Compound Multiple Access Channels with Partial Cooperation ^{†}^{†}thanks: This work was supported by NSF under grants CNS0626611 and CNS0625637, the DARPA ITMANET program under grant 11057411TFIND and the ARO under MURI award W911NF0510246. The work of S. Shamai has been supported by the Israel Science Foundation and the European Commission in the framework of the FP7 Network of Excellence in Wireless COMmunications NEWCOM++.
Abstract
A twouser discrete memoryless compound multiple access channel with a common message and conferencing decoders is considered. The capacity region is characterized in the special cases of physically degraded channels and unidirectional cooperation, and achievable rate regions are provided for the general case. The results are then extended to the corresponding Gaussian model. In the Gaussian setup, the provided achievable rates are shown to lie within some constant number of bits from the boundary of the capacity region in several special cases. An alternative model, in which the encoders are connected by conferencing links rather than having a common message, is studied as well, and the capacity region for this model is also determined for the cases of physically degraded channels and unidirectional cooperation. Numerical results are also provided to obtain insights about the potential gains of conferencing at the decoders and encoders.
I Introduction
In today’s complex communication networks there are often multiple “signal paths” to utilize in delivering data between a given transmitter and receiver. Such signal paths may take the form of (generalized) feedback from the channel to the transmitters or additional (orthogonal) communication links between either the transmitters or the receivers. The first case corresponds to scenarios in which the additional signal paths share the spectral resources with the direct transmitterreceiver links (inband signalling), while the second case refers to scenarios in which orthogonal spectral resources are available at the transmit and/or the receive side (outofband signalling).
In this work, we focus on the latter case discussed above and model the outofband signal paths as finitecapacity directed links. This framework is typically referred to as “conferencing” (or “partial cooperation”) in the literature to emphasize the possibly interactive nature of communication on such links. Conferencing encoders in a twouser multiple access channel (MAC) have been investigated in [1] [2]^{1}^{1}1It is noted that a MAC with conferencing encoders can be seen as a special case of a MAC with generalized feedback. and in [3] for a twouser interference channel. These works show that conferencing encoders can create dependence between the transmitted signals by coordinating the transmission via the outofband links, thus mimicking multiantenna transmitters. Conferencing decoders have been studied in [4] for a relay channel and in [5]  [9] for a broadcast channel. Such decoders can use the outofband links to exchange messages about the received signals so as to mimic a multiantenna receiver (see also [10]).
This work extends the state of the art described above by considering the compound MAC with conferencing decoders and a common message (see Fig. 1) and then with both conferencing encoders and decoders (see Fig. 5). These models generalize the setup of a singlemessage broadcast (multicast) channel with two conferencing decoders studied in [5]^{2}^{2}2Reference [5] also considers a broadcast channel with private messages to the two users.  [9], in that there are two transmitters that want to broadcast their messages to the conferencing receivers. Moreover, the transmitters can have a common message (Fig. 1) or be connected by conferencing links (Fig. 5). The model also generalizes the compound MAC with common messages studied, among other models, in [3], by allowing conferencing among the decoders. The main contributions of this work are summarized as follows:

The capacity region is derived for the twouser discretememoryless compound MAC with a common message and conferencing decoders in the special cases of physically degraded channels and unidirectional cooperation (Sec. IV);

Extension to the corresponding Gaussian case is provided, establishing the capacity region with unidirectional cooperation and deriving general achievable rates. Such achievable rates are also shown to be within some constant number of bits of the capacity region in several special cases (Sec. VI);
Finally, numerical results are also provided to obtain further insight into the main conclusions.
Ii System Model and Main Definitions
We start by considering the model in Fig. 1, which is a discretememoryless compound MAC with conferencing decoders and common information (here, for short, we will refer to this channel as the CM channel). The CM channel is characterized by with input alphabets and output alphabets The th encoder, , is interested in sending a private message of rate [bits/channel use] to both receivers and, in addition, there is a common message of rate to be delivered by both encoders to both receivers. The channel is memoryless and timeinvariant in that the conditional distribution of the output symbols at any time satisfies
(1) 
with being a given triplet of messages. Notationwise, we employ standard conventions (see, e.g., [11]), where the probability distributions are defined by the arguments, uppercase letters represent random variables and the corresponding lowercase letters represent realizations of the random variables. The superscripts identify the number of samples to be included in a given vector, e.g., It is finally noted that the channel defines the conditional marginals and similarly for Further definitions are in order.
