# 6.1 Interpolation, decimation, and rate changing by integer fractions

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## Interpolation: by an integer factor l

Interpolation means increasing the sampling rate, or filling in in-between samples. Equivalent to sampling abandlimited analog signal $L$ times faster. For the ideal interpolator,

${X}_{1}(\omega )=\begin{cases}{X}_{0}(L\omega ) & \text{if \left|\omega \right|< \frac{\pi }{L}}\\ 0 & \text{if \frac{\pi }{L}\le \left|\omega \right|\le \pi }\end{cases}$
$y(m)=\begin{cases}{X}_{0}(\frac{m}{L}) & \text{if m=\{0, ±(L), ±(2L), \dots \}}\\ 0 & \text{otherwise}\end{cases}$
The DTFT of $y(m)$ is
$Y(\omega )=\sum_{m=-\omega }$ y m ω m n x 0 n ω L n n x n ω L n X 0 ω L
Since ${X}_{0}({\omega }^{\prime })$ is periodic with a period of $2\pi$ , ${X}_{0}(L\omega )=Y(\omega )$ is periodic with a period of $\frac{2\pi }{L}$ (see [link] ). By inserting zero samples between the samples of ${x}_{0}(n)$ , we obtain a signal with a scaled frequency response that simply replicates ${X}_{0}({\omega }^{\prime })$ $L$ times over a $2\pi$ interval!

Obviously, the desired ${x}_{1}(m)$ can be obtained simply by lowpass filtering $y(m)$ to remove the replicas.

${x}_{1}(m)=(y(m), {h}_{L}(m))$
Given ${H}_{L}(m)=\begin{cases}1 & \text{if \left|\omega \right|< \frac{\pi }{L}}\\ 0 & \text{if \frac{\pi }{L}\le \left|\omega \right|\le \pi }\end{cases}$ In practice, a finite-length lowpass filter is designed using any of the methods studied so far ( [link] ).

## Decimation: sampling rate reduction (by an integer factor m)

Let $y(m)={x}_{0}(Lm)$ ( [link] )

That is, keep only every $L$ th sample ( [link] ) In frequency (DTFT):
$Y(\omega )=\sum_{m=()}$ y m ω m m x 0 M m ω m n M m n x 0 n k δ n M k ω n M ω ω M n x 0 n k δ n M k ω n DTFT x 0 n DTFT δ n M k
Now $\mathrm{DTFT}(\sum \delta (n-Mk))=2\pi \sum_{k=0}^{M-1} X(k)\delta ({\omega }_{\prime }-\frac{2\pi k}{M})$ for $\left|\omega \right|< \pi$ as shown in homework #1, where $X(k)$ is the DFT of one period of the periodic sequence. In this case, $X(k)=1$ for $k\in \{0, 1, \dots , M-1\}$ and $\mathrm{DTFT}(\sum \delta (n-Mk))=2\pi \sum_{k=0}^{M-1} \delta ({\omega }_{\prime }-\frac{2\pi k}{M})$ .
$(\mathrm{DTFT}({x}_{0}(n)), \mathrm{DTFT}(\sum \delta (n-Mk)))=({X}_{0}({\omega }^{\prime }), 2\pi \sum_{k=0}^{M-1} \delta ({\omega }_{\prime }-\frac{2\pi k}{M}))=\frac{1}{2\pi }\int_{-\pi }^{\pi } {X}_{0}({\mu }^{\prime })2\pi \sum_{k=0}^{M-1} \delta ({\omega }_{\prime }-{\mu }_{\prime }-\frac{2\pi k}{M})\,d {\mu }^{\prime }=\sum_{k=0}^{M-1} {X}_{0}({\omega }_{\prime }-\frac{2\pi k}{M})$
so $Y(\omega )=\sum_{k=0}^{M-1} {X}_{0}(\frac{\omega }{M}-\frac{2\pi k}{M})$ i.e. , we get digital aliasing .( [link] ) Usually, we prefer not to have aliasing, so the downsampler is preceded by a lowpass filter to remove all frequencycomponents above $\left|\omega \right|< \frac{\pi }{M}$ ( [link] ).

## Rate-changing by a rational fraction l/m

This is easily accomplished by interpolating by a factor of $L$ , then decimating by a factor of $M$ ( [link] ).

The two lowpass filters can be combined into one LP filterwith the lower cutoff, $H(\omega )=\begin{cases}1 & \text{if \left|\omega \right|< \frac{\pi }{\max\{L , M\}}}\\ 0 & \text{if \frac{\pi }{\max\{L , M\}}\le \left|\omega \right|\le \pi }\end{cases}$ Obviously, the computational complexity and simplicity of implementation will depend on $\frac{L}{M}$ : $2/3$ will be easier to implement than $1061/1060$ !

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
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da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
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Giriraj
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
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Alexandre
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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 ?
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?
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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