider this model in our analysis.
and no Channel State Information (CSI) is available at the
transmitter. The second one deals with the number of N
R
that
is considered to approach infinity (high number of receive
antennas) while N
T
has a finite value and the transmitter
has no information about channel knowledge. The third case
deals with the assumption of uniform allocation maximizing
the rate in no CSI case. Unfortunately this last implies that
we are not able to evaluate an average value for the rate but
the algorithm for the allocation should proceed by considering
different values for H. On the other side the problem can be
solved easily by reducing it to a scalar one. Before to proceed
let us consider the properties retained by the considered rate
function with respect to the B matrix.
Property 1: the maximum rate R
(B) is increasing in
Tra{B}.
Proof: Once supposed to have a set of B matrices
{B
0
, ...B
N−1
} that represents the optimal allocation al-
lowing the maximum rate R
when the constraint is
Tra{
i
B
i
}≤P
budget
, if we consider a new constraint given
by Tra{
i
B
i
}≤bP
budget
with b>1, then a sub-optimal
allocation {(b)B
0
, ...(b)B
N−1
obtained simply by multiplying
the matrices by (b) gives arise to R(bB
) > R(B
).
Property 2: the allocation able to maximize the total rate
consists into allocate power so to have the same rate over
all hops.
Proof: Since the network is composed by a cascade of links,
if one of them presents a rate that is lower with respect to
the others (assumed equal), this represents a bottleneck for
the system so, in order to maximize the total rate we need to
require that the rates of all hops are equal.
Property 3: the power allocation allowing to achieve the
maximum rate is unique.
Proof: Since, given H (also in the case of perfect CSI) the
power allocation to maximize is a SVD problem, we have
that the singular values allow us to achieve the maximum for
the general l-th hop. If we consider as solution a generalized
waterfilling approach performed over all the users and trans-
mitting antennas, we arrive at a solution that is unique, so the
proof.
Property 4: by considering the powers allotted on the
antennas of the l-th user and for all the users, we
have {P
0,0
(P
budget
),P
0,N
T
(P
budget
), ..., P
N−1,N
T
(P
budget
)}
that gives R
(P
budget
) and each P
l,i
(P
budget
) is no decreas-
ing in P
budget
.
Proof: in property 1 we show that R(bB) ≥R
(B) is b ≥ 1
so, by considering property 3, we prove property 4.
In the following we assume that the transmitter has no infor-
mation about channel coefficients so the best choice for the
single link is to allocate power uniformly over the antennas.
A. Case of large number of transmit antennas with no CSI
When we consider a large number of transmit antennas, the
rate function in eq.(5) becomes, according to the results in
[20],
R
l
=
N
R
K
log
2
(1 + ρ) (9)
where the term ρ indicates the signal to interference noise
ratio (SINR) that can be also be expressed as in the following
(under the assumption of Gaussian interference)
ρ =
P
(l)
a
W N
0
+
I
l
P
(s)
|1+l − s|
−α
a
(10)
so the eq.(9) can be equivalently rewritten as
R
l
=
N
R
K
log
2
1+
P
(l)
a
W N
0
+
I
l
P
(s)
|1+l − s|
−α
a
.
(11)
An optimal solution for eq.(11), by considering that all
nodes are required to allocate power so to force the rates to be
the same, is not analytically computable but it can be found
via an iterative algorithm reported in Table 1.
Before to proceed let us give some explanations about eq.(11).
First, since the transmitter has no information about the chan-
nel, then the allocation able to achieve capacity (maximum
rate over the antennas) is the isotropic-unitary one [21], so,
by fact, the matrix B
l
can be replaced by its trace, that is,
P
(l)
representing the total power allotted for the l-th node.
