Cooperative Diversity: Capacity Bounds and Code Designs - Adaptive Signal Processing in Wireless Communications - page 342

On the other hand, when cooperation is not allowed, i.e., c 21 = c 12 = c 43 = c 34 = 0, the
channel degenerates to the interference channel [9]. Figure 12.3 depicts three of these
four simplifications.
When we restrict the channels to be quasi-static, then all channel coefficients are
constant during transmission of each block of N symbols. In the synchronous model of
(12.1)-(12.4), we assume that the nodes are perfectly synchronized and have full chan-
nel state information (CSI), i.e., each node knows instantaneous values of all channel
coefficients and their statistics. While it is relatively simple to achieve symbol/time
synchronization between nodes, carrier synchronization, which requires phase-locking
separated microwave oscillators, is challenging in practice. Therefore, we also consider
the asynchronous model , where random phase offsets due to oscillator fluctuations are
added to the transmitted signals. We include these random phases in the channel coeffi-
cients, so that the model stays the same as (12.1)-(12.4). Under the asynchronous model
for receiver cooperation, the transmitters do not have any CSI, whereas the receivers
need to know only the magnitudes of all channel coefficients, not their phases. Thus,
receiver cooperation is suitable in the systems with simple transmitters. On the other
hand, under the asynchronous model for transmitter cooperation, the transmitters
must know the magnitudes of all channel coefficients.
We discuss the diversity and data rate gains achievable by node cooperation, while
focusing on the high-signal-to-noise ratio (SNR) regime, where the data rates are mainly
limited by interference. (In the low-SNR regime, the influence of channel noise prevails,
and hence the gain from cooperation is reduced.) The diversity gain [10], defined as

log( P e SNR

SNR

)

d =− lim ,

log

SNR →∞

shows how fast the probability of decoding error P e decays with SNR. A higher d means
lower P e at the same SNR, and thus a more reliable system. The data rate gain is usu-
ally decoupled into a multiplexing gain and an additive gain . The multiplexing gain (or
degree of freedom) [10] shows how fast the rate increases with SNR and is given by

R ( SNR

SNR

)

r = lim ,

log

SNR →∞

where R (SNR) denotes the sum of data rates of transmitting nodes for a given SNR. The
additive gain (or the high-SNR power offset) [11, 12] is a shift of the R (SNR) function
from the origin at high SNRs, i.e.,

a = lim( R SNR )log( − r SNR ) .

SNR →∞

If all the limits exist, then R (SNR) in the high-SNR regime can be approximated by a line
of slope r and SNR offset a , i.e.,

R ( SNR ) ≈ r log( SNR ) + . a