# 0.3 Signal processing in processing: sampling and quantization

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Fundamentals of sampling, reconstruction, and quantization of 1D (sounds) and 2D (images) signals, especially oriented at the Processing language.

## Sampling

Both sounds and images can be considered as signals, in one or two dimensions, respectively. Sound can be described as afluctuation of the acoustic pressure in time, while images are spatial distributions of values of luminance or color, thelatter being described in its RGB or HSB components. Any signal, in order to be processed by numerical computingdevices, have to be reduced to a sequence of discrete samples , and each sample must be represented using a finite number of bits. The first operationis called sampling , and the second operation is called quantization of the domain of real numbers.

## 1-d: sounds

Sampling is, for one-dimensional signals, the operation that transforms a continuous-time signal (such as, for instance,the air pressure fluctuation at the entrance of the ear canal) into a discrete-time signal, that is a sequence ofnumbers. The discrete-time signal gives the values of the continuous-time signal read at intervals of $T$ seconds. The reciprocal of the sampling interval is called sampling rate ${F}_{s}=\frac{1}{T}$ . In this module we do not explain the theory of sampling, but we rather describe its manifestations. For a amore extensive yet accessible treatment, we point to the Introduction to Sound Processing . For our purposes, the process of sampling a 1-D signal canbe reduced to three facts and a theorem.

• The Fourier Transform of a discrete-time signal is a function (called spectrum ) of the continuous variable $\omega$ , and it is periodic with period $2\pi$ . Given a value of $\omega$ , the Fourier transform gives back a complex number that can be interpreted as magnitude and phase(translation in time) of the sinusoidal component at that frequency.
• Sampling the continuous-time signal $x(t)$ with interval $T$ we get the discrete-time signal $x(n)=x(nT)$ , which is a function of the discrete variable $n$ .
• Sampling a continuous-time signal with sampling rate ${F}_{s}$ produces a discrete-time signal whose frequency spectrum is the periodic replication of the originalsignal, and the replication period is ${F}_{s}$ . The Fourier variable $\omega$ for functions of discrete variable is converted into the frequency variable $f$ (in Hertz) by means of $f=\frac{\omega }{2\pi T}$ .

The [link] shows an example of frequency spectrum of a signal sampled with sampling rate ${F}_{s}$ . In the example, the continuous-time signal had all and only the frequency components between $-{F}_{b}$ and ${F}_{b}$ . The replicas of the original spectrum are sometimes called images .

Given the facts , we can have an intuitive understanding of the Sampling Theorem,historically attributed to the scientists Nyquist and Shannon.

## Sampling theorem

A continuous-time signal $x(t)$ , whose spectral content is limited to frequencies smaller than ${F}_{b}$ (i.e., it is band-limited to ${F}_{b}$ ) can be recovered from its sampled version $x(n)$ if the sampling rate is larger than twice the bandwidth (i.e., if ${F}_{s}> 2{F}_{b}$ )

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
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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
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