Definition 1: A code for the CM channel consists of two encoding functions ()
(2) 
a set of “conferencing” functions and corresponding output alphabets ():
(3a)  
(3b)  
and decoding functions: 
(4a)  
(4b)  
Notice that the conferencing functions (3) prescribe conferencing rounds between the decoders that start as soon as the two decoders receive the entire block of output symbols and Each conference round, say the th, corresponds to a simultaneous and bidirectional exchange of messages between the two decoders taken from the alphabets and , similarly to [1], [14]. It is noted that other works have used slightly different definitions of conferencing rounds [7], [16]. After the conferencing rounds, the receivers perform decoding with functions (4) by capitalizing on the exchanged conferencing messages. Due to the orthogonality between the main channel and the conferencing links, the transmission from the users on one channel and conferencing/ decoding on the other can take place simultaneously.  
Definition 2: A rate triplet () is said to be achievable for the CM channel with decoders connected by conferencing links with capacities (see Fig. 1) if for any there exists, for all sufficiently large, a code with any such that the probability of error satisfies 
(5) 
and the conferencing alphabets are such that
(6) 
The capacity region is the closure of the set of all achievable rates ().
Iii Preliminaries and an Outer bound
Similarly to [3], it is useful to define the rate region for the MAC seen at the th receiver () as the set of rates
(7a)  
(7b)  
(7c)  
(7d)  
where the joint distributions of the involved variables is given by 
If the capacity region is given by [3]:
(8a)  
(8b)  
(8c)  
(8d)  
(8e)  
(8f)  
where the union is taken over all joint distributions of the form 
It is remarked that no convex hull operation is necessary in (8) as the region is convex [3] (see also [1], Appendix A).
We now derive an outer bound on the capacity region . To this end, it is useful to define the capacity region achievable when the two receivers are allowed to fully cooperate (FC), thus forming a twoantenna receiver. In this case, we have
(9a)  
(9b)  
(9c)  
(9d)  
(9e)  
where the joint distributions of the involved variables is given by 
(10) 
Proposition III.1
We have where (dropping the dependence on to simplify the notation)
(11a)  
(11b)  
(11c)  
(11d)  
(11e)  
(11f)  
(11g)  
in which the union is taken over all the joint distributions that factorize as (10). 
See Appendix A.
Iv Capacity Region with Physically Degraded Channels and Unidirectional Cooperation
The next proposition establishes the capacity region in the case of physically degraded outputs.
Proposition IV.1
If the CM channel is physically degraded in the sense that forms a Markov chain, then the capacity region is obtained as
(12a)  
(12b)  
(12c)  
(12d)  
(12e)  
(12f)  
Notice that here due to degradedness. 
See Appendix B.
Remark IV.1
A symmetric result clearly holds for the physically degraded channel
Establishment of the capacity region is also possible in the special case where only unidirectional cooperation is allowed, that is or This result is akin to [9] where a broadcast channel with two receiver under unidirectional cooperation was considered.
Proposition IV.2
In the case of unidirectional cooperation ( or ), the capacity region of the CM channel is given by
(13) 
or
(14) 
Achievability follows by using the same scheme as in the proof of Proposition IV.1. The converse is immediate.
V General Achievable Rates
Achievable rates can be derived for the general CM channel, extending the analysis of [5] from the broadcast setting with one transmitter to the CM channel. Notice that [5] uses a different definition for the operation over the conferencing channels but this turns out to be immaterial for the achievable rates discussed below.
Proposition V.1
The following region is achievable with oneround conferencing, i.e., :
(15a)  
(15b)  
(15c)  
(15d)  
(15e)  
subject to 
(16a)  
(16b)  
with , and the union is taken over all the joint distributions that factorize as 
(Sketch): The proof is similar to that of Theorem 3 in [5] and is thus only sketched here. A onestep conference () is used. Encoding and transmission are performed as for a MAC with common information (see proof of Proposition IV.1). Each receiver compresses its received signal using WynerZiv compression exploiting the fact that the other receiver has a correlated observation as well. The compression indices are exchanged during the one conferencing round via symbols and Decoding is then carried out at each receiver using joint typicality: For instance, receiver 1 looks for jointly typical sequences with where is the compressed sequence received by the second decoder.
The achievable strategy of Proposition V.1 is based on round of conferencing. It is easy to construct examples where such a strategy fails to achieve the outer bound (11) as discussed in the example below.
