erfc(x) = 1 - erf(x). For Q > 2, 14.1 Gaussian channel -1.6 dB 21.6 dB BER = 10-9 SNRbit (dB) Figure 14.3 Bit error rate (BER) as a function of bit signal-to-noise ratio (SNRbit) for NRZ- or OOK-modulated signals. Figure 14.3 shows the plot BER = f(SNRbit), along with the Shannon limit. It is seen from the figure that the specific value SNRbit = +21.6 dB yields a BER of 10-9, , corresponding to one bit error (on average) for every one billion bits transmitted. For a long time in the modern history of telecommunications, such a BER has been con- sidered to represent the standard for "error-free" transmission. In the limit of vanishing signals, or SNRbit 0 or SNRbit(dB) -00, the erf function vanishes and we have BER = 1/2. Since the BER is a probability, this result means that the bits 0 and 1 have equal probabilities of being detected, regardless of the input message sequence, which corresponds to a useless communication channel. The shaded region in Fig. 14.3 defines the impossibility of finding error-correction codes (ECC) able to improve the BER at a given SNRbit value, while right-hand side region indicates that the possibility exists. A detailed introduction to the subject of ECC is provided in Chapter 11. Here, I shall provide a heuristic example illustrating the actual effect of error correction on the (otherwise uncorrected) BER performance. Figure 14.4 shows an example of ECC implementation where the physically achiev- able signal-to-noise ratio is SNRbit = 16.6 dB. At the receiver level, and before error- correction decoding, the bit error rate is about BER = 10-3.5. After correction, the bit error rate is seen to improve BER = 10-9 As the figure indicates, the horizontal translation between the two BER curves (uncorrected on top, and corrected at bottom) corresponds to a virtual, but effective SNR shift of +5 dB. As formally described in Chapter 11, this SNR shift is referred to as the coding gain. the BER can be approximated by BER = exp(-Q2/2)/(QV2)T), which yields BER = 10-9 for Q = 6 or SNRbit = 4Q2