ether, where the kth copy is shifted by kfs and multiplied by Ck . Figure 4.11 shows this in one simple picture. Now, here is the important part. As with ideal sampling, if you'll take a look at Figure 4.11, you can see that as long as we keep fs - -fm>fsi-e., fs > -then the signal X(f) is contained perfectly in Xs(f) It can be recovered exactly by simply passing Xs(f) through a low-pass filter (LPF) that gets rid of everything but C.X(f), and introducing a gain of 1/c in the LPF. Xs(f) X(f+fs) Coox(f) .X(f-fs) -fs+fm fs-fm Figure 4.11 X(f), Fourier transform of the output signal xs(t) 4.2 Quantization You've made it to the second part of source coding. Take a moment to pause and congratulate yourself on getting this far. Now, let's continue our talk on source coding (making an analog signal a digital one). As you saw in the last section, the first part of source coding involves the mapping of the analog signal into samples of itself, a pro- cess suitably named sampling. The next operation carried out in source coding is called quantization, and the device which does it is called a quantizer. 4.2.1 Meet the Quantizer A quantizer is a fancy word for a very simple device. It is actually just an "amplitude changer"; it takes the incoming signal X's (t) and changes its amplitude (at every time) to the closest of one of N allowed values. This is most easily explained by example, SO let's jump right into one. To illus- trate, I'll need your help. Every time I say a number, you turn it into the nearest integer between 1 and 10. So I yell out 7.36, and you give me back the number 7. I yell out 3.9, you shout back 4. I pipe out 9.22, and you say 9. If you understand this, you totally understand quantizers. All they do is output the closest amplitude (among N possible amplitudes), given the amplitude of the incoming signal. What could be simpler? Chapter Four Xs(t) Xs(t) Quantizer Allowed output 3.3322 amplitudes are {0,1 9} Figure 4.12 Illustration of a quantizer and how it works To make this even clearer, take a look at 