<< Chapter < Page Chapter >> Page >
Proof of Shannon's sampling theorem

In order to recover the signal x t from it's samples exactly, it is necessary to sample x t at a rate greater than twice it's highest frequency component.

Introduction

As mentioned earlier , sampling is the necessary fundament when we want to apply digital signalprocessing on analog signals.

Here we present the proof of the sampling theorem. The proof is divided in two. First we find an expression for the spectrum of the signal resulting fromsampling the original signal x t . Next we show that the signal x t can be recovered from the samples. Often it is easier using the frequency domain when carrying out a proof,and this is also the case here.

    Key points in the proof

  • We find an equation for the spectrum of the sampled signal
  • We find a simple method to reconstruct the original signal
  • The sampled signal has a periodic spectrum...
  • ...and the period is 2 π F s

Proof part 1 - spectral considerations

By sampling x t every T s second we obtain x s n . The inverse fourier transform of this time discrete signal is

x s n 1 2 π ω π X s ω ω n
For convenience we express the equation in terms of the real angular frequency Ω using ω Ω T s .We then obtain
x s n T s 2 Ω π T s π T s X s Ω T s Ω T s n
The inverse fourier transform of a continuous signal is
x t 1 2 Ω X Ω Ω t
From this equation we find an expression for x n T s
x n T s 1 2 Ω X Ω Ω n T s
To account for the difference in region of integration we split the integration in into subintervals of length 2 π T s and then take the sum over the resulting integrals to obtain the complete area.
x n T s 1 2 π k Ω 2 k 1 T s 2 k 1 T s X Ω Ω n T s
Then we change the integration variable, setting Ω η 2 π k T s
x n T s 1 2 π k η T s π T s X η 2 π k T s η 2 π k T s n T s
We obtain the final form by observing that 2 π k n 1 , reinserting η Ω and multiplying by T s T s
x n T s T s 2 π Ω π T s π T s k 1 T s X Ω 2 π k T s Ω n T s
To make x s n x n T s for all values of n , the integrands in and have to agreee, that is
X s Ω T s 1 T s k X Ω 2 k T s
This is a central result. We see that the digital spectrum consists of a sum of shifted versions of the original, analog spectrum. Observe the periodicity!

We can also express this relation in terms of the digital angular frequency ω Ω T s

X s ω 1 T s k X ω 2 π k T s
This concludes the first part of the proof. Now we want to find a reconstruction formula, so that we can recover x t from x s n .

Proof part ii - signal reconstruction

For a bandlimited signal the inverse fourier transform is

x t 1 2 Ω T s T s X Ω Ω t
In the interval we are integrating we have: X s Ω T s X Ω T s . Substituting this relation into we get
x t T s 2 Ω T s T s X s Ω T s Ω t
Using the DTFT relation for X s Ω T s we have
x t T s 2 Ω T s T s n x s n Ω n T s Ω t
Interchanging integration and summation (under the assumption of convergence) leads to
x t T s 2 n x s n Ω T s T s Ω t n T s
Finally we perform the integration and arrive at the important reconstruction formula
x t n x s n T s t n T s T s t n T s
(Thanks to R.Loos for pointing out an error in the proof.)

Summary

X s Ω T s 1 T s k X Ω 2 k T s

x t n x s n T s t n T s T s t n T s

Go to

  • Introduction
  • Illustrations
  • Matlab Example
  • Hold operation
  • Aliasing applet
  • System view
  • Exercises
?

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
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, Information and signal theory. OpenStax CNX. Aug 03, 2006 Download for free at http://legacy.cnx.org/content/col10211/1.19
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Information and signal theory' conversation and receive update notifications?

Ask