Because of the higher datarate imposed by the channel coder, the probability of bit error occurring in the digital channel **increases** relative to the value obtained
when no channel coding is used. The bit interval duration must be reduced by in comparison to the no-channel-coding situation, which means the energy per bit *E*_{b} goes **down** by the same amount. The bit interval must decrease by a factor of three if the transmitter is to keep up with the data stream, as illustrated here (Figure 6.21: Repetition Code).

POINT OF INTEREST: It is unlikely that the transmitter's power could be increased to compensate.

Such is the sometimes-unfriendly nature of the real world.

Because of this reduction, the error probability *p _{e}*

**of the digital channel goes up. The question thus becomes does channel coding**

**really**help: Is the effective error probability lower with channel coding even though the error probability for each transmitted bit is larger? The answer is

**no**: Using a repetition code for channel coding cannot ultimately reduce the probability that a data bit is received in error. The ultimate reason is the repetition code's inefficiency: transmitting one data bit for every three transmitted is too inefficient for the amount of error correction provided.

**Exercise 6.26.1**

Using MATLAB, calculate the probability a bit is received incorrectly with a three-fold repetition code. Show that when the energy per bit *E _{b}* is reduced by 1/3 that
this probability is larger than the no-coding probability of error.

Repetition Codes represents a special case of what is known as ** block
channel coding.** For every

*K*bits that enter the block channel coder, it inserts an additional

*N - K*error-correction bits to produce a block of

*N*bits for transmission. We use the notation (N,K) to represent a given block code's parameters. In the three-fold Repetition Codes,

*K*= 1 and

*N*= 3. A block code's coding efficiency E equals the ratio , and quantifies the overhead introduced by channel coding. The rate at which bits must be transmitted again changes: So-called data bits b (n) emerge from the source coder at an average rate and exit the channel at a rate higher. We represent the fact that the bits sent through the digital channel operate at a different rate by using the index l for the channel-coded bit stream

*c (l)*. Note that the blocking (framing) imposed by the channel coder does not correspond to symbol boundaries in the bit stream

*b (n)*, especially when we employ variable-length source codes.

Does any error-correcting code reduce communication errors when real-world constraints are taken into account? The answer now is yes. To understand channel coding, we need to develop first a general framework for channel coding, and discover what it takes for a code to be maximally efficient: Correct as many errors as possible using the fewest error correction bits as possible (making the efficiency as large as possible).

- 899 reads