Second, the noise is represented by diagonal matrix and,
considerations carried out about the statistical features of
interference, allow us to consider the noise plus interference
matrices as diagonal so we resort, for the link power allocation,
to a scalar problem that results to be analogous to a single
antenna case. This justifies that eq.(9) has been rewritten as
eq.(11). Before to proceed some words about the effect of the
presence of interference. If we assume that the antennas are
strongly directive, then we can assume that the SINR can be
rewritten as ρ = P
i
a/W N
0
so the rate can be equivalently
rewritten as
R
l
=
N
R
K
log
2
1+
P
(l)
a
W N
0
(12)
so the power able to guarantee the rate R
l
is
P
(l)
=
W N
0
a
2
R
l
K/N
R
− 1
(13)
and by considering that all the hops present similar conditions,
we have that the power required for the rate has to satisfy the
constraint
P
(l)
=
P
budget
N
≤
W N
0
a
2
R
l
K/N
R
− 1
. (14)
By dealing with the algorithm of Table 1, at step 1 the reuse
parameter K is set, then the for loop starting at step 2 is
considered so to initialize al the powers to zero and to define
the sets I
l
in eq.(7). After this loop ends at step 7, a while
control instruction out the total power alloted is considered
at step 8 and an increment of power Δ is considered for the
nodes not experiencing interference while at step 13 power is
allocated for nodes that receive interference levels different
1. set K
2. for i =1, ..., N
3. P
i
=0;
4. m = idivK
5. q = |i|
K
6. I(q, m)=i;
7. end
8. while
P
i
≤ P
budget
9. for i =1, .., N
10. if i ≥ K
11. P
i
= P
i
+Δ
12. else
13. P
i
=
P
k
a
W N
0
W N
0
+
I
l
P
l
|1+l − s|
−α
14. end
15. end
16.end
TABLE I
A
PSEUDO-PROGRAM FOR POWER ALLOCATION
from zero. The allocation criterion is based on property 3
where it is stated that all the rate must be the same so the
SINR ρ, ratio must be same.
B. Case of large number of receive antennas with no CSI
When a large number of receive antennas is considered, the
expression in eq.(5) reduces itself to the following one
R
l
=
N
T
K
log
2
1+
N
R
N
T
ρ
(15)
where, differently from eq.(9), we have that the term multi-
plying the SINR ρ is now given by the ratio N
R
/N
T
that is,
due to the assumptions, greater than unit. Since the SINR ρ
contains, as for its definition given in eq.(10), the power terms
both at numerator and denominator, this makes the problem
not solvable in closed form. The expression in eq.(12) can be
justified by observing that H
∗
H/N
T
= H
∗
H/N
R
(N
R
/N
T
)
and the (N
T
× N
T
) matrix H
∗
H/N
R
converges to I
N
T
if
N
R
→∞. Furthermore, since in eq.(12), once assigned the
number of antennas, the only one parameter able to influence
the performance is the SINR ρ, so in order to proceed to
the allocation (uniform allocation) we can solve the problem
of over the users by employing the algorithm of Table 1.
In this situation we resort to considerations similar to those
carried out in the previous sub-section where interference can
be considered negligible so to arrive at the following simple
expression
R
l
=
N
T
K
log
2
1+
N
R
N
T
P
(l)
a
W N
0
(16)
that can be considered to evaluate the power needed for the
required rate as constraint leading to the following expression
P
(l)
=
P
budget
N
≤
W N
0
N
T
aN
R
2
R
l
K/N
T
− 1
. (17)
C. Case of finite number of antennas with no CSI
When the number of transmit and receive antennas is
limited, the problem of allocation can be solved by simply
allot power uniformly so the expression for the rate of the l-th
hop becomes [19]
R
l
=
1
K
E{log det{I
N
R
+
P
(l)
N
T
K
(l)
v
−1/2
H
(l)T
H
(l)∗
K
(l)
v
−1/2
}
(18)
since the B is diagonal at equates B
l
= P
(l)
I
N
T
where P
(l)
is the power of the l-th hop.
The differences between eqs.(9), (15) and (18) can be summa-
rized in the expectation that in (9), (15) do not appear while
in (18) has to be evaluated with respect to H variable so
distributed p(H
(l)
)=π
−N
R
N
T
exp
−Tra[H
(l)†
H
(l)
]
.
