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Forward error correction methods are fundamental in the transmitting and storing of information. They allow increasing the size of the message using so-called redundancy to be able to repair errors of assumed statistics, for instance that each bit can be changed with probability . In the standard approach we encode the message in independent blocks.[more]
This Demonstration simulates the creation of pseudorandom trees while using a new approach to error correction, in which we connect redundancies stored in blocks. Intuitively, this connection allows the transfer of surpluses of redundancy to cope with local error concentrations. Nodes of the tree correspond to corrections of the message up to a given point—they create a tree because each bit can be corrected or not.
For a given probability that bits are damaged ), we choose the parameter that corresponds to the amount of redundancy we add—the larger this parameter, the more we increase the size of the message, the easier it is to correct. The role of this parameter is that when we are not on the branch corresponding to the proper correction, at each step we have probability of spotting that, and so we can try to search somewhere else. At each step of the correction process we know the current tree and we use it to find a leaf that looks to be most probably correct and try to expand it.
The controls allow you to choose , , the lengths of bit blocks used in the process, and the number of steps to make. Choosing "rescale " makes that the slider for works in the most interesting range: . Using the "correction" or "errors" buttons, you can generate new corrections for the same error distribution or some random new one.
There is a variety of information at the top-right corner: the boundaries of this range, how much time the size of the message increases (theoretically it would be enough to increase to repair all damage (Shannon's limit)), and how many bits we would have in fact decoded or as an average for one step.
The simulator presents the correction process (tree) in three ways: how the local width of the tree behaves, the weights used to choose the most probable node in succeeding steps, or the tree itself.[less]
This new correction mechanism (that when trying a wrong correction at each step, the probability of realising it is ), can be constructed using an entropy coder like an asymmetric numeral system. The correction mechanism can be implemented by inserting an additional forbidden symbol with probability and rescaling the probabilities of the original symbols. Now after an error the decoder will produce a random symbol sequence—at each step the decoder has probability of spotting that something was wrong. This approach does not work for all block lengths—for very large noise, methods from fig. 3 of the paper below can be used.
If before adding redundancy, each step was encoded with an average bits, adding this forbidden symbol makes the average bits: the size of the message grows by times.
To expand the tree, we have to choose the most probable node. To do this we calculate for each node some weight and at each step we choose the leaf with the largest weight.
The weight of the given correction (using Bayesian analysis) is , where is the number of changed bits, is the number of unchanged bits, and is the number of steps. That means the longer the correction and the less bits it changes, the better.
We assume symmetric memoryless statistics of errors. That means each bit has probability of being changed. The control slider for this parameter is logarithmic.
The value corresponds to the theoretical (Shannon's) limit. Below this limit it is impossible to correct, given the statistics of errors. In the approach presented here, below this value the weight of the proper correction no longer grows statistically and so the tree would immediately grow exponentially. The approach presented here should work well asymptotically above . In the range the tree has generally small width, but rare large error concentrations implies that the expected width will be asymptotically infinite.
In one step we process one block of bits—their lengths are generally chosen randomly from some set. Each split of the tree corresponds to some correction of the currently processed block. We assume that the block lengths are chosen equally probable. The average number of bits processed in one step ) is the average length of a block.
To be able to compare different parameters, the distribution of errors is changed only if is changed or the "errors" button is pressed.
In "widths" mode, the positions of the errors are marked with yellow spikes. There are two lines: the blue line shows the width of the whole tree. The red vertical line shows the last position of proper correction. This red line shows the width assuming that it was the end of the correction. The difference between these lines represents subtrees of wrong corrections created while trying to correct the last error. We see that literally redundancy is transferred to help with local error concentrations.
In "weight" mode we see the best weights chosen in succeeding steps. The red dots with filling are proper corrections and the blue ones correspond to steps of wrong correction. We see that the weight generally grows, but locally can dramatically drop what require a very large number of steps/nodes to repair.
In "tree" mode yellow lines represent errors and the red path is the proper correction. Proceeding step by step we can see how the algorithm tries to find the proper branch.
More details can be found in the section of Asymmetric numeral systems.
Wolfram Demonstrations Project
Published: March 7 2011