pulse with a weight that equals the area underneath the original pulse. This equivalence is depicted in Fig. 4.10. Note that the pulses in the upper plot were nor- malized to have magnitude 1. The effective charge-pump current Icp can, for example, be taken into account in the loop filter model. Note that this condition is always satisfied in the small-signal limit. PLL small-signal analysis, in prin- ciple, is concerned with phase deviations 0 whose magnitude in the limit goes to zero. Hence, by choosing 0 small enough, the pulses produced by the PFD will always be sufficiently narrow for the equivalence in Fig. 4.10 to hold. 4.2 PHASE-LOCKED LOOP ANALYSIS Xin(t) Oin 0 1/N X1/N (t) Figure 4.9: Sampling phase-frequency detector. The input and (frequency- divided) VCO phase are compared using sequential digital logic. This logic steers a charge pump which produces a sequence of current pulses whose width is proportional to the instantaneous phase error Oin(t) - 01/N(t). Oin - 01/N Oin - 01/N Figure 4.10: If the PFD produces pulses whose width is small compared to the time constant of the loop filter-VCO combination, they will have the same effect as Dirac impulses with a weight that equals the area underneath the original pulses. 4.2.4 HTM-based PLL building block models Assuming the equivalence in Fig. 4.10 to hold, the input-output relation of a sampling PFD can be modeled as a multiplication of the phase error Oin - 01/N with a Dirac impulse train, or (4.32) (4.33) with T = 2/wo the sampling period. This multiplication can be stated in terms of HTMs as (4.34) (4.35) (4.36) ]r. Note that (4.34) is a special case of (4.26) with HPPD(s) = HPFD(s). Equation (4.36) reveals an important property corresponding to the HTM HPFD (s) of a sampling-PFD: it is a rank-one matrix. This should come as no suprise, since sampling maps all input signal content to the frequency band [-wo/2,000/2], a phenomenon known as aliasing. This baseband spectrum is then periodically repeated over the re- mainder of the frequency axis. So, kn