se error between the input and the frequency-divided VCO sig- nal. It turns out that, often, this PFD is the main responsible for a PLL's time-varying behavior. Two common PFD implementations are based on mixing and sampling [Lee98]. Mixing-PFDs operate by multiplying the input signal with the frequency- divided VCO signal. Sampling-PFDs produce a sequence of pulses the width of which equals the instantaneous phase error. In what follows, we derive HTMs for both types of PFDs. Mixing PFDs: A mixing PFD, as illustrated in Fig 4.8, multiplies the output of the frequency divider X1/N(t) = S1/N(t++op,1/N(t) + 01/N(t)) with the input signal Xin(1 = Sin (+Oin(t)). In what follows, we assume the mixing operation to be ideal. Using, for example, the algorithm later on outlined in section 4.3, it would also be pos- sible to derive models for more complex structures. This, however, only complicates the analysis without contributing to the discussion below. With both Oin(t) and 01/N(t) being small-signal excursions, we find the mixer output to equal YPFD(t) = (4.23) (4.24) In the lower equality, the upper term represents the steady-state part of the mixer's output while the lower terms are due to the small-signal phase variations Oin(t) and 01/N(t). Note that this latter part takes the form ypfd(t) (4.25) with ypfd(t) denoting the small-signal excursion of the mixer's output. Hence, the mixer small-signal behavior can be modeled as a linear combination of 01/N(t) and Such a normalization involves a change of the time variable tnorm = Wot. Note that proper normalization is important as it often helps us to simplify the resulting analysis while also improving the interpretability of the results. 4.2 PHASE-LOCKED LOOP ANALYSIS Xin(t) Xin(t)/ X1/N (t) Figure 4.8: A multiplying mixer as a phase frequency detector. Oin(t) with coefficients that vary periodically over time. The minus sign in (4.25) is introduced to emphasize the role of the mixer as a phase comparator. Using the HTM formalism, we can express (4.25) in the Laplace 