# 4.2 Interpolation

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Introduction of interpolation and it's application.

## Interpolation

Interpolation is the process of upsampling and filtering a signal to increase its effective sampling rate. To be more specific, say that $x(m)$ is an (unaliased) $T$ -sampled version of ${x}_{c}(t)$ and $v(n)$ is an $L$ -upsampled version version of $x(m)$ . If we filter $v(n)$ with an ideal $\frac{\pi }{L}$ -bandwidth lowpass filter (with DC gain $L$ ) to obtain $y(n)$ , then $y(n)$ will be a $\frac{T}{L}$ -sampled version of ${x}_{c}(t)$ . This process is illustrated in .

We justify our claims about interpolation using frequency-domain arguments. From the sampling theorem, we know that $T$ - sampling ${x}_{c}(t)$ to create $x(n)$ yields

$X(e^{i\omega })=\frac{1}{T}\sum_{k} {X}_{c}(i\frac{\omega -2\pi k}{T})$
After upsampling by factor $L$ , implies $V(e^{i\omega })=\frac{1}{T}\sum_{k} {X}_{c}(i\frac{\omega L-2\pi k}{T})=\frac{1}{T}\sum_{k} {X}_{c}(i\frac{\omega -\frac{2\pi }{L}k}{\frac{T}{L}})$ Lowpass filtering with cutoff $\frac{\pi }{L}$ and gain $L$ yields $Y(e^{i\omega })=\frac{L}{T}\sum_{\frac{k}{L}\in \mathbb{Z}} {X}_{c}(i\frac{\omega -\frac{2\pi }{L}k}{\frac{T}{L}})=\frac{L}{T}\sum_{l} {X}_{c}(i\frac{\omega -2\pi l}{\frac{T}{L}})$ since the spectral copies with indices other than $k=lL$ (for $l\in \mathbb{Z}$ ) are removed. Clearly, this process yields a $\frac{T}{L}$ -shaped version of ${x}_{c}(t)$ . illustrates these frequency-domain arguments for $L=2$ .

anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
Introduction about quantum dots in nanotechnology
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Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
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how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
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for teaching engĺish at school how nano technology help us
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Do somebody tell me a best nano engineering book for beginners?
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what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
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what is the actual application of fullerenes nowadays?
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is Bucky paper clear?
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so some one know about replacing silicon atom with phosphorous in semiconductors device?
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Do you know which machine is used to that process?
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how to fabricate graphene ink ?
for screen printed electrodes ?
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What is lattice structure?
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or in general
Ebrahim
in general
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Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
what's the easiest and fastest way to the synthesize AgNP?
China
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types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
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many many of nanotubes
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what is the k.e before it land
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what is the function of carbon nanotubes?
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I'm interested in nanotube
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what is nanomaterials​ and their applications of sensors.
how did you get the value of 2000N.What calculations are needed to arrive at it
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