46 Point-to-Point Protocols and Links Chap.2 in time,and this correspondingly expands S(f)in frequency (see Problem 2.5).The equalization at the receiver would then be required either to equalize H(f)over a broader band of frequencies,thus increasing the noise,or to allow more intersymbol interference. 2.2.3 The Sampling Theorem A more precise way of looking at the question of signaling rates comes from the sampling theorem.This theorem states that if a waveform s(t)is low-pass limited to frequencies at most W [i.e..S(f)=0,for f>Wl,then [assuming that S(f)does not contain an impulse at f =W],s(t)is completely determined by its values each 1/(2W)seconds; in particular, s(t)= 立() sin[2W(t-i/(2W))] (2.11) 2πW[t-i/(2W)] Also,for any choice of sample values at intervals 1/(2W),there is a low-pass waveform, given by Eq.(2.11),with those sample values.Figure 2.6 illustrates this result. The impact of this result,for our purposes,is that incoming digital data can be mapped into sample values at a spacing of 1/(2W)and used to create a waveform of the given sample values that is limited to f<W.If this waveform is then passed through an ideal low-pass filter with H(f)=1 for f<W and H(f)=0 elsewhere, the received waveform will be identical to the transmitted waveform (in the absence of noise);thus,its samples can be used to recreate the original digital data. The NRZ code can be viewed as mapping incoming bits into sample values of s(t), but the samples are sent as rectangular pulses rather than the ideal(sin )/r pulse shape of Eq.(2.11).In a sense,the pulse shape used at the transmitter is not critically important s(0)sin 2mWt 2xWt s(t) 3 2w 2W 2W 2W sin 2xW (t 1 2W 2W 2xW t- 2W Figure 2.6 Sampling theorem,showing a function s(t)that is low-pass limited to frequencies at most W.The function is represented as a superposition of(sin)/z functions.For each sample,there is one such function,centered at the sample and with a scale factor equal to the sample value.46 Point-to-Point Protocols and Links Chap. 2 r in time, and this correspondingly expands S(n in frequency (see Problem 2.5). The equalization at the receiver would then be required either to equalize H(f) over a broader band of frequencies, thus increasing the noise, or to allow more intersymbol interference. 2.2.3 The Sampling Theorem A more precise way of looking at the question of signaling rates comes from the sampling theorem. This theorem states that if a waveform set) is low-pass limited to frequencies at most W [i.e., S(n = 0, for IfI > W], then [assuming that S(n does not contain an impulse at f = W], set) is completely determined by its values each 1/(2W) seconds; in particular, set) = s (_7_') sin[21rW(t - i/(2W))] . 2W 21rW[t - i/(2W)] l=-CXJ (2.11 ) Also, for any choice of sample values at intervals 1/(2W), there is a low-pass waveform, given by Eq. (2.11), with those sample values. Figure 2.6 illustrates this result. The impact of this result, for our purposes, is that incoming digital data can be mapped into sample values at a spacing of 1/(2W) and used to create a waveform of the given sample values that is limited to IfI :::; w. If this waveform is then passed through an ideal low-pass filter with H(f) = 1 for IfI :::; Wand H(f) = ° elsewhere, the received waveform will be identical to the transmitted waveform (in the absence of noise); thus, its samples can be used to recreate the original digital data. The NRZ code can be viewed as mapping incoming bits into sample values of set), but the samples are sent as rectangular pulses rather than the ideal (sinx)/x pulse shape of Eq. (2.11). In a sense, the pulse shape used at the transmitter is not critically important 5(0) sin 21TWt 21TWt 2W sIt) 7 / / / -1'- / -........... ---.r'!!!'::=--::O::"""'/~-::::"---------::'>.~/'<:--'r--~:-----~"'::'--_::':::"-"-'....-==-===-="", t 1----~ '2' o 2W sin 21TW (t - 2W) 21TW (t - 2~ ) Figure 2.6 Sampling theorem, showing a function s(t) that is low-pass limited to frequencies at most W. The function is represented as a superposition of (sin x)/ x functions. For each sample, there is one such function, centered at the sample and with a scale factor equal to the sample value