Recall the decision output signal is
(8.36)
Let us start with the error estimator and use the joint PDF between the signal and noise components,
(8.37)
After invoking the independence assumption of the signal and noise, we have
(8.38)
(8.39)
Using the IS biasing technique produces
(8.40)
Lastly, using the sample mean estimator gives us the familiar form:
(8.41)
The only difference is that now, the weighting is based on the noise samples rather than the
received signal, as given earlier. Therefore, the noise PDF can now be replaced with it biased
version .
Thus far we have been able to mathematically show we can derive a BER estimator where the simulation
noise PDF is biased, that is, and the BER estimator remove this biasing through weighing
of the noise samples. Let us slightly increase the complexity of the digital communication system
simulation block diagram with the help of Fig. 8.7.
Next we discuss how to pick the biasing noise PDF, specifically . If is Gaussian, then
is typically chosen also to be Gaussian, as shown in Fig. 8.8.
(8.42)
Above we noticed the mean stays the same, but the variance is made larger.
fN (n) N(0, s2)
f *
N (n) N(0, s2*
)
f *
N(n)
fN(n) f *
N(n)
f *
N(n)
f *
N(n)
fN(n)
P ^
0*
1
N*
# aN
*
i1
h0(vi) # w(ni)
3
`
`
h0 [g(s n)] # fN(n)
f *
N(n)
# f *
N(n)dn 3
`
`
h0[g(s n)] # w(n) # f *
N(n) dn
P0 3
`
`
h0[g(s n)] # fN(n) dn
P0 3
`
`
3
`
`
h0[g(s n)] # fS(s) # fN(n) ds dn
P0 3
`
`
h0[g(x)] # fX(x)dx 3
`
`
3
`
`
h0[g(s n)] # fS, N (s,n) ds dn
vi g(xi) g(si ni)
fN(n) fX(x)
P^
0*
1
N*
# aN
*
i1
h0(vi) # w(xi)
P0 EX*5h0[g(x)] # w(x)6
466 CHAPTER EIGHT
Here we see that ; thus the error events occur more frequently, allowing us to accurately
count them.
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