# 0.9 Carrier recovery  (Page 8/19)

 Page 8 / 19

## The phase locked loop

Perhaps the best loved method of phase tracking is known as the phase locked loop (PLL). This section shows that the PLLcan be derived as an adaptive element ascending the gradient of a simple performance function. The key idea is to modulatethe (processed) received signal ${r}_{p}\left(t\right)$ of [link] down to DC, using a cosine of known frequency $2{f}_{0}$ and phase $2\theta +\psi$ . After filtering to remove the high-frequency components, the magnitude of the DC term can be adjusted by changing the phase. The value of $\theta$ that maximizes the DC component is the same as the phase $\Phi$ of ${r}_{p}\left(t\right)$ .

To be specific, let

${J}_{PLL}\left(\theta \right)=\frac{1}{2}\text{LPF}\left\{{r}_{p}\left(k{T}_{s}\right)cos\left(4\pi {f}_{0}k{T}_{s}+2\theta +\psi \right)\right\}.$

Using the cosine product relationship [link] and the definition of ${r}_{p}\left(t\right)$ from [link] under the assumption that ${f}_{c}={f}_{0}$ , ${J}_{PLL}\left(\theta \right)$ becomes

$\begin{array}{ccc}& =& \frac{1}{2}\text{LPF}\left\{cos\left(4\pi {f}_{0}k{T}_{s}+2\Phi +\psi \right)cos\left(4\pi {f}_{0}k{T}_{s}+2\theta +\psi \right)\right\}\hfill \\ & =& \frac{1}{4}\text{LPF}\left\{cos\left(2\Phi -2\theta \right)+cos\left(8\pi {f}_{0}k{T}_{s}+2\theta +2\Phi +2\psi \right)\right\}\hfill \\ & =& \frac{1}{4}\text{LPF}\left\{cos\left(2\Phi -2\theta \right)\right\}\hfill \\ & & \phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}+\frac{1}{4}\text{LPF}\left\{cos\left(8\pi {f}_{0}k{T}_{s}+2\theta +2\Phi +2\psi \right)\right\}\hfill \\ & \approx & \frac{1}{4}cos\left(2\Phi -2\theta \right),\hfill \end{array}$

assuming that the cutoff frequency of the lowpass filter is well below $4{f}_{0}$ . This is shown inthe middle plot of [link] and is the same as ${J}_{SD}\left(\theta \right)$ , except for a constant and a sign. The sign change implies that,while ${J}_{SD}\left(\theta \right)$ needs to be minimized to find the correct answer, ${J}_{PLL}\left(\theta \right)$ needs to be maximized. The substantive difference between the SD and the PLLperformance functions lies in the way that the signals needed in the algorithm are extracted.

Assuming a small stepsize, the derivative of [link] with respect to $\theta$ at time $k$ can be approximated as

$\begin{array}{ccc}& & {\left(\frac{d\text{LPF}\left\{{r}_{p}\left(k{T}_{s}\right)cos\left(4\pi {f}_{0}k{T}_{s}+2\theta +\psi \right)\right\}}{d\theta }|}_{\theta =\theta \left[k\right]}\hfill \\ & & \phantom{\rule{1.em}{0ex}}\approx \text{LPF}\left\{{\left(\frac{d{r}_{p}\left(k{T}_{s}\right)cos\left(4\pi {f}_{0}k{T}_{s}+2\theta +\psi \right)}{d\theta }|}_{\theta =\theta \left[k\right]}\right\}\hfill \\ & & \phantom{\rule{1.em}{0ex}}=\text{LPF}\left\{-{r}_{p}\left(k{T}_{s}\right)sin\left(4\pi {f}_{0}k{T}_{s}+2\theta \left[k\right]+\psi \right)\right\}.\hfill \end{array}$

$\theta \left[k+1\right]=\theta \left[k\right]-\mu \text{LPF}\left\{{r}_{p}\left(k{T}_{s}\right)sin\left(4\pi {f}_{0}k{T}_{s}+2\theta \left[k\right]+\psi \right)\right\},$

is shown in [link] . Observe that the sign of the derivative is preserved in the update(rather than its negative), indicating that the algorithm is searching for a maximum of the error surface rather than a minimum.One difference between the PLL and the SD algorithms is clear from a comparison of [link] and [link] . The PLL requires oneless oscillator (and one less addition block). Since the performance functions ${J}_{SD}\left(\theta \right)$ and ${J}_{PLL}\left(\theta \right)$ are effectively the same, the performance characteristics of the two are roughly equivalent.

Suppose that ${f}_{c}$ is the frequency of the transmitter and ${f}_{0}$ is the assumed frequency at the receiver (with ${f}_{0}$ close to ${f}_{c}$ ). The following program simulates [link] for time seconds. Note that the firpm filter creates an h with a zero phase at the center frequency and so $\psi$ is set to zero.

Ts=1/10000; time=1; t=Ts:Ts:time;       % time vector fc=1000; phoff=-0.8;                    % carrier freq. and phaserp=cos(4*pi*fc*t+2*phoff);              % simplified received signal fl=100; ff=[0 .01 .02 1]; fa=[1 1 0 0];h=firpm(fl,ff,fa);                      % LPF design mu=.003;                                % algorithm stepsizef0=1000;                                % assumed freq. at receiver theta=zeros(1,length(t)); theta(1)=0;   % initialize vector for estimatesz=zeros(1,fl+1);                        % initialize buffer for LPF for k=1:length(t)-1                     % z contains past fl+1 inputs  z=[z(2:fl+1), rp(k)*sin(4*pi*f0*t(k)+2*theta(k))];  update=fliplr(h)*z';                  % new output of LPF   theta(k+1)=theta(k)-mu*update;        % algorithm updateend pllconverge.m simulate Phase Locked Loop (download file) 

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
Got questions? Join the online conversation and get instant answers!