# 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).

where we get a research paper on Nano chemistry....?
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
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