oise voltage. tions (de Belin, 1991). So far, it has been assumed that the objective of equalisation is to eliminate intersymbol interference. An alternative is to have a large P(0)=P(1)=1 (19.11) but well defined ISI. This is known as partial response equalisation (Lender, 1963; Kretmer, 1966). It effectively turns a binary signal into a 3-level signal and enables a smaller bandwidth to be used. s - 1/2 (19.12) "s (t) Decision level "s (t) Decision level Figure 19.2 Detection of digital signals: (a) unipolar binary signal; (b) bipolar binary signal Digital transmission 19/5 (19.13) (19.24) The receiver compares the signal voltage V with a threshold erfc x = 1 - erf x (19.25) voltage of 1/2V, giving an output '0' when V < 1/2V and '1' when V 1/2V. If a noise voltage V is added, an error occurs if v.<-1/2V (19.26) when Vs = +V, = or if v_ + +1/2V when v_=0. Thus, the probability of error, given by Equation 19.14.S (19.27) (9.14) Tables of erf x have been published (e.g. Jahnke and Emde, 1960; But Equation 19.15 holds and, for white noise, Equation 19.16 is Betts, 1970). However, for x > 3, erf x is so close to unity that it is obtained. Therefore Equation 19.17 is also obtained. usually not tabulated. In these cases, the asymptotic expression for the error function (Beckmann, 1967) should be used as in Equation P(0)+P(1)=1 (19.15) 19.28. (19.16) erfx - 1 - expl-x3 (19.28) (19.17) This expression enables calculations to be made of the very low error probabilities obtained with good signal to noise ratios. The variation of error probability over a range of signal to noise ratios is White noise has a normal or Gaussian probability density distribu- shown in Figure 19.3. tion, given by Equation 19.18 where o is the standard deviation of If the signal to noise ratio is calculated for the peak power (v2/02) vn and is thus the r.m.s. noise voltage. Therefore Equation 19.19 is instead of the mean power (v2/202), then Equation 19.29 is ob- obtained. tained. (19.18) (19.29) If a bipolar binary signal is used, as shown in Fi