Download - SINR Diagram with Interference Cancellation
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!