Sec.2.2 The Physical Layer:Channels and Modems 41 the digital data into s()so as to minimize the deleterious effects of this distortion.Note that a black box viewpoint of the physical channel is taken here,considering the input- output relation rather than the internal details of the analog channel.For example,if an ordinary voice telephone circuit (usually called a voice-grade circuit)is used as the analog channel,the physical path of the channel will typically pass through multiple switches.multiplexers,demultiplexers,modulators,and demodulators.For dial-up lines, this path will change on each subsequent call,although the specification of tolerances on the input-output characterization remain unchanged. 2.2.1 Filtering One of the most important distorting effects on most analog channels is linear time- invariant filtering.Filtering occurs not only from filters inserted by the channel designer but also from the inherent behavior of the propagation medium.One effect of filtering is to "smooth out"the transmitted signal s(t).Figure 2.3 shows two examples in which s(f)is first a single rectangular puise and then a sequence of rectangular pulses.The defining properties of linear time-invariant filters are as follows: 1.If input s(t)yields output r(t).then for any r,input s(t-7)yields output r(t-T). 2.If s(t)yields r(t).then for any real number a,as(t)yields ar(t). 3.If s(t)yields ri(t)and s2(t)yields r2(t),then s(t)+s2(t)yields ri(t)+r2(t). The first property above expresses the time invariance:namely,if an input is de- layed by T,the corresponding output remains the same except for the delay of 7.The second and third properties express the linearity.That is.if an input is scaled by a, the corresponding output remains the same except for being scaled by a.Also,the s(t (al 6 Figure 2.3 Relation of input and output waveforms for a communication channel with filtering.Part (a)shows the response r(t)to an input s(t)consisting of a rectangular pulse,and part (b)shows the response to a sequence of pulses.Part(b)also illustrates the NRZ code in which a sequence of binary inputs (11 0 1 00)is mapped into rectangular pulses.The duration of each pulse is equal to the time between binary inputs.Sec. 2.2 The Physical Layer: Channels and Modems 41 the digital data into s(t) so as to minimize the deleterious effects of this distortion. Note that a black box viewpoint of the physical channel is taken here, considering the input￾output relation rather than the internal details of the analog channel. For example, if an ordinary voice telephone circuit (usually called a voice-grade circuit) is used as the analog channel, the physical path of the channel will typically pass through multiple switches, multiplexers, demultiplexers, modulators, and demodulators. For dial-up lines, this path will change on each subsequent call, although the specification of tolerances on the input-output characterization remain unchanged. 2.2.1 Filtering One of the most important distorting effects on most analog channels is linear time￾invariant filtering. Filtering occurs not only from filters inserted by the channel designer but also from the inherent behavior of the propagation medium. One effect of filtering is to "smooth out" the transmitted signal set). Figure 2.3 shows two examples in which set) is first a single rectangular pulse and then a sequence of rectangular pulses. The defining properties of linear time-invariant filters are as follows: 1. If input s(t) yields output r(t), then for any T, input set - T) yields output ret - T). 2. If set) yields ret), then for any real number Ct, ns(t) yields ctr(t). 3. If 81 (t) yields 1'1 (t) and se(t) yields re(t), then Sl (t) + se(t) yields rj (t) + re(t). The first property above expresses the time invariance: namely, if an input is de￾layed by T, the corresponding output remains the same except for the delay of T. The second and third properties express the linearity. That is, if an input is scaled by ct, the corresponding output remains the same except for being scaled by ct. Also, the o __.....Dl..:_(t_)__ o T 1 0 1 0 0 __ 1 _ 1s(t) LDI---__- o T 2T 3T 4T 5T 6T -,U Il..-_I (a) (b) o T 2T Figure 2.3 Relation of input and output waveforms for a communication channel with filtering. Part (a) shows the response r(t) to an input 8(t) consisting of a rectangular pulse, and part (b) shows the response to a sequence of pulses. Part (b) also illustrates the NRZ code in which a sequence of binary inputs (I 1 0 I 00) is mapped into rectangular pulses. The duration of each pulse is equal to the time between binary inputs