ECEN 661 Modulation Theory
Spring 2006
Practice Midterm Exam #2
Problem 1.
(a) For a full response CPFSK modulation format with a modulation index,
,
prove
that the
asymptotic relative power efficiency is bounded by
.
(b) Since
optimizes the above expression, suppose we choose
as our modu
lation index (since it is a simple rational number close to the optimal value).
Prove that for
this case, the upper bound in (a) is tight, that is, for
,
.
Problem 2.
Consider a binary signalling format where the two possible transmitted signals, whose complex
envelopes are
and
, satisfy:
(i)
, and
(ii)
.
The signal is transmitted over a channel the adds white Gaussian noise and also exhibits Rayleigh
fading so that the complex envelope of the received signal,
, is given by
,
where
is complex white Gaussian noise with
,
is a Ray
leigh random variable with PDF,
, and
is a Random variable uni
formly distributed over
.
Derive
the probability of error of the optimum noncoherent
detector for this signalling scheme over this channel.
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 Fall '11
 Miller
 Signal Processing, Probability theory, Rayleigh fading

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