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Friday, November 18, 2011

Mathematics of Mariner 9

In 1971 (!), Mariner 9 transmitted pictures of the planet Mars over a distance of 135 million km, with a transmitter that had a power of only 20 watts (!). With such low power and taking into to account the radio noise of space it is remarkable that data was received at all. Yet, the pictures were near-perfect. How is this possible?

The first pictures of Mars were taken at a resolution of 700 × 832 pixels. The images used 64 different shades of grey. The pictures were transmitted back to Earth by sending one pixel at a time, so we can think of a message as a single number between 0 and 63. The channel could send the two binary digits 0  and 1. Encoding each pixel as a string of 6 bits would not have worked well. A survey gave a probability of 0.05 that a bit would be flipped by the channel, implying that about 26% of the image would be wrong.

It was acceptable for each pixel to be encoded by up to 32 bits, so Mariner sent a volume of more than five times of the original data. But by doing so error-detection and correction techniques could be used so that finally a picture had less than 100 incorrect pixels. Mariner transmitted over 7200 pictures of Mars like this.

Coding Theory is a branch of mathematics that uses Galois fields, polynomials, vector spaces and matrices.

135,000,000 km far, sent with a 20 watt transmitter

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(Raumpatrouille – Die phantastischen Abenteuer des Raumschiffes Orion, colloquially aka Raumpatrouille Orion was the first German science fiction television series. Its seven episodes were broadcast by ARD beginning September 17, 1966. The series has since acquired cult status in Germany. Broadcast six years before Star Trek first aired in West Germany (in 1972), it became a huge success.)