42 Point-to-Point Protocols and Links Chap.2 output due to the sum of two inputs is simply the sum of the corresponding outputs. As an example of these properties,the output for Fig.2.3(b)can be calculated from the output in Fig.2.3(a).In particular,the response to a negative-amplitude rectangular pulse is the negative of that from a positive-amplitude pulse (property 2),the output from a delayed pulse is delayed from the original output (property 1),and the output from the sum of the pulses is the sum of the outputs for the original pulses (prop- erty 3). Figure 2.3 also illustrates a simple way to map incoming bits into an analog waveform s(t).The virtual channel accepts a new bit from the DLC module each T seconds;the bit value 1 is mapped into a rectangular pulse of amplitude +1,and the value 0 into amplitude-1.Thus,Fig.2.3(b)represents s(t)for the data se- quence 110100.This mapping scheme is often referred to as a nonreturn to zero (NRZ) code.The notation NRZ arises because of an alternative scheme in which the pulses in s(t)have a shorter duration than the bit time T,resulting in s(t)returning to 0 level for some interval between each pulse.The merits of these schemes will become clearer later. Next,consider changing the rate at which bits enter the virtual channel.Figure 2.4 shows the effect of increasing the rate by a factor of 4(i.e.,the signaling interval is reduced from T to T/4).It can be seen that output r(t)is more distorted than before. The problem is that the response to a single pulse lasts much longer than a pulse time,so that the output at a given t depends significantly on the polarity of several input pulses; this phenomenon is called intersymbol interference. From a more general viewpoint,suppose that h(t)is the channel output correspond- ing to an infinitesimally narrow pulse of unit area at time 0.We call h(t)the impulse response of the channel.Think of an arbitrary input waveform s(t)as a superposition of very narrow pulses as shown in Fig.2.5.The pulse from s()to s(T+6)can be viewed as a small impulse of area os()at time T:this gives rise to the output 8s()h(t-T)at 0T/4 0T4 (a) r(t) 3T/2 3T/2 (b) Figure 2.4 Relation of input and output waveforms for the same channel as in Fig.2.3. Here the binary digits enter at 4 times the rate of Fig.2.3,and the rectangular pulses last one-fourth as long.Note that the output r(t)is more distorted and more attenuated than that in Fig.2.3.42 Point-ta-Point Protocols and Links Chap. 2 output due to the sum of two inputs is simply the sum of the corresponding outputs. As an example of these properties, the output for Fig. 2.3(b) can be calculated from the output in Fig. 2.3(a). In particular, the response to a negative-amplitude rectangular pulse is the negative of that from a positive-amplitude pulse (property 2), the output from a delayed pulse is delayed from the original output (property 1), and the output from the sum of the pulses is the sum of the outputs for the original pulses (prop￾erty 3). Figure 2.3 also illustrates a simple way to map incoming bits into an analog waveform s(t). The virtual channel accepts a new bit from the DLe module each T seconds; the bit value I is mapped into a rectangular pulse of amplitude + I, and the value 0 into amplitude -1. Thus, Fig. 2.3(b) represents s(t) for the data se￾quence 110100. This mapping scheme is often referred to as a nonreturn to zero (NRZ) code. The notation NRZ arises because of an alternative scheme in which the pulses in s(t) have a shorter duration than the bit time T, resulting in s(t) returning to 0 level for some interval between each pulse. The merits of these schemes will become clearer later. Next, consider changing the rate at which bits enter the virtual channel. Figure 2.4 shows the effect of increasing the rate by a factor of 4 (i.e., the signaling interval is reduced from T to T /4). It can be seen that output r(t) is more distorted than before. The problem is that the response to a single pulse lasts much longer than a pulse time, so that the output at a given t depends significantly on the polarity of several input pulses; this phenomenon is called intersymbol interference. From a more general viewpoint, suppose that h(t) is the channel output correspond￾ing to an infinitesimally narrow pulse of unit area at time O. We call h(t) the impulse response of the channel. Think of an arbitrary input waveform s(t) as a superposition of very narrow pulses as shown in Fig. 2.5. The pulse from S(T) to S(T +6) can be viewed as a small impulse of area 6S(T) at time T; this gives rise to the output IiS(T)h(t - T) at o T/4 r(t) o T/4 (a) 3T/2 o T/2 T~ (b) Figure 2.4 Relation of input and output waveforms for the same channel as in Fig. 2.3. Here the binary digits enter at 4 times the rate of Fig. 2.3. and the rectangular pulses last one-fourth as long. Note that the output r(t) is more distorted and more attenuated than that in Fig. 2.3