SINR Diagram withInterference Cancellation

Merav Parter

Joint work withChen Avin, Asaf Cohen, Yoram Haddad,Erez Kantor, Zvi Lotker and David Peleg

SODA 2012, Kyoto

Stations with radio device

Synchronous operation

Wireless channel

No centralized control

S1

S2

S3

S4

S5

Wireless Radio Networks

d

Reception Map - Who Hears Me?

Vor(i) := Voronoi Cell of station si∈S.

Each point ``hears” the nearest station

Voronoi Diagram

M. Shamos and D. Hoey, FOCS '75

Reception Map - Physical Model

Point ``hears” a station if it can decode its massage

Attempting to model signal attenuation and interference explicitly

Signal to Interference plus Noise Ratio (SINR)

Physical Models: Signal to interference & noise ratio

),( psSINR i

Npsdistij j

j

),(

),( psdist i

i

Transmission Energy

Path-loss exponent

Received signal strength of

station si

Receiver point

Station si

NoiseInterference

Station si is heard at point p ∈d - S iff

),( psS I N R i

Fundamental Rule of the SINR model

Reception Threshold (>1)

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

Successive Interference Cancelation (SIC)

Received Signal

Interference

0 ,0 , 0 ,0 , 0 , 1 ,0 ,1 , 0 , 1 ,0 , 1, 1 , 1, 1 , 0 ,0 , 1 ,1 ...

Signal

0 ,0 ,0 ,1 ,0 ,0 ,0 ,0 ,0 ,1 ,1 ,0 ,1 ,0 ,0 ,1,1….

Decode the Strongest

InterferenceReceived SignalDecode(Rounding)

0 ,0 , 0 ,0 , 0 , 1 ,0 ,1 , 0 , 1 ,0 , 1, 1 , 1, 1 , 0 ,0 , 1 ,1 ...

and subtract (cancel) it …

Signal

Interference

Received Signal

Amplify

0 ,0 ,0 ,1 ,0 ,0 ,0 ,0 ,0 ,1 ,1 ,0 ,1 ,0 ,0 ,1,1….

Can you hear me? (Uniform Powers)

s3

s2

s1

s4

Without SIC: 𝑆𝐼𝑁𝑅 (𝑠1 ,𝑝 )≥ 𝛽pWith SIC:

To ``hear” s1, receiver p should cancel

signals of closer stations in order 𝑑𝑖𝑠𝑡 (𝑠3 ,𝑝 )<𝑑𝑖𝑠𝑡 (𝑠2 ,𝑝)<𝑑𝑖𝑠𝑡 (𝑠1 ,𝑝)

Can you hear me? (with IC)

s3

s2

s1

s4

p𝑺𝑰𝑵𝑹(𝒔𝟑 ,𝒑 )≥ 𝜷

𝑺𝑰𝑵𝑹 (𝒔𝟏 ,𝒑|{𝒔𝟏 , 𝒔𝟒 }¿≥ 𝜷

No

No

No

Yes

Yes

Yes

The Power of SIC: Exponential Node Chain

Q: How many slots are required to schedule all links?

Without SIC: n slots are required for any power assignment

With SIC: A single slot is required using uniform powers

Can receive multiple messages at once!

SINR Diagram

A map characterizing the reception

zones of the network stations

𝐻 (𝑠2)

𝐻 (𝑠1)

)

Reception Zone of Station si

psSINRSRpssH id

ii ,| )(

ise v e r y fo r ,),(| psS I N RSRpH id

𝐻 ∅

Null Zone

Cell := Maximal connected

component within a zone.

Cell

All stations transmit with power 1 (Ψi=1 for every i)

SINR Diagram with Uniform Power

Theorem:

Every reception zone H(si)

is convex and fat.

H(si) Vor(i).⊆

[Avin, Emek, Kantor, Lotker, Peleg, Roditty, PODC 2009]

[Avin et al. PODC09]

Reception Map for Physical Model with Interference Cancellation

Can I receive s1??

How hears me?

Goal: Uniform Power with SIC

SIC-SINR Diagram

No Cancellation 1- Cancellation 2- Cancellation

A map characterizing the reception zone of

stations in the SINR where receivers employ

IC

Reception Map for Station s1

Complexity of SIC-SINR Diagram

The polynomial depends on cancellation ordering.Aprioi there are exponentially many.