So, the evaluation of the rate cannot proceed as in the
previous case since the power to be allotted is strictly channel
dependent. The algorithm of Table 1 can be still considered
adequate to solve the problem of allocation but some changes
have to be considered. At step 13, the algorithm has to take
into account for the expression fot K
(l)
v
matrix that is given
by
K
(l)
v
≡
W N
0
+
I
l
1
|1+s − l|
−α
a
E{χ
2
s
}
1+k
s
P
(s)
I
N
R
+
I
l
P
(s)
|1+s − l|
−α
a
k
s
E{χ
2
s
}
N
T
(1 + k
s
)
11
T
1
T
1, (19)
so that the ”new” next step (13) consists into equate the
power to
P
(l)
=
P
(k)
a
W N
0
aK
−1/2
v
H
T
H
∗
K
−1/2
v
−1
(20)
where P
(k)
is the power allotted to the node l-th that is
in the set {0, ..., K − 1} collecting the nodes (first K) not
experiencing interference. Furthermore, it results clear that
the power to be allotted for the l-th user directly depends on
the channel state. Since the transmitter has no information
about the channel, the power allocation requires that side
information should arrive at destination so, by considering
half-duplex transmission, is sends to original source the power
to be allotted
3
. In this regard is simple to see that in this case, if
we consider the absence of interference, the expression for the
rate does not reduce to a simple form since the dependence on
the channel coefficients collected into matrix H still remains.
IV. I
NTERFERENCE MITIGATION
When interference suppression/mitigation is considered as
possible block preceding the receiver chain, it is important to
consider which kind of interference is present at the receiver
for decoding and power allocation. The ultimate goal of
interference cancellation is to assure a rate that is higher with
respect to the case of no cancellation. The approach is based
on an estimation and successive subtraction from the received
sequence of a version (estimated) of interference. The rate
3
This assumption requires to transmit over signaling channel only the power
level to transmit without sending all channel coefficients that, when M
R
and
M
T
presents high values, becomes prohibitive.
function, in this case, is quite similar to that in (18) since the
only difference consists into considering the matrix K
v
that is
the residual interference covariance matrix in place of K
v
.By
resorting to the approach carried out in [22], the receiver can
proceed with a spatial signal processing of received sequence.
In particular, the transmitter should provide a time interval
T
learning
during which no transmission is active so the receiv-
er can proceed to estimate the interference covariance matrix
K
(l)
v
. After, by dealing with channel estimator in [19] we
assume that, after a training symbol sequence emitted during
a T
training
period, the receiver acquires information about
channel state. This last information is of paramount importance
since the interference cancellation requires knowledge both
of K
(l)
v
and H
(l)
as detailed in the following. For sake of
simplicity, for the general reader, now we omitted the apex
(l).
By taking into account a linear estimator for this problem
(that is Gaussian), it can be observed that this last is efficient
since, as known from estimation theory [23], it approaches the
Cramer Rao Bound (CRB). Hence, the general expression for
the estimator is
˜
V = YA (21)
where A is the N
R
×N
R
matrix obtained from the following
equation derived from the Orthogonal Projection Lemma
E{(
˜
V − V)
†
Y} = 0
N
R
×N
R
. (22)
By substituting the expression eq.(21) in eq.(22) and by
solving with respect to A we obtain
A
†
E{Y
†
Y} = E{V
†
Y} (23)
that leads to
A =
E{V
†
Y}
E{Y
†
Y}
−1
†
. (24)
The above expression is not explicit so in order to give details
we express the term E{V
†
Y} as
E{V
†
Y} = K
v
(25)
since the terms V and
√
aΦH are statically independent. In
addition the term
E{Y
†
Y}
−1
can be rewritten as
E{Y
†
Y}
= E
√
aH + V
†
√
aH + V
=
= aE{H
†
Φ
†
ΦH} + K
v
=
= aP
(l)
E{H
†
H} + K
v
(26)
where the term Φ
†
Φ is P
(l)
I
N
T
since we consider uniform
allocation. So the expression for A is given by
A =
K
v
aP
(l)
E{H
†
H} + K
v
−1
+
†
(27)
so the estimated version of V is given by
˜
V = Y
K
v
aP
(l)
E{H
†
H} + K
v
−1
†
(28)
So in order to evaluate the residual interference covariance
matrix we have to evaluate
K
v
= E{(V−
˜
V)
†
} = K
v
+K
˜v
−2Re{E{(V)
†
Y}}A (29)
where the term K
˜v
is given (and not reported cause of tedious
algebra) by
K
˜v
= K
v
{aP
(l)
E{H
†
H} + K
v
}
−†
K
†
v
(30)
so the eq.(29) can be rewritten as
K
v
= K
v
to17.5I
N
R
− 2
K
v
{aP
(l)
E{H
†
H} + K
v
}
−1
†
+{aP
(l)
E{H
†
H} + K
v
}
−†
K
†
v
. (31)
Let us consider now what are the ”limit” conditions for in-
terference. When the term aP
(l)
E{H
†
H} becomes negligible
with respect to matrix K
v
the matrix K
v
approaches the
null one, that is (theoretically) infinite rate. On the other
hand, if we consider high level of term aP
(l)
, we obtain that
no interference is mitigated, since K
v
=K
v
. The effect of
interference mitigation will be more clear in the following
section and, at this stage it is useful to anticipate that the
diversity processing we introduce gives arise to an increment
of system complexity but, at the same time, it allows to achieve
higher rates.
