Noiseless coding theorem in .NET Integrated QR Code in .NET Noiseless coding theorem

3.3 Noiseless coding theorem use visual .net qrcode writer toproduce quick response code on .net QR Code This theorem i Denso QR Bar Code for .NET s one of Shannon"s basic theOretical results from [Shannon 1948] showing that the entropy of a source gives a fundamental limitation on how efficiently the information from that source can be transmitted. The immediate issue is minimizing the average encoded word length.

That is, minimizing the expected value of the length of encoded words, depending upon the probability distribution of the words emitted by the source. Let! : W -+ r;* be a code with m source words WI, ..

. , Wm in W, with encoded words ! (WI), ..

. , ! (w m ) of lengths iI, . , i m .

Let PI, ...

, Pm be the probabilities that the respective words WI, ...

, Wm are emitted by the source. Then the average length of an encoded word is average length ! =. Pi ii Note that this VS .NET QR is the expected value of the random variable which returns the length of the codewords. Example: Let the source words be cat with probability 1/4, dog with probability 1/8, elephant with probability 1/8, and zebra with probability 1/2.

Let the code alphabet be r; = {O, I}, and let the encoding! be !("cat") = "011" !("dog") = "01" !("elephant") = "0" !("zebra") = "111" Then the average length of an encoded word is, by definition, average length. = P("cat") le ngth(f("cat")) + P("dog") length(f("dog")). + P("elephant" qr barcode for .NET ) length(f("elepbant")) + P("zebra") length(f("zebra")) = P("cat") . length("Ol1") + P("dog") .

length("l1") + P("elephant") length("O") + P("zebra") length("l11"). 3 . Noiseless Coding = P("cat") 3+ P("dog") . 2 + P("elephant") . 1 + P("zebra") 3.

4 . 3 + 8" . 2 qr bidimensional barcode for .

NET + 8" . 1 + 2 . 3 = 8" = 2.

625. That is, the a verage codeword length with this encoding is 2.625. Note that the lengths of the source words play no role in this computation.

Let lEI denote the number of elements in a finite set E (such as an alphabet of symbols). Theorem: For a memoryless source X with entropy H(X), a uniquely decipherable code f : W ~ E* into strings made from an alphabet E (with lEI> 1) must have average length satisfying. Further, there exists a code f with Remark: This theorem describes the best achievable performance, measured in terms of avera qrcode for .NET ge word length, of any encoding of a given "vocabulary" W of source words. The adjective noiseless refers to the fact that we are still ignoring errors.

. ~ E* be a uniq uely decipherable code with m source words in W, with encoded words !(WI),."" !(wm ) of lengths fIt ..

", fm. Let PI, ..

. ,Pm be the probabilities that the respective words WI, ..

. ,Wm are emitted by the source. By the Kraft-McMillan inequality, letting n be the cardinality of the alphabet E.

WI, ... , Wm Proof: Let f :" W Define = n- li /. L ni Since (by cons truction) the sum of the qiS is 1, and since they are non-negative, the collection of numbers qI, ...

,qm fits the hypotheses of the Fundamental Inequality above, and we conclude that. - LPi log2 Pi :::; - LPi log2 qi By its definition 3.3 Noiseless coding theorem Therefore, substituting the right hand side for log2 qi gives By the Kraft-M cMillan inequality,. Ei n- li :5 1, so log2 Ei n- li :5 0, and thus - LPi log2 Pi :5log2 n LPili That is, the e ntropy of the source is less than or equal to the average length of the encoded words t~mes logz of the size of the alphabet. This proves the lower bound for the average word length. For the other half of the theorem, we will try to cleverly choose the word lengths according to the rule that li is the smallest integer such that.

Of course, it qrcode for .NET is not immediately clear that this is possible, but the fact that the probabilities Pi add up to 1 gives. so by the Kraf t-McMillan theorems there exists a uniquely decipherable code with these encoded word lengths. Taking logarithms base 2, and using the fact that the logarithm function is increasing, the condition.
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