Cooperative Diversity: Capacity Bounds and Code Designs - Adaptive Signal Processing in Wireless Communications - page 348

given in [24]. However, in practice, it is more convenient to use outage capacity [33, 34].
Since in the fading channel case, the source does not know the exact value of chan-
nel coefficients ( c = [ c sr , c sd , c rd ]) for a particular channel realization, it cannot determine
the maximum achievable rate; consequently, the sending rate R can be higher than the
capacity of the instantaneous channel realization C ( c ), in which case the destination
cannot decode. The probability of this event, P ( R > C ( c )), is called the outage proba-
bility. The outage capacity is then the maximum achievable rate with the outage prob-
ability less than a certain level p ; it can be computed as the (1 - p ) percentile of the rate
for the specific value of c .
In the following, we summarize the outage capacity bounds of the Gaussian half-
duplex relay channel, which can also be used for computing the outage capacity of a
wireless quasi-static flat-fading channel (as done in [24]). The bounds for the full-duplex
Gaussian channel can be found in [8, 23, 24].
Under the assumption that the nodes are synchronized and have perfect CSI, i.e.,
each node knows instantaneous values of all channel coefficients and their statistics, an
upper bound on the capacity of the Gaussian half-duplex relay channel (although chan-
nel coefficients are in general assumed to be complex, in this case they are positive real
constants) is derived in [24, 27] and given by

C = max in{ C , , C } (12 .9)

ub ub 1 ub 2

010 ≤≤,≤≤,≤≤ ρ α 01 k

where


 

 

 


  ,

α 

 


 

kP

  + −

1 α ( 1

1

−
−

kP )

2 2 2

2

C = log 1 + + c c

s

log 11 +− ( ρ ) c

s

ub 1 rs ds ds

2 α 2 α


 


 

α kP

2

C = log 1 + c

s

ub 2 ds

2 α

 

2 2

2

2 ρ cc ( 1 − kPP )









+ −

1 α ( 1

1

−
−

kP )

ds dr sr

2 2

log 1 + c

s

+ ccP + ,

ds dr r

2 α 1 − α

 

and P s and P r are the average source and relay power constraints, respectively. The
parameter ρ reflects the correlation between the source and relay signals, and it can be
written in closed form [24, 27]. It is clear from the bound above that the highest multi-
plexing gain r is 1. However, the full-diversity gain of 2 can be achieved with a simple
AF scheme [2, 35].
The rate bound of DF is [24, 27]:

R ≤ max in{ R , , R } (12.10)

DF DF 1 DF 2

010 ≤≤,≤≤,≤≤ ρ α 01 k