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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

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Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
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Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
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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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