Transport Capacity and Spectral Efficiency of Large Wireless CDMA - - PowerPoint PPT Presentation

Transport Capacity and Spectral Efficiency of Large Wireless CDMA Ad Hoc Networks

Yi Sun Department of Electrical Engineering The City College of City University of New York Acknowledgement: supported by ARL CTA Program

Wireless Ad Hoc Network

A Fundamental Question

What is the information-theoretical limit

Transport capacity (packet-meters/slot/node) Spectral efficiency (bit-meters/Hz/second/m2)

Gupta-Kumar Model (2000)

Assumption

Achievable rate on each link is fixed Effective communications are confined to

nearest neighbors

Gupta-Kumar Model (2000)

For an ad hoc network on a unit square, if node density

is D, the number of nodes on a path equals about D½

Gupta-Kumar Scaling Law (2000)

Scaling law

As node density D → ∞, transport capacity

converges to zero at rate O(1/D½)

Large scale wireless ad hoc networks are

incapable of information transportation – a pessimistic conclusion

Can Scaling Law be Overcome?

Gupta-Kumar Model

Communications are confined in nearest

neighbors

Radio frequency bandwidth is not considered

in the model

Spectral efficiency is unknown

Observation I

If communications are not confined to nearest

neighbors, transport capacity can be increased

Observation II

If CDMA channel is considered and spreading gain (or

bandwidth) is large compared with node density, then communications are not necessary to be confined in nearest neighbors

A wireless CDMA ad hoc network may overcome the scaling law

Our Model

Large Wireless CDMA Ad Hoc Networks

CDMA

Nodes access each other through a common

CDMA channel

Spreading sequences are random, i.i.d. (long

sequences)

Spreading gain N = WTb All nodes have same transmission power P0 No power control is employed

Power Decay Model

Power decays in distance r

P0 is transmission power, r0 > 0, β > 2

β

) 1 / ( ) ( + = r r P r P

Network Topology

Nodes are distributed on entire 2-D plane Node locations can be regular or arbitrary

Node Distributions

Nodes are uniformly distributed At any time t, a percentage ρ of nodes are

sending

Sending nodes are also uniformly distributed For each N, node density is dN, or

dN/N (nodes/Hz/second/m2)

Traffic intensity

ρdN/N (sending nodes/Hz/second/m2)

W f

Limiting Network

dN → ∞, N → ∞, dN/N → α

W f

Limiting Network

dN → ∞, N → ∞, dN/N → α

W f

Limiting Network

dN → ∞, N → ∞, dN/N → α

W f

Limiting Network

dN → ∞, N → ∞, dN/N → α

W f

Limiting Network

dN → ∞, N → ∞, dN/N → α

Objective

For the limiting network as dN → ∞, N → ∞,

dN/N → α, we derive

Transport capacity (bit-meters/symbol

period/node)

Spectral efficiency (bit-meters/Hz/second/m2)

Received Signal in a node

Chip matched filter output in a receiving node

r is link distance b, P(r), and s are for desired sending node bx, P(||x||), and sx are for interference nodes n ~ N(0,σ2I)

n s x s y

x x x

+ + =

∑

∈ ) (

||) (|| ) (

t BN

P b r P b

r

MF Output

MF outputs an estimate of b Unit-power SIR

y sT y = ) ( 1

2

I E

N ≡

η I b r P + = ) ( n s s s x

x x x T T t BN

P b b r P + + =

∑

∈ ) (

||) (|| ) (

Asymptotics

Theorem: Interference I is asymptotically independent

Gaussian, and unit-power SIR ηN converges a.s. to

where total interference power to a node is finite

(watts/Hz/second)

Include all interference of the network Limit network is capable of information transportation

) ( 1

2

∞ + = P σ η ) 1 )( 2 ( 2 ) (

2

− − = ∞ β β αρ π P r P

Limit Link Channel

From sending b to MF output, there is a link

channel, which is memoryless Gaussian

z ~ N(0,1), i.i.d.

SIR = ηP(r) depends only on link distance Same result can be obtained if a decorrelator

• r MMES receiver is employed

z b r P y + = ) ( η

r b y

Link Channel Capacity

For a link of distance r, the link capacity is

(bits/symbol period)

)) ( 1 ( log 2 1 ) (

2

r P r C η + =

Packet delivery

A packet is delivered from source node to

destination node via a multihop route ϕ(x) = {xi, i = 1, …, h(x), x1 + x2 + … + xh(x) = x}

A packet is coded

with achievable rate

The code rate of a packet

to be delivered via route ϕ(x) must be not greater than the minimum link capacity on the route

x1 S D x2 x3 x4 x

Route Transport Capacity

Via route ϕ(x), bits per symbol

period are transported by a distance of ||x|| meters

h(x) nodes participate in transportation Route transport capacity is

(bit-meters/symbol period/node)

