Block Error

Module by: Don Johnson

Summary: Channel coding's error correction capability compensates for the increased error probability due to the necessitated reduced bit energy if the signal-to-noise ration is large enough.

We must question whether a (7,4) code's error correction capability compensates for the increased error probability due to the necessitated reduced bit energy. (For example, the repetition code does not meet this requirement.) Figure 1 shows that if the signal-to-noise ratio is large enough that channel coding indeed yields a smaller overall error probability.

Figure 1: The probability of an error occurring in transmitted K=4 K 4 data bits equals 1-1- p e 4 1 1 p e 4 as 1- p e 4 1 p e 4 equals the probability that the four bits are received without error. The upper curve displays how this probability of an error anywhere in the four-bit block varies with signal-to-noise ratio. When a (7,4) single-bit error correcting code is used, the transmitter reduces the energy it expends during a single-bit transmission by 4/747 , appending three extra bits for error correction. Now the probability of any bit in the seven-bit block being in error after error correction equals 1-1- p e ′ 7-7 p e ′ 1- p e ′ 6 1 1 p e ′ 7 7 p e ′ 1 p e ′ 6 , where p e ′ p e ′ is the probability of a bit error occurring in the channel when channel coding occurs. Here, 7 p e ′ 1- p e ′ 6 7 p e ′ 1 p e ′ 6 equals the probability of exactly one of seven bits emerging from the channel in error; The channel decoder corrects this type of error, and all data bits in the block are received correctly.

Because the bit stream emerging from the source coder is segmented into four-bit blocks, the fair way of comparing coded and uncoded transmission is to compute the probability of a block error: the probability that any bit in a block remains in error despite error correction and regardless of whether the error occurs in the data or coding bits. Clearly, our (7,4) channel code does yield smaller error rates, and is worth the additional systems required to make it work.

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This work is licensed by Don Johnson under a Creative Commons Attribution License (CC-BY 1.0).

Last edited by Elizabeth Gregory on Aug 4, 2004 3:33 pm GMT-5.