<< 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.


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.)


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

how can chip be made from sand
Eke Reply
are nano particles real
Missy Reply
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
Lale Reply
no can't
where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
has a lot of application modern world
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
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.
yes that's correct
I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
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
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Berger describes sociologists as concerned with
Mueller Reply
what is hormones?
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now

Source:  OpenStax, Fundamentals of signal processing. OpenStax CNX. Nov 26, 2012 Download for free at http://cnx.org/content/col10360/1.4
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Fundamentals of signal processing' conversation and receive update notifications?