/~pat/coding/essays/turbo.pdf http://www.csee.wvu.edu/~mvalenti/ documents/valenti01.pdf Error correction Internet), and in global positioning systems (GPS, Galileo), to quote a few representative examples. Corrected bit-error-rate As we have seen in this chapter, ECCs can only correct error patterns having up to w = {(dmin - 1)/2} bit errors. It is clear, then, that in the general case, the bit error rate (BER) is always nonzero. This is despite the fact that it can be made arbitrarily close to zero through the appropriate ECC and under certain channel limiting conditions, as will be described in Chapter 12. Here, I will not detail the optimum receiver decision techniques, usually referred to as hard-decision and soft-decision decoding, respectively. Suffice it to state that in the soft-decision decoding approach, the receiver decision is optimized for each 1/0 symbol type (e.g., use of different matching filters in multilevel signaling), which eventually minimizes symbol errors. On the other hand, hard-decision decoding consists of making a single choice between 1 and 0 values for each of the received symbols. Here, we shall first derive a simple expression for the hard-decision-decoding BER, which represents a general upper bound, regardless of the type of code used. The uncorrected BER is defined by the function p(x), which is the probability of mis- reading 1 or 0 symbols. In the modulation and detection format referred to as intensity- modulation/direct-detection (IM-DD) or, equivalently, ON-OFF keying (OOK), the parameter x is the Q-factor.12 Simply defined, the Q-factor is a function of the mean received signal powers, (Po) and (P1), which are associated with the 0 and 1 bit symbols, respectively, and of the corresponding Gaussian noise powers with standard deviations V02,01 = for respectively. The Q-factor thus takes the simple form:13 (11.26) See, for instance: E. Desurvire, Survival Guides Series in Global Telecommunications, Signaling Principles, Network Protocols and Wireless Systems (New York: J. Wi