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# Error Probability

Module by: Don Johnson. E-mail the author

Summary: No matter how large the SNR, no other receiver can provide a smaller probability of error than the matched filter receiver.

How small should the error probability be? Out of NN transmitted bits, on the average N p e N p e bits will be received in error. Do note the phrase "on the average" here: Errors occur randomly because of the noise introduced by the channel, and we can only predict the probability of occurrence. Since bits are transmitted at a rate RR, errors occur at an average frequency of R p e R p e . Suppose the error probability is an impressively small number like 10-6 10 -6 . Data on a computer network like Ethernet is transmitted at a rate R=10 Mbps R 10  Mbps , which means that errors would occur roughly 100 per second. This error rate is very high, requiring a much smaller p e p e to achieve a more acceptable average occurrence rate for errors occurring. Because Ethernet is a wireline channel, which means the channel noise is small and the attenuation low, obtaining very small error probabilities is not difficult. We do have some tricks up our sleeves, however, that can essentially reduce the error rate to zero without resorting to expending a large amount of energy at the transmitter. We need to understand digital channels and Shannon's Noisy Channel Coding theorem.

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