Wireless Network Information Theory L-L. Xie and P. R. Kumar Dept. of Electrical and Computer Engineering, and Coordinated Science Lab University of Illinois, Urbana-Champaign Email prkumar@uiuc.edu DIMACS Workshop on Network Information Theory, Web http://black.csl.uiuc.edu/~prkumar March 17-19, 2003
Wireless Networks u Communication networks formed by nodes with radios u Ad Hoc Networks – Current proposal for operation: Multi-hop transport » Nodes relay packets until they reach their destinations – They should be spontaneously deployable anywhere » On a campus » On a network of automobiles on roads » On a search and rescue mission – They should be able to adapt themselves to » the number of nodes in the network » the locations of the nodes » the mobility of the nodes » the traffic requirements of the nodes Sensor webs u
Current proposal for ad hoc networks u Multi-hop transport Interference Interferenc Interference Interference + e + + Noise + Noise Noise – Packets are relayed from node to node Noise – A packet is fully decoded at each hop – All interference from all other nodes is simply treated as noise u Properties – Simple receivers – Simple multi-hop packet relaying scheme – Simple abstraction of “wires in space” u This choice for the mode of operation gives rise to – Routing problem – Media access control problem – Power control problem – …..
Three fundamental questions u If all interference is treated as noise, then how much information can be transported over wireless networks? u What is unconditionally the best mode of operation? u What are the fundamental limits to information transfer? u Allows us to answer questions such as – How far is current technology from the optimal? – When can we quit trying to do better? » E.g.. If “Telephone modems are near the Shannon capacity” then we can stop trying to build better telephone modems – What can wireless network designers hope to provide? – What protocols should be designed?
If interference is treated as noise ...
If all interference is regarded as noise … u … then packets can collide destructively or u A Model for Collisions – Reception is successful if Receiver not in vicinity of two transmissions r 1 r 2 (1+ D ) r 1 u Alternative Models (1+ D )r 2 – SINR ≥ b for successful reception – Or Rate depends on SINR
Scaling laws under interference model A square meters u Theorem (GK 2000) n nodes – Disk of area A square meters – i n nodes – Each can transmit at W bits/sec ( ) bit-meters/second Q W An u Best Case: Network can transport u Square root law – Transport capacity doesn’t increase linearly, but only like square-root c – Each node gets bit-meters/second n Ê ˆ 1 u Random case: Each node can obtain throughput of ˜ bits/second Q Á ˜ Á n log n Ë ¯
Optimal operation under “collision” model u Optimal operation is multi-hop Bit-Meters c Per Second Per Node n – Transport packets over many c hops of distance 1 n n 0 Range No Multi-hop u Optimal architecture connectivity Networks Broadcast – Group nodes into cells of size about log n – Choose a common power level for all nodes » Nearly optimal – Power should be just enough to guarantee network connectivity » Sufficient to reach all points in neighboring cell – Route packets along nearly straight line path from cell to cell
But interference is not interference u Excessive interference can be good for you – Receiver can first decode loud signal perfectly – Then subtract loud signal – Then decode soft signal perfectly – So excessive interference can be good u Packets do not destructively collide u Interference is information! u So we need an information theory for networks to determine – How to operate wireless networks – How much information wireless networks can transport – The information theory should be able to handle general wireless networks
Towards fundamental limits in wireless networks
Wireless networks don’t come with links u They are formed by nodes with radios – There is no a priori notion of “links” – Nodes simply radiate energy
Nodes can cooperate in complex ways A … while Nodes in Nodes in Group A while F Group D amplify and cancel …. X C interference of forward packets B D Group B at from Group E to E Group F Group C Signal One strategy: Decode and forward … SINR = Interference + Noise Instead, why not: Amplify and Forward One strategy: Increase Signal for Receiver Instead, why not: Reduce Interference at Receiver
How should nodes cooperate? Some obvious choices u – Should nodes relay packets? – Should they amplify and forward? – Or should they decode and forward? – Should they cancel interference for other nodes? – Or should they boost each other’s signals? – Should nodes simultaneously broadcast to a group of nodes? – Should those nodes then cooperatively broadcast to others? – What power should they use for any operation? – … Or should they use much more sophisticated unthought of strategies? u Decode and forward – Tactics such as may be too simplistic Amplify and Forward Interference cancellation ... Broadcast Multiple-access – Cooperation through does not capture all possible modes of operation Relaying ...
