Iterative Error Correction: Turbo, Low-Density Parity-Check by Sarah J. Johnson

By Sarah J. Johnson

Iterative mistakes correction codes have stumbled on frequent software in mobile communications, electronic video broadcasting and instant LANs. This self-contained remedy of iterative blunders correction provides the entire key rules had to comprehend, layout, enforce and examine those robust codes. rapid, low-density parity-check, and repeat-accumulate codes are given equivalent, distinctive assurance, with specified shows of encoding and deciphering techniques. labored examples are built-in into the textual content to light up every one new inspiration and pseudo-code is incorporated for vital algorithms to facilitate the reader's improvement of the ideas defined. for every topic, the remedy starts with the easiest case ahead of generalizing. there's additionally insurance of complex subject matters comparable to density-evolution and go out charts for these readers drawn to gaining a deeper realizing of the sphere. this article is perfect for graduate scholars in electric engineering and computing device technological know-how departments, in addition to practitioners within the communications undefined.

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Extra resources for Iterative Error Correction: Turbo, Low-Density Parity-Check and Repeat-Accumulate Codes

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6 For a random variable Z with a continuous Gaussian distribution, the (differential) entropy of Z is H (Z ) = E[I ( p(z))] 1 2 2 e−z /2σ = −E log2 √ 2π σ 2 1 z2 = −E log2 √ + (log2 e) − 2 2σ 2π σ 2 = −log2 √ 1 2π σ 2 − (log2 e) −1 E(z 2 ) 2σ 2 1 1 log2 2π σ 2 + (log2 e) σ2 2 2 2σ 1 1 = log2 2π σ 2 + log2 e 2 2 1 = log2 2π eσ 2 . 2 = We can also define the joint entropy of two random discrete variables, X and Y : H (X, Y ) = p(x, y) log2 1 . p(x, y) p(x, y) log2 1 p(x|y) The conditional entropy H (X |Y ) = can be thought of as the amount of uncertainty about X that remains after Y is known.

Q. Then q H (X ) = j=1 1 p j log2 = pj q i=1 1 log2 q = log2 q. 585 bits per symbol. 5 bits per symbol. Thus the equiprobable random variable has a higher entropy. To obtain an intuitive feel for the above result we return to our contest but this time each competitor has a different random variable. The winner of the contest is still the first to correctly name both symbols, that is, the first to have complete information about their random variable. 2 Entropy, mutual information and capacity Suppose the first competitor has a binary random variable that is equiprobable.

Indeed, classical block codes will work well with iterative decoding algorithms if they can be represented by a sparse parity-check matrix. Generally, however, finding such a matrix for an existing code is not practical. Instead LDPC codes are designed by constructing a suitable sparse parity-check matrix first and then determining an encoder for the code afterwards. The biggest difference between LDPC codes and classical block codes is how they are decoded. Classical block codes are generally decoded with MLlike decoding algorithms and so are usually short and designed algebraically to make this task less complex.

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