A variety of models has been considered to study cooperative capacity and diversity. For example, the cognitive radio channel shown in Figure 3.9 lets the source nodes cooperate by having one of the nodes aware of the message of the other node. The capacity region of this channel is known in some cases [89, 189]. The best cooperative schemes use both superposition coding and a sophisticated coding method known as dirty paper coding.
Cooperative diversity strategies can, of course, be extended to networks with many nodes. For example, each source-destination pair might use a two-phase listen-and-transmit scheme [113]. In the first phase, a source node transmits and several relay nodes listen; in the second phase, the relays amplify symbols or decode-and-forward. One can further identify scaling laws that characterize network performance in the limit of a large number of relays [20, 37, 58, 137, 138]. A two-hop cooperative scheme where the relays use AF achieves capacity as the number of nodes increases (see Section 4.2.2 and [58]). Cooperation also improves the energy efficiency as the number of nodes increases [37]; it allows for non-zero ergodic capacity for each source-destination pair when the number of relays scales as the square of the number of source-destination pairs; and it can "crystallize" a network, meaning that the source-destination links effectively appear as non-fading links [138].
Scaling law analyses can be done in several other ways. For instance, increasing the number of nodes in an area of fixed size increases the density of the network and one can achieve good scaling. The results are more pessimistic for extended networks where the node density is held constant while the area grows with the number of nodes. The throughput for n nodes decreases as
in a two-dimensional extended network [127]. It is only after disconnecting some fraction of nodes that a constant growth rate can be achieved [41]. Interestingly, it suffices to disconnect an arbitrarily small fraction of nodes. An excellent review of the throughput scaling laws in ad hoc networks is presented in [192]. More recently, the scaling laws for both dense and extended networks have been characterized [145].
The traffic scenarios considered in this topic have each message destined for a unique node, i.e., we considered unicast transmission. The multicast problem has a message destined for several destination nodes and is called a broadcast problem if the set of destination nodes includes all network nodes except for the source node. For both multicast and broadcast problems, cooperative strategies improve energy efficiency [132, 133] and network lifetime [134]. The asymptotic behavior of cooperative broadcast was analyzed in [170]. The simple two-phase protocol described above achieves the multicast capacity if the channels exhibit Rayleigh fading [96].
Finally, we note that clustering nodes at two locations lets one solve several relay problems. Examples of such results are listed below:
• CF and DF achieve capacity if the relays form a cluster with the destination and source, respectively [106]. When one set of relays is clustered with the source and another set with the destination, one can achieve MIMO-like rates. However, one cannot achieve the MIMO diversity-multiplexing rates [196, 197] unless the clustering gets tighter with SNR.
• Clustering increases the diversity gain of the systems with one source-destination pair [195].
• Cooperative schemes based on CF provide large throughput gains [92].
• If orthogonal channels are available between sources, one can exchange messages between the encoders to allow them to use dirty paper coding [87].
• For networks with n source-destination pairs, one can form hierarchies of clusters that cooperate and form virtual MIMO arrays. One can thereby achieve linear scaling of the total capacity in dense networks. In extended networks, the capacity scales as
