# 0.4 Analog (de)modulation  (Page 2/9)

 Page 2 / 9

## Amplitude modulation with large carrier

Perhaps the simplest form of (analog) transmission system modulates the message signal by a high frequency carrier in a two-stepprocedure: multiply the message by the carrier, then add the product to the carrier.At the receiver, the message can be demodulated by extracting the envelope of the received signal.

Consider the transmitted/modulated signal

$\begin{array}{ccc}\hfill v\left(t\right)& =& {A}_{c}w\left(t\right)cos\left(2\pi {f}_{c}t\right)+{A}_{c}cos\left(2\pi {f}_{c}t\right)\hfill \\ & =& {A}_{c}\left(w\left(t\right)+1\right)cos\left(2\pi {f}_{c}t\right)\hfill \end{array}$

diagrammed in [link] . The process of multiplying the signal in timeby a (co)sinusoid is called mixing . This can be rewritten in the frequency domainby mimicking the development from [link] to [link] . Using the convolution property of Fourier Transforms [link] , the transform of $v\left(t\right)$ is

$\begin{array}{ccc}\hfill V\left(f\right)& =& \mathcal{F}\left\{{A}_{c}\left(w\left(t\right)+1\right)cos\left(2\pi {f}_{c}t\right)\right\}\hfill \\ & =& {A}_{c}\mathcal{F}\left\{\left(w\left(t\right)+1\right)\right\}*\mathcal{F}\left\{cos\left(2\pi {f}_{c}t\right)\right\}.\hfill \end{array}$

The spectra of $\mathcal{F}\left\{w\left(t\right)+1\right\}$ and $|V\left(f\right)|$ are sketched in [link] (a) and (b). The vertical arrows in (b) represent the transform of the cosine carrierat frequency ${f}_{c}$ (i.e., a pair of delta functions at $±{f}_{c}$ ) and the scaling by $\frac{{A}_{c}}{2}$ is indicated next to the arrowheads.

If $w\left(t\right)\ge -1$ , the envelope of $v\left(t\right)$ is the same as $w\left(t\right)$ and an envelope detector can be used as a demodulator. One way to find the envelope of a signal is to lowpass filterthe absolute value. To see this analytically, observe that

$\begin{array}{ccc}\hfill \mathcal{F}\left\{|v\left(t\right)|\right\}& =& \mathcal{F}\left\{|{A}_{c}\left(w\left(t\right)+1\right)cos\left(2\pi {f}_{c}t\right)|\right\}\hfill \\ & =& |{A}_{c}|\mathcal{F}\left\{|w\left(t\right)+1||cos\left(2\pi {f}_{c}t\right)|\right\}\hfill \\ & =& |{A}_{c}|\mathcal{F}\left\{w\left(t\right)+1\right\}*\mathcal{F}\left\{|cos\left(2\pi {f}_{c}t\right)|\right\}\hfill \end{array}$

where the absolute value can be removed from $w\left(t\right)+1$ because $w\left(t\right)+1>0$ (by assumption). The spectrum of $\mathcal{F}\left\{|cos\left(2\pi {f}_{c}t|\right\}$ , shown in [link] (c), may be familiar from Exercise  [link] . Accordingly, $\mathcal{F}\left\{|v\left(t\right)|\right\}$ is the convolution shown in [link] (d). Low pass filtering this returns $w\left(t\right)+1$ , which is the envelope of $v\left(t\right)$ offset by the constant one.

An example is given in the following M atlab program. The “message” signal is a sinusoid with a drift in the DC offset,and the carrier wave is at a much higher frequency.

time=.33; Ts=1/10000;                         % sampling interval and time t=0:Ts:time; lent=length(t);                  % define a "time" vectorfc=1000; c=cos(2*pi*fc*t);                    % define carrier at freq fc fm=20; w=10/lent*[1:lent]+cos(2*pi*fm*t);     % create "message" > -1 v=c.*w+c;                                     % modulate with large carrierfbe=[0 0.05 0.1 1]; damps=[1 1 0 0]; fl=100;  % low pass filter design b=firpm(fl,fbe,damps);                        % impulse response of LPFenvv=(pi/2)*filter(b,1,abs(v));               % find envelope AMlarge.m large carrier AM demodulated with “envelope” (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!