fically, G(x) is a constant multiplied by x). At times when X is far from 0 (specifi- cally u/x /Xmay<<1), G(x) in this case reduces to (4.51) So, when x is far from 0, G(x) is a value proportional to the logarithm of X, as in Figure 4.22. The u-law equation for G(x), then, is just another way of saying "see Figure 4.22." Telephone quantizers are built using the compandor (Figure 4.18), and we've just seen the G(x) that telecommunication engineers decided to use. The final decision was how many levels to use in the uniform quantizer of the compandor. After some long days of work and one tired slide-rule later it was decided that 256 levels would be used in the uniform quantizer. Both the Europeans and Americans agreed, and with that, the compandor used in telephones was specified. 4.3 Source Coding: Pulse Code Modulator (PCM) You now have a solid understanding of sampling, the first part of source coding. You also have a good understanding of quantizers, the second part of source coders. In this section we'll put samplers and quantizers together, and throw in a third device, to build a source coder. Because there are other ways to build source coders, as we'll see later, this source coder is given a very particular name-the pulse code modulator (PCM). 4.3.1 Introducing the PCM You'll probably recall that the source coder is a device that maps an analog input into a digital output. One way to build it, called the PCM, is shown in Figure 4.23 where a sam- pler is used, followed by a quantizer, which is followed by a third device called a symbol-to-bit mapper. Here, an analog signal, which we call x(t), comes in at the input side of Figure 4.23. A sampler creates samples of this original signal, which we'll call X's (t) ; you can see this in Figure 4.23. A quantizer takes each sample that comes in and creates a new sample that comes out; this new sample has an amplitude that takes on one of N allowed levels. We'll call this signal xs(t), and you can see an example of it in Figure 4.23. Source Coding and Decoding: Ma