mohatta2015.pdf
signal processing from power amplifier operation control point of view
signal processing from power amplifier operation control point of view
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40 MATCHED FILTERING<br />
which simplifies to<br />
/<br />
oo<br />
•OO<br />
\h{t)\ 2 dt. (2.43)<br />
Output SNR. is maximized when equality is achieved, which occurs when<br />
The resulting output SNR, is<br />
f(t) = h(t). (2.44)<br />
/ \h{t)\ 2 dt. (2.45)<br />
-oo<br />
A similar analysis of QPSK would reveal that the matched filter response would<br />
be the same. In general, for an arbitrary modulation, we can define a complex<br />
decision variable z such that<br />
oo<br />
f*(t)r(t) dt. (2.46)<br />
/<br />
-oo<br />
As discussed before, we usually determine the SNR of resulting complex decision<br />
variable, rather than just the real part.<br />
2.3.4.1 Final detection Once we have the decision variable z, we need to determine<br />
the detected symbol s. Here we will consider the more general case in which s<br />
is one of M possible values, drawn from set S. With ML symbol detection (which<br />
minimizes symbol error rate), we find the hypothetical value of s, denoted Sj, that<br />
maximizes the likelihood of z given s = Sj. As z is a continuous r.v., we will use<br />
its PDF for likelihood. Mathematically,<br />
oo<br />
s = arg max Pr{z\s = Sj}, (2-47)<br />
where "arg" means taking the argument (the Sj value).<br />
We can model complex-valued z as<br />
z \= As + e, (2.48)<br />
where e is complex, Gaussian noise with PDF given in (1.27). Thus, z is complex<br />
Gaussian with mean As. While MF leads to a value for A that is purely real and<br />
positive, let's consider the general case where A is some arbitrary complex number.<br />
The likelihood of z given s — Sj is then<br />
Pr {^}^exp{^d!}. (2.49)<br />
where the real and imaginary parts of e both have variance σ\. Since the likelihood<br />
function is positive and increasing, we can maximize over its log instead and ignore<br />
terms that do not depend on Sj. As a result, (2.49) becomes<br />
s = arg max -1« - ASj\ 2 — arg min \z - ASj\ 2 . (2.50)<br />
h£S<br />
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