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Results of work in errors of transmission


Errors of transmission in telephone channels
Benoît Mandelbrot Mathematician
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In 1961, '62, I was shown by a friend of mine, J. Berger, some data about errors of transmission. And these were very, very strange data because first of all, again: what is the average number of errors of transmission per day, per minute, whatever? Every number was being discussed. Peoples' opinions varied by several orders of magnitude. Then everybody was worried about distribution between blocks and error. Blocks and error meant either one error because you were transmitting at, for example, one microsecond signals, was either well or incorrect, correctly or incorrectly transmitted, but sometimes it was a second or a minute or an hour in which at least one error occurred, they didn't want to record all the errors. The folklaw, it was simply folklore said that if you see the statistics of blocks and error of any size, you couldn't tell which size it is. Now that was taken by everybody as a catastrophic finding, and all kinds of pyramids of simple explanations were created to account for it. So a very special, in fact charming, fellow at Bell Labs, our sister laboratory, was speaking of channels being error-prone or not error-prone. In the error-prone they are very error-prone or not very error-prone. If they are very error-prone, they are extremely error-prone - and a pyramid of explanations, which if you put in enough parameters, was giving anything you wanted. But he never faced the main fact which was it didn't matter whether the block was a short one or a long one. So, I was struck by that, and then by my memory of esoterica and my visual feeling for them became important because I'd been looking at the Cantor sets quite carefully, even though "they were just esoteric constructions of no possible significance," so my teachers were saying. So how do you take an interval: you take off the middle third, again the middle third of each part, the middle third, middle third. What are left is, in a certain sense, if you look carefully, little events which are put in bursts, bigger burst, bigger burst and bigger bursts. You put in the burst structure and the hierarchy, by hand. And then Paul Levy had introduced a set that was not very important in his work but later on I called Levy Dust to honour his memory, in which the same thing was true, but randomly. And the main thing was that I knew by this experience, with those mathematical and probabilistic esoterica that you can get bunching at all levels without having to put in by hand each time deliberately. So I told Berger, "Let's try this." We tried. Well, the success was absolutely fantastic, with one number we got a great deal of the reality, but more so we didn't have enough long intervals between errors, and that's a question of what probability of error occurs. So Berger said, "Well, you know very well there are all kinds of errors, there's also some white noise exponent which is independent, and so you shouldn't expect this thing to hold over a long length." I said, "Look, you shouldn't expect it, but what does it cost us to check?" That was one of the most important things I did. He wrote to the man who provided us with data saying we don't find enough very long-term intervals. The man responded instantly saying, "I'm sorry. I didn't send you any of the tapes which had less than nine errors because I thought it was statistically insignificant." Well that added two orders of magnitude to our data. As a matter of fact this fit was beyond belief, nine orders of magnitude, but it was very unsatisfactory because it still didn't tackle this matter of why is it so, for the errors and for the other things, and then there were other discrepancies. Berger went his way, and I went on worrying about this phenomenon. Well, what did I do? I did a block renormalization. which distribution of inter-block intervals, under some conditions that I felt were sensible, would be obtained independent in respect to the block size. The distribution was slightly more complicated than the one that Berger and I had found, and the complications did account for the discrepancies. Again the dates are early '60s. I mention them because the technique became routine in the late '60s and early '70s. Again, I didn't push this technique very far but I certainly had a gut feeling that by following the folklore of experienced and hard-boiled engineers I was doing better than by listening to simple statistical rules.

Benoît Mandelbrot (1924-2010) discovered his ability to think about mathematics in images while working with the French Resistance during the Second World War, and is famous for his work on fractal geometry - the maths of the shapes found in nature.

Listeners: Daniel Zajdenweber Bernard Sapoval

Daniel Zajdenweber is a Professor at the College of Economics, University of Paris.

Bernard Sapoval is Research Director at C.N.R.S. Since 1983 his work has focused on the physics of fractals and irregular systems and structures and properties in general. The main themes are the fractal structure of diffusion fronts, the concept of percolation in a gradient, random walks in a probability gradient as a method to calculate the threshold of percolation in two dimensions, the concept of intercalation and invasion noise, observed, for example, in the absorbance of a liquid in a porous substance, prediction of the fractal dimension of certain corrosion figures, the possibility of increasing sharpness in fuzzy images by a numerical analysis using the concept of percolation in a gradient, calculation of the way a fractal model will respond to external stimulus and the correspondence between the electrochemical response of an irregular electrode and the absorbance of a membrane of the same geometry.

Duration: 5 minutes, 20 seconds

Date story recorded: May 1998

Date story went live: 24 January 2008