On optimizing low SNR wireless networks using network coding
Abstract
The rate optimization for wireless networks with low SNR is investigated. While the capacity in the limit of disappearing SNR is known to be linear for fading and nonfading channels, we study the problem of operating in low SNR wireless network with given node locations that use network coding over flows. The model we develop for low SNR Gaussian broadcast channel and multiple access channel respectively operates in a nontrivial feasible rate region. We show that the problem reduces to the optimization of total network power which can be casted as standard linear multicommodity mincost flow program with no inherent combinatorially difficult structure when network coding is used with non integer constraints (which is a reasonable assumption). This is essentially due to the linearity of the capacity with respect to vanishing SNR which helps avoid the effect of interference for the degraded broadcast channel and multiple access environment in consideration, respectively. We propose a fully decentralized PrimalDual Subgradient Algorithm for achieving optimal rates on each subgraph (i.e. hyperarcs) of the network to support the set of traffic demands (multicast/unicast connections).
Index Terms  Low SNR Gaussian broadcast channel, network coding, rate optimization, PrimalDual Subgradient Method.
I Introduction
Wideband fading channels have been studied since the early 1960’s. Kennedy showed that for the Rayleigh fading channel at the infinite bandwidth limit, the capacity is similar to the capacity of the infinite bandwidth AWGN channel with the same average received power [1, 2]. The robustness of this result in the case of with or without channel state information helps us model the low SNR wideband wireless networks in a general manner. It should be noted that when the band grows large, the transmitting power is shared among large degrees of freedom. This results in smaller SNR per degree of freedom. Using this as our underlying informationtheoretic assumption to approximate the capacity over a link, we model the general traffic for this network and show that the linearity of capacity for disappearing SNR makes for the fundamental reason for simplicity in our model. Hence, we claim, it is possible to do networking over such a model with simplistic and essentially linear approach.
In the context of wideband multipath fading relay channel, it was shown in [3] that in the noncoherent multipath fading relay channel, the same lower bound on the rate can be achieved as in the frequency division AWGN relay channel with the same received SNR, by using a peaky binning scheme. In this paper, we use a relaying scheme based on superposition coding. The rates achieved by peaky binning [3] are higher than the rates achieved by the relaying scheme that we propose. However, our relaying scheme has the advantage to extend easily to large networks due to hyperarc decomposability. We would like to mention here that number of hyperarcs for Gaussian broadcast channel using superposition coding is equal to for receivers, instead of .
The traffic model we use is quite general. It is divided into two classes: unicast and multicast (broadcast is considered as a special case of multicast), where each pair of source and receiver group in the network form a session for a particular class of traffic. The problem of successfully establishing multicast connections in wireline or wireless networks has been long thought to be NPComplete using arbitrary directed and undirected network models. With the advent of network coding (ref., [4], [5], [6]), and in turn breaking of the fluid model for data networks i.e. by performing coding over incoming packets, this approach has been able to intrinsically circumvent the combinatorial hardness of the multicast flow problem. It was also shown that establishing minimum cost multicast connections boils down to optimizing subgraph over coded packet networks [7].
In this paper, we consider a low SNR wireless network with Gaussian broadcast channel and MAC. The problem we consider is to optimize the rates for a given set of demands that needs to be met by this network. We show that this problem can be casted as a minimum cost multicommodity flow problem with intrasession network coding.
This paper is organized as follows. Section II is composed of general problem formulation. In section III we propose a decentralized solution. We present our results in section IV and finally, we mention concluding remarks in section V.
Ii SetUp and Problem Formulation
In this section we introduce the hypergraph models for the low SNR Gaussian broadcast and multiple access channel.
A. Low SNR physically degraded Gaussian broadcast channel [14].
Consider a general wideband fading channel where the input waveform
is x and the output waveform is y, the fading coefficient matrix is given by h and
n is the additive white noise. The channel is given by:
(1) 
The capacity of the channel, for both Gaussian channels and fading channels increase sublinearly with the increase in signal to noise ratio but in the low SNR regime the capacity in the limit is linear in SNR for fading and nonfading channels:
(2) 
Clearly at low SNR, the signaltonoise ratio per degree of freedom (SNR) approaches zero in the limit [2, 8, 9]. For such a case the point to point capacity boils down to:
(3) 
where and is the distance between the transmitter and receiver. Let us now look at the standard model of a single sender and two receivers with noise variances and respectively (ref. Fig 1(a)). The capacity region is given by:
(4) 
where , , and is the total power (ref. Fig. 1(b)), the transmission scheme is superposition coding [15].
The rate region defined in (4), when looked under the low SNR lens comes across as a rather simpler picture. For the power limited low SNR regime, the effect of the power allocated for the better receiver, as the contribution to the total noise experienced by the worse receiver is negligible (ref. Fig. 1(b), for the rate region for low SNR in the limit). So, for the low SNR physically degraded Gaussian broadcast channel, the rate for the worst receiver can be approximated as
(5) 
Generalizing the same idea for the case of a given source with power and receiver nodes, where the receiver set can be broken into subsets as for ordered in decreasing order of reliability. The rate region defined for each hyperarc in the low SNR limit is given as
(6)  
(7) 
where, , which when combined appropriately gives the rate region of the set . The equation (7) comes from the fact that capacity is linear in the limit of disappearing SNR, where for all is the location of the node and is the loss exponent. We formalize the above mentioned concepts and motivate our next definition. Let , .
Definition. 1: For a given sender with total power and a receiver set in low SNR physically degraded Gaussian broadcast channel, the set can be decomposed into hyperarcs where each hyperarc is defined as the connection from the sender to the receiver set , where . The rate over each hyperarc is defined as where, , and the set ranges from best to worst receiver (ref. Fig 1 (c)).
B. Interference issues in multiple access at low SNR.
Now, let’s consider the case of multiple access where more
than one node tries to access the channel at the given instance. Let
there be nodes in the system at an instance, and
of them are trying to access the channel at this instance, if node
intends to communicate with node among others in , the signal to interference and noise ratio
(SINR, denoted as ) experienced at node is given
by:
(8) 
