Cooperative Diversity: Capacity Bounds and Code Designs - Adaptive Signal Processing in Wireless Communications - page 352

In a similar manner, DF can be extended to receiver cooperation. The DF strategy for
receiver cooperation can be found in [36].

12.3.5 Capacity Bounds of the Gaussian Full-Duplex Receiver
Cooperative Channel
In this section we summarize the results of [36], where the upper and lower bounds
on capacity of receiver cooperation are derived for both CF and DF. The derived upper
bounds are tighter than the standard max-flow-min-cut bound. It is given by:


 


 

2

11 ++ c P

4 1 1


 


 +

2 2

RR +≤ + log 1 cPPcP + + log

1 2 41 1 2 43 3

2

1 + cP

41 1

in the asynchronous case, and


 


 

2

11 ++ c P


 


  +

41 1

2 2 2 2 2

RR +≤ + log 1 cPPcP + + + 2 c PP cc + PP log

1 2 41 1 2 43 3 43 23 43 41 13

2

1 + cP

41 1

in the synchronous case. Note that we get a symmetric set of rate bounds if nodes 1 and 2
are exchanged with nodes 3 and 4. Achievable (lower) rate bounds for CF and DF can be
found in [36]. Figure 12.7 shows the sum-rate R 1 + R 2 as a function of the received SNR
on the direct link between nodes 1 and 3. The received SNR at the link between nodes 3
and 4 is 30 dB higher than that from the direct link—an indication that the cooperating
nodes are close together. The average powers of all four nodes are the same. All channels
are independent Rayleigh flat fading, meaning that each c ji is an i.i.d. Gaussian random
variable, and the results are averaged over simulated ensembles of channel realizations.
For comparison purposes, the rate limits of the two-user MAC with two antennas at the
receiver (with perfect cooperation) and the interference channel (without cooperation)
are included.
It is seen from Figure 12.7 that receiver cooperation with CF gives an additive gain
that can be up to 20 dB higher than no cooperation, CF always performs close to the
upper bound, and there is no gain from synchronization. Interestingly, receiver coop-
eration performs close to using two receiver antennas at low and medium SNRs, thus
providing a multiplexing gain of 2. However, at the high SNRs, the multiplexing gain
drops to 1, and the rate gain over the noncooperative case boils down to a high additive
gain.
We can see that, in the high-SNR regime, receiver cooperation gives a multiplexing
gain of only r = 1, as opposed to the two-user MAC with two receiver antennas, which
results in r = 2. However, the additive gain with receiver cooperation, which is upper
bounded by