V. N
UMERICAL RESULTS
Let us analyze the performance of a linear network as
representative of a WMN backhaul.
The first result to show deals with the behavior of the resulting
strategy for a network where the number of hops is assigned
(i.e., N =8), the distance is 10Km and, for different
reuse factor K values by considering different α values. By
observing Fig.2, we can appreciate that, when the reuse factor
is within the value K =3, then the system presents higher
rate when path loss is heavier (α =4) than when it is mild
(α =2). This result can be justified by considering that, when
2 3 4 5 6 7 8
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
spatial reuse parameter K
throughput link (bits/s/Hz)
N=8 d=10Km W=40MHz
alfa = 2
alfa = 3
alfa = 4
Fig. 2. System performance for different α values.
the K parameter is low (e.g., K =2) we have that the sets
I
l
are composed by the half of the number of nodes so, in
the α =4case, the interference is heavily faded and the only
parameter to dominate performance is the presence of noise.
On the contrary, when we consider high values of K, since
we have the I
l
sets containing few terms, the parameter ρ is
higher for α =2so the power of reference transmitter is less
attenuated.
Passing now to analyze the results shown in Fig.3, we can
appreciate how, when large number of transmit antennas N
T
is considered, by evaluating the rate for N
R
ranging from 1
to 8m this gives arise to a rate increment. In particular, when
the plot for an assigned value of N
R
is considered, then it
is possible to observe that when N
R
=1, the maximum is
obtained for K =1, while, when N
R
=8is considered, the
maximum is obtained for K =3. For sake of brevity we do
5 10 15 20 25
0
2
4
6
8
10
12
number of hops K=N
throughput link (bits/s/Hz)
Ptot=10W,d=10Km,alfa=4,W=40MHz,K = N(uniform allocation)
MIMO channel
N
T
=1 N
R
=30
N
T
=2 N
R
=30
N
T
=3 N
R
=30
N
T
=4 N
R
=30
N
T
=5 N
R
=30
N
T
=6 N
R
=30
N
T
=7 N
R
=30
N
T
=8 N
R
=30
Fig. 3. System performance for large number of transmit antennas and high
level of noise.
not report simulations for different α values but, we state that
when α =2the rate expression can be considered
-convex
that is not the same for α =4.
In Fig. 4 a similar behavior to that shown in Fig.3 is depicted.
In fact, in this particular case, we detail the case for N
T
ranging from 1 to 8 and very large N
R
. In this situation,
the system behavior shows, in a more clear way, how the
maximum value for different K changes by taking into account
for different N
T
values. In fact, when N
T
=1transmit
antenna is considered, the maximum value is obtained for
K =2while when N
T
=8the maximum is achieved for
K =3.
A. Single hop -vs- multi hop routing
End-to-end transport delay is a key performance metric
for multi-hop networks. As known, transport delay is due
to the waiting time at the various relaying nodes plus the
transmission delay. In turn, this last is composed by delay due
to contention among different users (re-scheduling delay) and
packet transmission time. By considering a multi-hop network
5 10 15 20 25
0
2
4
6
8
10
12
number of hops K=N
throughput link (bits/s/Hz)
Ptot=10W,d=10Km,alfa=4,W=40MHz,K = N(uniform allocation)
MIMO channel
N
T
=1 N
R
=30
N
T
=2 N
R
=30
N
T
=3 N
R
=30
N
T
=4 N
R
=30
N
T
=5 N
R
=30
N
T
=6 N
R
=30
N
T
=7 N
R
=30
N
T
=8 N
R
=30
Fig. 4. System performance for large number of receive antennas and high
level of noise.
