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/m 2 )
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 D ½
Gupta-Kumar Scaling Law (2000) � Scaling law � As node density D → ∞ , transport capacity converges to zero at rate O (1/ D ½ ) � 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 = WT b � All nodes have same transmission power P 0 � No power control is employed
Power Decay Model � Power decays in distance r P = 0 ( ) P r β + ( / 1 ) r r 0 � P 0 is transmission power, r 0 > 0, β > 2
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 d N , or (nodes/Hz/second/m 2 ) d N / N � Traffic intensity ρ d N / N (sending nodes/Hz/second/m 2 )
Limiting Network � d N → ∞ , N → ∞ , d N / N → α W f
Limiting Network � d N → ∞ , N → ∞ , d N / N → α W f
Limiting Network � d N → ∞ , N → ∞ , d N / N → α W f
Limiting Network � d N → ∞ , N → ∞ , d N / N → α W f
Limiting Network � d N → ∞ , N → ∞ , d N / N → α W f
Objective � For the limiting network as d N → ∞ , N → ∞ , d N / N → α , we derive � Transport capacity (bit-meters/symbol period/node) � Spectral efficiency (bit-meters/Hz/second/m 2 )
Received Signal in a node � Chip matched filter output in a receiving node ∑ = + + ( ) (|| ||) b P r b P y s x s n x x ∈ ( ) B N t x r � r is link distance � b , P ( r ), and s are for desired sending node � b x , P (|| x ||), and s x are for interference nodes � n ~ N ( 0 , σ 2 I )
MF Output � MF outputs an estimate of b y = s T y ∑ = + + T T ( ) (|| ||) P r b b P x s s s n x x ∈ ( ) B N t x = + ( ) P r b I � Unit-power SIR 1 η N ≡ 2 ( ) E I
Asymptotics � Theorem: Interference I is asymptotically independent Gaussian, and unit-power SIR η N converges a.s. to 1 η = σ + ∞ 2 ( ) P � where total interference power to a node is finite π αρ 2 2 r P ∞ = 0 0 ( ) P (watts/Hz/second) β − β − ( 2 )( 1 ) � Include all interference of the network � Limit network is capable of information transportation
Limit Link Channel � From sending b to MF output, there is a link channel, which is memoryless Gaussian y = η + r ( ) y P r b z b z ~ N (0,1), i.i.d. � SIR = η P ( r ) depends only on link distance � Same result can be obtained if a decorrelator or MMES receiver is employed
Link Channel Capacity � For a link of distance r , the link capacity is 1 = + η ( ) log ( 1 ( )) C r P r 2 2 (bits/symbol period)
Packet delivery � A packet is delivered from source node to destination node via a multihop route ϕ ( x ) = { x i , i = 1, …, h ( x ), x 1 + x 2 + … + x h ( x ) = x } � A packet is coded D with achievable rate � The code rate of a packet x 4 x to be delivered via route x 3 ϕ ( x ) must be not greater x 2 than the minimum link x 1 S capacity on the route
Route Transport Capacity � Via route ϕ ( x ), bits per symbol min (|| ||) C x i ≤ ≤ 1 ( ) i h x period are transported by a distance of || x || meters � h ( x ) nodes participate in transportation D � Route transport capacity is x 4 || || min (|| ||) C x x x i ≤ i ≤ Γ = 1 ( ) h x ϕ ( ) x ( ) h x 3 x x 2 (bit-meters/symbol period/node) x 1 S
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 V u [ ϕ ( x )]
Transport Throughput � Transport throughput achieved under routing protocol u ( ) Γ = ρ Γ ( ) u E ϕ u ( ) r || || min ( ) C x x ∫ ∫ ≤ ≤ = ρ ϕ 1 ( ) i h x ( ( )) ( ) dV dF x x u ( ) h x ϕ ∈ Ω ℜ 2 ( ) ( ) x x u (bit-meters/symbol period/node) � F ( x ) – distribution of S-D vector x � V u [ ϕ ( x )] – route distribution
Transport Capacity � Each achievable routing protocol attains a transport throughput � Transport capacity is defined as Γ = Γ sup ( ) u ∈ Ψ u Ψ – collection of all achievable routing � protocols
Spectral Efficiency � Given transport capacity Γ , spectral efficiency is Π = α Γ (bit-meters/Hz/second/m 2 )
Main Result � Theorem: Transport capacity equals ∞ ( / ( )) C r h r ∫ Γ = ρ * max ( ) r dF r ≥ ( ) h r ( ) 1 h r 0 � r – S-D distance; F ( r ) – distribution of r � Spectral efficiency equals ∞ ( / ( )) C r h r ∫ Π = αρ * max ( ) r dF r ≥ ( ) h r ( ) 1 h r 0
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 d N → ∞ ), then Γ = α ( 1 / ) O Π = ( 1 ) O � 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
Scaling Law � The “scaling law” can be overcome, provided spreading gain N (or bandwidth) increases at the same rate as node density d N increases Γ = constant > 0 Π = constant > 0 � A large wireless CDMA ad hoc network is capable of information transportation!
Transport Capacity vs. Traffic Intensity Transport capacity vs. alpha 4 GIGC BIGC 3.5 BSC MMSE Γ (bit-meters/symbol period/node) 3 Dec MF 2.5 2 1.5 1 0.5 0 -4 -3 -2 -1 0 5x10 0 10 10 10 10 10 α (nodes/Hz/second/m 2 ) - node density / processing gain � Transport capacity monotonically decreases with α
Spectral Efficiency vs. Traffic Intensity -5 x 10 1.6 GIGC BIGC 1.4 BSC MMSE Π (bit-meters/Hz/second/m 2 /watt) 1.2 Dec MF 1 0.8 0.6 0.4 0.2 0 -4 -3 -2 -1 0 5x10 0 10 10 10 10 10 α (nodes/Hz/second/m 2 ) � Π monotonically increases with α
Transport Capacity vs. Transmission Power 0.25 GIGC BIGC BSC 0.2 MMSE Γ (bit-meters/Hz/second/m 2 /watt) Dec MF 0.15 0.1 0.05 0 -20 -10 0 10 20 30 40 P 0 (dB) � Transport capacity monotonically increases with P 0
Spectral Power Efficiency vs. Transmission Power -3 x 10 3.5 GIGC BIGC 3 BSC MMSE Π (bit-meters/Hz/second/m 2 /watt) Dec 2.5 MF 2 1.5 1 0.5 0 -20 -10 0 10 20 30 40 P 0 (dB) � Π monotonically decreases with P 0
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