Note that, since every node in is interested only in a common receiver, we allocate the whole power of the node over this single hyperarc, so and for every transmitter. But as we are operating in the low SNR regime, the intuition suggests that the effect of the interference should be negligible. We straightforwardly include it in our assumption, thus we define the rate (denoted with ) experienced at the receiver as:
(9)  
(10)  
(11) 
The approximation (10) comes from the fact that the contribution of other signals being transmitted from other sources in the system with low SNR channel to the interference is negligible and the approximation (11) comes from the linearity of capacity in the limit of disappearing SNR (ref. Fig 2(a) and 2(c)). In Fig. 2(b), we can see that the SNR curve approaches the capacity curve in the limit, corroborating our assumption that the SNR equals capacity in the limit of disappearing SNR per degree of freedom.
C. Low SNR network rate optimization.
Let us represent the wireless network as a directed hypergraph
, where is the set of nodes and
is the set of hyperarcs, where each hyperarc emanates
from a node and a terminates at a group of nodes, which we also
refer to as the broadcast group of the hyperarc. Note that we
consider multicast in our multicommodity flow optimization model (as
opposed to only unicast), thanks to network coding.
It’s important to note that the common rate associated with each hyperarc , is the capacity of the hyperarc, because this is the rate that can be guarantied to all the receivers in this hyperarc. Also, , is a nonnegative function of the transmit power of the hyperarc . Now that we consider a network with more than one sender, update of notations is required. For a sender , that is capable of reaching nodes, where each , the hyperarcs are denoted by , .
Imagine a set of traffic demands where sessions need to be established, each with set of receivers, in a given wireless network that experiences low SNR and that is represented by the hypergraph . We know from the definition of hyperarc that a single node can lie on multiple hyperarcs, therefore, we need a way to carefully count the incoming information and outgoing information to apply the law of flow conservation to the hypergraph and finally be able to cast the problem as a flow optimization problem. For that, we define another graph , which is simply the equivalent directed graph of with arcs instead of hyperarcs. This graph can be easily obtained by decomposing the hypergraph appropriately. Let’s define the term (ref. [7] for detailed notation explanation):
(12) 
which simply describes the way to add all the flow entering a node on all incoming hyperarcs, corresponding to the graph . Notice that is not the same as defined in the previous section, can be interpreted as the flow between and receiver of the hyperarc , and it cannot exceed the common rate () associated to the hyperarc which is also the hyperarc capacity, for each .
Let, . Then, the minimum cost optimization problem for the low SNR network can be formulated as:
(A) 
subject to:
(13)  
(14)  
(15)  
(16) 
where is given , , and a bounded polyhedron made of flow conservation constraints:
(17)  
(18)  
(19) 
As opposed to standard multicommodity flow problem in which flows are simply added over a link, the constraint in fact catches the essence of network coding by taking only the maximum among all the flows of a session (note that we only consider intrasession network coding). Since is the polyhedron formed by the laws of flow conservation, constraint translates the flow conservation laws from the underlying directed graph to the hypergraph (the wireless network) by adding the flows on all hyperarcs between node and i.e. flow in is the sum of all the flows on the hyperarcs , .
As we can see, the above mentioned problem is a convex optimization problem. The only nonlinear constraint is (13), and could be readily replaced by the set of linear inequality constraints , . The modified problem results in a standard linear multicommodity flow problem with linear objective and linear constraint set.
(B) 
subject to:
(20)  
(21)  
(22)  
(23) 
where , and is a bounded polyhedron made of flow conservation constraints. Note that we optimize the power over each hyperarc, to determine the optimal rates for each hyperarc that satisfies the network demands, we simply need to multiply the optimal power with . We will prefer to solve the problem by proposing a decentralized algorithm for generally understood and appreciated reasons.
Iii Decentralized Algorithm
For developing a decentralized solution for problem we need to understand the structure of the primal problem first and transform it into a separable form. We know that the objective function is a linear and increasing in its domain and so are the constraints.
Taking the Lagrangian dual of the problem we get the dual optimization problem as:
(C) 
subject to:
(24) 
where,
(25) 
The dual problem is clearly hyperarc separable and could be solved in a decentralized manner. But the dual problem is not differentiable at all the points in the dual domain, this is due to the fact that there might not be a unique minimizer of for every dual point as the objective function is a minimum over sum of linear functions for fixed dual variables. To solve the dual problem , we need to solve its subproblem . The subproblem (and the dual problem ) could be solved with a lot of techniques, [10, Chapters 810], [11Chapters 56, 12Chapters 6] using some subgradient based technique but they do not necessarily yield the primal solution (which is of our interest here). There are however, methods for recovering primal solutions from the dual optimizers.
We will take a different technique than the above mentioned approaches but before lets look into some interdependence characteristics of the dual and primal problem structures. Simply having convex primal problem in hand does not guarantee strong duality, but with some constraint qualifications we can assert that strong duality holds or not. One such simple constraint qualification technique is called Slater’s condition.
In our case it can be easily seen for constraints (or ) of problem (or ), there exist a vector for which the inequality can be strict.
Let us represent the set of primal vectors as where is the feasible set for the primal problem, and similarly we can do it for the dual problem, . As we can see that the primal and dual optimal are equal (thanks to strong duality), we can express our problem in the standard saddle point form where function is the Lagrangian dual of the problem . This implies that for we get the hyperarc separable saddlepoint form
(26) 
Now we are in the position where we can solve the problem, separable
in hyperarcs using any saddlepoint optimization method for
nonsmooth functions. For our problem set up, we propose a
PrimalDual Subgradient Algorithm by Nesterov for nonsmooth
optimization [ref. 13]. Nesterov’s method generates a subgradient
scheme intelligently based on DualAveraging method which beats the
lower case complexity bound for any blackbox subgradient scheme.
The algorithm works in both primal and dual spaces, generating a
sequence of feasible points, and ultimately squeezing the duality
gap to zero by finally approaching the optimal solution. A positive
consequence of the PrimalDual approach is that at each iteration we
get a pair of points (p,d) which are primal and dual
feasible, hence, we get the primal feasible solution with
essentially no extra effort. As opposed to many subgradient type
methods where there needs to be a method for primal recovery,
specially for large and illposed problems.
A. PrimalDual Subgradient Algorithm.
Since the dual function is hyperarc separable, we can optimize the power
over each hyperarc separately and add each of the optimal solutions
to construct the optimal solution of the dual problem , ultimately achieving the primal optimal solution for problem . The algorithm is as follows:

