Wireless Communication using

Directional Antennas

Directional Antennas

• Focus the energy in a desired direction:

Antenna Array

Desired Communication

Point

• Geometric representation:

r

Communication Links

A

Requirement #1

A

B

Strong Connectivity

Requirement #2

B

A

HopSpanner

|AB| ≤ 1: minpath(A, B) has a constant #hops.

Requirements #3

B

A

MinimumRadius

How can A reach B?

B

A

Problem Statement

• Given: – plane point set S, fixed angle – one -antenna per point

• Find: – orientations of -antennas – a minimum radius r

such that the induced communication graph is a strongly connected hop spanner.

B

A

Case = 180o

REuclideanMST

180o Antennas - Clustering

RBFS Traversal

Select non-adjacent edges of the MST.

180o Antennas - Clustering

R

Select non-adjacent edges of the MST.

BFS Traversal

180o Antenna Orientations

R

Basic Observation

R

Basic Observation

R

B

A

180o Antennas of Radius 2

R

B

A

How can A reach B now?

B

A

(180o, 2)-Communication Graph

RCommunicationGraph for: = 180o

Radius = 2

Strongly connectedHop factor = 3

What about < 180o ?

• Next: ≥ 120o

K2

K1

120o Antennas – 3 Point Connectivity

A

C

B

Want: Strong Connectivity Plane Coverage

120o Antennas – 3 Point Connectivity

A

C

B

Want: Strong Connectivity Plane Coverage

A

C

B

Radius = Maximum Pairwise Distance

120o Antennas – 3 Point Connectivity

A

C

B

Radius = Maximum Pairwise Distance ≤ 2

A

C

B

Observation: Radius = 3 also covers the unit disk around each point

120o Antennas – 3 Point Connectivity

A

C

B

Radius = Maximum Pairwise Distance ≤ 2

A

C

B

Observation: Radius = 3 also covers the unit disk around each point

120o Antennas – 3 Point Connectivity

A

C

B

Radius = Maximum Pairwise Distance ≤ 2

A

C

B

Observation: Radius = 3 also covers the unit disk around each point

Connecting 3-Point Clusters

A

C

B

Assumption: Clusters at unit distance.

X

Z

Y

X

Z

Y

A

C

B

Connecting 3-Point Clusters

A

C

B

Assumption: Clusters at unit distance.

X

Z

Y

X

Z

Y

A

C

B

Connecting 3-Point Clusters

A

C

B

Assumption: Clusters at unit distance.

X

Z

Y

X

Z

Y

A

C

B

120o Antennas - Clustering

REuclideanMST(S)

Partition S intoclusters of ≥ 3 nodes

(120o, 5)-Communication Graph

Rr = 5:Each cluster isstrongly connected

and

covers the enclosing unit halo

(120o, 5)-Communication Graph

Rr = 5:Each cluster isstrongly connected

and

covers the enclosing unit halo B

A

120o Antennas - Summary of Results

• Given: – plane point set S– fixed angle ≥ 120o

• There exist– orientations of -antennas

• Such that– radius r = 5 establishes a communication graph

that is a strongly connected, 5-hop spanner.

• Lower bound: r = 2

K1K2

K3

≥ 90o : Similar Approach

• 4 Point Connectivity

D

K4

A

B

C

K1

K2

K3

≥ 90o : Similar Approach

• 4 Point Connectivity

D

K4

A

B

C

D

A

B

C

Communication Graph

Radius =

Second longest pairwise distance

≥ 90o : Similar Approach

• Radius + 1: Unit halo coverage

D

A

B

C

• Allows us to connect clusters at unit distance

90o Antennas - Clustering

REuclideanMST

Partition S intoclusters of ≥ 4 nodes

r = 7:Each cluster isstrongly connected

and

covers the enclosing unit halo

(90o, 7)-Communication Graph

R

B

A

90o Antennas - Summary of Results

• Given: – plane point set S– fixed angle ≥ 90o

• There exist– orientations of -antennas

• Such that– radius r = 7 establishes a communication graph

that is a strongly connected, 6-hop spanner.

• Lower bound: r = 2

OPEN

• What about < 90 ?• This approach does not work:

• Strong connectivity:– each antenna must cover at least one point

• Plane coverage:– some antennas cover no points

• Conflicting criteria!

Restriction

too strong!