# 0.9 Carrier recovery  (Page 5/19)

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The squaring nonlinearity is only one possibility in the pllpreprocess.m routine.

1. Try replacing the ${r}^{2}\left(t\right)$ with $|r\left(t\right)|$ . Does this result in a viable method of emphasizing the carrier?
2. Try replacing the ${r}^{2}\left(t\right)$ with ${r}^{3}\left(t\right)$ . Does this result in a viable method of emphasizing the carrier?
3. Can you think of other functions that will result in viable methods of emphasizing the carrier?
4. Will a linear function work? Why or why not?

Determine the phase shift $\psi$ of the BPF when

1. fl=490, 496, 502 .
2. Ts=0.0001, 0.000101 .
3. M=19, 20, 21 . Explain why $\psi$ should depend on fl , Ts , and M .

## 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

${r}_{p}\left(t\right)=cos\left(4\pi {f}_{c}t+2\Phi \right),$

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

$\begin{array}{ccc}\hfill {J}_{SD}\left(\theta \right)& =& \text{avg}\left\{{e}^{2}\left(\theta ,k\right)\right\}\hfill \\ & =& \frac{1}{4}\text{avg}\left\{{\left({r}_{p}\left(k{T}_{s}\right)-cos\left(4\pi {f}_{0}k{T}_{s}+2\theta \right)\right)}^{2}\right\}\hfill \end{array}$

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 ${r}_{p}\left(k{T}_{s}\right)\ne cos\left(4\pi {f}_{0}k{T}_{s}+2\theta \right)$ , $e\left(\theta ,k\right)\ne 0$ , 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] .

$\theta \left[k+1\right]=\theta \left[k\right]-\mu {\left(\frac{d{J}_{SD}\left(\theta \right)}{d\theta }|}_{\theta =\theta \left[k\right]},$

which is the same as [link] with the variable changed from $x$ to $\theta$ .  Thus,

$\begin{array}{ccc}\hfill \frac{d{J}_{SD}\left(\theta \right)}{d\theta }& =& \frac{d\text{avg}\left\{{e}^{2}\left(\theta ,k\right)\right\}}{d\theta }\hfill \\ & \approx & \text{avg}\left\{\frac{d{e}^{2}\left(\theta ,k\right)}{d\theta }\right\}\hfill \\ & =& \frac{1}{2}\text{avg}\left\{e,\left(\theta ,k\right),\frac{de\left(\theta ,k\right)}{d\theta }\right\}\hfill \\ & =& \text{avg}\left\{\left({r}_{p},\left(k{T}_{s}\right),-,cos,\left(4\pi {f}_{0}k{T}_{s}+2\theta \right)\right)\hfill \\ & & \phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}sin\left(4\pi {f}_{0}k{T}_{s}+2\theta \right)\right\}.\hfill \end{array}$

Substituting this into [link] and evaluating at $\theta =\theta \left[k\right]$ gives Recall the convention that $\theta \left[k\right]=\theta \left(k{T}_{s}\right)=\theta \left(t\right){|}_{t=k{T}_{s}}$ .

$\begin{array}{cc}\hfill \theta \left[k+1\right]=\theta \left[k\right]-\mu \text{avg}\left\{\left(& {r}_{p}\left(k{T}_{s}\right)-cos\left(4\pi {f}_{0}k{T}_{s}+2\theta \left[k\right]\right)\right)\hfill \\ & sin\left(4\pi {f}_{0}k{T}_{s}+2\theta \left[k\right]\right)\right\}.\hfill \end{array}$

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).

what is the stm
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nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
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research.net
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how did you get the value of 2000N.What calculations are needed to arrive at it
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