Initialization: Set . Choose .

Iteration :

Compute .

Choose and set .

Choose
Set

where is the set of primal and dual subgradients and and are aggregated sequence of points.
Iv Simulations
We now show the results of our simulations that support the claims of the algorithm presented. We solved the dual problem in a decentralized way by solving it for every hyperarc separately and then adding up the respective solutions to construct the dual optimal solution of the problem , when this is optimal, it is the primal optimal solution for problem in our case.
The setup consists of uniformly placed nodes on a chosen area of , with given node locations. We start our simulations with smaller networks of only 4 nodes on a area with the area size increasing as the number of nodes in the network increase to keep the node density/area in a controlled range. Each node has a single hyperarc and it can communicate with all the nodes in the network, this is just a simple generalization of our case where a node can communicate with only a subset of total nodes in the network. For each network we randomly choose a set of multicast sessions and set of receivers for each session respectively with the required rate demand associated with each session that need to be established, but making sure the the traffic demands are the respective mincut for each session to make the problem feasible.
In Figure 3, we compare the optimal solution approximations of the PrimalDual Subgradient Method for problem with the standard infeasible path following method for problem . It can be seen that the our proposed algorithm gives close approximations of the primal solution of the problem . Note that the path following method is directly applied to the primal problem and the PrimalDual subgradient method is applied to the dual problem, to compute the dual solution of the problem , which will be give us the close approximation to the primal solution of problem .
V Conclusion
We develop an efficient optimization model that provides an achievable rate region for the low SNR broadcast channel and MAC. We do this by showing that rate optimization for the Low SNR physically degraded broadcast wireless network can be formulated as a standard linear multicommodity flow problem for optimizing power over each hyperarc using network coding. Our model is relieved from interference related issues, this is due to the fact that the capacity of the low SNR wideband channel is essentially linear in SNR per for vanishing SNR in the limit, which relieves the system from interference and related issues. Our model operates in the nontrivial feasible rate region that achieves capacity in the limit of disappearing SNR with appropriate encoding scheme.
We use a primaldual algorithm to construct a decentralized solution for solving the problem, which has apparent advantages for recovering the primal solution than standard projected subgradient methods. In the simulation results shown, we don’t present the gains of routing using network coding over simple routing. But is already a vast literature establishing this fact.
Finally, we believe that realizing low SNR networks is a worthwhile attempt as the linearity of capacity in the limit provides a fundamental simplicity for networking to be done. Insights reveal interesting and promising work could be build up and blended with our simple model (e.g. mobility, reliability etc), which remains to be explored in this scenario.
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