Multicasting over Multiple-Access Networks

Transcription

1 ing oding apacity onclusions ing Department of Electrical Engineering and omputer Sciences University of alifornia, Berkeley May 9, 2006 EE 228A

2 Outline ing oding apacity onclusions oding 5 apacity 6 onclusions

3 Wireless Network apacity ing oding apacity onclusions omputing the capacity region of an arbitrary wireless network is an extremely difficult (and unsolved) problem Even with simplifying assumptions, most results are Gupta-Kumar style scaling laws Full characterization of capacity will lead to design insights for wireless networks We will focus on multicasting over a general model that incorporates interference and show that sometimes collisions are helpful.

4 ing oding apacity onclusions One simplifying assumption is to assume our wireless network is well-represented as a graph, G = (V,E), where the users are the vertices and the labelled edges represent point-to-point channels between users In this case, we know the capacity region for: Unicast (single source, single destination): routing is sufficient (single source, multiple destinations): network coding may be necessary However, this is just a wireline network: we have made the problem too simple.

5 Previous Work ing oding apacity onclusions Since we have tools for finding the capacity region of networks modelled as graphs, we should try to incorporate wireless aspects into these graphs. Dana et al. find the multicast capacity for the case when nodes must broadcast the same information on all outgoing edges. They use a packet erasure model and assume that the erasure pattern is known at all receivers. However, they exclude the possibility of intererence. Our approach will be to add an interference componen tto the original graph model.

6 Network oding: Example ing oding apacity onclusions a b a b a b a a b b a b a b a b a b Ahlswede et al. showed that pure routing is suboptimal for multicast, mixing packets is necessary. Butterfly example: Each link has unit capacity. One network use needed to send a and b to both receivers. Routing needs more than one network use.

7 Network oding ing oding apacity onclusions Let G = (V,E) be a network where the vertices are the users and the links between users are given by the labelled, directed edges in the graph. One source must transmit messages to multiple receivers (multicasting). The multicast capacity is given by the max-flow min-cut theorem. Original achievable scheme: randomly-generated codebook at each node that randomly mixes incoming information Of course, this is not a practical solution. Ho et al. have shown that a linear solution exists so long as the field size is larger than the number of receivers.

8 Standard Multiple- ing oding bits S 1 EN1 S EN1 S 2 EN2 S EN2... S M EN S M EN M X 1 X 2 X M P Y X1X 2...X M Y DE bits DE S Ŝ 1 Ŝ 2 Ŝ Ṁ. apacity onclusions M users with possibly correlated sources, S i Must perfectly recover the sources Source encoders do distributed compression

9 ing oding apacity onclusions Assume we have an acyclic network of point-to-point and multiple-access channels. Each user is represented as a vertex in a graph with incoming and outgoing links. The output of a multiple-access channel in a single time step can be written as the weighted sum of its inputs: Y = α 1 X 1 + α 2 X α K X K + Z where α i F q \ {0}, X i F q, and Z is an iid noise process. (All multiple-access channels operate over the same finite field.) Assume there is one sender and multiple receivers. We would like to characterize the multicast capacity of any network of this form.

10 Motivating Example ing S S R1 MA R2 MA MA oding apacity onclusions Ŝ (a) Ŝ Ŝ (b) Ŝ We have converted our problem into a graph of point-to-point links. capacity given by network coding. Is this optimal?

11 Motivating Example ing oding apacity onclusions S Example: multiple-access channel performs mod-2 addition. Separation-based scheme requires > H(S) 2 and > H(S). Using the channel as part of the network code only requires > H(S) 2 and 1 + > H(S) Separation is not optimal as we can use the interference as part of our network code. Ŝ Ŝ Not that interesting without noise.

