# 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 ${F}_{s}> 2{F}_{b}$ )

can someone help me with some logarithmic and exponential equations.
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
Commplementary angles
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Sherica
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Sherica
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Tamia
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a perfect square v²+2v+_
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algebra 2 Inequalities:If equation 2 = 0 it is an open set?
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
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Kristine 2*2*2=8
Differences Between Laspeyres and Paasche Indices
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
is it 3×y ?
J, combine like terms 7x-4y
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Asali
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Samantha
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Asali
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Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
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Porter
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Yasmin
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Cesar
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Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
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Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
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Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
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Prasenjit
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Damian
silver nanoparticles could handle the job?
Damian
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Azam
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At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
how did you get the value of 2000N.What calculations are needed to arrive at it
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