zer (a quantizer that minimizes the mse) is rather creative. We'll use Figure 4.18 to help. Here, we see a device that's made up of three parts: (1) a block G() that maps the input x to the output G(x); (2) a uniform quantizer; and (3) a block G-1.) that maps the signal y to the output G-1 (). =G(x) Uniform = G-1(s) G-1(.) Quantizer Figure 4.18 A new way to create the best quantizer y = 3,6, or 9 Uniform X = 2,4, or 5 G-1(.) Quantizer maps 3 [2,3.5) [2,4.5) Figure 4.19 Illustration example of the workings of the new quantizer of Eq. 4.18 Source Coding and Decoding: Making it Digital The three parts of Figure 4.18 can work together to create any type of quantizer we want. This happens by simply choosing the G(.) and G-1-) carefully. Let me illustrate that by example. Let's say I call out a value between 2 and 10, and you, a quantizer, call back either 2, 4, or 5, whichever minimizes the error (whichever is closest). So I yell 2.3 and you say 2; I shout 7.9 and you say 5. However, we can do this differently. Let's say that you, our quantizer, are tired and only want to play the quan- tizer game if you can be a uniform quantizer and call back either 3, 6, or 9. Well, I still want a quantizer that maps inputs to outputs of either 2, 4, or 5. Can we both get what we want? Yes. Take a look at Figure 4.19, which helps explain how this can happen. As we see in this figure, we'll need to introduce two new functions: first, looking at the right of the figure, we've introduced which maps the output values you say-3, 6, or 9-to the output values I want to hear: 2, 4, or 5. That's half of it; the other half is shown on the left of Figure 4.19, where we find the function G(.). This function changes the inputs you hear me calling out. Consider this: if what I want are outputs either 2, 4, or 5, then I'll want inputs [2, 3.5) to go to 2 (because those numbers are closest to 2); but, because the outputs you say are 3, 6, or 9, then for the numbers you hear between [2, 4.5) you'll say 3 (because these numbers are closest to 3)-so 