lutely zero error" event, the latter being defined as the sum of all possible "no error" events (yi,xj) of associated probability p(y,,) According to Eq. (12.19), the SER of the binary symmetric channel is defined as SER = =p(y1/x2)p(x2)+p(y2/x1)p(x1) (12.20) The noise parameter E, which we used in the earlier section to characterize the binary symmetric CC noise, thus, corresponds to the channel SER. Given any string of N input symbols and the corresponding string of N output symbols YiYjYk, yn' the SER gives the probability of any count of errors. For with N suf- ficiently large, the average error count effectively measured is close to the SER. For instance, SER = 0.001 = 10-3 corresponds to an average of one error in 1000 trans- mitted symbols, two errors in 2000 transmitted symbols, etc. This estimation becomes accurate for sufficiently long sequences, such that N >> 1/SER. More accurately, the SER should refer to a "mean error ratio" rather than an "error rate." The term comes from communication systems where symbols are transmitted at a certain symbol rate, i.e., the number of symbols transmitted per unit time. Given a symbol transmission rate of N symbols per unit time, the corresponding channel error rate (or mean error count per unit time) is, thus, N X SER. In binary channels, the SER is referred to as bit error rate or BER. For sufficiently long and random bit sequences, the bit probabilities become very nearly equal, i.e., p(x1="0")~p(x2="1")=1/2.Accordingto = Eq. (12.20), we have, in this case, (12.21) Most real-life binary channels are asymmetric, meaning that usually, p(0|1) # p(1|0). Then the smallest BER is achieved when the sum p(0|1) + p(1|0) is minimized. As an easy illustration of SER (or BER) minimization, consider the case of the "Z channel," which was described earlier. To recall for convenience, the Z channel has the transition matrix (12.22) According to Eq. (12.19), the corresponding symbol error rate is SER = ep(x2) +0x p(x1) = ep(x2). The SER, thus, only depends on the probability p(x