0.9 Carrier recovery  (Page 5/19)

Squared difference loop

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).