domain as YPFD(s) = - (4.26) Here, Oin(s) and O1/N(s) are vectors of equivalent baseband components as defined by (4.1) and (4.2). The HTMs HPFD(S) and HPFD(s) are obtained by respectively inserting the Fourier coefficients of and hpfd(t) in (4.9). Example (Sinusoidal oscillations): Again, consider the system depicted in Fig. 4.7. Note that in this case, X1/N(t) = Xvco(t) as there is no frequency divider. After a little algebra, we then obtain (4.27) 1/2 [(1+cos(2t)) - sin(2t)Oop(t)]Oin(t - (4.28) with Oop(t) given by (4.21). The lower (first-order) approximation holds for Oop(t) << 1. This typically holds true as long as the feedback loop does not become exceedingly fast. Straightforward conversion of (4.28) into HTM language yields AHop(s) (4.29) 4.2.4 HTM-based PLL building block models HPFD( (4.30) as the HTMs that characterize the behavior of the mixing PFD. Here, Re {H(2j)} AHop(s)=1 Re {H(2j)} (4.31) Re {H(2j)} with H(s) specified by (4.18). Note that the name (s) was chosen to emphasize the fact that it represents an increment that is typically smaller than the leading terms in (4.29) and (4.30). Note, however, that it still may be of importance in analyzing the details of the time-varying components of a PLL's behavior. A similar analysis as the one above is possible when the input signal and/or the VCO signal are square waves. This is equivalent with the mixer in Fig. 4.8 being imple- mented as an EXOR gate. Sampling PFD: In sampling PFDs, the phase error Oin(t) - 01/N(t) between the in- put and the frequency-divided VCO signal is measured (sampled) once per period of the input signal. Typical sampling circuits, like the charge-pump-based topology in Fig. 4.9 [Lee98], code this error as the width of a sequence of digital pulses. These pulses are triggered by the zero-crossings of the input and the frequency-divided VCO signal. Hence, they occur at equidistant instances in time. If the width of such a pulse is small compared to the PLL's settling time constant5, it will have the same effect as a Dirac im