where the inter-node distances are assigned and each relaying
terminal uses an assigned transmission power, let us denote by
T
h
the corresponding single-hop delay. Thus, the total delay
T
tot
of the considered route can be evaluated as
T
tot
= T
h
N. (32)
Now, at least when the network load is light, T
h
may be
considered almost independent the hop number N
h
, so that in
this case T
tot
scales linearly with N. Under networking heavy
load, contention phenomena may induce additional latency, so
that T
h
may depend on N. However, since also in this case T
tot
typically grows for increasing values of N , we conclude that,
in principle, low routing delay T
tot
demands for single-hop
routes. However, although appealing in terms of routing delay,
till now the single-hop approach has been somewhat neglected
on the basis of the (partially incorrect) consideration that
single-hop routing requires high power levels to be radiated
by the transmit nodes that, in turns, increase Multi-user levels
and, then, lower the final QoS experienced at the destination
nodes.
By considering the results reported in Fig.5, it is possible
to appreciate how the gain offered (for low noise level) by
interference cancellation is greater at low K values, when the
nodes experience much interference. In fact, when K =1the
gain for N
T
= N
R
=4ranges from 8 bit/sec/Hz to 10, that
is an improvement of 25%. The gain offered by cancellation
for N
T
= N
R
=2, is shown in Fig.5 and, in this case, it is
less than the previous case.
When K = N =24then no interference is experienced so
cancellation is no more needed even if the achieved rate is, in
this case, considerably lower.
VI. C
ONCLUSIONS
By basing on the above analysis and results, we can argue
that routing for linear networks, able to model WMN backhaul,
can be seen as a power allocation procedure and, at the same
0
2
4
6
8
10
12
1 3 5 7 9 11 13 15 17 19 21 23
K
Throughput (bit/sec/Hz)
NT=4, NR=4, cancellation
NT=4, NR=4, no cancellation
NT=2, NR=2, cancellation
NT=2, NR=2, no cancellation
Fig. 5. System performance for limited number of transmit/receive antennas
and low level of noise with cancellation.
time, as ana access problem by considering the reuse factor K.
Furthermore, since interference cancellation aided by spatial
diversity is performed, the pursued analysis may be considered
as a time (K) and space (N
T
,N
R
) hybrid access to mitigate
interference and improve performance with the goal of reduce
the number of hops, and consequently, the delay.
The possibility to mitigate the interference effect allows to
take into account for possible mistiming effects in the reuse
scheme. The price to be paid for the performance improve-
ment consists into setup stations with N
R
,N
T
antennas and,
which is the most, the receiver complexity increases with an
algorithm that requires matrix inversion so the cost is of the
order of O(N
3
R
), that can be acceptable for the obtained gain
offered by the pursued spatial signal processing.
R
EFERENCES
[1] E. Baccarelli, M. Biagi, R. Bruno, M. Conti, E. Gregori, Broadband
Wireless Access Networks: a Roadmap on Emerging Trends and Standards,
in Broadband Services: Business Models and Technologies for Community
Networks, I. Chlamtac, A. Gumaste, C. Szabo (Editors), John Wiley Sons
(Publisher), Mar. 2005.
[2] ETSI, Broadband Radio Access Network (BRAN)HIPERACCESS, Type
2: Physical Layer Specification / DL Specification.
[3] ETSI, Broadband Radio Access Network (BRAN) HiperLAN, Type 2:
Physical Layer Specification / DL Specification.
[4] ETSI Broadband Radio Access Network (BRAN) HIPERMAN, Type 2:
Physical Layer Specification / DL Specification.
[5] IEEE Std 802.11b-2001, (Supplement to ANSI/IEEE Std 802.11-1999):
Wireless LAN Medium Access Control (MAC) and Physical Layer (PHY)
specifications: Higher-Speed Physical Layer Extension in the 2.4 GHz
Band, 7 November 2001.
[6] IEEE 802.11 WG, Draft Supplement LAN/MAN Specific Requirements
- Part 11: Wireless MAC and PHY Specifications: Specification for
Enhanced Security, IEEE Std 802.11i/D7.0, Dec.2003
[7] IEEE Standard 802.16a-2003, IEEE Standard for Local and metropolitan
area networks . Part 16: Air Interface for Fixed Broadband Wireless
Access Systems . Amendment 2: Medium Access Control Modifications
and Additional Physical Layer Specifications for 2-11 GHz, January 29,
2003.
