6.25 Block channel coding

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 bereduced by $\frac{K}{N}$ 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 transmitteris to keep up with the data stream, as illustrated here .

Using MATLAB, calculate the probability a bit is received incorrectly with a three-fold repetitioncode. 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.

With no coding, the average bit-error probability $p_e$ is given by the probability of error equation : $p_e=Q(\sqrt{\frac{2\alpha ^2{E}_b}{N_0}})$ . With a threefold repetition code, the bit-error probability is given by $3p_e^{\prime }^2(1-p_e^{\prime })+p_e^{\prime }^3$ , where $p_e^{\prime }=Q(\sqrt{\frac{2\alpha ^2{E}_b}{3N_0}})$ . Plotting this reveals that the increase in bit-error probability out of the channel because of theenergy reduction is not compensated by the repetition coding.

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The repetition code 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'sparameters. In the three-fold repetition code , $K=1$ and $N=3$ . A block code's coding efficiency $E$ equals the ratio $\frac{K}{N}$ , and quantifies the overhead introduced by channel coding. The rate at which bits must be transmitted againchanges: So-called data bits $b(n)$ emerge from the source coder at an average rate $\langle B(A)\rangle$ and exit the channel at a rate $\frac{1}{E}$ higher. We represent the fact that the bits sent through the digital channel operate at a different rate by usingthe 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 thebit 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 isyes. To understand channel coding, we need to develop first a general framework for channel coding, and discover what it takesfor a code to be maximally efficient: Correct as many errors as possible using the fewest error correction bits as possible(making the efficiency $\frac{K}{N}$ as large as possible).