.21 Graph illustrating a G(x) with consideration of negative inputs Source Coding and Decoding: Making it Digital But, alas, that same eagle-eyed engineer now spotted a new problem in the G(x) curve of Figure 4.21: "What shall we do when a zero (or number close to it) comes into the G(x) end of the compandor?" Another good point, because a look at G(x) in Figure 4.21 indi- cates that it doesn't tell us what to do when the input of X equal or close to 0 comes into the quantizer. "Well," piped out another engineer, "I've got a simple solution. Let's make G(x) G(x) =-log(-x) G(x) =X G(x) = log(x) map an input of x=0 to an output of 0, and have it map Figure 4.22 Shape of G(x) used in telephone inputs close to 0 to outputs system to maintain constant SQNR close to 0. And we'll let the other two curves of Figure 4.21 apply whenever x is not SO close to 0. The new G(x) will look like this." And he pointed to a figure that looked like Figure 4.22. Next, engineers decided they needed a careful mathematical description to fully describe the goings-on in the G(x) of a telephone compandor (Figure 4.22). It was here that the Europeans and the Americans had a falling out. The Americans used a math- ematical description of G(x) called the u-law description, while the Europeans made up their own description of this G(x) called the A-law description. Since both are just mathematical descriptions of the same G(x) of Figure 4.22, I'll present just one of these descriptions. Being North American, I'll provide the u-law description. The -law description of the G(x) of Figure 4.22 is the long equation (4.49) where typically =255, log(x) is the natural logarithm of X (also written as In(x)), and sgn(x) is +1 if X is positive and -1 if X is negative. Now, we'll see how the above G(x) equation describes the curve of Figure 4.22. Chapter Four At times when x is close to 0 (specifically 1/1 / x may<<1), the G(x) equation simplifies to sgn x (4.50) So, at times when x is close to 0, G(x) is just a value close to x, as in Figure 4.22 (speci