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1.0 0.8 0.6 0.4 0.2 0.0 0.6481 0.2 0.4 0.6 0.8 h hBP generate, create qr codes none with .net projects iReport Figure . : EXI qr codes for .NET T chart for transmission over the BEC(h metric (big-numerator) parallel concatenated ensemble.

. 0.6481) for an asym-. Input this per .net vs 2010 Denso QR Bar Code muted sequence of length 2(n + m) into the convolutional encoder. Because we have m zeros at the end of xs as well as at the end of (xs ) the output of the convolutional encoder is simply the concatenation of the output of the standard parallel concatenated code with two component codes, i.

e., x p = (x p1 , x p2 ). We therefore see that this new encoder structure is simply an alternative way of.

xs repeat G(D) xp Figure . : Alt QR Code 2d barcode for .NET ernative view of an encoder for a standard parallel concatenated code.

accomplishing the encoding. Assume next that we li the restriction on the permutation on 2(n + m) letters (except that the last m positions should be xed). e corresponding FSFG is shown in Figure .

. A moment s thought shows that the density evolution analysis for this new (larger) ensemble of codes is identical to the one of the corresponding parallel concatenated code, i.e.

, the thresholds are identical. In this FSFG every systematic bit has degree exactly 2, i.e.

, it appears exactly twice at the input to the encoder. is corresponds to the fact that the code is composed of two concatenated component codes. It is now straightforward to generalize this picture by allowing a varying degree of repetition.

is is shown in Figure . . As always, let L(x) denote the (normalized) degree distribution from a node perspective.

If we assume that we puncture a fraction p of all parity bits we see that the design rate ful lls the relation 1 = 1 + pL (1). r. Figure . : Alternative view of the FSFG of a standard parallel concatenated code. degree 2 degree 3 degree dlmax Figure . : FSFG of an irregular parallel concatenated turbo code. E . (I T E ). qr bidimensional barcode for .

NET Consider the turbo code ensemble with a degree distribution from an edge perspective equal to (x) = 55 100x + 45 100x9 . is correspond to a node degree distribution of L(x) = 55 64x2 + 9 64x10 . Further assume that the puncturing is random and that we puncture parity bits at a rate p = 68 100.

From the previous formula we can then compute the rate 68 of the code to be equal to 1 = 1 + (1 100 ) 25 , which yields r = 1 2. To complete the r 8 speci cation of the code we choose G = 11 13. A density evolution analysis for the case that transmission takes place over the BEC( ) shows that BP 0.

4825, which is close to the Shannon threshold of onehalf. Further, we can compute the upper bound on the threshold MAP 0.495.

. Our approach t o convolutional codes presented in Section . is non-standard. Usually, a convolutional code is de ned as the set of all output streams resulting from the set of all input streams, where these streams are taken to be either semi-in nite,.

starting at ti VS .NET QR Code 2d barcode me zero, or bi-in nite. e classical reference is the work by Forney [ , , , ].

We also recommend the article by McEliece [ ]. In-depth references are the book by Johanneson and Zigangirov [ ] as well as the book by Piret [ ]. Considering in nite sequences is convenient for the purpose of analysis, but it is also meaningful since the length of a convolutional code has little impact on its performance.

For applications in iterative decoding systems, to the contrary, one usually considers terminated convolutional codes. e termination rule which we introduced on page (removing the feedback for the last m steps) leads to a simple analysis but is not the best in terms of the resulting error oor. A popular solution proposed by Divsalar and Pollara [ ] is the following: for the last m steps the input is chosen in such a way that the rst encoder terminates.

is can be accomplished by choosing the input equal to the feedback signal. is choice e ectively cancels the feedback and pushes zeros into the shi register. A er m steps the register is cleared and we are back to the all-zero state.

is method has the same e ect on the parity bits as the method we presented but the systematic bits are in general not zero; hence the weight of the codeword is increased. Problem . discusses how the weight distribution can be calculated for this termination method.

ere is a large number of alternatives. ey di er in which of the component encoders is terminated, how the termination is achieved, and whether or not tail bits are transmitted: Joerssen and Meyr [ ]; Barbulescu and Pietrobon [ ]; Blackert, Hall, and Wilson [ ]; Reed and Pietrobon [ ]; Hattori, Murayama, and McEliece [ ]; Khandani [ ]; van Dijk, Egner, and Motwani [ ]; Le Dantec and Piret [ ]; Tanner [ ]; McEliece, Le Dantec, and Piret [ ]; Huang, Vucetic, Feng, and Tan [ ]; Le Bars, Le Dantec, and Piret [ ]; and Anderson and Hladik [ ]. A summary and comparison of many of the proposed termination schemes was written by Hokfelt, Edfors, and Maseng [ ].

