# 0.3 Modelling corruption  (Page 7/11)

 Page 7 / 11
$w\left(t\right)=\frac{A}{2j}\left[{e}^{j2\pi {f}_{0}t},-,{e}^{-j2\pi {f}_{0}t}\right].$

Applying the linearity property [link] and the result of Exercise  [link] gives

$\begin{array}{ccc}\hfill \mathcal{F}\left\{w\left(t\right)\right\}& =& \frac{A}{2j}\left[\mathcal{F},\left\{{e}^{j2\pi {f}_{0}t}\right\},-,\mathcal{F},\left\{{e}^{-j2\pi {f}_{0}t}\right\}\right]\hfill \\ & =& j\frac{A}{2}\left[-,\delta ,\left(f-{f}_{0}\right),+,\delta ,\left(f+{f}_{0}\right)\right].\hfill \end{array}$

Thus, the spectrum of a sine wave is a pair of $\delta$ functions with opposite signs, located symmetrically about zero frequency. The corresponding magnitude spectrum,shown in [link] , is at the heart of one importantinterpretation of the Fourier transform: it shows the frequency content of any signal by displayingwhich frequencies are present (and which frequencies are absent) from the waveform. For example, [link] (a) shows the magnitude spectrum $W\left(f\right)$ of a real-valued signal $w\left(t\right)$ . This can be interpreted as saying that $w\left(t\right)$ contains (or is made up of) “all the frequencies” up to $B$ Hz, and that it contains no sinusoids with higher frequency. Similarly,the modulated signal $s\left(t\right)$ in [link] (b) contains all positive frequencies between ${f}_{c}-B$ and ${f}_{c}+B$ , and no others.

Note that the Fourier transform in [link] is purely imaginary, as it must be because $w\left(t\right)$ is odd (see  [link] ). The phase spectrum is a flat line at $-{90}^{\circ }$ because of the factor  $j$ . The magnitude spectrum of a sinusoid with frequency f 0 and amplitude A contains two δ function spikes, one at f = f 0 and the other at f = - f 0 .

What is the magnitude spectrum of $\mathrm{sin}\left(2\pi {f}_{0}t+\theta \right)$ ? Hint: Use the frequency shift property [link] . Show that the spectrum of $\mathrm{cos}\left(2\pi {f}_{0}t\right)$ is $\frac{1}{2}\left(\delta \left(f-{f}_{0}\right)+\delta \left(f+{f}_{0}\right)\right)$ . Compare this analytical result to the numerical resultsfrom Exercise  [link] . The magnitude spectrum of a message signal w ( t ) is shown in (a). When w ( t ) is modulated by a cosine at frequency f c , the spectrum of the resulting signal s ( t ) = w ( t ) cos ( 2 π f c t + Φ ) is shown in (b).

Let ${w}_{i}\left(t\right)={a}_{i}\mathrm{sin}\left(2\pi {f}_{i}t\right)$ for $i=1,2,3$ . Without doing any calculations, write down the spectrum of $v\left(t\right)={w}_{1}\left(t\right)+{w}_{2}\left(t\right)+{w}_{3}\left(t\right)$ . Hint: Use linearity. Graph the magnitude spectrum of $v\left(t\right)$ in the same manner as in [link] . Verify your results with a simulation mimicking that in  [link] .

Let $W\left(f\right)=\mathrm{sin}\left(2\pi f{t}_{0}\right)$ . What is the corresponding time function?

## Convolution in time: it's what linear systems do

Suppose that a system has impulse response $h\left(t\right)$ , and that the input consists of a sum of three impulses occurring at times ${t}_{0}$ , ${t}_{1}$ , and ${t}_{2}$ , with amplitudes ${a}_{0}$ , ${a}_{1}$ , and ${a}_{2}$ (for example, the signal $w\left(t\right)$ of [link] ). By linearity of the Fourier transform, property [link] , the output is a superpositionof the outputs due to each of the input pulses. The output due to the first impulse is ${a}_{0}h\left(t-{t}_{0}\right)$ , which is the impulse response scaled by the size of the input and shifted to beginwhen the first input pulse arrives. Similarly, the outputs to the second and thirdinput impulses are ${a}_{1}h\left(t-{t}_{1}\right)$ and ${a}_{2}h\left(t-{t}_{2}\right)$ , respectively, and the complete output is the sum ${a}_{0}h\left(t-{t}_{0}\right)+{a}_{1}h\left(t-{t}_{1}\right)+{a}_{2}h\left(t-{t}_{2}\right)$ .

Now suppose that the input is a continuous function $x\left(t\right)$ . At any time instant $\lambda$ , the input can be thought of as consisting of an impulse scaled by the amplitude $x\left(\lambda \right)$ , and the corresponding output will be $x\left(\lambda \right)h\left(t-\lambda \right)$ , which is the impulse response scaled by thesize of the input and shifted to begin at time $\lambda$ . The complete output is then given by integrating over all $\lambda$

$y\left(t\right)={\int }_{-\infty }^{\infty }x\left(\lambda \right)h\left(t-\lambda \right)d\lambda \equiv x\left(t\right)*h\left(t\right).$

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
Berger describes sociologists as concerned with
what is hormones?
Wellington
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