gure 19.2(b), the mean signal power is given by Equation 19.30 and the signal noise ratio by Equation 19.31. s = V (19.30) (19.19) (19.31) Now the probability integral is defined as in Equation 19.20 and the complementary function is as in Equation 19.21. Therefore Equation 19.22 is obtained. Substituting from Equation 19.13 gives The receiver gives an output 'O' when v < 0 and '1' when v>0. Equation 19.23. An error will occur if when v. = +V, or if v. > +V when = = V. Thus, the probability of error is given by Equation 19.32. (19.20) P(v,>+V) (19.21) (19.22) (19.32) 19.23) Consequently, the same error rate is obtained with a 3dB lower signal to noise ratio. Alternatively, a much lower error rate can be obtained for the same signal to noise ratio. Many authors use the form of error function defined as in Equa- The above analysis can be extended to multilevel digital signals, tion 19.24 and 19.25. Consequently Equations 19.26 and 19.27 are by replacing the pulse amplitude with the spacing between adjacent obtained. signal levels. If the number of levels is large, the error rate is nearly 19/6 Digital modulation doubled. This is because all the intermediate levels can be misinter- preted in either direction, owing to noise voltages of either polarity. q r r For the case of a unipolar binary signal disturbed by white noise, (19.33) the bit error rate, calculated from Equation 19.23, varies with the signal to noise ratio as shown in Figure 19.3. For example, if the This suggests that an ideal receiver for the signal would be a bank signal noise ratio is 20dB, less than one digit per million is received of correlators, as shown in Figure 19.4. Each consists of a multiplier in error. For telephone transmission, an error rate of 1 in 10³ is followed by an integrator. The signal is applied to one input of each intolerable, but an error rate of 1 in 105 is acceptable. Lower error multiplier and the waveform corresponding to one of the symbols is rates are required for data transmission; if the error rate of the applied to