1. 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 TexPoint fonts used in EMF: AAA A AA A A A A

2. 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?

3. How much can cooperation help? Gupta-Kumar 00 ? Courtesy: David Reed of MIT

4. 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) = n 1- .

5. Channel Model Baseband channel gain between node k and l: where r kl is the distance apart and  kl is the random phase (iid across nodes). is the path loss exponent (in power)

6. 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.

7. Gupta-Kumar Capacity is Interference-Limited Long-range transmission causes too much interference. Nearest-neighbor transmission means each packet is transmitted 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).

8. 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.

9. 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?

10. 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.

11. Parallelization across S-D Pairs Phase 1: Clusters work in parallel. Sources in each cluster take turn distributing their bits. Total time = M 2 Phase 2: 1 MIMO trans. at a time. Total time = n Phase 3: Clusters work in parallel. Destinations in each cluster take turn collecting their bits. Total time = kM 2

12. Back-of-the-Envelope Throughput Calculation total number of bits transferred = nM total time in all three phases = M 2 +n + kM 2 throughput: bits/second Optimal cluster size Best throughput:

13. Further Parallelization In phase 1 and 3, M 2 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 M 2. 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.

14. Recursion Lemma: A scheme with thruput M b for the smaller network yields for the original network a thruput: with optimal cluster size:

15. 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 n 1-  => near network-wide MIMO cooperation.

16. 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 The hierarchical scheme is nearly information theoretically optimal.

17. 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!

18. 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 n 1/2 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.

19. Extended Networks:  >2 For  > 2, even when each node transmits at full power in the MIMO phase, total received SNR per node = n 1-  /2 -> 0 n by n MIMO transmission is now power-limited: C MIMO ~ total Rx power = n 2-  /2 Can the hierarchical scheme achieve arbitrarily close to this scaling?

20. 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 power- limited and degrees-of-freedom-limited regimes.

21. Bursty Hierarchical Scheme Runs the hierarchical scheme of the time with power P/n per node. Satisfies the power constraint P/n  /2 per node. Achieves an aggregate throughput of This achieves arbitrarily close to the total energy transferred between a size n transmit cluster and a size n receive cluster at distance n 1/2 apart.

22. Is Our Scheme Optimal for Extended Networks? We show: for all , the cutset bound scales like the total received power under no Tx cooperation. A dichotomy:  > 3: this total power is, dominated by transfer between the few boundary users. Multihop is optimal.  <3: total power is n 2-  /2, dominated by transfer between the many interior users. Our scheme is optimal. Tx cluster: size n 1-  Rx cluster: size n 1-  Distance Achievable (Top Level of Hierarchical Scheme) Cutset Bound

23. Calculation of Cutset Bound: Subtleties Cutset bound is the capacity of a MIMO link with n/2 transmit nodes on the left and n/2 receive nodes on the right. Subtlety: individual nodal power constraint rather than total power constraint. Relaxing to total transmit power constraint is too loose. Right thing to do is to relax to total weighted power constraint, Weight assigned to a node proportional to the total power received on the other side of cut per transmitted watt.

24. Generalization to d-dimensional networks Capacity scaling exponent: For dense networks For extended networks:

25. 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.

26. 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 network covers an increasing area. Here power limitation comes into play in addition to interference limitation.

27. Main Results for Extended Networks For path loss exponent  between 2 and 3: a modification of the hierarchical scheme can achieve: T(n) = n 2-  /2 We also showed this is the best that one can do. For  > 3: Multihop is optimal:

28. 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. Also yields optimal scaling law for extended networks.