Wireless Network Design: Optimization Models and Solution ...

20 Dinesh Rajan
Perror ≤ e −nE random(R)
(2.10)
where, Erandom(R) is the random coding exponent and R is the rate of transmission.
This random coding exponent can be calculated based on the channel characteristics.
For the Gaussian noise channel,
Erandom(R) = 1 − β + SNR
SNR
+ 0.5log(β − ) + 0.5log(β) − R, (2.11)
2 2
where β = 0.5[1 + SNR
2 +
�
1 + SNR2
].
4
This expression
�
for the error exponent is valid for transmission rates R < 0.5log[0.5+
SNR
4 +0.5 1 + SNR2
4 ]. For rates above this threshold and below the capacity, the expression
for the error exponent is slightly different and is given in [18]. Further, a
lower bound on the decoding error probability for any code over a particular channel
is given by the sphere packing error exponent, which can be computed in a manner
similar to the random coding error exponent [18].
Fig. 2.6 Illustration of a random codebook used to prove achievability of Shannon’s channel capacity.