Example 1. Consider a symmetric scenario with and equal private rates (i.e., ). Fix to a constant without loss of generality (given the absence of a common message) and the input distribution to We are interested in finding the maximum achievable equal rate Assume that the conferencing capacities satisfy and In this case, it can be seen that the maximum equal rate is upper bounded as by the outer bound (11) which corresponds to the maximum equal rate of a system with full cooperation at the receiver side This bound can be achieved if both receivers have access to both outputs and With the oneround strategy, since receiver 1 can provide to receiver 2 via SlepianWolf compression, but receiver 2 cannot do the same with receiver 1 since Therefore, rate cannot be achieved by this strategy, which in fact attains equal rate (recall (16)).
We now consider a second strategy that generalizes the previous one and is based on two rounds of conferencing As will be shown below, this strategy is able to improve upon the oneround scheme, while still failing to achieve the outerbound (11) in the general case.
Proposition V.2
The following rate region is achievable with two rounds of conferencing, i..e., :
(17) 
where “co” indicates the convex hull operation, and we have
(18a)  
(18b)  
(18c)  
(18d)  
(18e)  
is similarly defined: 
(19a)  
(19b)  
(19c)  
(19d)  
(19e)  
subject to 
with , and the union is taken over all the joint distributions that factorize as
(Sketch): The proof is quite similar to Theorem 4 in [5], and here we only sketch the main points. Conferencing takes place via conferencing rounds. Moreover, two possible strategies are considered, giving rise to the convex hull operation in (17) by timesharing. The two corresponding rate regions in (18) and in (19) are obtained as follows. Consider Receiver 2 randomly partitions the message sets and into , and subsets respectively, for a given and as in the proof of Proposition IV.1. Encoding and transmission are performed as for the MAC with common information. Receiver 1 compresses its received signal using WynerZiv quantization as for the scheme discussed in the proof of Proposition V.1. This index is sent in the first conferencing round (notice that and ). Upon reception of the compression index , receiver 2 proceeds to decoding via joint typicality and then sends the subset indices (see proof of Proposition IV.1) to receiver 1 via (now, and ). The latter decoder performs jointtypicality decoding on the subsets of messages left undecided by the conferencing message received by 1. The rate region is obtained similarly by simply swapping the roles of decoder 1 and decoder 2.
Example 1 (cont’d): To see the impact of the tworound scheme, here we reconsider Example 1 discussed above. It was shown that, for the scenario discussed therein, the oneround scheme is not able to achieve the outer bound . However, it can be seen that the tworound scheme does indeed achieve the outer bound. In fact, receiver 1 can provide to receiver 2 via SlepianWolf compression as for the oneround case, while receiver 2 does not send anything in the first conferencing round ( is a constant). Now, receiver 2 decodes and sends the bin index of the decoded messages to receiver 1 in the second conferencing round according to the tworound strategy discussed above (receiver 1 is silent in the second round). Since by assumption, it can be seen from Proposition V.2 that the maximum equal rate achieved by the two round scheme is
We finally remark that it is possible in principle to extend the achievable rate regions derived above to more than two conferencing rounds, following [6] [7]. This is generally advantageous in terms of achievable rates. While conceptually not difficult, a description of the achievable rate region would require cumbersome notation and is thus omitted here.
Vi Gaussian Compound MAC
Here we consider the Gaussian version of the CM channel:
(20a)  
(20b)  
with channel gains independent white zeromean unitpower Gaussian noise and persymbol power constraints Notice that the channel described by (20) is not physically degraded.  
The outer bound of Proposition III.1 can be extended to (20) by using standard arguments. In particular, the capacity region of the Gaussian CM, satisfies the following. 
Proposition VI.1
We have where:
(21a)  
(21b)  
(21c)  
(21d)  
with 
(22a)  
(22b)  
(22c)  
and 
Similarly to Proposition III.1, one can prove that the rate region (11) is an outer bound on the achievable rates. It then remains to be proved that a Gaussian joint distribution with where is and are independent Gaussian zeromean unitpower random variables, is optimal. This can be done following the steps of [2], where the proof is given for a single MAC channel with common information (see also [15]). The proof is concluded with some algebra.
The achievable rates in Proposition V.1 (for ) and Proposition V.2 (for ) can also be extended to the Gaussian CM channel. In so doing, we focus on jointly Gaussian auxiliary random variables for WynerZiv compression. While no general claim of optimality is put forth here, some conclusion on the optimality of such schemes can be drawn as discussed later in Sec. VIA.
Proposition VI.2
The following rate region is achievable with oneround conferencing, :
(23a)  
(23b)  