) ( ||) (|| min || ||

) ( 1 ) (

x x x

x x

h C

i h i≤ ≤

= Γ

ϕ

x1 S D x2 x3 x4 x

||) (|| min

) ( 1 i h i

C x

x ≤ ≤

Routing Protocol

A global routing protocol schedules routes of

all packets

Consider achievable routing protocols that

schedule routes without traffic conflict

Let distribution of S-D vector x be F(x) For the same S-D vector x, different routes

ϕ(x) may be scheduled

Under routing protocol u, let route ϕ(x) for S-

D vector x have distribution Vu[ϕ(x)]

Transport Throughput

Transport throughput achieved under routing

protocol u

(bit-meters/symbol period/node)

F(x) – distribution of S-D vector x Vu[ϕ(x)] – route distribution

( )

) (

) (

r ϕ

ρΓ = Γ

u

E u

∫ ∫

ℜ Ω ∈ ≤ ≤

=

2

) ( )) ( ( ) ( ) ( min || ||

) ( ) ( ) ( 1

x x x x x

x x x

dF dV h C

u

u h i ϕ

ϕ ρ

Transport Capacity

Each achievable routing protocol attains a

transport throughput

Transport capacity is defined as

• – collection of all achievable routing

protocols

) ( sup u

u

Γ = Γ

Ψ ∈

Ψ

Spectral Efficiency

Given transport capacity Γ, spectral efficiency

is

(bit-meters/Hz/second/m2)

Γ = Π α

Main Result

Theorem: Transport capacity equals

r – S-D distance; F(r) – distribution of r

Spectral efficiency equals

∫

∞ ≥

= Γ

1 ) ( *

) ( ) ( )) ( / ( max r dF r h r h r C r

r h

ρ

∫

∞ ≥

= Π

1 ) ( *

) ( ) ( )) ( / ( max r dF r h r h r C r

r h

αρ

Outline of Proof

Step 1: Show that Γ* is an upper bound Step 2: Show that Γ* is the lowest upper

bound

Need to find an achievable routing protocol to

attain Γ* − ε for any ε > 0

Scaling Law

If α → ∞ (or N fixed but dN → ∞), then

Transport capacity goes to zero at rate 1/α -

“scaling law” behavior

Spectral efficiency converges to a constant

This scaling law is due to that radio

bandwidth does not increases as fast as node density increases

– different from that of Gupta-Kumar model

) / 1 ( α O = Γ ) 1 ( O = Π

Scaling Law

The “scaling law” can be overcome, provided

spreading gain N (or bandwidth) increases at the same rate as node density dN increases

A large wireless CDMA ad hoc network is

capable of information transportation!

constant > = Γ constant > = Π

Transport capacity monotonically decreases with α

10

• 4

10

• 3

10

• 2

10

• 1

10 0.5 1 1.5 2 2.5 3 3.5 4 5x100 Transport capacity vs. alpha α (nodes/Hz/second/m2) - node density / processing gain Γ (bit-meters/symbol period/node) GIGC BIGC BSC MMSE Dec MF

Transport Capacity vs. Traffic Intensity

Π monotonically increases with α

Spectral Efficiency vs. Traffic Intensity

10

• 4

10

• 3

10

• 2

10

• 1

10 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 x 10

• 5

5x100 α (nodes/Hz/second/m2) Π (bit-meters/Hz/second/m2/watt) GIGC BIGC BSC MMSE Dec MF

Transport capacity monotonically increases with P0

• 20
• 10

10 20 30 40 0.05 0.1 0.15 0.2 0.25 P0 (dB) Γ (bit-meters/Hz/second/m2/watt) GIGC BIGC BSC MMSE Dec MF

Transport Capacity vs. Transmission Power

Π monotonically decreases with P0

Spectral Power Efficiency vs. Transmission Power

• 20
• 10

10 20 30 40 0.5 1 1.5 2 2.5 3 3.5 x 10

• 3

P0 (dB) Π (bit-meters/Hz/second/m2/watt) GIGC BIGC BSC MMSE Dec MF

Sensor Networks:

Sensor Density vs. Transmission Power

Sensor network is low powered, P0 → 0 Question: with given total power per square meter

αρP0 = ω,

should we increase node density and decrease node

transmission power?

• r converse?

Answer:

we should increase node density and decrease node

transmission power in terms of increase of spectral power efficiency

∫

∞ ∈ = →

+ = Π

+

) ( ,

) ( ] 1 )) ( ( )[ ( min lim r dR r h r r r h r c

Z r h P P β ω ω αρ

αρη

Conclusions

If radio bandwidth increases slower than

node density increases, transport capacity decreases to zero – “scaling law”

The scaling law is essentially different from

that of Gupta-Kumar model

The scaling law can be overcome, provided

radio bandwidth increases as fast as node density increases

A large wireless CDMA ad hoc network is

capable of information transportation