mohatta2015.pdf

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