mohatta2015.pdf
signal processing from power amplifier operation control point of view
signal processing from power amplifier operation control point of view
You also want an ePaper? Increase the reach of your titles
YUMPU automatically turns print PDFs into web optimized ePapers that Google loves.
36 MATCHED FILTERING<br />
converts the continuous-time r(t) into a set of one or more discrete variables referred<br />
to as statistics. Each statistic is the result of integrating r(i) with a particular basis<br />
function. A set of sufficient statistics is one in which no pertinent information is lost<br />
in reducing the continuous time r(t) into the set of statistics. Once these statistics<br />
are obtained, MLD involves finding the hypothetical value for s that maximizes the<br />
likelihood of what is observed (the statistics).<br />
We are going to take a less rigorous approach which leads to the same result in<br />
a more intuitive way. Let's hypothesize a value of s, denoted Sj. Also, suppose for<br />
the moment we only have a single sample of r(t) to work with, r(t\). With MLD,<br />
we want to find the value of Sj that maximizes the likelihood of r(t\) given that<br />
s = Sj.<br />
From our model, we know that r{t\) is complex Gaussian with mean h{t\)s.<br />
Because r{t\) is a continuous r.v., its "likelihood" will refer to its PDF value. The<br />
PDF value conditioned on s — Sj is given by<br />
1 f-|rfti)-5^(*i)l a \ (2. 23)<br />
Now suppose we have a second sample r(Í2). It will have a similar likelihood form.<br />
Since the noise on these two samples are uncorrelated (white noise assumption),<br />
the likelihood of both occurring is simply the product of their likelihoods.<br />
Maximizing the likelihood is equivalent to maximizing the log-likelihood. The<br />
product of two likelihoods becomes the sum of two log likelihoods. Thus, given two<br />
received samples r(ii) and r(