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This module builds on the intuition developed in the sampling module to discuss the Nyquist-Shannon sampling theorem, including a full statement and a proof.

Introduction

With the introduction of the concept of signal sampling, which produces a discrete time signal by selecting the values of the continuous time signal at evenly spaced points in time, it is now possible to discuss one of the most important results in signal processing, the Nyquist-Shannon sampling theorem. Often simply called the sampling theorem, this theorem concerns signals, known as bandlimited signals, with spectra that are zero for all frequencies with absolute value greater than or equal to a certain level. The theorem implies that there is a sufficiently high sampling rate at which a bandlimited signal can be recovered exactly from its samples, which is an important step in the processing of continuous time signals using the tools of discrete time signal processing.

Nyquist-shannon sampling theorem

Statement of the sampling theorem

The Nyquist-Shannon sampling theorem concerns signals with continuous time Fourier transforms that are only nonzero on the interval ( - B , B ) for some constant B . Such a function is said to be bandlimited to ( - B , B ) . Essentially, the sampling theorem has already been implicitly introduced in the previous module concerning sampling. Given a continuous time signals x with continuous time Fourier transform X , recall that the spectrum X s of sampled signal x s with sampling period T s is given by

X s ( ω ) = 1 T s k = - X ω - 2 π k T s .

It had previously been noted that if x is bandlimited to ( - π / T s , π / T s ) , the period of X s centered about the origin has the same form as X scaled in frequency since no aliasing occurs. This is illustrated in [link] . Hence, if any two ( - π / T s , π / T s ) bandlimited continuous time signals sampled to the same signal, they would have the same continuous time Fourier transform and thus be identical. Thus, for each discrete time signal there is a unique ( - π / T s , π / T s ) bandlimited continuous time signal that samples to the discrete time signal with sampling period T s . Therefore, this ( - π / T s , π / T s ) bandlimited signal can be found from the samples by inverting this bijection.

This is the essence of the sampling theorem. More formally, the sampling theorem states the following. If a signal x is bandlimited to ( - B , B ) , it is completely determined by its samples with sampling rate ω s = 2 B . That is to say, x can be reconstructed exactly from its samples x s with sampling rate ω s = 2 B . The angular frequency 2 B is often called the angular Nyquist rate. Equivalently, this can be stated in terms of the sampling period T s = 2 π / ω s . If a signal x is bandlimited to ( - B , B ) , it is completely determined by its samples with sampling period T s = π / B . That is to say, x can be reconstructed exactly from its samples x s with sampling period T s .

The spectrum of a bandlimited signals is shown as well as the spectra of its samples at rates above and below the Nyquist frequency. As is shown, no aliasing occurs above the Nyquist frequency, and the period of the samples spectrum centered about the origin has the same form as the spectrum of the original signal scaled in frequency. Below the Nyquist frequency, aliasing can occur and causes the spectrum to take a different than the original spectrum.

Questions & Answers

Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
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
what is the stm
Brian Reply
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?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
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
what is Nano technology ?
Bob Reply
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?
Damian Reply
what king of growth are you checking .?
Renato
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
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
hi
Loga
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
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Source:  OpenStax, Signals and systems. OpenStax CNX. Aug 14, 2014 Download for free at http://legacy.cnx.org/content/col10064/1.15
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