e time domain is convolution in the frequency domain. With this in mind, we have (4.6) where * denotes convolution. Next, applying simple properties of convolution, we're able to move X(f) into the sum; that is, Xs(f) becomes (4.7) Finally, we get an even simpler equation for X (f) by applying the shifting property of the delta function. This gets us to (4.8) Let's write out the summation term by term to help us better understand Xs(f). Doing this, we get Xs(f) described by (4.9) X(f-f)+X(f-2f)+...} Figure 4.2 (a) X(f), the Fourier transform of the input signal x(t) (b) P(f), the Fourier transform of the impulse train p(t) Chapter Four Now that we've got X's (f) described mathematically, let's understand what this means using words and a picture. In words, this last equation indicates that Xs (f) consists of a number of X(f)'s, shifted by multiples of fs, and all added together. Using pictures, for X(f) shown in Figure 4.2(a), Xs (f) is shown in Figure 4.3. Xs(f) 1 x(f+fg) 1x(f-fs) 1 X(f-2fs) -fs+fm f5-fm Figure 4.3 X (f), the Fourier transform of the ideal sampling output (t) Here's where it gets really interesting, SO read on carefully. Take a look at Figure 4.3. Here, we see that as long as A<B and D<C, then the X(f) term in Xs(f) does not overlap with its neighbors X(f-f) and X(f+f). This means that, as long as A<B and D<C, X(f) is preserved perfectly in Xs(f). In other words, as long as A<B and D<C, then all of Carl's incoming speech signal (t) is completely safe in the sampled signal X's (t). Now A<B means fm<f-fm-that is, >2fm Similarly, D<C means -fs+ fm <-fm; that is, f>>2fm. So, as long as f>>2fm, then A<B and D<C, and X(f) is preserved perfectly in Xs(f). WOW. All of the information of x(t), the incoming speech signal, is in the sampled signal (t) if we just insure that fs > 2fm. That's always seemed like an incredible result to me. It seems logical to think that at least some information would be lost when we sample an incoming signal, but not so, as we just saw. Getting Back All the Information from 