<< Chapter < Page | Chapter >> Page > |
The squaring nonlinearity is only one possibility in
the
pllpreprocess.m
routine.
Determine the phase shift $\psi $ of the BPF when
fl=490, 496, 502
.Ts=0.0001, 0.000101
.M=19, 20, 21
.
Explain why
$\psi $ should depend on
fl
,
Ts
,
and
M
.The problem of phase tracking is to determine the phase $\Phi $ of the carrier and to follow any changes in $\Phi $ using only the received signal. The frequency ${f}_{c}$ 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 $\psi $ into account is given in Exercise [link] . the dependence on the known phase shift $\psi $ of the BPF and sets the constant $\frac{{s}_{avg}^{2}}{2}$ to unity (compare with [link] ). The form of ${r}_{p}\left(t\right)$ implies that there is an essential ambiguity in the phase since $\Phi $ can be replaced by $\Phi +n\pi $ for any integer $n$ without changing the value of [link] . What can be done to recover $\Phi $ (modulo $\pi $ ) from ${r}_{p}\left(t\right)$ ?
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 ${r}_{p}\left(t\right)$ and a sinusoid generated, using an estimate of the phase; that is, seek to minimize
by choice of $\theta $ , where ${r}_{p}\left(k{T}_{s}\right)$ is the value of ${r}_{p}\left(t\right)$ sampled at time $k{T}_{s}$ and where ${f}_{0}$ is presumed equal to ${f}_{c}$ . (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 $\theta $ could be found so that $\theta =\Phi +n\pi $ , then the value of the performance functionwould be zero. When $\theta \ne \Phi +n\pi $ , then ${r}_{p}\left(k{T}_{s}\right)\ne cos(4\pi {f}_{0}k{T}_{s}+2\theta )$ , $e(\theta ,k)\ne 0$ , and so ${J}_{SD}\left(\theta \right)>0$ . Hence, [link] is minimized when $\theta $ has correctly identified the phase offset, modulo the inevitable $\pi $ ambiguity.
While there are many methods of minimizing [link] , an adaptive element that descends the gradient of the performance function ${J}_{SD}\left(\theta \right)$ 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 $x$ to $\theta $ . Thus,
Substituting this into [link] and evaluating at $\theta =\theta \left[k\right]$ gives Recall the convention that $\theta \left[k\right]=\theta \left(k{T}_{s}\right)=\theta \left(t\right){|}_{t=k{T}_{s}}$ .
This is implemented in
pllsd.m
for a phase offset of
phoff=-0.8
(i.e.,
$\Phi $ of
[link] is
$-0.8$ , 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
$-0.8+\pi $ (about
$2.3$ ) and to
$-0.8-\pi $ (about
$-$ 4).
Notification Switch
Would you like to follow the 'Software receiver design' conversation and receive update notifications?