It is worth pointing out that in classical coding the code itself plays the central role, whereas for iterative systems more prominence is due to the encoder. Turbo codes were introduced together with the corresponding turbo decoding algorithm by Berrou, Glavieux, and itimajshima [ ]. is paper set o a revolution in coding and, more generally, communications and led to the rediscovery of Gallager s thesis.

e importance of the introduction of turbo codes on the development of coding cannot be overstated. Very soon a er the introduction of their original scheme for binary transmission, Le Go , Glavieux, and Berrou showed [ ] that the same principle can be applied for non-binary modulation and it quickly became clear that the turbo principle, as Hagenauer termed it in [ ], had wide applicability. Wiberg introduced the notion of a support tree, which is our computation tree [ ].

An early paper that recognized the role of the computation tree is by Gelblum, Calderbank, and Boutros [ ]. e method of analysis (concentration around en-. semble average and asymptotic analysis via density evolution) presented in Section . is due to Richardson and Urbanke [ ]. An alternative route for the analysis is the geometric interpretation of the turbo decoding algorithm introduced by Richardson [ ] and followed up by Agrawal and Vardy [ ] and Kocarev, Lehmann, Maggio, Scanavino, Tasev, and Vardy [ ].

Turbo codes can also be analyzed from a statistical mechanics point of view. is was accomplished by Montanari and Sourlas [ ]. e basis for this analysis is the observation by Sourlas that codes can be phrased in the language of spin-glass systems [ ].

e weight distribution of turbo codes has been studied by a large set of authors. Probably one of the most important steps was the realization by Benedetto and Montorsi that, although the weight distribution of individual codes is hard to determine, the average weight distribution of the ensemble is relatively easy to compute [ , ]. is was the beginning of the ensemble average analysis.

With few exceptions, most analytic results for iterative decodes are due to this ensemble point of view. Much of what we discuss in Section . has been the topic of investigation in [ ].

Similar concepts were discussed around the same time by Perez, Seghers, and Costello [ ]. In particular, [ ] as well as [ ] both contain the error oor expressions for the parallel case. e average value analysis was re ned by Baligh and Khandani [ ] as well as Richardson and Urbanke [ ] to include the distribution of low-weight codewords.

e limiting Poisson distribution of the number of minimal codewords stated in Lemma . is similar in spirit to the distribution of cycles in the turbo graph which was studied by Ge, Eppstein, and Smyth [ ]. In the derivation of the average weight distribution of convolutional codes in Section .

. we follow the description of McEliece [ ] and employ the transfer matrix method. It was shown by Sason, Telatar, and Urbanke [ , ] how to e ciently compute the growth rate of the weight distribution.

In this respect it is interesting to note that Bender, Richmond, and Williamson showed [ ] that one can also derive central and local limit theorems for the growth of the components of the power of a matrix. is allows one in principle to apply Hayman-like techniques to the problem of the weight distribution of turbo codes in a similar manner as we discuss in Appendix D for the weight distribution of LDPC ensembles. Much material on the weight distribution problem (including Problem .

) can be found in the Ph.D. thesis by P ster [ ].

e average weight distribution was used by a large number of authors to derive upper bounds on the performance of maximum-likelihood decoders. Since the corresponding list of references is considerable we cite Sason and Shamai [ ], where the reader can nd an extensive literature review. e minimum distance was investigated by Kahale and Urbanke in a probabilistic setting [ ].

In particular, it was shown that the minimum distance of typical parallel concatenated turbo codes with two component codes grows at most logarithmically. e rst worst case upper bound was the O( n) bound proposed by Breil-. ing and Huber VS .NET qr-codes [ ]. It has since been improved to O(n1 3 ) and, nally, to O(log n) [ , ].

is is the result we present in Section . . .

A similar logarithmic bound was also shown by Perotti and Benedetto [ ]. e material in Section . .

follows closely the results of Breiling and Huber. Conversely, Truhachev, Lentmaier, Wintzell, and Zigangirov give [ ] an explicit construction of a permutation that leads to a minimum distance that grows like (log n) (see also the work of Boutros and Z mor [ ]). is is quite pleasing since the combination of the upper bound on the lower bound shows that the optimum growth of the minimum distance as a function of the blocklength is (log n).

