which is plotted in Figure E4.1(a). To determine the output in the frequency domain, we turn the pages back until we reach equation (4.8), which tells us simply (E4.5) X,(f)=2 XX(f-2k) (E4.6) =2.[...+X(f+2)+X(f)+X(f-2)+.. (E4.7) and this fellow is plotted in Figure E4.1(b). xs(t) Xs(f) x(t) = sinnt Figure E4.1 (a) Signal in time (b) in frequency Source Coding and Decoding: Making it Digital 4.1.2 Zero-order Hold Sampling We'll now talk about a method of sampling that can be physically built, a method called zero-order hold sampling. First, I'll provide you with a brief description using only a picture. The input signal-we'll use Carl's speech signal, x(t)-and the resulting output sampled signal, xs(t), are both shown in Figure 4.6. Although this type of sampling method is not physically built in this way, Figure 4.7 gives some insight into what goes on in zero-order hold sampling. Figure 4.7 shows the incoming signal, Carl's speech signal x (t), first going into an ideal sampler. Here, it gets multiplied by a pulse train p(t). As we saw previously, this leads to impulses of height s(kTs) once every kT's seconds; that is, it leads to (4.10) k=-00 Next, the output of the ideal sampler, Xi (t), enters into a linear time invariant (LTI) system. The LTI system is described by the impulse response h(t) shown in Figure 4.7 (a rectangular impulse response of amplitude 1 and duration T). This leads to the output: for each incoming sample of height x(kTs) (in x,(t)), the output corresponds to "holding on" to the value x(kT) for a duration of Ts. The total output xs(t) is shown in Figure 4.7, and it's described by the words: xs(t) is a signal with height xs(kTs) in each time interval [kT,,(k+1)7). The Information in the Samples We saw earlier that for ideal sampling all the information contained in the input signal x (t) is preserved in the output samples (t), , as long as we sample at a rate fs >2fm Telecommunication engineers suspected that the same thing was true for zero order hold sampling. We'll now see that their sus