1), pY in .NET Paint QR-Code in .NET 1), pY

1), pY using barcode printing for .net control to generate, create qr code image in .net applications. CBC 1), pY (pY (1), pY ( 1 )) = ( , 1 ), . pX(1)= . X ( 1. X ( 1. 1)) = ( , ), . 1)) = (0, 1),. so that the inf ormation rate I (X; Y) = I(X; Y) ( . ). for a xed equals I (X; Y) = H(Y ) H(Y X) = h( ) h( ). . Some calculus r QR Code ISO/IEC18004 for .NET eveals (see Problem . ) that the optimal choice for is ( .

) so that ( ) =. , 1 + CZC( ) = h( qr bidimensional barcode for .NET ( ) ) ( )h( ) = log(1 + ), . where the last step requires several lines of calculus. Figure . compares CZC( ) with I = 1 (X; Y).

is is the highest rate that can be achieved with a uniform input 2 distribution. Only little is lost by insisting on the uniform input distribution. As discussed in more detail in Problem .

, the rate which is achievable by using a uniform input distribution is at least a fraction 1 e ln(2) 0.942 of capacity over the entire 2 range of (with equality when approaches 1): from this perspective the Z channel is the extremal case the information rate of any binary-input memoryless channel when the input distribution is the uniform one is at least a fraction 1 e ln(2) of its 2. 0.8 0.6 0.

4 0.2 QR Code for .NET 0.

0 0.2 0.4 0.

6 0.8 . Figure . : Comp arison of CZC( ) (solid curve) with I = 1 (X; Y) (dashed curve), 2 both measured in bits. capacity.

From the preceding discussion we conclude that, when dealing with asymmetric channels, not much is lost if we use a binary linear coding scheme (inducing a uniform input distribution). Consider the density evolution analysis. It seems at rst that we have to analyze the behavior of the decoder with respect to each codeword.

Fortunately this is not necessary. First note that, because we consider an ensemble average, only the "type" of the codeword matters. More precisely, let us say that a codeword has type if the fraction of zeros and ones is and , respectively.

For x C, let (x) be its type. Let us assume that we use a low-density parity-check (LDPC) ensemble whose dominant type is one-half. is means that typical codewords contain roughly as many zeros as ones.

Although it is possible to construct degree distributions which violate this constraint, most degree distributions that we encounter do ful ll it (see proof of Lemma . ). Under this assumption there exists some strictly positive constant such that ( .

) P (X) [1 2 n, 1 2 + n] e .. We can therefor QR Code for .NET e analyze the performance of such a system in the following way: determine the error probability assuming that the type of the transmitted codeword is close to the typical one. Since sublinear changes in the type do not gure in the density analysis, this task can be accomplished by a straightforward density evolution analysis.

Now add to this the probability that the type of a random codeword deviates signi cantly from the typical one. e second term can be made arbitrarily small (see right-hand side of ( . )) by choosing su ciently large.

Consider therefore the density evolution with respect to the typical type. is means that half the nodes have initial density a+ ( ) (y) and the remaining nodes ZC have initial density a ( ) (y). Proceed with a density evolution analysis which has ZC two types of messages (namely those that are connected to a variable node with transmitted value +1 and those that are connected to a variable node with transmitted value 1).

Fortunately we can do even better. We can factor out the sign. of the received message. More precisely, assume that for all nodes with associated value 1, call them minus nodes, we ip the sign of the received message. By using the symmetry of the processing rules as discussed in Section .

. , one can check that the signs of all those messages which enter or exit minus nodes are ipped as well (with respect to the identical decoder which is fed with the original input) but that their magnitude is identical. Further, for this modi ed decoder the message densities owing into the variable nodes are the same regardless of the sign of the variable node.

In short, density evolution for an asymmetric channel with respect to the typical type is equivalent to density evolution with respect to the symmetrized channel (a+ ( ) (y) + a ( ) ( y)) 2. ZC ZC at this density is indeed symmetric is quickly checked by direct computations. More generally, as discussed in Problem .

, this is the case for any binary-input memoryless channel. From this observation the derivation of the stability condition as well as the methods of optimization follow in a straightforward fashion. It is the goal of Problem .

to show that (under the uniform input distribution) the Bhattacharyya constant associated with this channel is B(aZC( ) ) = , so that the stability condition for this channel reads (0) (1) < 1 . As a nal remark: if it is crucial to approach capacity very closely, so that a uniform input distribution is not su cient, one can combine the linear code with a nonlinear mapper in order to induce a non-uniform input distribution..

. .C.

Consider an ins tance in which the factor graph methodology can help in answering an information-theoretic question. We want to compute the information rate (maximal rate at which information can be transmitted reliably for a given input distribution) of a channel with memory. Assuming that the memory has a Markov structure, this problem can be solved in a computationally e cient manner using the factor graph framework.

is is of interest in itself, but it also forms the starting point in our investigation of low-complexity coding schemes for channels with memory. More precisely, assume we are interested in computing the information rate.
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