<< Chapter < Page Chapter >> Page >

Proof of the sampling theorem

The above discussion has already shown the sampling theorem in an informal and intuitive way that could easily be refined into a formal proof. However, the original proof of the sampling theorem, which will be given here, provides the interesting observation that the samples of a signal with period T s provide Fourier series coefficients for the original signal spectrum on ( - π / T s , π / T s ) .

Let x be a ( - π / T s , π / T s ) bandlimited signal and x s be its samples with sampling period T s . We can represent x in terms of its spectrum X using the inverse continuous time Fourier transfrom and the fact that x is bandlimited. The result is

x ( t ) = 1 2 π - π / T s π / T s X ( ω ) e j ω t d ω

This representation of x may then be sampled with sampling period T s to produce

x s ( n ) = x s ( n T s ) = 1 2 π - π / T s π / T s X ( ω ) e j ω n T s d ω

Noticing that this indicates that x s ( n ) is the n th continuous time Fourier series coefficient for X ( ω ) on the interval ( - π / T s , π / T s ) , it is shown that the samples determine the original spectrum X ( ω ) and, by extension, the original signal itself.

Perfect reconstruction

Another way to show the sampling theorem is to derive the reconstruction formula that gives the original signal x ˜ = x from its samples x s with sampling period T s , provided x is bandlimited to ( - π / T s , π / T s ) . This is done in the module on perfect reconstruction. However, the result, known as the Whittaker-Shannon reconstruction formula, will be stated here. If the requisite conditions hold, then the perfect reconstruction is given by

x ( t ) = n = - x s ( n ) sinc ( t / T s - n )

where the sinc function is defined as

sinc ( t ) = sin ( π t ) π t .

From this, it is clear that the set

sinc ( t / T s - n ) | n Z

forms an orthogonal basis for the set of ( - π / T s , π / T s ) bandlimited signals, where the coefficients of a ( - π / T s , π / T s ) signal in this basis are its samples with sampling period T s .

Practical implications

Discrete time processing of continuous time signals

The Nyquist-Shannon Sampling Theorem and the Whittaker-Shannon Reconstruction formula enable discrete time processing of continuous time signals. Because any linear time invariant filter performs a multiplication in the frequency domain, the result of applying a linear time invariant filter to a bandlimited signal is an output signal with the same bandlimit. Since sampling a bandlimited continuous time signal above the Nyquist rate produces a discrete time signal with a spectrum of the same form as the original spectrum, a discrete time filter could modify the samples spectrum and perfectly reconstruct the output to produce the same result as a continuous time filter. This allows the use of digital computing power and flexibility to be leveraged in continuous time signal processing as well. This is more thouroughly described in the final module of this chapter.

Psychoacoustics

The properties of human physiology and psychology often inform design choices in technologies meant for interactin with people. For instance, digital devices dealing with sound use sampling rates related to the frequency range of human vocalizations and the frequency range of human auditory sensativity. Because most of the sounds in human speech concentrate most of their signal energy between 5 Hz and 4 kHz, most telephone systems discard frequencies above 4 kHz and sample at a rate of 8 kHz. Discarding the frequencies greater than or equal to 4 kHz through use of an anti-aliasing filter is important to avoid aliasing, which would negatively impact the quality of the output sound as is described in a later module. Similarly, human hearing is sensitive to frequencies between 20 Hz and 20 kHz. Therefore, sampling rates for general audio waveforms placed on CDs were chosen to be greater than 40 kHz, and all frequency content greater than or equal to some level is discarded. The particular value that was chosen, 44.1 kHz, was selected for other reasons, but the sampling theorem and the range of human hearing provided a lower bound for the range of choices.

Sampling theorem summary

The Nyquist-Shannon Sampling Theorem states that a signal bandlimited to ( - π / T s , π / T s ) can be reconstructed exactly from its samples with sampling period T s . The Whittaker-Shannon interpolation formula, which will be further described in the section on perfect reconstruction, provides the reconstruction of the unique ( - π / T s , π / T s ) bandlimited continuous time signal that samples to a given discrete time signal with sampling period T s . This enables discrete time processing of continuous time signals, which has many powerful applications.

Questions & Answers

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
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
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
Daniel
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
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
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
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
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.
Tarell
what is the actual application of fullerenes nowadays?
Damian
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.
Tarell
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.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
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.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
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
Good
Berger describes sociologists as concerned with
Mueller Reply
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, Signals and systems. OpenStax CNX. Aug 14, 2014 Download for free at http://legacy.cnx.org/content/col10064/1.15
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Signals and systems' conversation and receive update notifications?

Ask