23)
where is the biased PDF necessary to reduce the variance of the estimator, . We can define the
IS weight as
(8.24)
Then we have
(8.25)
which is equivalent to the following, where the expectation operation has been assumed to be with
respect to the biased PDF:
(8.26) P0 E*5h0(v) # w(v)6
P0 3
`
`
h0(v) # w(v) # f *
0 (v) dv
w(v)
f0(v)
f *
0 (v)
P^
0 f *
0 (v)
P0 3
`
`
h0(v) # f0(v)
f *
0 (v)
# f *
0 (v) dv
P0 3
`
`
h0(v) # f0(v) dv
464 CHAPTER EIGHT
The IS BER can be estimated by the sample mean estimator as follows:
(8.27)
Notice we are removing the effects of biasing the noise statistics in the error counter itself by the multiplication
of the IS weight.
Here are some properties of the IS BER estimator .
Mean:
(8.28)
Variance:
(8.29)
Comparing these results to the MC given earlier can be done by using the ratio of the variance of
the two estimators. The addition of the IS weight w(v) allows us to reduce the variance of the IS
estimator. Using Eq. 8.30, we can define the variance reduction factor by setting N N*
(8.30)
At this point, we can take two paths: The first involves equating the variances and enjoying the benefits
of the smaller sample size N* N, while the second equates the sample sizes and benefits from the
smaller variance 2 (?†P
0*) 2( ?†P
0).
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