f [-xmaxXmax], (4.43) where q is the size of one cell in the uniform quantizer and G(.) is the derivative of G(.). Example 4.5 Determine the optimal compression characteristic when the input to a compandor is described by Figure E4.6. Solution: To figure out the G(x), we turn to equation (4.42), which tells us (E4.10) Now, for X in the range of [0,3], we have p(x)=1/6. = Applying this to (E4.10) leads us to (E4.11) (E4.12) Figure E4.6 The pdf of the input Source Coding and Decoding: Making it Digital (E4.13) When X is larger than 3, we use equation (E4.10), which tells us (E4.14) (E4.15) (E4.16) When X is in the range [-3,0], we have p(x) = =1/6. Applying this to (E4.10), we get (E4.17) (E4.18) When X is less than -3, we use (E4.10) to discover G(x)=]:/Kp(x) dx, X<-3 (E4.19) (E4.20) (E4.21) Putting it all together, we end up with (E4.22) which is plotted in Figure E4.7. Chapter Four G (x) Figure E4.7 The compressor The value of K is traditionally determined by deciding that we want the following: when the input is at its maximum value x max' the output should be G(xmax) = Xmax In this case, that means we want the value of G(3) to be 3. Requiring this leads to 3.(K/Y) (E4.23) K ==6 (E4.24) 4.2.3 The Quantizer and the Telephone Let's say we want to build a quantizer for a telephone, one that's going to be applied to a sampled speech signal. This section discusses how we build such a quantizer and explains the standard quantizer (for a sampled speech signal) used in telephone systems in the United States and Europe. The Idea Since all kinds of people use the telephone-little people and big ones, fast talkers and slow ones, loud speakers and quiet ones-telecommunication engineers sat down and said, "We'll have to come up with a quantizer that does a good job for most any pos- sible px(x) (the input amplitude distribution), since SO many different people will talk into our phone." And with this, the quest for the best telephone quantizer begins. Let's say we decide, just like those early telecommunications engineers d