“There are more things in heaven and earth, Horatio, Than are dreamt of in your philosophy.” — Hamlet
A plethora of choices u The strategy space is infinite dimensional u Problem has all the complexities of – team theory – partially observed systems u We want Information Theory to tell us what the basic strategy should be – Then one can develop protocols to realize the strategy
Key Results: A dichotomy u If absorption in medium u If there is no absorption, and attenuation is very small – Transport capacity grows like Q ( n ) – Transport capacity can grow » when nodes are separated by superlinearly like Q ( n q ) for q > 1 distance at least r min – Coherent multi-stage relaying with – Square-root law is optimal interference cancellation can be ( ) = Q n An ( ) » i Q optimal » Since area A grows like W ( n) – Multi-hop decode and forward is order optimal u Along the way – Total power used by a network bounds the transport capacity – Or not – A feasible rate for Gaussian multiple relay channels
A quick review of information theory and networks
Shannon’s Information Theory u Shannon’s Capacity Theorem Channel x y – Channel Model p ( y | x ) p(y|x) » Discrete Memoryless Channel – Capacity = Max p ( x ) I ( X ; Y ) bits/channel use Ê p ( X , Y ) ˆ Á ˜ I ( X ; Y ) = Â p ( x , y ) log p ( X ) p ( Y ) Ë ¯ x , y N(0, s 2 ) u Additive White Gaussian Noise (AWGN) Channel P ¯ , where S ( z ) = 1 Ê ˆ x y – Capacity = S 2 log 1 + z ( ) s 2 Ë Ex 2 ≤ P
Shannon’s architecture for communication Encode Channel Source code Source decode for the Decode (Compression) (Decompression) channel
Network information theory: Some triumphs u Gaussian scalar broadcast channel u Multiple access channel
Network information theory: The unknowns u The simplest relay channel u The simplest interference channel u Systems being built are much more complicated and the possible modes of cooperation can be much more sophisticated – How to analyze? – Need a general purpose information theory
The Model
Model of system: A planar network n nodes in a plane u j r ij = distance between nodes i and j u r ij ≥ r min i Minimum distance r min between nodes u e - gr Signal attenuation with distance r : u r d – i g ≥ 0 is the absorption constant » Generally g > 0 since the medium is absorptive unless over a vacuum » Corresponds to a loss of 20 g log 10 e db per meter – d > 0 is the path loss exponent » d =1 corresponds to inverse square law in free space
Transmitted and received signals ik } {1,2,3, K ,2 TR W i = symbol from some alphabet to be sent by node i u x i y j t - 1 , W i ) x i ( t ) = signal transmitted by node i time t = f i , t ( y i u - gr ij n e  y j ( t ) x i ( t ) + z j ( t ) = signal received by node j at time t = u d r ij i = 1 N(0, s 2 ) i ≠ j T , W j ) ˆ Destination j uses the decoder W i = g j ( y j u ˆ Error if W i ≠ W i u ( R 1 , R 2 ,..., R l ) is feasible rate vector if there is a sequence of codes with ( u Pr( ˆ Max W i ≠ W i for some i W 1 , W 2 ,..., W l ) Æ 0 as T Æ• W 1 , W 2 ,..., W l Individual power constraint P i ≤ P ind for all nodes i u n Or Total power constraint  P £ P i total i = 1
The Transport Capacity: Definition u Source-Destination pairs – ( s 1 , d 1 ), ( s 2 , d 2 ), ( s 3 , d 3 ), … , ( s n(n-1) , d n(n-1) ) u Distances – L r 1 , r 2 , r 3 , … , r n(n-1) distances between the sources and destinations u Feasible Rates – ( R 1 , R 2 , R 3 , … , R n(n-1) ) feasible rate vector for these source-destination pairs u Distance-weighted sum of rates – S S i R i r i u Transport Capacity n ( n - 1) Â C T = sup R i ⋅ r i bit-meters/second or bit-meters/slot 1 , R 2 , K , R n ( n - 1) ) ( R i = 1
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