SIC-SINR reception zones are not convex in d nor hyperbolically convex in d+1

.

Challenges

Nice Properties of

Zones

Bounding#Cells

(how many?)

Map Drawing (which are they?)

Point LocationTask

(how to store them?)

Algo

rithm

sTo

polo

gy

Reception Region of Cancellation Ordering

SsssS iii 121 ,,

Ordering of stations

The reception region of points that share the same order of cancellation

ik

iik

ii ssHssSsHSH 11111 ,,,,|)(

H(s1)

H(s2, s1)

H(s2, s3, s1) H(s3, s2, s1)

H(s3, s1)

For =1 psSINRSRpssH idii ,| )( 111

For >1

Every region is characterized by a polynomial.Thus it can be drawn.

Reception Region of Cancellation Ordering (CO)

21212 |),( sSsHsHssH

∩H(s2) H(s1|S-{s2}) H(

High-Order Voronoi Diagram

Extension of the ordinary Voronoi diagram

Cells are generated by more than one point in S.

Every region consists of locations having the same closest points in S.

Order k=2

Cell’s generators

M. Shamos and D. Hoey, FOCS '75

Ordered order-k Voronoi Diagram

ikii ssS ,,1

iij

ij

ik

iidi SspsdistpsdistpsdistpsdistpSVor

),,(),(),(),(| 21

ik

iik

iiki ssVorssSsVorSVor 1111 ,,,\|

Or, inductively

Sorted list of generators

For k=2

Lemma [High-Order cells]There are O(n2d) orderings such that

Lemma [distinct cells] Consider , uch that ), )Then ) )=.

High-Order Voronoi Diagram and SIC-SINR Diagram

Lemma [High Order and SIC-SINR]

H(i)

SsssS iii 121 ,,

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

SsssS iii 121 ,,

iSVor

Drawing Map: Introducing Hyperplane Arrangement

𝐻𝑃 (𝑠𝑖 ,𝑠𝑗 )={𝑝∈d  |𝑑𝑖𝑠𝑡 (𝑠𝑖 ,𝑝 )=𝑑𝑖𝑠𝑡 (𝑠𝑗 ,𝑝 )}Separating hyperplane of station

The set of hyperplanes of pairs in Sdissects d into connected regions

We call this dissection the arrangement of S

Theorem [Arrangments, Edelsbrunner 1987]

𝐴𝑟 (𝑆)𝐻𝑃 (𝑠2 ,𝑠3)

𝐻𝑃 (𝑠1 ,𝑠2)

𝐻𝑃 (𝑠1 ,𝑠3)

Labelled Arrangement

Label each cell with ordered list of S in non-decreasing distance from

(1,2,3) (1, 3,2)Stations ordering is required only

once

Observe:𝐻𝑃 (𝑠2 ,𝑠3)

𝐻𝑃 (𝑠1 ,𝑠3)

𝐻𝑃 (𝑠1 ,𝑠2)Subsequent labels are computed in

𝑓

(3, 1, 2)

(3 , 2, 1)(2, 3 , 1)

(2, 1, 3 )

p

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.

Drawing SIC-SINR Reception Map

For station s1

(1,2,3) (1, 3,2)

( 3,1,2)

( 3,2,1)( 2,3,1)

( 2,1,3)

1 1

2,1 3,1

3,2,12,3,1

Bounding #Cells – The Compactness Parameter

Compactness Parameter of network

𝐶𝑃 (𝐴 )=𝛼√𝛽Reception

Threshold (>1)

Path-loss exponent

Bounding #Cells

Theorem [Linear #Cells]If >5 then C is composed of O(1) cells forany dimension .

,

Challenges

Nice Properties of

Zones

Bounding#Cells

(how many?)

Map Drawing (which are they?)

Point LocationTask

(how to store them?)

Algo

rithm

sTo

polo

gy

Challenges – What We Know

#Cells General- O(n2d)

Compact network- O(1)

Map DrawingEfficient constructionof labeled arrangement O(n2d+1)

Point Location TaskPoly-time construction.

Logarithmic time per query

Algo

rithm

sTo

polo

gy Nice Properties of ZonesCollection of convex cellsEach related to high-order

Voronoi cell.

Questions?

Thanks for listening!