# 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.

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is this allso about nanoscale material
Almas
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yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
no can't
Lohitha
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William
currently
William
where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
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da
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Bhagvanji
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Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
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ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
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Damian
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I think
Professor
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Damian
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LITNING
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Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
how did you get the value of 2000N.What calculations are needed to arrive at it
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