12 oding ing oding apacity onclusions P(S 1, S 2 ) S 1 S p 2 p 1 2 p 2 1 p 2 S 1 EN 1 X 1 S 2 EN 2 X 2 P Y X1,X2 Y DE U = S 1 S 2, H(U) = h B (p) Must losslessly transmit the mod-2 sum, U = S 1 S 2, across a multiple-access channel Û Sources have uniform marginals, independent when p = 1 2 S 2 looks like S 1 passed through a BS

13 Optimal Source oding: Körner-Marton ing oding apacity onclusions Then what is the optimal source code? Idea: use a linear source code. Any iid B(p) source can be compressed to the entropy rate, h B (p), with a linear code This code can be written as a matrix H u = [U(1)U(2) U(k)] w = uh Example: [1 0] = [1 0 0]

14 Optimal Source oding: Körner-Marton ing oding apacity onclusions Now use this source coding matrix at both terminals w 1 = s 1 H w 2 = s 2 H w 1 and w 2 are provided to the decoder. Neither source can be reconstructed but U comes through perfectly by decoding from the mod-2 sum of the compressed bits w = w 1 w 2 = (s 1 s 2 )H = uh Körner-Marton scheme achieves: R 1 > h B (p) R 2 > h B (p)

15 An Appropriately Matched Multiple- hannel ing oding S 1 EN 1 X 1 S 2 EN 2 X 2 W BSq Y DE Û apacity onclusions hannel takes a mod-2 sum of the inputs: Y = X 1 X 2 This is followed by a BS with crossover probability q Sum rate capacity, MA = 1 h B (q) apacity region is a simplex, time-sharing is optimal

16 Separation-Based Scheme ing oding apacity onclusions Use Körner-Marton scheme to compress U: R S1 + R S2 > 2h B (p) Then use a multiple-access channel code: R X1 + R X2 < 1 h B (q) Achieves any computation rate satisfying: κ SEP < 1 h B(q) 2h B (p) an we do better?

17 Linear hannel ode for the BS ing oding apacity onclusions Random coding does not take advantage of the channel structure We need a linear channel code and a linear source code For any BS, there is a linear channel code that can achieve capacity. Any linear channel code can be written as a generator matrix: G x = wg [ Example: [1 1 0] = [1 1] ]

18 oding ing oding apacity onclusions Theorem There exists a linear code that can approach the computation rate κ = 1 h B(q) h B (p). This is the best available computation rate for lossless transmission of U = S 1 S 2 over this channel. Achievability. hoose G for the BS and H for compressing U to entropy. Set x 1 = s 1 HG x 2 = s 2 HG After the channel, it looks as if U was jointly encoded. onverse. Relax to joint encoding of U. By the data processing inequality, I(U; Û) I(X 1, X 2 ; Y ).

19 Separation vs. Joint Source-hannel oding ing Separation Slepian Wolf oding apacity onclusions κ p Our scheme dominates separation by a factor of 2, even when the sources are independent. We can get the benefits of uncoded transmission and maintain reliable communication.

20 apacity of Finite-Field Multiple- ing oding apacity onclusions Theorem If the field size of the multiple-access channels in the network, q, is larger than the number of receivers, L, then the multicast capacity is given by the max-flow min-cut bound. Proof Sketch: We begin by transforming our multiple-access network into a purely point-to-point network. We replace each multiple-access channel with a node whose output has a capacity equal to the original sum-rate capacity. Each incoming link to the original channel is replaced with an infinite capacity link to the new node. We now need the result of Ho et al.: A linear solution for this network exists so long as the field size is larger than the number of receivers.

21 Proof Sketch ontinued ing oding apacity onclusions Each new node now performs a computation for the overall network code and sends this out on the outgoing link at the original sum-rate capacity. We duplicate each of these computations on the original multiple-access channels with computation coding. Achieves multicast capacity of transformed network, which is guaranteed to be the same or larger than our original multicast capacity.

22 Future Work ing oding apacity Incorporate random fading model. If each node knows the fading process of the nodes before it, this shouldn t really change the capacity region. Attempt to incorporate a broadcast constraint. onclusions

23 onclusions ing oding apacity onclusions Large gains are possible for reliable computation with joint source-channel codes. Very simple codes, such as linear codes can completely achieve these gains. Interference is not necessarily a hindrance to wireless communication. In certain scenarios, it can be made part of an optimal code. Techniques such as computation coding may be necessary to characterize more general networks. Unfortunately, computation capacity becomes very difficult to analyze outside of the field addition case. (Even Gaussian case not completely solved.)