A very general and fundamental bound which relates the complexity of the encoding process with the resulting minimum distance is due to Bazzi and Mitter [ ]. Algorithms to compute the low-weight terms of the weight distribution for a speci c turbo code were discussed by Perez, Seghers, and Costello [ ]; Breiling and Huber [ ]; Berrou and Vaton [ ]; Garello, Pierleoni, and Benedetto [ ]; Garello and Casado [ ]; Crozier, Guinand, and Hunt [ , ]; and Rosnes and Ytrehus [ ]. e stability of speci c turbo code ensembles was rst studied by Montanari and Sourlas [ ].

e general stability condition as stated in Conjecture . is due to Richardson and Urbanke [ ]. Lentmaier, Truhachev, Zigangirov, and Costello state [ ] a su cient condition for a turbo code that the bit error probability converges to zero by tracking the evolution of the Bhattacharyya parameter.

An important practical question which we have not discussed is the choice of the permutation(s) . is is typically referred to as the interleaver design problem. It is probably fair to say that there are more papers written on this issue than on any other topic in iterative decoding.

We will not even attempt to summarize the state of a airs, but simply give a number of pointers to the literature. e aim of most interleaver designs is to construct permutations with a large minimum distance. Under MAP decoding such an interleaver guarantees a low error oor.

A popular class of interleavers are the so-called S-random interleavers which require that for the permutation the sum i j + (i) ( j) is lower bounded for all distinct pairs (i, j) (there are several variants of this de nition in the literature). e intuition for this restriction is simple. Consider a convolutional encoder and inputs of weight 2.

If the two non-zero positions are in a general position then the encoder produces an output weight which is proportional to the length of the encoded sequence and we do not need to worry. However, if we place the inputs a multiple of the period of the encoder apart, then the corresponding output has a weight which is proportional to the length i j. e same argument is true for the permuted tuple ( (i), ( j)).

e lower bound on i j + (i) ( j) therefore translates directly into a lower bound on the weight due to inputs of weight 2. As we have seen in our discussion on the weight distribution, for a random permutation the number of low-weight codewords is asymptotically determined by such input pairs. of weight 2. I t is therefore intuitive that the preceding restriction should lead to an increased (as compared to random interleavers) minimum distance. e condition on the permutation does not depend on the component codes (in particular the period of the component codes).

S-random permuters are therefore universal. On the other hand, they are not optimized with respect to the particular components that are used. S-random permuters were introduced by Divsalar and Pollara [ ], with follow-up work by Fragouli and Wesel [ ], Crozier [ ], Dinoi and Benedetto [ ], Breiling, Peeters, and Huber [ ], and Sadjadpour, Sloane, Nebe, and Salehi [ ].

Dithered relative prime interleavers were introduced by Crozier and Guinand [ ]. Quadratic interleavers were proposed by Sun and Takeshita [ ]. Alternatively, one can directly optimize the minimum distance for a given component code.

As samples of this approach we refer to the papers by Yuan, Vucetic, and Feng [ ]; Daneshgaran and Mondin [ ]; Le Ruyet, Sun, and ien [ ]; Abbasfar and Yao [ ]; and Ould-Cheikh-Mouhamedou, Crozier, and Kabal [ ]. Explicit constructions were given by Takeshita and Costello [ ] and Le Bars, Le Dantec, and Piret [ ]. A word of warning: Under iterative decoding the error oor may not be dominated by the contributions due to codewords (i.

e., pseudo codewords might dominate). As discussed in the text, there are uncountable many avors of the basic theme.

Serially concatenated codes were introduced by Benedetto, Divsalar, Montorsi, and Pollara [ ]. Asymmetric turbo codes were suggested by Takeshita, Collins, Massey, and Costello [ ]. Big numerator codes were introduced by Massey, Takeshita, and Costello [ ] and independently discovered by Huettinger, ten Brink, and Huber [ ].

It was suggested by Hokfelt and Maseng [ ] to optimize the energy which is assigned to the various streams (systematic and parity) of turbo codes. e same idea was investigated also by Duman and Salehi [ ]. Irregular turbo codes were rst suggested by Frey and MacKay [ ] and optimized versions were presented by Boutros, Caire, Viterbo, Sawaya, and Vialle [ ].

Examples . and . are due to M asson and appeared in [ , ].

Implementation issues concerning turbo codes are discussed by Pietrobon [ ].. . (M G C E ). VS .

NET qrcode Consider the convolutional encoder in Figure . . is convolutional encoder is a rate one-half non-recursive encoder that has m = 2.

For any memory m, let i denote the state vector at time i and p,q p q let xi = (xi , xi ) be the length 2 output vector at time i. Write down the general form of the state equations ( . ) and specify the corresponding matrices in terms of the lter coe cients.

Draw one labeled trellis section..
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