# 0.4 Analog (de)modulation  (Page 4/9)

 Page 4 / 9

As illustrated in [link] , the received signal can be demodulated by mixingwith a cosine that has the same frequency and phase as the modulating cosine, and the original message canthen be recovered by low pass filtering. But, as a practical matter,the frequency and phase of the modulating cosine (located at the transmitter)can never be known exactly at the receiver.

Suppose that the frequency of the modulator is ${f}_{c}$ but that the frequency at the receiver is ${f}_{c}+\gamma$ , for some small $\gamma$ . Similarly, suppose that the phase of the modulator is 0but that the phase at the receiver is $\Phi$ . [link] (b) shows this downconverter, which can be described by

$x\left(t\right)=v\left(t\right)\phantom{\rule{4pt}{0ex}}\mathrm{cos}\left(2\pi \left({f}_{c}+\gamma \right)t+\Phi \right)$

and

$m\left(t\right)=\phantom{\rule{4pt}{0ex}}\text{LPF}\left\{x\left(t\right)\right\},$

where LPF represents a lowpass filtering of the demodulated signal $x\left(t\right)$ in an attempt to recover the message. Thus, the downconversion describedin [link] acknowledges that the receiver's local oscillator may not have thesame frequency or phase as the transmitter's local oscillator.In practice, accurate a priori information is available for carrier frequency, but (relative) phase could be anything, since it dependson the distance between the transmitter and receiver as well as when the transmission begins.Because the frequencies are high, the wavelengths are small and even small motions canchange the phase significantly.

The remainder of this section investigates what happens when the frequency and phase are not known exactly,that is, when either $\gamma$ or $\Phi$ (or both) are nonzero. Using the frequency shift property of Fourier transforms [link] on $x\left(t\right)$ in [link] produces the Fourier transform $X\left(f\right)$

$\begin{array}{cc}\hfill \frac{{A}_{c}}{4}\left[& {e}^{j\Phi }\left\{W\left(f+{f}_{c}-\left({f}_{c}+\gamma \right)\right)+W\left(f-{f}_{c}-\left({f}_{c}+\gamma \right)\right)\right\}\hfill \\ \hfill +& {e}^{-j\Phi }\left\{W\left(f+{f}_{c}+\left({f}_{c}+\gamma \right)\right)+W\left(f-{f}_{c}+\left({f}_{c}+\gamma \right)\right)\right\}\right]\hfill \\ & =\frac{{A}_{c}}{4}\left[{e}^{j\Phi }W\left(f-\gamma \right)+{e}^{j\Phi }W\left(f-2{f}_{c}-\gamma \right)\hfill \\ & +{e}^{-j\Phi }W\left(f+2{f}_{c}+\gamma \right)+{e}^{-j\Phi }W\left(f+\gamma \right)\right].\hfill \end{array}$

If there is no frequency offset (i.e., if $\gamma =0$ ), then

$\begin{array}{cc}\hfill X\left(f\right)=\frac{{A}_{c}}{4}\left[\left({e}^{j\Phi }& +{e}^{-j\Phi }\right)W\left(f\right)+{e}^{j\Phi }W\left(f-2{f}_{c}\right)\hfill \\ & +{e}^{-j\Phi }W\left(f+2{f}_{c}\right)\right].\hfill \end{array}$

Because $\mathrm{cos}\left(x\right)=\left(1/2\right)\left({e}^{jx}+{e}^{-jx}\right)$ from [link] , this can be rewritten

$\begin{array}{ccc}\hfill X\left(f\right)& =& \frac{{A}_{c}}{2}W\left(f\right)\mathrm{cos}\left(\Phi \right)\hfill \\ & & \phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}+\frac{{A}_{c}}{4}\left[{e}^{j\Phi },W,\left(f-2{f}_{c}\right),+,{e}^{-j\Phi },W,\left(f+2{f}_{c}\right)\right].\hfill \end{array}$

Thus, a perfect lowpass filtering of $x\left(t\right)$ with cutoff below $2{f}_{c}$ removes the high frequency portions of the signal near $±2{f}_{c}$ to produce

$m\left(t\right)=\frac{{A}_{c}}{2}w\left(t\right)\phantom{\rule{4pt}{0ex}}\mathrm{cos}\left(\Phi \right).$

The factor $\mathrm{cos}\left(\Phi \right)$ attenuates the received signal (except for the special case when $\Phi =0±2\pi k$ for integers $k$ ). If $\Phi$ were sufficiently close to $0±2\pi k$ for some integer $k$ , then this would be tolerable. But there is no way to know the relative phase, and hence $\mathrm{cos}\left(\Phi \right)$ can assume any possible value within $\left[-1,1\right]$ . The worst case occurs as $\Phi$ approaches $±\pi /2$ , when the message is attenuated to zero!A scheme for carrier phase synchronization, which automatically tries toalign the phase of the cosine at the receiver with the phase at the transmitter is vital. This is discussedin detail in [link] .

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
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!