the Samples Telecommunication engineers jumped up and down with joy, ate ice cream, and had a party, all the while shouting, "It's great that all of Carl's speech x(t) is found in his sampled signal xs(t), because that means I can sample his speech and not lose any information." Then, one engineer piped up, "Now, let's say at some later time I want to get Carl's speech (t) back from that sampled signal (t). How do I do that?" And all the engineers scratched their heads, went back to their chalkboards, and 5 minutes later came back and said, "Easy!" Take a look at Figure 4.3, which shows us Xs(f) From this figure, it's clear that if we simply get rid of everything except the stuff between - fm M and fm, , and add a gain of Ts, then we'll have regained Carl's speech. How do we get rid of everything except the stuff between - fm and fm? Easy again-just use a low-pass filter (LPF) to cut off everything outside of -fmand fm, add a gain of Ts, and voila, Carl's sound is back. Source Coding and Decoding: Making it Digital One last note. We have some choice regarding exactly what frequen- xs(f) LPF H(f) cies the LPF will cut out. Take a look at Figure 4.4. As long as the cutoff frequency of the filter, fc, is between fm and fs - fm, then this LPF will do a fine job in recovering x(t) exactly from its sampled signal xs(t). One fs-fm common choice for the cutoff fre- quency of the filter is f==/g./2. This Figure 4.4 puts the cutoff frequency f smack The use of an LPF with frequency dab in the middle of the X(f) and its response H(f) to recover X(f) from X (f) shifted replica X(f-f). Some Commonly Used Words A number of terms are commonly used when talking about sampling. First, there's the sampling theorem. The sampling theorem simply states that a signal can be recovered from its samples as long as it is sampled at f>22M We know that, and now we've got a name for it. Next, there's the Nyquist rate, fn. The Nyquist rate is the smallest sampling rate fs that can be used if you want to recover the original signal from its sa