44 Point-to-Point Protocols and Links Chap.2 on sinusoids of different frequencies.It is convenient analytically to take a broader viewpoint here and allow the channel input s(t)to be a complex function of t;that is, s(t)=Rels(t)]+j Im[s(t)],where j=v-1.The actual input to a channel is always real,of course.However,if s(t)is allowed to be complex in Eq.(2.1),r(t)will also be complex,but the output corresponding to Re[s(t)]is simply Relr(t)][assuming that h(t) is reall,and the output corresponding to Im[s(t)]is Imfr(t)].For a given frequency f,let s()in Eq.(2.1)be the complex sinusoid =cos(2f)+j sin(f).Integrating Eq.(2.1)(see Problem 2.3)yields r(t)=H(f)e (2.2) where H0= h(r)e-i2xidr (2.3) Thus,the response to a complex sinusoid of frequency f is a complex sinusoid of the same frequency,scaled by the factor H(f).H(f)is a complex function of the frequency f and is called the frequency response of the channel.It is defined for both positive and negative f.Let H(f)be the magnitude and H(f)the phase of H(f) li.e..H(f)=H The response r(t)to the real sinusoid cos(2ft)is given by the real part of Eq.(2.2),or r1(t)=|H(f)川cos2πft+∠H(f] (2.4) Thus,a real sinusoidal input at frequency f gives rise to a real sinusoidal output at the same freqency:H(f)gives the amplitude and H(f)the phase of that output relative to the phase of the input.As an example of the frequency response,if h(t)=ae-a for t≥0，then integrating Eq.(2.3)yields Hf)= (2.5) a+j2πf Equation(2.3)maps an arbitrary time function h(t)into a frequency function H(f); mathematically,H(f)is the Fourier transform of h(t).It can be shown that the time function h(t)can be recovered from H(f)by the inerse Fourier transform H(edf (2.6) Equation (2.6)has an interesting interpretation;it says that an (essentially)arbi- trary function of time h(t)can be represented as a superposition of an infinite set of infinitesimal complex sinusoids,where the amount of each sinusoid per unit frequency is H(f).as given by Eq.(2.3).Thus.the channel input s(t)(at least over any finite interval)can also be represented as a frequency function by s(= s(t)e-j2r'dt (2.7) stt)-stpezrdr (2.8)44 Point-to-Point Protocols and Links Chap. 2 on sinusoids of different frequencies. It is convenient analytically to take a broader viewpoint here and allow the channel input set) to be a complex function of t; that is, set) = Re[s(t)] + j Im[s(t)], where j = ;=T. The actual input to a channel is always real, of course. However, if set) is allowed to be complex in Eq. (2.1), ret) will also be complex, but the output corresponding to Rels(t)] is simply Relr(t)] [assuming that h(t) is real], and the output corresponding to Im[s(t)] is Im[r(t)]. For a given frequency f, let s( T) in Eq. (2.1) be the complex sinusoid ej27r j T = cos(2rrf T) +j sin(2rrf T). Integrating Eq. (2.1) (see Problem 2.3) yields r(t) = H(f)ej27rjf (2.2) where (2.3) Thus, the response to a complex sinusoid of frequency f is a complex sinusoid of the same frequency, scaled by the factor H(j). H(j) is a complex function of the frequency f and is called the frequency response of the channel. It is defined for both positive and negative f. Let [H(j)[ be the magnitude and LH(j) the phase of H(f) [i.e., H(j) = [H(f)\ejLH1fl]. The response rl(t) to the real sinusoid cos(2rrft) is given by the real part of Eq. (2.2), or rl (t) = [H(j)[ cos[27rft + UI(j)] (2.4) (2.6) (2.8) (2.7) Thus, a real sinusoidal input at frequency f gives rise to a real sinusoidal output at the same freqency; IH(j)1 gives the amplitude and LH(j) the phase of that output relative to the phase of the input. As an example of the frequency response, if h(t) = ae-of for t 0, then integrating Eq. (2.3) yields n H(j) = +'2 f (2.5) a J rr. Equation (2.3) maps an arbitrary time function h(t) into a frequency function H(j); mathematically, H(j) is the Fourier transform of h(t). It can be shown that the time function h(t) can be recovered from H(f) by the inverse Fourier transform h(t) = .l: H(j)ej27rjf df Equation (2.6) has an interesting interpretation; it says that an (essentially) arbi￾trary function of time hU) can be represented as a superposition of an infinite set of infinitesimal complex sinusoids, where the amount of each sinusoid per unit frequency is H(f), as given by Eq. (2.3). Thus, the channel input set) (at least over any finite interval) can also be represented as a frequency function by S(f) = .l: S(tje-j 2TC j l dt set) = .l: S(j)ej27rjfdf