# Dr. Wilhelm, Claus: New models - symmetric binary channels with Memory. Update March 2014

Welcome to the homepage of channels-networks. This website provides details of

results from my own research concerning disturbed binary channels and digital

networks, all of which are available for download.

First, a brief history of Channel Models - see downloads

In symmetric binary channels, bit errors occur in bundles.

Beginning in 1960 when **E. N. Gilbert** presented the first such model, today exists a

variety of model proposals which have arisen as a result of measured error

sequences.

The channel memory does not simply end with the last bit error, but rather extends

further into the past of the error bursts.

Thereby were the Hidden Markov Models (HMM) developed, which were then

perfected by **William Turin.**

What has however interested communication protocol developers and coding

theorists, are the probabilities of error structures in any finite time interval such as

the block length or the cycle length of a transmission procedure.

These probabilities are typically difficult to present analytically.

Many of my own studies, confirmed in the work of other authors, have found

that the block error probability (pb) dependent on the block length (n) in the initial part

is linear when presented double-logarithmically.

With this approach, the author **C. Wilhelm** developed the gap **L-model** and

distance **A-model** with complete sets of formulae concerning the probabilities of error structures occurring in bursts, and in blocks.

So you can calculate analytically residual error probabilities occurring in codes and

transmission procedures.

In the same way, you can generate artificial error sequences.

One can vary the degree of bundling as well as the bit error rate.

The claim that HMM models are "accurate" speaks against the fact that the

Markov state space is constructed from finite measurements and is therefore no

more than statistically accurate.

After enough error-free steps, a "gap" is generated, which can be described as a

renewal process. This renewal process concludes with the occurrence of the next

error bit, whereby the HMM model begins over. For all of these gaps occurring

between bursts, the distance **A-model** should be applied, otherwise the Markov state

space will be too large.

Interestingly, short tests have shown that the distance A-model might function as a

time gap model when applied to sequences of motor vehicles. For example, the

probability that in n time intervals at least one car will pass by.

The basic formulas for the distance models are developed in Part 1:

“Channel models Part 1 distance model (A-model) and gap model (L-model)”-see downloads

The calculation of error probabilities for structures that are needed to evaluate

transmission procedures and codes is included in Part 2, supplemented with

examples.

“Channel models Part 2 error structures (A-model), with examples” - see downloads

The content of this homepage was updated at least **March 2014.**

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