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


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 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 ω , and it is periodic with period 2 π . Given a value of ω , 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 n T , 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 ω for functions of discrete variable is converted into the frequency variable f (in Hertz) by means of f ω 2 π 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 .

Frequency spectrum of a sampled signal

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 )

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
for teaching engĺish at school how nano technology help us
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
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.
what is the actual application of fullerenes nowadays?
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get the best Algebra and trigonometry course in your pocket!

Source:  OpenStax, Media processing in processing. OpenStax CNX. Nov 10, 2010 Download for free at http://cnx.org/content/col10268/1.14
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Media processing in processing' conversation and receive update notifications?