I am simulating an 8-QAM communication system over an AWGN channel for various signal to noise ratios, in order to plot the average bit error. However I have not been able to derive an equation for the QAM system in terms of the Q-function.
The 8-QAM system is rectangular with 2 possible values for the x basis vector and 4 possible values for the y basis vector such that the constellation consists of 8 symbols:
(-d/2,3d/2), (d/2,3d/2) (-d/2,d/2), (d/2,d/2) (-d/2,-d/2), (d/2,-d/2) (-d/2,-3d/2), (d/2,-3d/2)
where d is the distance between symbols.
The symbols are coded using grey coding for the x and y component as follows:
(0,10), (1,10) (0,11), (1,11) (0,01), (1,01) (0,00), (1,00)
My attempt to derive an expression for bit error rate was based on the symmetry of the constellation as it consists of only two unique error scenarios, since the received vectors are:
r1 = s1 + n1, r2 = s2 + n2
The probability of error in each scenario is $$ P(bit error scenario 1) = P(r1<0|s1) + P(r2>d|s1) + P(r2<0|s1) = 3Q(d/2\sigma) $$ $$ P(bit error scenario 2) = P(r1<0|s1) + P(r2<d|s1) = 2Q(d/2\sigma) $$ Average probability of bit error is then:
$P(bit error) = (5/2)Q(d/2\sigma)$
Assuming the transmit signals are equi-probable. and:
$d = \sqrt(2E_s/3), $ $\sigma =\sqrt(N_0/2)$
however there is significant disparity between this theoretical error rate and the simulated error rate. I would appreciate it if someone could point me in the right direction, thanks.