[8] P. Gupta and P. R. Kumar, ”The capacity of wireless networks,” IEEE
Trans. Inf. Theory, vol. 46, no. 2, pp. 388-404, Mar. 2000.
[9] P. Gupta and P. R. Kumar ”Toward an information theory of large
networks: An achievable rate region,” IEEE Trans. Inf. Theory, vol. 49,
no. 8, pp. 1877-1894, Aug. 2003.
[10] L.-L. Xie and P. R. Kumar, ”A network information theory for wireless
communication: Scaling laws and optimal operation,” IEEE Trans. Inf.
Theory, vol. 50, no. 5, pp. 748-767, May 2004.
[11] M. Gastpar and M. Vetterli, ”On the capacity of wireless networks: The
relay case,” in Proc. IEEE INFOCOM02, vol. 3, New York, Jun. 2002, pp.
1577-1586.
[12] M.Gastpar and M. Vetterli, ”On the capacity of wireless networks: The
relay case,” in Proc. IEEE INFOCOM02, vol. 3, New York, Jun. 2002, pp.
1577-1586.
[13] L.-L. Xie and P. R. Kumar, ”An achievable rate for the multiple-level
relay channel,” IEEE Trans. Inf. Theory, vol. 51, no. 4, pp. 1348-1358,
Apr. 2005.
[14] G. Kramer, M. Gastpar, and P. Gupta, ”Capacity theorems for wireless
relay channels,” in Proc. 41st Allerton Conf. Communication, Control and
Computing, Monticello, IL, Oct. 2003, pp. 3037-3063.
[15] A. B. Carleial, ”Multiple-access channels with different generalized
feedback signals,” IEEE Trans. Inf. Theory, vol. IT-28, no. 6, pp. 841-850,
Nov. 1982.
[16] E.Baccarelli, M.Biagi, C.Pelizzoni, N.Cordeschi, F.Garzia, ”When does
interference not reduce capacity in multi-antenna networks?”, IEEE Signal
Processing Advances in Wireless Communications, 2005 pp.298-302,
2005.
[17] M.Sikora, J.N. Laneman, M.Haenggi, D.J.Costello, T.E.Fuja,
”Bandwidth- and Power-Efficient Routing in Linear Wireless Networks”,
IEEE Trans. on Inf. Theory, vol.52, no.6, pp.2624-2633, June 2006.
[18] L.Schumacher, K.I.Pedersen, P.E. Mogensen, P.E., ”From antenna spac-
ings to theoretical capacities - guidelines for simulating MIMO systems”,
The 13th IEEE International Symposium on Personal, Indoor and Mobile
Radio Communications, Vol. 2, pp.587-592, 15-18 Sept.2002.
[19] E.Baccarelli, M.Biagi, C.Pelizzoni, ”A Distributed Power-Allocation
and Signal-Shaping Game for the Competitively Optimal Throughput-
Maximization of Multiple-Antenna ”ad hoc” Networks Vehicular Tech-
nology, IEEE Transactions on Volume 55, no.6, pp.1862-1876, Nov.
2006.
[20] B.M.Hochwald, T.L.Marzetta, V.Tarokh, ”Multiple-Antenna Channel
Hardening and Its Implications for Rate Feedback and Scheduling”, IEEE
Trans. on Inf. Theory, vol.50, no.9, pp.1893-1909, Sept.2004.
[21] E.Baccarelli, M.Biagi, ”Power-allocation policy and optimized design of
multiple-antenna systems with imperfect channel estimation”, IEEE trans.
on Vehic. Tecn., vol. 53, no.1, pp.136-145, Jan.2004.
[22] E.Baccarelli, M.Biagi, C.Pelizzoni, N.Cordeschi, F.Garzia, Interference
Suppression in MIMO systems for Throughput Enhancement and Er-
ror Reduction, IEEE International Wireless Communications and Mobile
Computing Conference, 3-6 July 2006, Vancouver, pp.611-616.
[23] H.V. Poor, An Introduction to Signal Detection and Estimation,Springer
Texts in Electrical Engineering
