 by the receiver. Notice that in a PCM signal, not all the bit errors have the same impact, but the effect of the bit errors depends on the position of the erroneous bits. Considering a binary codeword of length N(b1, b2, bn) in which the mth bit distinguishes between quantum levels spaced by 2m times the step size, 8, an error in the mth bit shifts the decoded level Em = + 2m X S. The average of E2 over the N bits equals the mean square error for random bit error location. (1.20) Assuming that the probability of bit error, Per is very small (Pe << 1), then the probability of a single error in N bits can be approximated by N.Pe and the probability of more than a single bit error can be considered to be 0. Therefore, the decoding channel noise power can be written as (1.21) The total noise power can be written as the summation of the quantization and the decoding channel noise. Therefore (1.22) (1.23) Figure 1.20 shows the graph of the total SNR versus the bit error rate for different values of 8 and for the case where Sx is set to 1. As the figure illustrates, the impact of the quantiza- tion noise is dominant when the bit error rate is very low. In this case the advantage of employing a smaller step size can be clearly observed. However, with an increase in the bit error rate, the impact of the channel noise becomes more pronounced and the curves start to converge. Eye Diagram Normally, digital transmission systems employ pulse-shaping techniques to limit the effec- tive bandwidth of the transmitting signal. This is very useful for controlling the effect of Intersymbol Interference (ISI) caused by the channel. The impact of the noise and other transmission impairments can be clearly observed by superimposing the various transitions Chapter 1 Voice Transmission S = 1/64 S = 1/32 S = 1/16 Fig. 1.20 Signal-to-noise ratio versus bit error rate for different values of the quantization step. M1 M2 Set no Fig. 1.21 Example eye diagrams: (a) wide-open diagram of a transmission system with a low bit error rate (2.45 X 10