# 1.1 Optimal encoding

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We shall consider now the encoding of signals on $\left[-T,T\right]$ 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 $\left(E,D\right)$ $E$ 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.

Define the distortion $d$ for this encoder-decoder by

$d\left(K,E,D,X\right):=\text{sup}{\phantom{\rule{4.pt}{0ex}}}_{f\in K}\parallel f-D\left(Ef\right){\parallel }_{\overline{\underline{X}}}.$
Let $n\left(K,E\right)=\text{sup}{\phantom{\rule{4.pt}{0ex}}}_{f\in K}#Ef$ where $#Ef$ is the number of bits in the bitstream $Ef$ . Thus $n$ is the maximum length of the bitstreams for the various $f\in K$ . 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 ${n}_{ϵ}\left(K,X\right):=\text{inf}{\phantom{\rule{4.pt}{0ex}}}_{\left(E,D\right)}\left\{n\left(K,E\right):d\left(K,E,D,X\right)\le ϵ\right\}$ . This is the smallest bit budget under which we could encode all elements of $K$ to distortion $ϵ$ .
• Prescribe $N$ : find the smallest distortion $d\left(K,E,D,X\right)$ over all $E,D$ with $n\left(K,E\right)\le N$ . 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.

Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
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?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
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?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
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
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
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?
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 ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
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
how did you get the value of 2000N.What calculations are needed to arrive at it
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