B. P. Lathi, Zhi Ding - Modern Digital and Analog Communication Systems-Oxford University Press (2009)

14
ERROR CORRECTING
CODES
14. 1 OVERVIEW
A
seen from the discussion in Chapter 13, the key to achieving error-free digital communication
in the presence of distortion, noise, and interference is the addition of
appropriate redundancy to the original data bits. The addition of a single parity check
digit to detect an odd number of errors is a good example. Since Shannon's pioneering paper, 1
a great deal of work has been carried out in the area of forward error correcting (FEC) codes.
In this chapter, we will provide an introduction; readers can find much more in-depth coverage
of this topic from the classic textbook by Lin and Costello. 2
Generally, there are two important classes ofFEC codes: block codes and convolutional codes.
In block codes, every block of k data digits is encoded into a longer codeword of n digits
(n > k). Every unique sequence of k data digits fully determines a unique codeword of n
digits. In convolutional codes, the coded sequence of n digits depends not only on the k data
digits but also on the previous N - l data digits (N > 1). Hence, the coded sequence for a
certain k data digits is not unique but depends also on N - l earlier data digits. In short, the
encoder has memory. In block codes, k data digits are accumulated and then encoded into an
n-digit codeword. In convolutional codes, the encoding is done on a continuous running basis
rather than by blocks of k data digits.
Shannon's pioneer work 1 on the capacity of noisy channels has yielded a famous result
known as the noisy channel coding theorem. This result states that for a noisy channel with a
capacity C, there exist codes of rate R < C such that maximum likelihood decoding can lead
to error probability
(14.1)
where E b (R) is the energy per information bit defined as a function of code rate R. This
remarkable result shows that arbitrarily small error probability can be achieved by increasing
the block code length n while keeping the code rate constant. A similar result for convolutional
codes was also shown in Ref. 1. Note that this result establishes the existence of good codes. It
does not, however, tell us how to find such codes. In fact, it is not simply a question of designing
good codes. Indeed, this result also requires large n to reduce error probability and requires
decoders to use large storage and high complexity for large codewords of size n. Thus, the key
problem in code design is the dual task of searching for good error correction codes with large
802