owledge of the output's signal content in one frequency band implies knowledge of its content in all other frequency bands. This explains the rank-one property of the HTM corresponding to the sampling operator. 4.2.4.2 Loop filter The PFD output is fed to the loop filter. The transfer function of this loop filter is an important design variable in obtaining good PLL stability and settling behavior. Typically, the loop filter is implemented as a time-invariant analog filter with transfer function HLF(s). Using (4.8), the HTM corrsponding to the time-invariant loop filter HLF(s) is found to 4.2 PHASE-LOCKED LOOP ANALYSIS frequency divider Figure 4.11: VCO-frequency divider model. The left-hand block represents a (noisy) VCO model. The right-hand block models the behavior of the frequency-divider. equal HLF (s - jwo) HLF(s) = HLF(s) (4.37) HLF(s+jwo) Although filter nonlinearities can cause time-varying small-signal behavior, and there- fore off-diagonal components in the loop filter HTM, we assume such effects to be negligible. This is justified by the fact that many loop filters are implemented as higly linear passive or opamp-RC based networks [Lee98]. Note, however, that this assump- tion of time invariance is not mandatory. 4.2.4.3 VCO-frequency divider Finally, we discuss the VCO-frequency divider combination. The frequency of the VCO is steered by the control signal coming from the loop filter. This frequency is then divided by a factor N. Fig. 4.11 illustrates the functional model used in this text to capture the behavior of the VCO and frequency divider. In what follows, we discuss the underpinnings of this model in more detail. VCO: By means of a procedure similar to the one for modeling the PLL behavior as a whole, it is possible to obtain a HTM that captures all aspects of the VCO's dy- namic behavior. Each of the VCO building blocks is linearized in the neighborhood of the steady-state operating point6. Using the rules (4.6)-(4.7) for computing HTMs of composed systems, we then obtain a HTM for the VCO 