SINR Diagram with Interference Cancellation

1. SINR Diagram withInterference Cancellation MeravParter Joint work with Chen Avin, Asaf Cohen, Yoram Haddad, Erez Kantor, ZviLotker and David Peleg SODA 2012, Kyoto

2. Wireless Radio Networks d S1 • Stations with radio device • Synchronous operation • Wireless channel • No centralized control S2 S3 S4 S5

3. Reception Map - Who Hears Me? Each point ``hears” the nearest station Voronoi Diagram Vor(i) := Voronoi Cell of station si∈S. M. Shamos and D. Hoey, FOCS'75

4. Reception Map - Physical Model Point ``hears” a station if it can decode its massage Attempting to model signalattenuationand interferenceexplicitly Signal to Interference plus Noise Ratio (SINR)

5. Physical Models: Signal to interference & noise ratio • Received signal strength of station si • Transmission • Energy • Station si • Receiver point • Path-loss exponent • Noise • Interference

6. Fundamental Rule of the SINR model Station si is heard at point p ∈d- S iff • Reception Threshold (>1)

7. Interference Cancellation Receivers decode interfering signals Cancel them from received message Decode their intended message The optimal strategy in several scenarios (strong interference, spread spectrum communication, etc.) J.G. Andrews. Wireless Communications, IEEE, 2005

8. Successive Interference Cancelation (SIC) Interference Received Signal Signal

9. Decode the Strongest Interference Decode (Rounding) Received Signal

10. and subtract (cancel) it … Interference Received Signal Amplify Signal

11. Can you hear me? (Uniform Powers) s3 Without SIC: With SIC: p To ``hear” s1, receiver p should cancel signals of closer stations in order s2 s1 s4

12. Can you hear me? (with IC) s3 p No Yes s2 No Yes s1 Yes No s4

13. The Power of SIC: Exponential Node Chain Q: How many slots are required to schedule all links? Can receive multiple messages at once! Without SIC: n slots are required for any power assignment With SIC: A single slot is required using uniform powers

14. SINR Diagram A map characterizing the reception zones of the network stations Reception Zone of Station si Null Zone ) Cell Cell := Maximal connected component within a zone.

15. SINR Diagram with Uniform Power All stations transmit with power 1 (Ψi=1for everyi) • Theorem: • Every reception zoneH(si) is convex and fat. • H(si) ⊆ Vor(i). [Avin et al. PODC09] [Avin, Emek, Kantor, Lotker, Peleg, Roditty, PODC 2009]

16. Reception Map for Physical Model with Interference Cancellation Goal: Uniform Power with SIC How hears me? Can I receive s1??

17. SIC-SINR Diagram A map characterizing the reception zone of stations in the SINR where receivers employ IC Reception Map for Station s1 2- Cancellation No Cancellation 1- Cancellation

18. Complexity of SIC-SINR Diagram SIC-SINR reception zones are not convex in d nor hyperbolically convex in d+1. The polynomial depends on cancellation ordering. Aprioi there are exponentially many.

19. Challenges Bounding #Cells (how many?) Nice Properties of Zones Topology Map Drawing (which are they?) Point Location Task (how to store them?) Algorithms

20. Reception Region of Cancellation Ordering • Ordering of stations • The reception region of points that share the same order of cancellation For =1 For >1 H(s1) Every region is characterized by a polynomial. Thus it can be drawn. H(s2, s1) H(s3, s1) H(s2, s3, s1) H(s3, s2, s1)

21. Reception Region of Cancellation Ordering (CO) H(s2) H(s1|S-{s2}) H(

22. High-Order Voronoi Diagram Order k=2 • Extension of the ordinary Voronoidiagram • Cells are generated by more than one point in S. • Every region consists of locations having the same closest points in S. Cell’s generators M. Shamos and D. Hoey, FOCS'75

23. Ordered order-k Voronoi Diagram Sorted list of generators Or, inductively For k=2 Lemma [High-Order cells] There are O(n2d) orderings such that

24. High-Order Voronoi Diagram and SIC-SINR Diagram Lemma [High Order and SIC-SINR] H(i) Lemma [distinct cells] Consider ,uch that ),) Then ) )=.

25. The Topology of Reception Zone with SIC • The Reception zone of every network’s station • is a collection of O(n2d) shapes. • Each shape is: • Convex • Correspond to distinct cancellation ordering • Fully contained in the high-order Voronoi cell

26. Drawing Map: Introducing Hyperplane Arrangement Separating hyperplane of station The set of hyperplanes of pairs in S dissects d into connected regions We call this dissection the arrangement of S Theorem [Arrangments, Edelsbrunner 1987]

27. Labelled Arrangement Label each cell with ordered list of S in non-decreasing distance from Observe: (1,2,3) (1, 3,2) Stations ordering is required only once p (2,1,3) (3, 1, 2) Subsequent labels are computed in (3 , 2, 1) (2, 3 , 1)

28. Drawing SIC-SINR Reception Map For station s1 • Construct the labelled arrangement • Extract cancellation ordering of non-empty cells. • Derive the Characteristic Polynomial of • H() (Intersection of |Si| polynomials). • Draw the resulting polynomial. • Efficiency: in time & place. 1 1 (1,2,3) (1, 3,2) ( 3,1,2) 2,1 ( 2,1,3) 3,1 ( 3,2,1) ( 2,3,1) 3,2,1 2,3,1

29. Bounding #Cells – The Compactness Parameter Compactness Parameter of network • Path-loss exponent • Reception Threshold (>1)

30. Bounding #Cells Theorem [Linear #Cells] If >5 then Cis composed of O(1) cells for any dimension . ,

31. Challenges Bounding #Cells (how many?) Nice Properties of Zones Topology Map Drawing (which are they?) Point Location Task (how to store them?) Algorithms

32. Challenges – What We Know #Cells General- O(n2d) Compact network- O(1) Nice Properties of Zones Collection of convex cells Each related to high-order Voronoi cell. Topology Map Drawing Efficient construction of labeled arrangement O(n2d+1) Point Location Task Poly-time construction. Logarithmic time per query Algorithms

33. Questions? Thanks for listening!