# 7.1 Digital receiver: carrier recovery  (Page 2/3)

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## Phase detector

The goal of the PLL is to maintain a demodulating sine and cosine that match the incoming carrier. Suppose ${\omega }_{c}$ is the believed digital carrier frequency. We can then represent the actual received carrier frequency as theexpected carrier frequency with some offset, $\stackrel{˜}{{\omega }_{c}}={\omega }_{c}+\stackrel{˜}{\theta }(n)$ . The NCO generates the demodulating sine and cosine with the expected digital frequency ${\omega }_{c}$ and offsets this frequency with the output of the loop filter. The NCO frequency can then be modeled as $\stackrel{^}{{\omega }_{c}}={\omega }_{c}+\stackrel{^}{\theta }(n)$ . Using the appropriate trigonometric identities $\cos A\cos B=1/2(\cos (A-B)+\cos (A+B))$ and $\cos A\sin B=1/2(\sin (B-A)+\sin (A+B))$ . , the in-phase and quadrature signals can be expressed as

${z}_{0}(n)=1/2(\cos (\stackrel{˜}{\theta }(n)-\stackrel{^}{\theta }(n))+\cos (2{\omega }_{c}+\stackrel{˜}{\theta }(n)+\stackrel{^}{\theta }(n)))$
${z}_{Q}(n)=1/2(\sin (\stackrel{˜}{\theta }(n)-\stackrel{^}{\theta }(n))+\sin (2{\omega }_{c}+\stackrel{˜}{\theta }(n)+\stackrel{^}{\theta }(n)))$
After applying a low-pass filter to remove the double frequency terms, we have
${y}_{1}(n)=1/2\cos (\stackrel{˜}{\theta }(n)-\stackrel{^}{\theta }(n))$
${y}_{Q}(n)=1/2\sin (\stackrel{˜}{\theta }(n)-\stackrel{^}{\theta }(n))$
Note that the quadrature signal, ${z}_{Q}(n)$ , is zero when the received carrier and internallygenerated waves are exactly matched in frequency and phase. When the phases are only slightly mismatched we can use therelation
$\forall \theta , \mathrm{small}\colon \sin \theta \approx \theta$
and let the current value of the quadrature channel approximate the phase difference: ${z}_{Q}(n)\approx \stackrel{˜}{\theta }(n)-\stackrel{^}{\theta }(n)$ . With the exception of the sign error, this difference is essentially how much we need to offset our NCOfrequency If $\stackrel{˜}{\theta }(n)-\stackrel{^}{\theta }(n)> 0$ then $\stackrel{^}{\theta }(n)$ is too large and we want to decrease our NCO phase. . To make sure that the sign of the phase estimate is right, in this example the phase detector issimply negative one times the value of the quadrature signal. In a more advanced receiver, information from boththe in-phase and quadrature branches is used to generate an estimate of the phase error. What should the relationship between the I and Q branches be fora digital QPSK signal?

## Loop filter

The estimated phase mismatch estimate is fed to the NCO via a loop filter, often a simple low-pass filter. For thisexercise you can use a one-tap IIR filter,

$y(n)=\beta x(n)+\alpha y(n-1)$
To ensure unity gain at DC, we select $\beta =1-\alpha$

It is suggested that you start by choosing $\alpha =0.6$ and $K=0.15$ for the loop gain. Once you have a working system, investigate the effects of modifying these values.

## Matlab simulation

Simulate the PLL system shown in [link] using MATLAB. As with the DLL simulation, you will have to simulate the PLL on a sample-by-sample basis.

Use [link] to implement your NCO in MATLAB. However, to ensure that the phase term does not grow toinfinity, you should use addition modulo $2\pi$ in the phase update relation. This can be done by setting $\theta (n)=\theta (n)-2\pi$ whenever $\theta (n)> 2\pi$ .

[link] illustrates how the proposed PLL will behave when given a modulated BPSK waveform. In this case thetransmitted carrier frequency was set to $\stackrel{˜}{{\omega }_{c}}=\frac{\pi }{2}+\frac{\pi }{1024}$ to simulate a frequency offset.

Note that an amplitude transition in the BPSK waveform is equivalent to a phase shift of the carrier by $\frac{\pi }{2}$ . Immediately after this phase change occurs, the PLL begins to adjust the phase to force the quadraturecomponent to zero (and the in-phase component to $1/2$ ). Why would this phase detector not work in a real BPSK environment? How could it be changed to work?

where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
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
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
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
how did you get the value of 2000N.What calculations are needed to arrive at it
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