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The squaring nonlinearity is only one possibility in
the
pllpreprocess.m
routine.
Determine the phase shift of the BPF when
fl=490, 496, 502
.Ts=0.0001, 0.000101
.M=19, 20, 21
.
Explain why
should depend on
fl
,
Ts
,
and
M
.The problem of phase tracking is to determine the phase of the carrier and to follow any changes in using only the received signal. The frequency of the carrier is assumed known, though ultimately it too must be estimated.The received signal can be preprocessed (as in the previous section) to create a signal that strips away the data, in essence fabricating a sinusoid which has twice the frequency at twice the phase of the unmodulated carrier. This can be idealized to
which suppresses An example that takes into account is given in Exercise [link] . the dependence on the known phase shift of the BPF and sets the constant to unity (compare with [link] ). The form of implies that there is an essential ambiguity in the phase since can be replaced by for any integer without changing the value of [link] . What can be done to recover (modulo ) from ?
Is there some way to use an adaptive element? [link] suggested that there are three steps to the creation of a good adaptive element: setting a goal,finding a method, and then testing. As a first try, consider the goal of minimizingthe average of the squared difference between and a sinusoid generated, using an estimate of the phase; that is, seek to minimize
by choice of , where is the value of sampled at time and where is presumed equal to . (The subscript SD stands for squared difference, and is used todistinguish this performance function from others that will appear in this and other chapters.)This goal makes sense because, if could be found so that , then the value of the performance functionwould be zero. When , then , , and so . Hence, [link] is minimized when has correctly identified the phase offset, modulo the inevitable ambiguity.
While there are many methods of minimizing [link] , an adaptive element that descends the gradient of the performance function leads to the algorithm Recall the discussion surrounding the AGC elements in Chapter [link] .
which is the same as [link] with the variable changed from to . Thus,
Substituting this into [link] and evaluating at gives Recall the convention that .
This is implemented in
pllsd.m
for a phase offset of
phoff=-0.8
(i.e.,
of
[link] is
, though this value is unknown to the algorithm).
[link] plots the estimates
theta
for 50 different initial guesses
theta(1)
. Observe that
many converge to the correct value at
0.8. Others converge
to
(about
) and to
(about
4).
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