values of X but corresponds to a straight line through the origin for small values of X, as shown in Figures 19.19(c) and (d). 19.4.5 Companding In typical telephone speech, the probability distribution of signal dy=(H)dx = (19.54) amplitudes has a maximum at zero amplitude and becomes pro- gressively smaller for high amplitudes, having a near Gaussian distribution. In addition, substantial level variations exist between y = b log cx (19.55) one conversation and another, depending on such factors as the line length between the subscriber and the exchange, the sensitivity of One compression law that is used is the A-law shown in Figure the subscriber's handset and the loudness of the talker. These factors 19.19(c). This is given by Equation 19.56. Figure 19.19 Compression characteristics: (a) linear law; (b) logarithmic law; (c) A-law; (d) u-law Differential pulse code modulation 19/17 bits for the mantissa, with the leading bit (which is always one) removed. The A-law characteristic is shown in Figure 19.20. (19.56) 1 + log A 111 1111 The section below X = 1/A is linear. The section between 1/A and 111 0000 1 is logarithmic and, for A 87.6, this gives constant SQNR over a 110 0000 range of 20 10g1087.6 = 38dB. For small signals, the A-law gives Equation 101 0000 (19.57) 100 0000 However, for uniform quantisation, dy/dx=1. Thus, if the A-law companding advantage is defined as the improvement in SQNR 010 0000 compared with uniform quantisation, it is as in expression 19.58. For A=87.6, this is 24dB. 000 0000 2010810(dx) (19.58) 0 16 32 64 128 Input amplitude Another compression law that is used is the u-law shown in Figure 19.19(d). This is given by Equation 19.59. (19.59) Figure 19.20 CCITT A-law characteristic 0sxs1 This approximates to the logarithmic law of a linear Differential pulse code modulation law forx<< s - since Equation 19.60 holds. 19.5.1 Basic differential PCM Conventional PCM permits any sample to have any value. The (19.60) system can therefore cope with adjacent samples having values at the extre