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Claim 2 The probability Q ( T ( P x ) of the type class T ( P x ) obeys,

( n + 1 ) - ( r - 1 ) · 2 - n D ( P x Q x ) Q ( T ( P x ) ) 2 - n D ( P x Q x ) .

Consider now an event A that is a union over T ( P x ) . Suppose T ( Q ) A , then A is rare with respect to (w.r.t) the prior Q . and we have lim n Q ( A ) = 0 . That is, the probability is concentrated around Q . In general, the probability assigned by the prior Q to an event A satisfies

Q ( A ) = Σ x A Q ( x ) = Σ T ( P x ) A Q ( T ( P x ) ) = ˙ Σ T ( P x ) A 2 - n D ( P x Q ) = ˙ 2 - n · min p A D ( P Q ) ,

where we denote a n = ˙ b n when 1 n log ( a n b n ) 0 .

Fixed and variable length coding

Fixed to fixed length source coding : As before, we have a sequence x of length n , and each element of x is from the alphabet α . A source code maps the input x n r n to a set of 2 R n bit vectors, each of length R n . The rate R quantifies the number of output bits of the code per input element of x . We assume without loss of generality that R n Z . If not, then we can round R n up to R n , where · denotes rounding up. That is, the output of the code consists of n R bits. If n and R is fixed, then we call this a fixed to fixed length source code.

The decoder processes the n R bits and yields x ˆ α n . Ideally we have that x ˆ = x , but if 2 n R < r n then there are inputs that are not mapped to any output, and x ˆ may differ from x . Therefore, we want Pr ( x ˆ x ) to be small. If R is too small, then the error probability will go to 1. On the other hand, sufficiently large R will drive this error probability to 0 as n is increased.

If log ( r ) > R and Pr ( x ˆ x ) is vanishing as n is increased, then we are compressing, because 2 log ( r ) n = r n > 2 R n , where r n is the number of possible inputs x and there are 2 R n possible outputs.

What is a good fixed to fixed length source code? One option is to map 2 R n - 1 outputs to inputs with high probabilities, and the last output can be mapped to a “don't care" input.We will discuss the performance of this style of code.

An input x r n is called δ -typical if Q ( x ) > 2 - ( H + δ ) n . We denote the set of δ -typical inputs by T Q ( δ ) , this set includes the type classes whose empirical probabilities are equal (or closest) to the true prior Q ( x ) . Note that for each type class T x , all inputs x ' T x in the type class have the same probability, i.e., Q ( x ' ) = Q ( x ) . Therefore, the set T Q ( δ ) is a union of type classes, and can be thought of as an event A ( [link] ) that contains type classes consisting of high-probability sequences. It is easily seen that the event A contains the true i.i.d. distribution Q , because sequences whose empirical probabilities satisfy P x = Q also satisfy

Q ( x ) = 2 - H n > 2 - ( H + δ ) n .

Using the principles discussed in [link] , it is readily seen that the probability under the prior Q of the inputs in T Q ( δ ) satisfies Q ( T p ( δ ) ) = Q ( A ) 1 when n . Therefore, a code C that enumerates T Q ( δ ) will encode x correctly with high probability.

The key question is the size of C , or the cardinality of T Q ( δ ) . Because each x T Q ( δ ) satisfies Q ( x ) > 2 ( - H + δ ) n , and x T Q ( δ ) Q ( x ) 1 , we have | T Q ( δ ) | < 2 ( H + δ ) n . Therefore, a rate R H + δ allows near-lossless coding , because the probability of error vanishes(recall that Q ( ( T p ( δ ) ) C ) 0 , where ( · ) C denotes the complement).

On the other hand, a rate R H - δ will not allow lossless coding, and the probability of error will go to 1. We will see this intuitively. Because the type class whose empirical probability is Q dominates, a type class T x whose sequences have larger probability, e.g., Q ( x ) > 2 - ( H - δ ) n , will have small probability in aggregate. That is,

x : Q ( x ) > 2 - n ( H - δ ) Q ( x ) n 0 .

In words, choosing a code C with rate R = H - δ that contains the words x with highest probability will fail, it will not cover enough probabilistic mass.We conclude that near-lossless coding is possible at rates above H and impossible below H.

To see things from a more intuitive angle, consider the definition of entropy, H ( Q ) = - a α Q ( a ) log ( Q ( a ) ) . If we consider each bit as reducing uncertainty by a factor of 2,then the average log-likelihood of a length- n input x generated by Q satisfies

E [ - log ( Pr ( x ) ) ] = E [ - log ( i = 1 n P r ( x i ) ) ] = - i = 1 n E [ log ( Q ( x i ) ) ] = - i = 1 n a α Q ( a ) · log ( Q ( a ) ) = n H .

Because the expected log-likelihood of x is n H , it will take n H bits to reduce the uncertainty by this factor.

Fixed to variable length source coding : The near-lossless coding above relies on enumerating a collection of high-probability codewords T Q ( δ ) . However, this approach suffers from a troubling failure for x T Q ( δ ) . To solve this problem, we incorporate a code that maps x to an output consisting of a variable number of bits. That is, the length of the code will be approximately n H on average, but could be greater or lesser.

One possible variable length code is due to Shannon. Consider all possible x α n . For each x , allocate - log ( Q ( x ) ) bits to x . It can be shown that it is possible to construct an invertible (uniquely decodable)code as long as the length of the code l ( x ) in bits allocated to each x satisfies

x 2 - l ( x ) 1 .

This result is known as the Kraft Inequality. Seeing that

x 2 - l ( x ) = x 2 - - log ( Q ( x ) ) x 2 - ( - log ( Q ( x ) ) ) = x Q ( x ) = 1 ,

we see that the length allocation we suggested satisfies the Kraft Inequality. Therefore, it is possible to construct an invertible (and hence lossless) codewith lengths upper bounded by

l x = - log ( Q ( x ) ) - log ( Q ( x ) ) + 1 ,

and we have

E [ l ( x ) ] E [ - log ( Q ( x ) ) ] + 1 = n H + 1 .

This simple construction approaches the entropy up to 1 bit.

Unfortunately, a Shannon code is impractical, because it requires to construct a code book of exponential size | α | n . Instead, arithmetic codes  [link] are used; we discussed arithmetic codes in detail in class, but they appear in all standard text books and so we do not describe them here.

Questions & Answers

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
sciencedirect big data base
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.
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
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
characteristics of micro business
for teaching engĺish at school how nano technology help us
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
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
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.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Universal algorithms in signal processing and communications. OpenStax CNX. May 16, 2013 Download for free at http://cnx.org/content/col11524/1.1
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