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We shall consider now the encoding of signals on [ - T , T ] where T > 0 is fixed. Ultimately we shall be interested in encoding classes of bandlimited signals like the class B A However, we begin the story by considering the more general setting of encoding the elements of any given compact subset K of a normed linear space X . One can determine the best encoding of K by what is known as the Kolmogorov entropy of K in X .

To begin, let us consider an encoder-decoder pair ( E , D ) E maps K to a finite stream of bits. D maps a stream of bits to a signal in X . This is illustrated in . Note that many functions can be mapped onto the same bitstream.

Illustration of encoding and decoding.

Define the distortion d for this encoder-decoder by

d ( K , E , D , X ) : = sup f K f - D ( E f ) X ̲ ¯ .
Let n ( K , E ) = sup f K # E f where # E f is the number of bits in the bitstream E f . Thus n is the maximum length of the bitstreams for the various f K . There are two ways we can define optimal encoding:

  • Prescribe ϵ , the maximum distortion that we are willing to tolerate. For this ϵ , find the smallest n ϵ ( K , X ) : = inf ( E , D ) { n ( K , E ) : d ( K , E , D , X ) ϵ } . This is the smallest bit budget under which we could encode all elements of K to distortion ϵ .
  • Prescribe N : find the smallest distortion d ( K , E , D , X ) over all E , D with n ( K , E ) N . This is the best encoding performance possible with a prescribed bit budget.

There is a simple mathematical solution to these two encoding problems based on the notion of Kolmogorov Entropy.

Questions & Answers

How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
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
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
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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.
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Damian Reply
absolutely yes
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Akash Reply
it is a goid question and i want to know the answer as well
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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
how can I make nanorobot?
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
While the American heart association suggests that meditation might be used in conjunction with more traditional treatments as a way to manage hypertension
Beverly Reply
in a comparison of the stages of meiosis to the stage of mitosis, which stages are unique to meiosis and which stages have the same event in botg meiosis and mitosis
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Source:  OpenStax, Compressive sensing. OpenStax CNX. Sep 21, 2007 Download for free at http://cnx.org/content/col10458/1.1
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