peech signal x (t) is natural sampling. The workings of natural sampling are shown in Figure 4.9. Here, we see Carl's incoming speech signal x(t) multiplied by a signal called p(t). This p (t) is made up of a bunch of tall, skinny, rectangular shapes of height 1/T and width T; these tall skinny rectangles are spaced T seconds apart. Figure 4.9 shows this more clearly than my description. xs(t) Figure 4.9 Natural sampling Chapter Four The output of the sampler is simply (t)=x(t).p(t). . This signal is just a sequence of pieces of x(t). It consists of tall, skinny shapes pieced together, with one shape coming every Ts seconds; each shape lasts for a time T seconds and has a height shaped by x(t). Again, Figure 4.9 shows this better than my description. The Information in the Samples Again, the question arises: how much Figure 4.10 information is in those samples? The same X(f), Fourier transform of input x(t) answer applies: if you sample at fs > 2fm, you can get back Carl's speech signal X (t) from the sampled signal. We turn to the frequency domain as a helpful tool to show that all the information is in the samples if fs > 2fm. We begin by assuming that X(f), the Fou- rier transform of Carl's speech signal, x(t), looks like Figure 4.10. Next, we want to figure out the Fourier transform of xs(t), Xs(f). This is done, using simple math, as follows: X((f)=F(x(t) p(t)} (4.11) As a way to simplify this, we'll come up with another equation for p(t). Because p(t) is a periodic signal, it can be written using a Fourier series according to (4.12) where Ck are the Fourier series coefficients. For the p(t) at hand, Ck can be computed to get the result (4.13) Plopping this p(t) equation in the Xs(f) of equation (4.9) gets us (4.14) Using some well-known Fourier transform stuff, we can simplify this equation to (4.15) (4.16) Source Coding and Decoding: Making it Digital Now we have Xs(f) described mathematically. Let me explain this Xs(f) to you in words and pictures. In words, Xs(f) consists of many copies of X(f) added tog