Hierarchical Cooperation Achieves Linear
Scaling in Ad Hoc Wireless Networks
David Tse
Wireless Foundations
U.C. Berkeley
MIT LIDS
May 7, 2007
Joint work with Ayfer Ozgur and Olivier Leveque at EPFL
Scaling of Ad Hoc Wireless Networks
• 2n nodes randomly located in a fixed area.
• n randomly assigned source-destination
pairs.
• Each S-D pair demands the same data rate.
• How does the total throughput T(n) of the
network scale with n?
How much can cooperation help?
Courtesy: David Reed of MIT
?
T(n) = £ (1)
p
T(n) = £ ( n)
Gupta-Kumar 00
??
Main Result
Linear capacity scaling is achievable with
intelligent cooperation.
More precisely:
For every  > 0, we construct a cooperative scheme
that can achieve a total throughput
T(n) = n1-.
Channel Model
• Baseband channel gain between node k and l:
hk l =
¡ ®2
Gr k l
ej µk l
where rkl is the distance apart and kl is the
random phase (iid across nodes).
• ® ¸ 2 is the path loss exponent (in power)
Dense networks
• Setting: many nodes all within communication
range of each other.
• Number of nodes are large but nearest nodes are
still far field from each other.
• Example:
–
–
–
–
Berkeley campus (1 square km)
n = 10,000 users (n >> 1)
Typical distance between nearest neighbors: 10m
Carrier frequency: 2.4 GHz => wavelength ~0.1m
• Will talk about extended networks later.
Gupta-Kumar Capacity is Interference-Limited
• Long-range transmission causes too much
interference.
• Nearest-neighbor transmission
means each
p
packet is transmitted n times (multi-hop).
• To get linear scaling, must be able to do many
simultaneous long-range transmissions.
• How to deal with interference?
• A natural idea: distributed MIMO (Aeron &
Saligrama 06).
MIMO:Multiple Transmit Multiple Receive Antennas
• The random MxM channel matrix allows
transmission of M parallel streams of data.
• Originally conceived for antennas co-located at
the same device.
Distributed MIMO
• MIMO effect can be simulated if nodes within
each cluster can cooperate.
• But cooperation overhead limits performance.
• What kind of architecture minimizes overhead?
A 3-Phase Scheme
• Divide the network into clusters of size M nodes.
• Focus first on a specific S-D pair. source s wants
to send M bits to destination d.
Phase 1 :
Setting up Tx
cooperation:
1 bit to each node in
Tx cluster
Phase 2:
Long-range
MIMO between
s and d clusters.
Phase 3:
Each node in Rx cluster
quantizes signal into k bits
and sends to destination d.
Parallelization across S-D Pairs
Phase 1:
Phase 2:
Clusters work in parallel.
1 MIMO trans.
Sources in each cluster take at a time.
turn distributing their bits.
Total time = n
2
Total time = M
Phase 3:
Clusters work in parallel.
Destinations in each cluster
take turn collecting their bits.
Total time = kM2
Back-of-the-Envelope Throughput
Calculation
total number of bits transferred = nM
total time in all three phases = M2 +n + kM2
throughput:
nM
bits/second
M 2 + n + kM 2
p
¤
Optimal cluster size
M = n
Best throughput:
p
n
Further Parallelization
• In phase 1 and 3, M2 bits have to be exchanged
within each cluster, 1 bit per node pair.
• Previous scheme exchanges these bits one at a
time (TDMA), takes time M2.
• Can we increase the spatial reuse ?
• Can break the problem into M sessions, each
session involving M S-D pairs communicating 1 bit
with each other:
cooperation = communication
• Any better scheme for the small network can build
a better scheme for the original network.
Recursion
Lemma: A scheme with thruput Mb for the smaller
network yields for the original network a thruput:
n
1
2¡ b
with optimal cluster size:
¤
M = n
f (b) =
1
2¡ b
1
2¡ b
MIMO + Hierarchical Cooperation
-> Linear Scaling
Setting up Tx
cooperation
Long-range
MIMO
Cooperate
to decode
.
At the highest level hierarchy, cluster size is of the order
n1- => near network-wide MIMO cooperation.
Upper Bound
• A simple upper bound: each source node has the benefit
of all other nodes in the network cooperating to receive
without interference from other nodes.
• Each source gets a rate of at most order log n.
• Yields an upper bound on network throughput
Cn · O(n logn)
• The hierarchical scheme is nearly information
theoretically optimal.
Transmit Power Requirement of Scheme
• At all levels of hierarchy, transmit powers in the
MIMO phase can be set such that the total
average received SNR at each node is 0 dB.
• This yields MIMO rate linear with the cluster size
in phase 2.
• This also explains why a fixed number of
quantization bits per sample suffices.
• At the total level of hierarchy, the transmit power
per node is P/n.
• We have power to spare!
From Dense to Extended Networks
• So far we have looked at dense networks, where
the total area is fixed.
• Another natural scaling is to keep the density of
nodes fixed and the networks covers an increasing
area.
• Distances are increased by a factor of n1/2 in
extended networks.
• Equivalently, an extended network is a dense
network with power constraint P/n/2 per node.
• Immediate result:
For  =2, linear scaling can be achieved for
extended networks.
Extended Networks: >2
• For  > 2, even when each node transmits at full
power in the MIMO phase,
total received SNR per node = n1-/2 -> 0
• n by n MIMO transmission is now power-limited:
CMIMO ~ total Rx power = n2-/2
• Can the hierarchical scheme achieve arbitrarily
close to this scaling?
Quantization is a Problem
• Subtle issue: information per received sample per
Rx antenna in MIMO goes to zero.
• If we use fixed number of bits to quantize each
sample, we are doomed.
• Cannot use vanishing number of bits either.
• Use bursty transmission so that during
transmission the SNR at each Rx antenna is again
0db.
• We are still power-efficient but Rx cooperation is
no longer onerous.
• We are operating at the boundary of powerlimited and degrees-of-freedom-limited regimes.
Is Our Scheme Optimal for Extended
Networks?
Achievable (Top Level of Hierarchical Scheme)
Cutset Bound
Rx cluster: size n1-
Distance
p
n
Tx cluster: size n1-
We show: for all , the cutset bound scales like the total
received power under no Tx cooperation.
A dichotomy:
p
 > 3: this total power is n , dominated by transfer
between the few boundary users. Multihop is optimal.
 <3: total power is n2-/2, dominated by transfer between
the many interior users . Our scheme is optimal.
Conclusion
• Hierachical cooperation allows network-wide
MIMO without significant cooperation overhead.
• Network wide MIMO achieves a linear number of
degrees of freedom.
• This yields a linear scaling law for dense networks.
• It also achieves maximum energy transfer in
extended networks when path loss exponent is less
than 3.
• Better than Gupta-Kumar scaling is possible in the
low attenuation regime.