mohatta2015.pdf

36 MATCHED FILTERING
converts the continuous-time r(t) into a set of one or more discrete variables referred
to as statistics. Each statistic is the result of integrating r(i) with a particular basis
function. A set of sufficient statistics is one in which no pertinent information is lost
in reducing the continuous time r(t) into the set of statistics. Once these statistics
are obtained, MLD involves finding the hypothetical value for s that maximizes the
likelihood of what is observed (the statistics).
We are going to take a less rigorous approach which leads to the same result in
a more intuitive way. Let's hypothesize a value of s, denoted Sj. Also, suppose for
the moment we only have a single sample of r(t) to work with, r(t\). With MLD,
we want to find the value of Sj that maximizes the likelihood of r(t\) given that
s = Sj.
From our model, we know that r{t\) is complex Gaussian with mean h{t\)s.
Because r{t\) is a continuous r.v., its "likelihood" will refer to its PDF value. The
PDF value conditioned on s — Sj is given by
1 f-|rfti)-5^(*i)l a \ (2. 23)
Now suppose we have a second sample r(Í2). It will have a similar likelihood form.
Since the noise on these two samples are uncorrelated (white noise assumption),
the likelihood of both occurring is simply the product of their likelihoods.
Maximizing the likelihood is equivalent to maximizing the log-likelihood. The
product of two likelihoods becomes the sum of two log likelihoods. Thus, given two
received samples r(ii) and r(