I'll introduce a function G() that changes my original inputs of [2, 3.5) to inputs in [2, 4.5). In this way, you always say 3 when I want you to say 2 (and that's great because the function G-1 on the right of the figure will map your 3 to my 2). By introducing these functions G(.) and as shown in Figure 4.19, I've turned you, Mr. or Ms. Uniform Quantizer saying 3, 6 or 9, into exactly the quantizer I want. So you see, going back to the original Figure 4.18, we can use this to get any quantizer we want, with the exact choice of G(.) and G-1.) determining the specific type of quantizer. In this section we'll find out what indeed is the best choice of G(.) and G-1.). By best choice, I mean: Given the distribution of the input amplitude, px(x), the G(.) and G-1-) that allows the quantizer of Figure 4.18 to have a minimum mse. Let me stop here to make some brief points and introduce some commonly used words. Briefly, the function G() and G-1 are always inverses of one another, SO once you know one you can always specify the other. Now, some notation. G(.) is typically called a compressor because in most cases of practical interest it ends up smooshing the original input into a smaller set of values. G-1 is typically called an expandor, because, being the inverse of G(.), it usually maps the quantizer output values into a larger set of values. Finally, the entire quantizer of Figure 4.18 is often dubbed the compandor, with the com coming from compressor and the pandor coming from expandor. Now, how do we create G(.) and G-1-) in such a way that we minimize the mse of the compandor of Figure 4.18? I'm just going to simply state the general result that some researching engineer came up with, because the details don't really add any great insight. The answer is: given an input with distribution p(x), the mse is minimized by choosing G(.) according to the equation Chapter Four G(x)=[V/Kp(x) (4.42) The mse of a quantizer using G(.) (as in Figure 4.18) is shown (not here though) to be: for inputs with values in the range o